Shrinkage Porosity Prediction Using Casting Simulation M. Tech. Dissertation submitted in partial fulfilment of the requirements for the degree of Master of Technology (Manufacturing Engineering) by Amit V. Sata (08310301) Guide Dr. B. Ravi Department of Mechanical Engineering INDIAN INSTITUTE OF TECHNOLOGY BOMBAY 2010
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Shrinkage Porosity Prediction Using
Casting Simulation
M. Tech. Dissertation
submitted in partial fulfilment of the requirements
for the degree of
Master of Technology
(Manufacturing Engineering)
by
Amit V. Sata (08310301)
Guide
Dr. B. Ravi
Department of Mechanical Engineering INDIAN INSTITUTE OF TECHNOLOGY BOMBAY
2010
Dissertation Approval Certificate
This is to certify that Mr. Amit V. Sata (08310301) has satisfactorily completed his
dissertation titled “Shrinkage Porosity Prediction using Casting Simulation” as a part
of partial fulfillment of the requirements for the award of the degree of Master of
Technology in Mechanical Engineering with a specialization in Manufacturing
Technology at Indian Institute of Technology Bombay.
“I declare that this written submission represents my ideas in my own words and where
others' ideas or words have been included, I have adequately cited and referenced the
original sources. I also declare that I have adhered to all principles of academic honesty
and integrity and have not misrepresented or fabricated or falsified any
idea/data/fact/source in my submission. I understand that any violation of the above will
be cause for disciplinary action as per the rules of regulations of the Institute”
Date: Signature
Place: Name: Amit V. Sata
i
Abstract
Shrinkage porosity is one of the most common defects in castings. Various existing techniques of shrinkage porosity prediction like modulus and equi-solidification time and criterion function have been reviewed. Various criteria functions including Niyama criterion, dimensionless Niyama criterion, Lee et al. criterion and Franco criterion for prediction of shrinkage porosity have been studied in this work. From literature, L shape casting has been analyzed for predicting location of shrinkage porosity using solidification simulation. Simulation result is comparable with available experimental result. Threshold values of Lee et al., Davis, Franco and Bishop criterion for cast steel have been established by comparing results with Niyama criterion. Benchmark casting, a combination of three T-Junction, has been cast and analyzed to understand dependency of shrinkage defect size on geometric parameters and thermal parameters. The experiments were carried out for Ductile iron (500/7), plain carbon steel (1005 steel) and stainless steel (SS 410). These experimental data are used to set limiting temperature gradient values in AutoCAST®. Further, simulation experiments were carried out by varying thickness ratio from 0.25 to 1.5. The result of experiments and simulations are used as input to regression analysis to evolve a set of empirical equations to predict shrinkage porosity defect size in T junction considering the effect of geometric parameter alongwith thermal parameters. Further, an empirical model of SS 410 is validated by casting of T junction which is having thickness ratio and length ratio of 1.75 and 5 respectively. The predicted size of shrinakge defect is approximately matching with observed size of defect. Keywords: Shrinkage porosity, Casting simulation, Criterion function, Plain carbon steel, Stainless steel, SG Iron, LM 6 (Al Alloy).
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Table of Contents
Abstract i Table of Contents ii List of Figures iv List of Tables vi Nomenclatures viii 1 INTRODUCTION 1 1.1 Porosity in Metal Casting 1 1.2 Need of Defect Prediction 3 1.3 Organization of Report 3
2 LITERATURE REVIEW 4 2.1 Classification and Formation of Porosity 4
2.2 Factors Affecting Shrinkage Porosity 8 2.3 Modeling of shrinkage porosity 10 2.4 Casting Solidification Simulation 12 2.4.1. Finite element method 15 2.4.2. Vector element method 16 2.5 Shrinkage Porosity Prediction 17 2.5.1. Modulus and equi-solidification time method 17 2.5.2.Criterion function method 19 2.6 Summary 31
3 PROBLEM DEFINITION 34 3.1 Motivation 34 3.2 Goal, Scope and Objectives 35 3.3 Approach to Project 35
4 SHRINAKGE DEFECT LOCATION 37 4.1 Approach to Predict Location of Shrinkage Porosity 37 4.1.1. Solidification simulation using FEM 38 4.1.2. Solidification simulation using VEM 44 4.2 Summary 46
6.1 Threshold Value of Various Criterion Function 90
viii
Nomenclatures
CMI Casting/mold interface V/A Modulus of casting D Diffusion co-efficient m Liquidus slop c
0 Alloy composition
k Equilibrium distribution coefficient, Ny Niyama threshold value Ny* Dimensionless Niyama threshold value LCC Lee et al. Criterion FRN Friction resistance number FCC Franco Chisea Criterion G Temperature gradient Vs Solidification velocity tf Local solidification time gl liquid volume fraction ul Shrinkage velocity Cλ Material constant dT/dt Cooling rate Pp Pressure inside the pore ∆Pcr Critical pressure drop β Total solidification shrinkage θ Dimensionless temperature = T – Tsol /∆Tf ∆Tf Freezing range
µl Liquid dynamic viscosity %P Percentage porosity P Probability of local porosity f Fraction of a phase; (fl - fraction liquid, fs - fraction solid) L Length of the mushy zone x Spatial co ordinate T Temperature K Permeability σ Surface tension it is between pore and surrounding liquid r0 Initial radius of curvature at pore formation µl Liquidus viscosity ρs Solidus density ρl Liquidus density λ2 Secondary dendrite arm spacing (SDAS)
Xcr, Position at which the melt pressure drops to Pcr and porosity begins to form Tcr, Temperature at which the melt pressure drops to Pcr and porosity begins to form gl,cr, , Liquid fraction at which the melt pressure drops to Pcr and porosity begins to form R1 Thickness ratio = t/T R2 Length ratio = l/T
1
Chapter 1
Introduction
Metal Casting is one of the oldest manufacturing processes and is still considered as an art,
rather than science. Casting is used to manufacture complex shape. The basic principle of
casting process is simple. The molten metal is poured into mould or cavity which is
similar to required finished shape.
