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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.
Chairman External Examiner Internal Examiner Guide Date: Mechanical Engineering Department, IIT Bombay - Mumbai
Declaration of Academic Integrity
“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).
ii
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
iii
5 SHRINAKGE DEFECT SIZE PREDICTION 47 5.1 Benchmark shape 47 5.2 Solidification Simulation of Benchmark Shape 49 5.2.1 Solidification simulation : Ductile iron 51 5.2.2 Solidification simulation : Plain carbon steel 53
5.2.3 Solidification simulation : Stainless steel 55 5.3 Casting Experiements and Results 57 5.3.1 Ductile iron 57 5.3.2 Plain carbon steel 63 5.3.3 Stainless steel 67 5.4 Empirical Model Development 71 5.4.1 Approach 71 5.4.2 Ductile iron 76 5.4.3 Plain carbon steel 79 5.4.4 Stainless steel 82 5.5 Summary 87 6 SUMMARY AND FUTURE WORK 89 8.1 Summary 89 8.2 Future work 91 Annexure I : Comparison of Casting Simulation Software 93 Annexure II : Data for Regression Analysis – Ductile iron 95 Annexure III : Data for Regression Analysis - Plain Carbon Steel (AISI 1005) 97 Annexure IV : Data for Regression Analysis – SS 410 99 References 101 Acknowledgement 105
iv
List of Figures
Figure Description Page
1.1 Porosity in Casting
2
2.1 Solidification of a bar casting
6
2.2 Representation of the origin of porosity as section thickness is increased.
8
2.3 Shrinkage prediction by modulus method 18
2.4 Shrinkage porosity prediction by equisolidification Method 19
2.5
Comparison of Gradient and equisolidification time method 20
2.6
The relation between the experimentally determined G and tf 21
2.7 The relation between the experimentally determined critical Niyama
criterion and the calculated tf
22
2.8 Schematic of a 1-D mushy zone solidifying with constant temperature gradient, G and isotherm velocity, R
24
2.9 (a) Relation of thermal gradient and porosity content (b) Relation of solidus velocity and porosity content
29 29
2.10 (a) Porosity content as a function of solidification time. (b) Prediction of porosity by feeding efficiency parameter.
29 29
4.1 Approach to locate shrinkage porosity 40
4.2 Geometric parameters of L shape casting 42
4.3 Modelling and meshing: cast steel L shape casting 42
4.4 Solidification simulation of L junction using FEM
43
4.5 Solidification simulation using VEM 45
5.1 Benchmark shape 48
5.2 3D model of Benchmark Shape 49
5.3 Temperature dependent (a) Specific heat (b) Density : Ductile iron 51
5.4 Solidification simulation using FEM and VEM: Ductile iron
52
5.5 Temperature dependent (a) Thermal conductivity (b) Specific heat (c) Density: Plain carbon steel
53
v
Figure Description Page
5.6 Solidification simulation using FEM and VEM: Plain carbon steel
54
5.7 Temperature dependent (a) Thermal conductivity (b) Specific heat (c) Density: Stainless steel
55
5.8 Solidification simulation using FEM and VEM: Stainless steel
56
5.9 Wooden Patterns for Casting 58
5.10 Layout, runner, gating and cavity of casting – Ductile iron 59
5.11 Setup of casting – Ductile iron 60
5.12 Benchmark casting – Ductile iron 60
5.13 Porosity in benchmark casting – Ductile iron 62
5.14 Layout, runner, gating and cavity of casting – Plain carbon steel 64
5.15 Setup of casting – Plain carbon steel 64
5.16 Benchmark casting – Plain carbon steel 65
5.17 Porosity in benchmark casting – Plain carbon steel 66
5.18 Layout, runner, gating and cavity of casting – Stainless steel 68
5.19 Setup of casting – Stainless steel 68
5.20 Benchmark casting – Stainless steel 69
5.21 Porosity in benchmark casting – Stainless steel 70
5.22 Maximum gradient 73
5.23 Adjustment of percent limiting value of gradient in AutoCAST® 73
5.24 Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, 2 and 3 – Ductile iron
77
5.25 Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, 2 and 3 - Plain Carbon Steel
80
5.26 Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, Junction 2 and Junction 3 – SS 410
83
5.27 T junction casting for validation – SS 410
85
5.28 Feed path : validation casting 86
5.29 Hotspot: Validation casting 86
vi
List of Tables
Table Description Page
2.1 Categories of approach on the basis of literature
13
2.2 Proposed and calculated critical values of several solidification parameters for centreline porosity prediction
21
2.3 Thermal parameters based criteria for porosity prediction 32
4.1 Nomenclature for L junction 41
4.2 Properties of Cast steel and sand mould 41
4.3 Input parameters for Cast steel 42
4.4 Comparison of various Criteria for case I 44
5.1 Variations in benchmark shape 48
5.2 Input parameters for solidification simulation using FEM 50
5.3 Chemical Composition: Ductile iron 59
5.4 Experimental details: Ductile iron 59
5.5 Surface sink and Shrinkage porosity distribution - Ductile iron 61
5.6 Chemical Composition: Plain carbon steel 63
5.7 Experimental details: Plain carbon steel 64
5.8 Porosity distribution - Plain carbon steel 67
5.9 Chemical Composition: Stainless steel 67
5.10 Experimental details: Stainless steel 68
5.11 Porosity distribution - Stainless steel 71
5.12 Limiting value of gradient for junction 1, 2 and 3 – Ductile iron 76
5.13 Regression Statistics – Ductile iron 78
5.14 Regression analysis – Ductile iron 79
5.15 Limiting value of gradient for junction 1, 2 and 3 - Plain carbon steel 80
5.16 Regression Statistics – Plain carbon steel 81
5.17 Regression analysis – Plain carbon steel 81
5.18 Limiting value of gradient for junction 1, 2 and 3 - Stainless steel 82
vii
Table Description Page5.19 Regression Statistics – Stainless steel 84
5.20 Regression analysis – Stainless steel 84
5.21 Co efficient of empirical model 88
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)
ix
Pliq Melt pressure Pcr Critical pressure ∆Pcr Critical pressure drop = Pliq - Pcr
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:
Improper closure: flash, mismatch
Incomplete filling: cold shut, misrun
Gaseous entrapments: blow holes, gas porosity
Solid inclusions: sand inclusions, slag inclusions
Solidification shrinkage: cavity, porosity, centerline, sink
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)
3
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.
5
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).
11
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
G/ (dT/dt)1/2 1.0 0.92 – 1.1 0.93 – 1.08 0.83- 0.98 0.94 – 1.07
22
Fig.2.7: The relation between the experimentally determined critical Niyama criterion value and the calculated t
f(Niyama et al. 1982)
D. The dimensionless Niyama criterion
Foundries use the Niyama criterion primarily in a qualitative fashion, to identify regions in
a casting that are likely to contain shrinkage porosity. The reason for such limited use is
twofold:
(1) 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 (e.g., sand mold vs. steel mold,
application of pressure, etc.); and
(2) 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).
Threshold Niyama values reported in the literature depend on the sensitivity of the method
with which the presence or absence of shrinkage porosity was determined. In another
study involving steel castings, Carlson et al. found that, in order to predict micro shrinkage
that is not detectable using radiography, a much higher threshold Niyama value should be
used.
There is no reason to use the Niyama criterion only for steel castings. It can also be
expected to predict shrinkage porosity in other alloys, such as those based on Ni, Mg, or
Al. Carlson et al. (2009) determined a threshold Niyama value for Ni-alloy sand castings
that is higher than the one found earlier for steel. Shrinkage porosity is also a widespread
problem in Mg-alloy castings, and at least one study has been performed in which the
23
3l
2K= Ko
g ........................................................................................................... .(2.2)(1 )− l
eqng
Niyama criterion was found to correlate qualitatively well with porosity measurements for
Mg-alloy castings. For Al-alloy castings, hydrogen-related gas porosity is often a major
factor, but if the Al alloy is well degassed, shrinkage porosity can also be a problem.
However, threshold Niyama criterion values for Mg and Al alloys have not been
established. Because the Niyama criterion is only a function of thermal parameters and
does not take into account the properties and solidification characteristics of an alloy, it is
not universal in nature; calculated Niyama criterion values for one alloy do not mean the
same as those for another alloy.
Carlson et al. (2009) had investigated the development of criteria function that can be used
to predict not only the presence of shrinkage porosity in castings but also the quantity of
shrinkage that forms. The ability to predict actual shrinkage pore volume fractions (or
percentages) completely avoids the need to know threshold values. The criterion function
developed by Carlson et al. is a 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. Once the dimensionless Niyama criterion is presented, it is
shown how it can be used to predict the shrinkage pore volume fraction knowing only the
solid fraction-temperature curve and the total solidification shrinkage of the alloy.
D.1 Model Development
Carlson et al. (2009) had derived the present criterion using directionally solidifying 1-D
system. It can be seen in fig 2.8. Darcy’s law can be written for this system as
Where gl is the liquid volume fraction, ul is the liquid velocity in the mushy zone (i.e.,
shrinkage velocity), µl is the liquid dynamic viscosity, P is the melt pressure, and x is the
spatial coordinate, as indicated in Figure. The permeability in the mushy zone, K, is
determined from the Kozeny–Carman relation
Where, K0= λ2/180; in which λ2 is the secondary dendrite arm spacing (SDAS).
l ( ) ..................................................................................................... .(2.1)= −g uldPK eqndxlμ
24
Fig.2.8: Schematic of a 1-D mushy zone solidifying with constant temperature gradient, G and isotherm velocity, R (Carlson et al., 2009)
Assuming that the liquid and solid densities (ρs and ρl) are constant during solidification,
one can define the total solidification shrinkage in terms of these densities as β = (ρs - ρl)/
ρl. Using β to simplify the 1-D mass conservation equation and then integrating the result,
it can be shown that the shrinkage velocity throughout the mushy zone is constant and can
be expressed as µl =- βR , where R is the constant isotherm velocity. If one further realizes
that R can be expressed in terms of the temperature gradient, G and cooling rate dT/dt,
shrinkage velocity in mushy zone can be written as
Substituting this expression into eqn.(2.1) yields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 . 4 )l ld P g d T e q nd x K g d t
μ β ⎡ ⎤= ⎢ ⎥⎣ ⎦
Fig. 2.8 illustrates that, as the solid fraction increases, the melt pressure decreases from the
value at the liquidus, Pliq, down to some critical pressure, Pcr, at which point shrinkage
[ ]1 / / ...................................................................................... .(2.3)= − = −u R dT dt G eqnβ β
25
cr 0
2P = - P = - ..................................................................................................... .(2.5)r
⎡ ⎤⎢ ⎥⎣ ⎦
eqnσσ
porosity begins to form. For convenience, the critical pressure drop can be defined as
∆Pcr = Pliq - