199 mathematical modeling of mold-filling and solidification - Acta ...

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 199
MATHEMATICAL MODELING OF MOLD-FILLING AND SOLIDIFICATION
OF CASTINGS: PART I – THEORETICAL BASIS
Pariona M. M. 1 , Bertelli F. 2 , Cheung N. 2 , Garcia A. 2
1 Department of Mathematics and Statistics, State University of Ponta Grossa, UEPG Campus
Uvaranas, Block CIPP, Laboratory LIMAC, CEP:84030900, Ponta Grossa, PR, Brazil
2 Department of Materials Engineering, University of Campinas - UNICAMP, PO Box 6122,
13083-970, Campinas, SP, Brazil.
MATEMATICKÉ MODELOVANIE PLNENIA FORMY A TUHNUTIA ODLIATKOV:
ČASŤ I – TEORETICKÉ ZÁKLADY
Pariona M. M. 1 , Bertelli F. 2 , Cheung N. 2 , Garcia A. 2
1 Department of Mathematics and Statistics, State University of Ponta Grossa, UEPG Campus
Uvaranas, Block CIPP, Laboratory LIMAC, CEP:84030900, Ponta Grossa, PR, Brazil
2 Department of Materials Engineering, University of Campinas - UNICAMP, PO Box 6122,
13083-970, Campinas, SP, Brazil.
Abstrakt
Numerická simulácia sa široko využíva a prijíma pri výrobe ako prostriedok zlepšenia
kvality výrobkov a optimalizácie výrobných procesov. Zlievárensky priemysel vo vzrastajúcej
miere využíva simuláciu s cieľom zlepšenia konštrukcie odliatku a kvality výrobku.
Zlievateľnosť možno stanoviť prostredníctvom simulácie v snahe zabezpečiť riadené plnenie
formy a postupné tuhnutie smerom k náliatku. Simulácia umožňuje vizualizovať dynamiku
plnenia a charakterizovať teplotné polia, stupeň tuhnutia, zmraštenie, ako aj riadiť proces
zmenou operačných parametrov. Cieľom prvej časti príspevku je prezentovať teoretické základy
pre rozvoj numerického modelu pre simuláciu plnenia formy a tuhnutia binárnej zliatiny. Pre
proces plnenia formy, modely turbulencie okrem algoritmu zlomkového objemu (Volume of
Fluid – VOF) sa používa na modelovanie výpočtovej dynamiky fluida (CFD). Prístup umožňuje
pracovať aspoň s presnosťou druhého poriadku, kontinuálne presúvať hranicu medzi
vstrekovaným fluidom a vzduchom, ktorý pôvodne je v dutine formy. Tekutý kov predpokladá
podmienky Newtonovho fluida a nestlačiteľného fluida. Pre proces tuhnutia, matematické
vyjadrenie prestupu tepla je založené na všeobecnej rovnici nestacionárneho stavu vedenia tepla.
Prezentované sú tri spôsoby uvoľnenia latentného tepla počas tuhnutia: lineárny, exponenciálny
a sinusový priebeh ako funkcia tuhej frakcie.
Abstract
Numerical simulation is widely used and accepted in manufacturing as a way to
improve the quality of products and to optimize the manufacturing processes. The foundry
industry is increasingly using computer simulation with a view to improve casting design and
quality objectives. The castability can be assessed via simulation in an attempt to ensure
controlled mold-filling behavior and progressive solidification toward feeders. The simulation
permits to visualize the filling dynamics and to characterize the thermal field, the cooling stage,
shrinkage as well as to control the process by varying the operational parameters. The aim of the
first part of this paper is to present the theoretical basis for the development of a numerical

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 200
model for simulation of mold-filling and solidification of a binary alloy. For the mold filling
process, turbulence models in addition to the Fractional Volume (Volume of Fluid - VOF)
algorithm is used to model the computational fluid dynamics (CFD). The approach allows one to
handle, with at least second order accuracy, the continuously moving boundary between the
injected fluid and the air that initially is in the mold cavity. The liquid metal assumes Newtonian
fluid and incompressible fluid conditions. For the solidification process, the mathematical
formulation of heat transfer is based on the general equation of unsteady state heat conduction.
Three forms of latent heat release during solidification are presented: linear, exponential and
sinusoid behavior as a function of the solid fraction.
Keywords: numerical simulation, finite-element method, mold-filling, solidification, heat
generation
1. Introduction
In the last years a perceptible evolution of numerical modeling in all science areas can
be easily detected [1]. Particularly for casting processes, a great progress happened due to the
evolution of both hardware and software used for simulation purposes. The stages of casting can
be defined as: heating-melting, mold-filling and solidification. Each stage has a great influence
on the final casting quality. Mold-filling can be classified as a problem of fluid mechanics with
many phenomena involved such as, molten metal turbulence; heat transfer along the metal/mold
system; pressure variation inside the casting, etc. [2]. The modeling of solidification associated
to the fluid dynamics permits many phenomena and parameters which are important to assure a
free defect product to be analyzed, e.g. [3]: heat transfer in the whole system (casting and mold),
thermal gradients, macrosegregation and microsegregation [4], thermal stresses in metal and
mold, porosity formation, etc. Despite its importance, the simulation of the casting process
becomes a challenge due to the complexity of the simultaneous fluid-thermal-mechanics
phenomena which are involved. Besides, they are transient phenomena and experimental
measurements permitting fluid motion to be characterized are not an easy task [2]. In this paper
a numerical model is developed permitting the simulation of both mold filling and solidification
of binary alloys castings. The moving boundary liquid metal/air is modeled using the VOF
method to trace the free surface during mold filling.
2. Governing equations
The mathematical modeling of mold-filling and solidification of metallic alloys will
be described in the present study by the following equations:
2.1 Mold filling
In the simulation of mold filling, the fluid was considered to be incompressible and
having Newtonian characteristics. The mass, momentum and energy conservation equations that
govern fluid flow and heat transfer can be expressed in vector calculus notation, as it follows:
a) Mass conservation equation (continuity) for incompressible fluids:
∂ u ∂ v ∂ w
+ + = 0
∂ x ∂ y ∂ z
(1)

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 201
where v r =(u, v, w) is the velocity.
b) The momentum conservation equation (Navier–Stokes) can be expressed as [5]:
r
⎧∂
v r r⎫
2 r r
ρ ⎨ + v ∇. v⎬
= -∇ P + µ ∇ v + ρg
⎩ ∂ t ⎭
(2)
where, ρ is the density assumed to be constant, P r is the pressure, µ is the dynamic viscosity
constant and g r is the gravity vector.
c) The energy conservation equation for the liquid region is given by [6]:
⎛ ∂T
r ⎞ 2 L ∂φ
ρc⎜
+ v.
∇T
⎟ = k∇
T −
⎝ ∂t
⎠ c ∂t
(3)
where c is the specific heat, k is the thermal conductivity, L is the latent heat, φ is the
volumetric liquid fraction and T is the temperature.
When the molten metal starts to fill the mold, the turbulence phenomenon begins.
Turbulence means that the instantaneous velocity is fluctuating at every point in the flow field.
Many mathematical models existing in the literature are used to simulate the turbulence,, i.e., a
total of eight turbulence models are available [5; 7, 8; 9; 10; 11; 12]. These models acronyms
and names are: Standard k-ε Model, Zero Equation Model, RNG - (Re-normalized Group
Model), NKE - (New κ-ε Model), GIR, SZL, Standard k-ω Model, SST. The κ-ε model and its
extensions entail solving partial differential equations for the turbulent kinetics energy κ and its
dissipation rate ε.
d) The turbulent kinetics energy equation for NKE - (New κ-ε Model) is given by:
∂(
ρκ ) ∂(
ρuκ)
∂(
ρvκ
) ∂(
ρwκ)
∂ ⎛ µ
+ + + = ⎜ t
∂t
∂x
∂y
∂z
∂x
⎝ σ k
C βµ ⎛ ∂T
∂T
∂T
⎞
− ρε +
4 t
⎜ g x + g y + g z
⎟
σ t ⎝ ∂x
∂y
∂z
⎠
∂κ
⎞ ∂ ⎛ µ
⎟ + ⎜ t
∂x
∂
⎠ y ⎝ σ k
∂κ
⎞ ∂ ⎛ µ
⎟ + ⎜ t
∂y
∂
⎠ z ⎝ σ k
∂κ
⎞
⎟
+ µ tΦ
∂z
⎠
(4)
The viscous dissipation term in tensor notation is:
⎛
⎜ ∂u
Φ = µ i
⎜
⎝
∂xk
∂u
⎞
k ∂u
+ ⎟ i
∂x
j ⎟
⎠
∂xk
where σ t , is the turbulent Prandtl (Schmidt) number; g x , g y and g z , are the components of
acceleration due to gravity and u i , the magnitude of the velocity vector.
The turbulent viscosity is calculated as a function of the turbulent kinetics energy
parameter, κ, and its dissipation rate ε, that is:
2
κ
µ t = ρCµ
ε
where, C µ , is the turbulent constant; κ is the turbulent kinetics energy parameter and ε is the
turbulent kinetics energy dissipation rate.

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 202
For the incompressible solution algorithm the following equation is used:
dρ 1
=
dP β
where β is the bulk modulus.
e) The dissipation rate equation for NKE - (New k-ε Model) is given by:
∂( ρε ) ∂( ρuε
) ∂( ρvε
) ∂( ρwε
) ∂ ⎛ µ ⎞ ⎛ ⎞ ⎛ ⎞
+ + + = ⎜ t ∂ε
∂ µ
⎟ +
+ ⎜ ⎟
⎜ t ∂ε
∂ µ
⎟ t ∂ε
+
∂t
∂x
∂y
∂z
∂x
⎝ σ ε ∂x
⎠ ∂y
⎝ σε ∂y
⎠ ∂z
⎝ σ ε ∂z
⎠
2
ε ε Cµ
(1 C3)
β1ρκ
⎛ ∂T
∂T
∂T
⎞
C1ε
µ t Φ C2ρ
+
⎜ g x + g y + g z
⎟
κ κ σ t ⎝ ∂x
∂y
∂z
⎠
(5)
The new functions; deformation tensor S ij , and the antisymetric tensor W ij , are based
on the velocity components u k in the flow field.
1
Sij
= ( ui,
j + u j,
i )
2
Wij
= 1
( ui,
j − u j,
i ) + Cµ Ω mε
mij
2
where, C µ , is the turbulent constant; Ω m is the angular velocity of the coordinate system and ε mij
is the alternating tensor operator.
The invariants are:
η = κ
2S ij S ij and κ
ξ =
ε
2W ij
ε
Wij
C 1
µ =
, ⎛ η
2 2
4 + 1.5 η + ξ
⎟ ⎞
C 1ε
= max ⎜C1M
⎝ η + 5 ⎠
where, C 1ε , C 2 , σ k , σ ε , σ t , C 3 , C 4 , C 1M and β 1 are constants.
2.2 Free surface modeling
In the last years due to software and hardware technological evolutions, intensive
studies and applications concerning the mold filling by metallic fluids can be found in the
literature. During the mold filling process, liquid metal and air coexist in the mold and the
interface position changes rapidly with time [13]. It is essential to introduce a free-surfacetracking
algorithm to analyze the filling process. The Volume of Fluid Method (VOF) is the
most widely used method for free-surface tracking during mold filling [13]. In a three
dimensional rectangular coordinate frame, the transport equation on the VOF, F(r,t), for an
incompressible fluid can be written as:
∂ F r
+ ∇ F.
v = 0
∂ t
(6)
The F(r,t) function governed by the above equation is unity in the region occupied by
the fluid and zero in the empty region. For the given computational domain, the F(r,t) field

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 203
obtained from Eq. (6) gives the information for the free surface. The cells with F(r,t) values
between 0 and 1 are the free surface cells. The VOF method has been applied successfully to
many engineering problems involving free surfaces including the mold filling problem.
However, since the original VOF method uses an explicit differencing method in time, large
computational time is required to analyze a typical filling problem.
Im et al. [13] state that the use a time-implicit VOF method to alleviate the severe
time step restriction is more efficient than the time-explicit VOF method. The time implicit VOF
method is based on the assumption that a cell being filled cannot transmit fluid to its neighboring
cells until it is completely filled. Once the cell is filled, it can transmit fluid to neighboring cells.
However, According to McBride et al. [14] almost all viable simulation tools for the simulation
of free-surface flow in shaped castings use a fixed grid approach as defined by the mold
geometry, and they typically employ the concept of volume of fluid (VOF) to track the interface,
as originally proposed by Hirt and Nichols [15].
The numerical method used to analyze fluid flow and thermal behavior during and
after filling in the mold is described as follows [13]. Continuity, momentum, energy and F(r,t)-
transport equations are discretized using the fully implicit method in time. Firstly, the velocity
and pressure fields are obtained from the momentum and continuity equations. Then the new
fluid configuration is computed from Eq. (6). The energy equation is solved for the fluidoccupied
region to determine the temperature and the liquid fraction. The updated values of
F(r,t) are utilized to modify the velocity and pressure, and then the temperatures and liquid
fractions are updated based on the modified velocity and pressure. This iteration procedure is
repeated until the total fluid volume and temperature changes are small enough to satisfy the
prescribed convergence criteria. If the convergence criteria are satisfied, the time is advanced.
This time-implicit VOF method is adopted to reduce the tremendous computation time that is
required when the original VOF method is used [13].
According to Kermanpur et al. [16], mold filling problems involve tracking free
surfaces that are the boundaries between liquid metal and the surrounding air. The most
commonly used method to describe free surfaces is the VOF. It enables the tracking of transientfree
surfaces with arbitrary topology and deformations (e.g., fluid surface break-up and
coalescence). The ‘true’ VOF method consists of three main components:
1. A fluid fraction function F(r,t) which is equal to 1.0 in fluid regions, and equal to 0.0 in
voids. Since the fluid configuration may change with time, F(r,t) is a function of time,
t, as well as space, r. Averaged over a computational control volume, the fluid fraction
function has a fractional value in cells containing a free surface.
2. Zero shear stress and constant pressure boundary conditions are applied at free
surfaces.
3. A special advection algorithm is used for tracking sharp free surfaces.
The boundary conditions at the free surface are; normal and tangential stresses equal
to zero.
A free surface advection method must preserve the sharpness of the interface having
minimal free surface distortion. Generally, such advection algorithms are based on geometric
reconstruction of the free surface using the values of F(r,t) at the grid nodes [16]. Sometimes, a
free surface is approximated by a density discontinuity between metal and air and flow equations
are solved for both fluids. In that case it is difficult to enforce correct boundary conditions at the
surface. This is because free surface pressure and velocities in the two-fluid approach are not set

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 204
explicitly, but are computed by solving the flow equations and these flow equations are solved in
terms of mixture variables. Since densities of liquid metal and air differ greatly (e.g., by a factor
of 7000 for steel), the mixture velocity may not always be an accurate measure of the relative
motion of metal and air [16].
Kim et al. [17] proposed an interesting flow chart of the mold filling computational
procedure, as shown in Fig. 1. In the first stage of the numerical analysis, an initial grid is
generated and values for the order of surface refinement, material properties, boundary
conditions, and initial volume fractions are the model input. Then, the time step is increased and
the filling pattern is selected for each element, and the free surface is predicted by the filling
pattern. In the next stage, through the refinement and mergence procedure, an adaptive grid is
generated. According to the volumetric fraction at a given time during filling, the domain
element for the FEM (Finite Element Method) analysis is created. Then, by the FEM analysis,
the velocity and pressure fields are obtained and the flow rate in each element is calculated.
Subsequently, the procedure of advection treatment is accomplished and the volume fraction in
each element is obtained. These procedures are iterated until the current filling time reaches the
total time.
Fig.1 Flow chart of the computational procedure [17]
One of the requirements that should be satisfied in the VOF method for mold filling,
is the conservation of the F(r,t) function even though with the presence of convection. This
method uses a minimum of stored information and it follows regions rather than boundaries,
which avoids the logic problems associated with intersecting surfaces. The derivatives of this
function can be used to estimate the orientation of the fluid surface, improving the
computational efficiency [18].

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 205
2.3 Equations for solidification
The mathematical formulation of heat transfer to predict the temperature distribution
during solidification is based on the general equation of unsteady state heat conduction, which is
given by [1; 15; 19; 20; 21; 22]:
∂ ( ρ . c.
T ) = ∇.(
k∇T
) + Q&
(7)
∂ t
where ρ is the density [kgm -3 ]; c is the specific heat [J kg -1 K -1 ]; k is the thermal conductivity
[Wm -1 K -1 ]; ∂T/∂t is the cooling rate [K s -1 ], T is the temperature [K], t is time [s], x, y and z are
the space coordinates [m] and Q & represents the term associated to the internal heat generation
due to the phase change. For this equation, it was assumed that the thermal conductivity, density
and specific heat vary with temperature [22].
The boundary condition at the mold external surface is given by:
∂T
− -k = h ( T - T∞
)
(8)
∂n
Where h is the ambient surface convection heat transfer coefficient associated with
free convection, n is the unit vector outward normal to the sand mold, T ∞ is the environment air
temperature and T is the temperature at the outer mold wall.
2.3.1. Phase change
When there is a phase change from liquid to solid, a heat source term arises, which is
given by an explicit solid fraction–temperature relationship. The solid fraction depends on a
number of parameters; however, it is quite reasonable to assume f s varying only with
temperature, where f s is the solid fraction and L is the latent heat of fusion [J kg -1 ]. So, we can
write:
c ’ = c–L.∂f s /∂T (9)
The term c’ can be considered a pseudo-specific heat. The sub-indices S and L
indicate solid and liquid, respectively [22].
In the current system, no external heat source is applied and the only heat generation
is due to the latent heat of solidification, L (Jkg -1 ) or ∆H (Jm -3 ). Q & is proportional to the rate of
change of the solid fraction, f s , as it follows [21; 22; 23]:
f f
Q & ∂
H
s ∂
= ∆ = ρL
s
(10)
∂t
∂t
In many systems, especially when the undercooling is small, the solid fraction may be
assumed as being dependent on temperature only. Different forms have been proposed to take
into account the evolution of solid fraction with temperature. The simplest approach is that given
by a linear relationship [23]:
f s
⎧0
⎪
= ⎨(
Tl
⎪
⎩
− T ) /( Tl
− Ts
)
1
T > Tl
Ts
≤ T ≤ Tl
T < Ts
(11)

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 206
where T l and T s are the liquidus and the solidus temperatures (K), respectively. Another widely
used expression is the Scheil equation, which assumes uniform solute concentration in the liquid
but no diffusion in the solid [23]:
−1/1−k ⎛ T
o
m − T ⎞
f s = 1 − ⎜ ⎟
Tm
T
⎝ − l ⎠
(12)
where T m is the melting point of the solvent metal (K),and k o is the equilibrium partition
coefficient of the alloy.
Eq. (10) defines the heat flux [3] which is released during liquid cooling and
solidification. In classical models, the heat generated after solidification is assumed to be zero,
i.e., for T< T s , Q & = 0. However, experimental investigations showed that lattice defects and
vacancies are condensed in the already solidified part of the crystal and the enthalpy of the solid
increases and thus the latent heat will decrease. Based on this fact, another way to represent the
change of the solid fraction during solidification can be written as [3]:
2
⎡ π ( T − T ) ⎤
( T − T ) + ( Ts
− T )(1 − cos
l
l
l ⎢ ⎥
π
⎣ 2( Ts
− Tl
)
f =
⎦
s
( Tl
− Ts
)(( 1−
2 / π )
(13)
Combining Eqs. (7), (9) and (10), the new unsteady state heat conduction equation
can be expressed in the form [4; 23]:
∂ ( ρ c´
T )
= ∇.( k∇
T)
∂ t
(14)
At the range of temperatures where solidification occurs for binary metallic alloys, the
physical properties will be evaluated taking into account the amount of liquid and solid that
coexists in equilibrium at each temperature, as shown by Eqs. (9), (10) and (14).
3. Conclusions
A comprehensive theoretical basis regarding equations of fluid mechanics and heat
flow has been shown. Like other CFD problems, the momentum equation, the continuity
equation, and the energy equation are the partial differential equations that are essential in the
numerical analysis of the casting process. The moving boundary liquid metal/air is modeled
using the VOF method to trace the free surface in mold filling processes. The latent heat release
during the unsteady solidification phenomenon is taken into account as a source term in the
energy equation. The implementation of such equations is aimed to furnish important
information such as filling dynamics and the thermal field along the casting process.
Acknowledgements
The authors acknowledge financial support provided by CNPq (The Brazilian
Research Council), FAPESP (The Scientific Research Foundation of the State of São Paulo,
Brazil) and CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior.

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 207
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