1.1 Porosity in Metal Casting
Sand castings are used to manufacture complex shapes. The castings are likely to have one
or more defect. The presence of defects leads to casting rejections. The metal casting
process suffers from the following types of defect:
Hindered cooling contraction: hot tear, crack, distortion
The improper tool design causes unacceptably high turbulence, unfilled thin sections,
solidification before complete filling and hindered heat flow. These cause the major three
defects viz. incomplete filling, solidification shrinkage and hindered cooling contraction.
Porosity is one of the regular problems which impact the quality of the castings and
worsen the mechanical properties, such as tensile strength and fatigue life. In case of
AS7G03 (Al- Si7- Mg0.3 cast Al alloy) 1% volume fraction porosity can lead to a
2
reduction of 50% of the fatigue life and 20% of the endurance limit compared with same
alloy with a similar microstructure but showing no pores(J.Y Buffiere et al., 2000).
Porosity is the most persistent and common complain of casting users. Forgings, machined
parts and fabrications are able to avoid porosity with ingot cast feedstock and mechanical
processing. Porosity in castings contributes directly to customer concerns about reliability
and quality. Controlling porosity depends on understanding its sources and causes.
Significant improvements in product quality, component performance, and design
reliability can be achieved if porosity in castings can be controlled or eliminated.
Porosity in castings can be grouped into one of two broad categories (macroporosity or
microporosity) on the basis of scale and mechanism of formation. Macroporosity is
generally large in scale and forms as a result of solidification of liquid that has been
enclosed by a solidified material. The size of the resulting pore or cavity in dependent
upon the volume of enclosed liquid and the volume shrinkage associated with the liquid-
to-solid phase transformation. Macroporosity is easily corrected by proper gates and risers
within the mould and /or using chills and and/or exothermic to control the progress of
solidification.
In contrast, microporosity forms interdendritically at the scale of the microstructure. Thus,
its formation is more complex mechanically, more difficult to predict, and generally more
difficult to correct. There are two primary sources of microporosity: solute gas
precipitation in the interdendritic liquid, and/or poor liquid feeding from volume shrinkage
within the mushy zone.
Fig. 1.1: Porosity in metal casting (Source: Greyduct Foundries - Ambala)
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1.2 Need of Defect Prediction
The task of a mold designer and foundry engineer is to make an optimized geometric
casting design and choose proper process parameters that eliminate or minimize porosity
development. But porosity formation is a complex phenomenon where the final sizes and
the distribution of porosity voids are determined by several strongly interacting process
and alloys variables. As a result, it is usually difficult to eliminate porosity completely
from metal castings, while reducing it or moving it to an unimportant area can be a choice.
So there is a need for some prediction technique which will predict the location and size of
the porosity.
1.3 Organization of Report:
This report is organized in the following manner.
Chapter 1 gives introduction of casting process and need of defect prediction
Chapter 2 gives detail literature review of shrinakge porosity formation, modeling
and various prediction methods.
Chapter 3 introduces the problem definition.
Chapter 4 gives information about location based predication method and
comparison of various criterion functions.
Chapter 5 includes benchmark shape and its solidifaction simulation. It also
includes experiements and results, development of empirical model using
regression technique and validation of empirical model.
Chapter 6 includes summary and future work.
4
Chapter 2
Literature Review
The properties of casting determine the quality of the final product. In particular porosity
or shrinkage voids are usually undesirable. It appears that one half to three quarters of
scrap castings are lost because of porosity (Lee et al., 2001).
This chapter includes the classification of porosity and its formation and modeling of
porosity. It also includes various numerical methods for casting solidification simulation.
The various methods for location based prediction of shrinakge porosity have also been
discussed. One of method of location based prediction of shrinakge porosity; the criterion
function method is studied in detail because of its wide use in existing simulation
software.
2.1 Classification and Formation of Shrinkage Porosity
A. Classification Shrinkage Porosity
Shrinkage related defects in shape casting are major cause of casting rejections and rework
in the casting industry. Lee at el., (2001) proposed the classification of shrinkage porosity
in castings by the size of the pores:
(i) macroporosity and
(ii) microporosity; and by the cause for the pores forming:
(i) shrinkage porosity and
(ii) gas porosity.
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Sabau et al.(2002) considered porosity is usually to be either “hydrogen porosity” or
“shrinkage porosity”. Hydrogen porosity is the term given to porosity that is generally
rounded, isolated, and well distributed. Porosity that is interconnected or clustered and an
irregular shape corresponding to the shape of the interdendritic region is usually termed
shrinkage. In general, the occurrence of microporosity in alloys is due to the combined
effects of solidification shrinkage and gas precipitation.
A. Reis et al.(2008) classified important defects that arise from shrinkage solidification are
External defects: pipe shrinkage and caved surfaces;
Internal defects: macroporosity and microporosity.
Generally short freezing alloys are more prone to internal defects, whereas long freezing
alloys are more prone to surface depressions.
B. Formation of shrinkage porosity
From a scientific point of view, the problem of porosity formation is complex and most
interesting. The thermal properties of the alloy being cast (latent heat of fusion and
thermal conductivity), the composition of the alloy (freezing range and dissolved gas
content), the mold properties, and the geometry of the casting are all important to the
properties of the final cast product. However, the relative effect of these variables is very
complicated. The problem has been studied in detail for nearly 20 years, but there appears
to be no clear agreement as to which mechanisms control the formation of porosity. In the
absence of a clear scientific understanding, foundrymen used empirical rules to design
their molds.
Despite of all these things, effort has been made to provide information regarding the
shrinkage porosity formation in this section because the objective of this project is limited
to predict shrinkage porosity for different metals. Starting with the definition of the first
cause, shrinkage is the term for obstruction of fluid flow coupled with a difference in the
specific volumes of liquid and solid metal.
6
As the casting solidifies, metal that is still fluid will try to flow to compensate for the
liquid/solid volume change; however, the flow may be hindered by the solid which has
already formed. If a poorly fed region is large and completely cut off from a source of
liquid metal, then a large void (generally greater than 5 mm in maximum length) is
formed. The resulting void is termed `macroporosity'. (Note that gas solubility differences
may contribute to macro pore formation as well). The area in which macro pores form
solidifies after the surrounding region, termed as a `hot spot' with reference to the islands
of hot metal completely surrounded by colder material.
Pellini's(1953) observations are of some importance to the theoretical thermal analysis.
The feeding length of a riser is best considered by examination of Figure 2.1. The data
presented are for a steel bar cast in green sand. The distance from the riser to the end of
the casting is sufficiently long that there is a central section which is "semi-infinite." In
this region, the solidification proceeds as if the bar had no ends and was infinitely long. In
other words, the temperature in this region is uniform along its length, so the entire section
freezes at the same time. Consider the experimental freezing velocity curve at the lower
right-hand section of the figure. Five minutes after pouring, a shell 1.5 inches (~40-mm)
thick from the end has formed at the centerline of the bar. At 10 minutes, there is a region
3 inches (~80-mm) thick which is completely solid.
Fig.2.1: Solidification of a bar casting (G.K.Sigworth and Chengming Wan,1993)
7
At 16 minutes, this shell has reached the right-hand end of the semi-infinite region, whose
entire section now freezes. The freezing "wave" then slows down as it approaches the hot
riser. Pellini(1953) also observed centerline shrinkage in these central "semi-infinite"
sections of plate and bar castings and in regions adjoining the semi-infinite region. An
analysis of his cooling curves showed that in 2-inch- (50-mm-) thick plates, shrinkage
porosity occurred in areas where the temperature gradient was less than 1 to 2 F/in. (20 to
40 0C/m). In 4 inch (100-mm) bars, a higher gradient was required to prevent centerline
shrinkage: 6 to 12 F/in (120 to 240 0C/m). Pellini made a number of steel plate castings
whose length from riser to end varied. He found that the total length of the plate could be
as much as 4.5 times the thickness of the plate. Longer plate sections developed centerline
shrinkage. In bar castings, the total feeding length was equal to six times the square root of
the thickness.
A. Reis et al. (2008) had shown in their research that this shrinkage related defect results
from the interplay of several phenomena such as heat transfer with solidification, feeding
flow and its free surfaces, deformation of the solidified layers and the presence of
dissolved gases.
P. D. Lee et al. (2001) believed that porosity formation in aluminium alloys has two
primary causes: (1) volumetric shrinkage; and (2) hydrogen gas evolution. Volumetric
shrinkage refers to the density difference between the solid and liquid alloy phases. As
solidification proceeds, the volume diminishes and surrounding liquid flows in to
compensate. Depending on the amount and distribution of solid, the fluid flow may be
impeded or even completely blocked. When sufficient liquid is not present to flow in
cavity, voids (pores) form. This shrinkage porosity can either be many small distributed
pores or one large void.
D.R. Gunasegarama et al.(2009) believed that shrinkage porosity defects occurring in
castings are strongly influenced by the time-varying temperature profiles inside the
solidifying casting. This is because the temperature gradients within the part would
determine if a region that is just solidifying has access to sufficient amounts of feed metal
at a higher temperature. Shrinkage pores will emerge in regions experiencing volume
reduction due to phase change with no access to feed metal.
8
J Campbell (1991) provided good idea about the initiation of the shrinkage porosity with
the help of pictorial view of solidification steps occurred during cooling of casting. It is
shown in fig. 2.2.
Other researchers have studied the formation of shrinkage porosity by offering theoretical
models or empirical prediction criteria like (G/(dT/dt) and G/√(dT/dt). A review of the
literature shows that consensus has emerged. Consequently, further study appears not to be
warranted.
2.2 Factors Affecting Shrinkage Porosity
Heat transfer rates at the casting/mold interface (CMI) play a significant role in
determining the temperature gradients in the solidifying casting in permanent molds
(Campbell, 1991; Gunasegaram, 2009).
Fig.2.2: Representation of the origin of porosity as section thickness is increased. (Campbell, 1969).
9
That is because CMI is the rate controlling factor due to the fact that it offers the largest
resistance to heat flowing out of the casting and into the metallic mold. Heat flux Q
(W/m2) across CMI is the product of the heat transfer coefficient h (Wm−2 K−1), which
quantifies the degree of thermal contact between the casting and the mold, and T (K),
which is the temperature difference between the surface of the casting at the CMI and that
of the mold at the same CMI. The thermal resistance h of the CMI is attributed to the mold
coat until an air gap (Gunasegaram et al., 2009); forms between the expanding mold and
the contracting casting. The insulating gap is thereafter the major contributor to the
resistance (Hallam and Griffiths, 2004; Hamasaiid et al., 2007).
Temperature gradients are also a function of the geometry of the casting and that of the
runner (Campbell, 1991). Since thinner sections would solidify sooner than thicker areas, a
temperature gradient exists from thinner to thicker sections.
Melt flow patterns inside the casting cavity and filling durations are also determinants of
the temperature gradients within the casting (Campbell, 1991). Depending on the flow
length of the melt inside the cavity between the instant it enters the cavity and the moment
it comes to rest, the amount of heat lost also will vary. Melt flowing longer distances
would be colder after losing greater amounts of heat to the mold.
As a general solution, directional solidification, where a temperature gradient is always
exists between a solidifying region and a large pool of molten liquid (Campbell, 1991).
This is carried out by ensuring that the casting section kept increasing towards the feed
metal or by using composite molds (Gunasegaram et al 2009) with or without forced
cooling (Gunasegaram et al., 2009). Composite molds are made of materials with vastly
differing thermal properties allowing differential heat extraction from various parts of the
casting and to thereby force favorable temperature gradients within a casting.
In all literature it is found that the complex shape of the commercial casting with
frequently varying cross sectional areas made the directional solidification solution
redundant. Consequently, either physical experimentation or numerical simulation was
required to isolate the critical factors.
10
To summarize, factors affecting shrinkage porosity formation in casting are generally
known but no work reported in the public domain appears to have identified the most
critical of those factors that would help manage the size and location of shrinkage porosity
in a casting with varying cross sectional areas.
2.3 Modeling of Shrinkage Porosity
Once porosity forms, the pores will grow until they have reached equilibrium between all
the forces acting on them including pressure and interfacial energy. Hence, to model pore
nucleation and growth, the following physics should be simulated (Lee et al., 2001):
(i) the thermal field;
(ii) the flow field (for pressure, heat and mass transport);
(iii)fraction solid (nucleation and growth of the solid grains and their interaction with
the thermal and solute concentration fields);
(iii)the impingement of pores upon growing grains (altering both the interfacial energy
and imposing curvature restrictions on the bubbles).
An ideal model would include all these phenomena. However, due to the complexity of
the problem, each of the models reviewed in this paper only considers a few of these
phenomena and assumes that the other effects are negligible. The validity of the model
assumptions is dependent upon the alloy, process and particular design.
Many different models of pore formation and growth have been proposed so far; however,
none of them takes into consideration all of the previously listed physical phenomena. A
model that did account for all of these phenomena may not be industrially viable, being so
computationally intense that it would not be cost effective. Additionally, such a model
might be so complex that the necessary boundary conditions and material properties could
not be obtained with sufficient accuracy, either experimentally or via theoretical
calculations. The methods that have been proposed to model pore formation are
categorized below into four different groups; each with its own benefits and drawbacks as
far as industrial application is concerned (Lee et al., 2001).
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1 Analytical solutions.
2 Criterion function models, based on empirical functions.
3 Numerical solutions of Stokes flow (Darcy's law), coupled with energy and mass
conservation, and continuity equations.
4 Models using a stochastic approach to nucleation of pores and grains in
combination with continuum solutions for diffusion, taking into consideration the
pore and microstructure interactions.
An extensive review on these models has been made by Lee et al. (2001) and Stefanescu
(2005). The first model that took into account feeding flow dates back to the early 1D
analytic work of Piwonka and Flemings (1966). This early analytical work formed the
basis of a other category of models based upon Darcy’s law. Darcy’s law relates the flow
through a porous media to the pressure drop across it. Kubo and Pehlke (1985) were the
pioneers in presenting a 2D numerical model by coupling Darcy’s law to the equations of
continuity estimating the fluid flow. The methodology proposed by Kubo et al. has been
used with little change in numerous studies, such as those of Combeau et al. and Rousset
et al.
Later other 2D model was presented by Zhu and Ohnaka (1991) and Huang et al. (1998).
In terms of 3D models, Bounds et al. (2000) presented a model that predicts
macroporosity, misruns and pipe shrinkage in shaped castings. Later Sabau and
Viswanathan (2002), Pequet et al. (2002) and Carlson et al. (2003) also presented 3D
models that included the concept of pore nucleation and growth. Some models came up
that were based on solving the heat transfer and mass conservation to predict the position
of the free surface and macro-shrinkage cavity. There was a model proposed to account for
shrinkage and consequently determine the shrinkage profile resulting from phase and
density change. It was a method presented for macro-shrinkage cavity prediction based on
a continuum heat transfer model which determines when an area will be completely cut off
from sources of liquid metal (such as risers) where a void will form to account for volume
deficit and its size is calculated through the mass conservation equation. Another
approach, and more complex one, is the one, which tries to consider the feeding flow
analysis.
12
The initial effort of casting simulation was to develop codes that only analyze the
solidification behavior by heat conduction models, solving the energy transport equations.
For defects prediction they use a criteria function, empirical models for evaluation of
shrinkage porosity defects, based on some relations of the local temperature gradient. The
most well known is the Niyama Criterion (Niyama et al., 1982), based on finding the last
region to solidify as most probable location for shrinkage defects. These and other
functions have been summarized by Overfelt et al. (1997), Spittle et al. (1997) and later
evaluated by Taylor and Berry (1998).
A. Reis et al., (2007) also presented a model of shrinkage for long and short freezing
metals by taking into consideration that volume deficit due to shrinkage can only be
compensated by two phenomena: depression of the outside surface or by creating internal
pores.
Typical published models from each category for the modeling of pore formation during
the solidification are discussed above and compared in Table 2.1.
2.4 Casting Solidification Simulation
Many solidification simulation programs now exist, but some require computers of a high
power not generally available to practical foundry men, while others take an unacceptably
long time to obtain meaningful results. The aim of casting simulation is to (T.R.
Vijayaram et al., 2005)
Predict the pattern of solidification, indicating where shrinkage cavities and
associated defects may arise.
Simulate solidification with the casting in various positions, so that the optimum
position may be selected.
Calculate the volumes and weights of all the different materials in the solid model.
Provide a choice of quality levels, allowing, for example, the highlighting or
ignoring of micro-porosity.
Perform over a range of metals, including steel, white iron, grey iron and ductile
iron and non- ferrous metals.
13
Table 2.1: Categories of approach on the basis of literature
Sr. No. Category Author Modeling Approach
1 Analytical solutions.
Walther et al. Piwonka and
Flemings
Focused on shrinkage driven pore growth, developing models that range from exact mathematical solutions to approximate asymptotic analytical solutions using 1D Darcy’s law.
2
Criterion function models, based on empirical functions.
Pellini, Niyama et al.
Empirical models for evaluation of shrinkage porosity defects, based on some relations of local temperature gradient using relationship between pressure drop and solidification conditions, assuming flow in a porous medium in cylindrical coordinates (Darcy's law).
3
Numerical solutions of Stokes flow (Darcy's law), coupled with energy and mass conservation, and continuity equations.
Kubo& Pehlke Combeau et al. Rousset et al.
Zhu et al. Huang et al. A. Reis et al
Presents 2D numerical model by coupling Darcy’s law to equations of continuity estimating the fluid flow.
4
Models using a stochastic approach to nucleation of pores and grains in combination with continuum solutions for diffusion, taking into consideration the pore and microstructure interactions.
Lee et al. Viswanathan et al.
Pequet et al. Carlson et al.
Presents 3D models that included the concept of pore nucleation and growth.
14
From the existing and recent literature citations it is found that the currently available
casting solidification simulation software’s have not taken all constraints and conditions
required for the realistic simulation process This matters more and influences critically on
the output results. Normally simulation is done for simple shape castings particularly
cylindrical and of slab type. Very limited complicated shape castings of real engineering
components have taken for this research work and yet not applied all constraints and
complete boundary conditions.
Solidification of castings varies for different metal-process combinations. The result of the
simulation process helps to design the castings effectively by identifying the defect
locations from the geometrical features of the components. By generating practical
conditions in the software, one can predict the optimum values like die/mold temperature,
molten metal or alloy pouring temperature and perform preheating temperature. This helps
us to identify whether complete infiltration has taken place or not during solidification
process.
Various numerical techniques have been extensively utilized for modeling the behaviors of
materials in processing in the past two decades. The behaviour of materials can be either
macroscopic or microscopic. In macroscopic, the concept of material continuum for which
the densities of mass, momentum, and energy exist in the mathematical sense of the
continuum is applied to study the physical behavior of materials. The continuum is a
mathematical idealization of the real world and is applicable to problems in which the
microstructure of matter can be ignored. When the microstructure is to be studied, the
concepts of micromechanics should be applied.
Based on either the continuum or micromechanics concept, partial differential equations
governing different material behaviors can usually be formulated. It is well known that in
macro modeling, the Navier-Stokes equation for the momentum field, the Fourier equation
for the temperature field, and the Maxwell equation for the electromagnetic field are the
respective governing equations. In general, these partial differential equations should be
considered simultaneously. Consequently, depending on the stiffness of the system,
advanced numerical coupling techniques which further complicate the already formidable
situation are often required. For example, in modeling the induction heating process, the
15
electromagnetic, heat transfer, and fluid flow behaviors are strongly coupled and should be
solved together.
Many numerical techniques, including the finite difference method (FDM), finite element
method (FEM), and boundary element method (BEM) etc. have been developed to solve
these differential equations with complex boundary conditions arising from material
processing. There is also a method called vector element method (VEM) for prediction of
hot spot in casting. Our discussion is limited to FEM and VEM because other methods are
beyond the scope of this project.
2.4.1 Finite element method
In the last almost four decades the finite element method (FEM) has become the prevalent
technique used for analyzing physical phenomena in the field of structural, solid, and fluid
mechanics as well as for the solution of field problems. The FEM is a useful tool because
one can use it to find out facts or study the processes in a way that other tool cannot
accomplish.
Finite element simulation of casting solidification process is one of the best ways to
analyze the process of solidification. It involves the physical approximation of the domain,
wherein the given domain is divided into sub-domains called as elements. The field
variable inside the elements is approximated using its value at nodes. Elemental matrices
are obtain using Galerkin’s weighted residual or variational principles and are assembled
in the same way, as the elements constitute the domain. This process results in the set of
simultaneous equations. The solution of these set of equations gives the field variables at
the nodes of the elements
With FEM we can solve simultaneously energy equation with advection and diffusion
term, momentum equation with advection, diffusion and buoyancy term and continuity
equation.
FEM Advantage:
• Ability to model complex domain. It is also capable to handle non-linear
boundaries and in implementing boundary condition.
16
FEM Complexities:
• Method requires the much effort for formulation of the problem and data
preparation
• Need long processing time and large memory space.
2.4.2 Vector element method
This method is based on determining the feed path passing through any point inside the
casting and following the path back to the local hot spot. Fourier law of heat conduction is
used to determine the gradient as follows:
Heat flux,STkq
ΔΔ
−=
Where, ST
ΔΔ is thermal gradient (G)
Hence, qk
G ⎟⎠⎞
⎜⎝⎛ −=
1
The feed path is assumed to lie along the maximum thermal gradient direction. Thermal
gradient is zero along the isothermal lines, and maximum normal to the isothermal lines.
The magnitude and direction of maximum thermal gradient at any point in side the casting
is proportional to the vector resultant of thermal vectors in all direction originating from
that point. Now casting volume is sub-divided into a number of pyramidal sectors
originating from the given point, each with a small solid angle. For each sector heat
content and cooling surface area is determined to compute the flux vector. Once resultant
vector is computed, we move along it, reach to the new location and repeat the
computation, until the resultant flux vector is less than some specifies limit. This final
location obtain is regarded as a hottest part of the casting under observation. Locus of the
points along which vector moved is the feed path of the casting, because metal will always
flow along the maximum thermal gradient.
The various methods of predicting locations of the shrinkage porosity will be discussed in
the following section.
17
2.5 Shrinkage Porosity Prediction
Although the phenomenon of porosity formation has been well understood, the time to
predict the defect precisely has not yet come. In the past fifty years, especially in the
recent twenty years, research efforts have been made to predict porosity with the help of
computer simulation. The studies made can be classified as the following three
approaches:
(1) Modulus and equisolidification time method, which determines the areas that
solidify last.
(2) Criterion function method, which calculates parameters to characterize resistance
to interdendritic feeding.
(3) Direct simulation method, which directly simulates the formation of porosity by
mathematically modeling the solidification process.
Among the approaches described above, direct numerical simulation gives insight into the
formation of dispersed porosity. But its application is mainly limited in research field for
its complexity in use so it will be omitted for further discussion.
2.5.1 Modulus and equi‐Solidification time method
A. Modulus method
The modulus method is based on Chvorinov’s rule
that solidification time, tf of a casting
area is proportional to the square of its volume to area ratio, V/A, named modulus.
tf = B (V/A)2
B in this eq. is a factor that depends on the thermal properties of the metal and mold material. This
experiment-based eq. has been testified by other researchers, and was incorporated to some
computer programs with which the solidification order of a 2 or 3- dimensional model can be
calculated. It can be shown in fig. 2.3.
18
Fig.2.3: Shrinkage prediction by Modulus Method (S. J. Neises et al., 1987)
B. Equisolidification time method
With the introduction of finite element/difference method to foundry field,
equisolidification time contours or other isochronal contours could readily be calculated.
The principles of the calculations are well established, and the results calculated are in
good agreement with the corresponding experimental results in showing the last
solidification area.
C. The deficiency of the modulus and equisolidification time method
To date, the determination of the areas that solidify last can be successfully carried on
either by the modulus calculation or equi-solidification time calculation based on
numerical simulation of heat transfer. In estimating solidification sequence, the later is
more accurate than the former, because modulus calculation does not take into account the
mold temperature variation and the metal material physical properties. Therefore, the
numerical simulation of heat transferring represents the most important application of
computer simulation in foundry industry currently.
But both methods have their limitation in predicting dispersed porosity, since they do not
consider such factors, as interdendritic feeding and gas evolution, which govern separately
or cooperatively the formation of dispersed porosity. This approach is, however, reliable in
predicting gross shrinkage.
19
Fig.2.4: Shrinkage porosity prediction by Equisolidification Method
(H. Iwahori et al., 1985) 2.5.2. Criterion function method
Criteria functions are simple rules that relate the local conditions (e.g., cooling rate,
solidification velocity, thermal gradient, etc.) to the propensity to form pores. The
application of criteria functions to micro porosity is not new, and can be traced back as
early as 1953, when Pellini extended the idea of a criterion for the size of risers to a
feeding distance criterion to prevent interdendritic centerline shrinkage in steel plates.
Since that time, many different criteria functions have been proposed; some were based
upon statistical analysis of experimental observations, whilst others were based upon the
physics of one of the driving forces
A. Parameters used for criterion function method
Due to the inefficiency of the modulus and equi-solidification time method in predicting
centerline and dispersed porosity, the criterion function approach has received
considerable attention in porosity prediction. These criteria reflect the limiting conditions
of interdendritic feeding. To predict the position of a possible location of porosity we need
following physical parameters as function of time and space:
Flow modeling: velocity vectors, pressures, and surface tracking
Heat Transfer modeling: temperature, temperature gradients (of filled metal and
mold both), and heat transfer process (conduction, convection, and radiation)
Solidification: change in physical properties (density, viscosity, coefficient of
conductivity, etc.).
A combination of these parameters can be easily obtained from numerical solutions.
20
Fig.2.5: Comparison of Gradient and equi-solidification time method (H. F.Bishop et al., 1951)
B. Temperature gradient criterion (Niyama et al. 1981)
The importance of temperature gradient was first proposed by Bishop et al. and developed
by Niyama et al. into a computer simulation method. This criterion gives information
directly related to interdendritic flow. Therefore, it can predict centerline porosity more
precisely than the equisolidification time method. The comparison between
equisolidification method and gradient criterion is shown in fig. 2.5.
C. The Niyama criterion (Niyama et al. 1981)
In 1982, Niyama et al.
found that the critical temperature gradient was inversely
proportional to the square root of the solidification time. Therefore, they proposed to use
G / (dT/dt)1/2
at the end of solidification as a criterion for porosity prediction. This criterion
was justified by Darcy’s Law because it included the physics behind the difficulty of
providing feed liquid in the last stages of solidification when the interdendritic liquid
channels are almost closed. The critical value of the criterion was proven to be
independent of casting size, first by Niyama et al. and later by other researchers.
This criterion has been widely integrated into current existing computer software to relate
the output of the numerical heat transferring calculations (temperature gradient,
solidification time, etc.) to empirical findings on porosity. The reasons of its popularity
can be explained as per followings:
21
Table 2.2: Proposed and calculated critical values of several solidification parameters for centerline porosity prediction (S. Minakawa et al., 1985)
The criterion itself simple and only requires data obtainable from temperature
measurements for verification.
G/ (dT/dt)1/2 = (G/Vs)1/2, while G/Vs is the most important parameter governing
the constitutional under cooling, and hence decide the range of mushy zone,
columnar or equiaxed growth in solidification. The crucial condition of columnar
growth is, G/Vs >= mc0(1/k-1)/D, in which m is liquidus slope, c
0 is alloy
composition, k is the equilibrium distribution coefficient, and D is diffusion
coefficient in liquid. Therefore, this criterion has essentially a close relation with
the solidification process, and hence porosity formation.
The final solidification areas usually have a lower value of G/R1/2, because these
areas usually has lower G but higher Vs. The former is caused by the deteriorated
heat transferring condition at a final solidification area, while the later occurs due
to the phenomenon named as the acceleration of solidification (fig.2.6).
The authors have proposed a critical value of 1.0 (deg1/2.sec. cm-1) and its
effectiveness has been verified with steel casting. There exist different values for
different materials since the value is influenced by material properties as declared
by the authors (fig.2.7).
Fig.2.6: The relation between the experimentally determined G and t f
(Niyama et al. 1982)
Parameters Proposed Critical Values
Calculated critical values for plate thickness listed.
50 mm 25 mm 12.5 mm 5 mm G 0.22 – 0.44 1.8 – 2.2 3.6 – 4.4 6.6 – 8.0 14.6 – 19.7
The use of eqn. (2.12) to approximate the final pore volume fraction in conjunction with
the dimensionless Niyama criterion is a novel concept that significantly enhances the
usefulness of this new criterion. Rather than having to compare criterion values to
generally unknown threshold values to determine whether porosity forms, the present
criterion allows one simply to compute the volume fraction of shrinkage porosity
throughout the casting.
To summarize, the dimensionless Niyama criterion, Ny*, can be calculated from local
casting conditions and material properties using eqn.(2.9), which also provides the value
of the integral I(gl,cr) This integral value is then used to determine the value of gl,cr using
eqn.(2.8). Finally, eqn.(2.12) is used to determine the shrinkage pore volume fraction, gp.
When the new criterion is incorporated into casting simulation software, the user need not
even be aware of it; the software can simply provide throughout the casting the volume
fraction of shrinkage porosity predicted by the present method.
E. Lee et al. criterion
Following the Niyama criterion, Lee et al. developed a criteria function for long freezing
range aluminum alloys, sometimes referred to a LCC after the authors. It also referred as
Feeding Efficiency Parameter (FEP). The most important difference between LCC and
the Niyama criterion lies in the way they correlate permeability to liquid fraction in the
mushy region. The criterion is given by
Where tf is solidification time and Vs is the solid front velocity. The work of Lee et al. (1989) focused on plate casting with varying lengths and riser sizes.
The porosity content was calculated down the length of the casting and correlated with the
thermal gradient (fig.2.9 a), solidus velocity (fig.2.9 b), solidification time (fig.2.10 a) and
LCC parameter (fig.2.10b). Their results demonstrated that areas in a casting with LCC
values of 10C min5/3 / cm2 or less tend to contain porosity.
29
(a) (b)
Fig. 2.9(a) Relation of thermal gradient and porosity content (b) Relation of solidus velocity and porosity content, where R2 is square of multiple correlation coefficient
(Lee et al.,1989)
(a) (b)
Fig. 2.10 (a)Porosity content as a function of solidification time. (b) Prediction of porosity by feeding efficiency parameter (Lee et al., 1989)
F. Feeding resistance number
In a different, more empirical approach, Suri et al. (1994) developed a porosity parameter
based on the feeding resistance number (FRN) defined in eqn.(2.14). Their criterion
attempts to account for such variables as liquid viscosity, solidification shrinkage, primary
dendrite arm spacing (in either columnar or equiaxed dendrites), and solidus velocity. The
result was a dimensionless number, which was used to assess the onset of microporosity
based on its magnitude relative to some critical threshold.
Nomenclature: G : Temperature gradient Vs: Solidification velocity ts: Local solidification time Cλ: Material constant dT/dt : Cooling rate ∆Pcr: Critical pressure drop β: Total solidification shrinkage ∆Tf: freezing range µl: Liquid dynamic viscosity
33
Both modulus and equi-solidification method not consider the effect of inter
dendrite and gas evolution.
Among the thermal-parameter based criterion, the Niyama criterion has the most
popularity for its well-accepted discriminability in predicting shrinkage and
porosity of casting steel and it is easy to verify this criterion with temperature
measurements.
Foundries use the Niyama criterion primarily in a qualitative fashion, to identify
regions in a casting that are likely to contain shrinkage porosity.
There are certain limitation of Niyama criterion namely
i. The threshold Niyama value below which shrinkage porosity forms is
generally unknown, other than for steel, and can be quite sensitive to the
type of alloy being cast and sometimes even to the casting conditions.
ii. The Niyama criterion does not provide the actual amount of shrinkage
porosity that forms, other than in a qualitative fashion (i.e., the lower the
Niyama value, the more shrinkage porosity forms).
The recently published dimensionless version of the Niyama criterion that accounts
for not only the thermal parameters but also the properties and the solidification
characteristics of the alloy. It can predict both qualitative and quantative prediction
of shrinkage porosity.
There are also other criteria functions available in literature like LCC, FRN,
Bishop, Davis etc. but every criterion function is having their own metal-process
combination from which they have derived.
It can be concluded by literature review that it is required to predict the size of
shrinkage defect accurately for major metal-process combination. It is also found
that criterion function is not cosidering the effect of geometric parameters along
with thermal parameters.
34
Chapter 3
Problem Definition
The manufacturing of most of castings was based on trial and error. Foundry plays with
process parameters to achieve desired quality level. Consideration of geometric parameters at
design stage itself would reduce these numbers of trials. As seen in previous chapter, criterion
function is not considered the effect of the geometric parameters along with thermal
parameters.
3.1 Motivation
Starting from the middle of 1980s’, due to the decreasing cost of computers and advances in
computing methods, computer simulation of foundry process has been developed and
improved by both academic and industry. Studies on porosity have then stepped forward from
experiment-based investigations to computer simulation aided research. Most research jobs
have been done to explore the mechanism of porosity formation and the ways to predict it.
There have been, however, very few publications whose results can be directly applied in
mass production because the results of the studies have not been confirmed with tests in
manufacturing scale.
Computer simulation with solidification software, to which various criterion functions is
integrated, is a useful tool in predicting porosity. Generally, they are predicting the location of
shrinkage porosity by considering thermal parameters like temperature gradeint, cooling rate,
solidification front velocity etc. But there are certain limitations of each criterion function and
they are also limited to particular metal- process combination. It will be very helpful to
35
develop empirical model which can predict size of the shrinkage poristy considering the
geometric parameters along with thermal parameters.
3.2 Goal, Scope and Objectives
Goal
Prediction of size and location of shrinkage defect considering geometric and thermal
Solidification simulation was carried out for benchmark casting to check the possibility of
shrinkage porosity in casting using FEM and VEM. The methodology is already defined in
chapter 4.
It is obvious that probable location of the shrinkage porosity is the region which solidifies
last. The region solidifies last is referred as hot spot. So, hotspot is the location at which
shrinkage porosity likely to occur. Solidification simulation using FEM gives the
solidification pattern while VEM directly gives the location of hotspot.
Temperature gradient, solidification time and temperature can be found out using FEM. The
threshold value of various criterion functions can be calculated using thermal gradient and
temperature. The calculation part of the various criterion functions has been omitted becasuse
some of the threshold values of metals selected for cassting are not available in literature. So,
solidification pattern of different metals will be shown. Also, the solidification simulation is
shown jointly for FEM and VEM for each metal but the particular details for FEM and VEM
are described seperately. The same input parameters are given to both simulation methods.
Case I Case II Case III Case IV
50
A. Solidification simulation using FEM
The basic methodolgoy for solidfication simulation is remained same as discussed in section
4.1.1. The only difference here is that the input parameters are different. The input parameters
for different metals are shown in table 5.2. The temperature dependent properties of different
metals are shown in follwing section
The variation in thermal properties of silica sand is assumed to be constant throughout
solidification. Thermal conductivity and specific heat of silica sand is 0.519 W/m2 K and 1170
J/kg. The density of silica sand is taken as 1490 kg/m3. IHTC for sand mould to atmosphere
is taken as 11.2 W/m2 K. The solidifcation simulations of ductile iron, plain carbon steel and
stainless steel are shown in follwing section.
Table 5.2: Input parameters for solidification simulation using FEM
Analysis Full Transient Thermal Element Plane 55element behavior Plane thicknessMaterial mode 1 CastingMaterial model 2 MouldMaterial model 3 Atmosphere
Mesh attributes Area 1- for MetalArea 2- for sand mould Area 3 – for Atmosphere
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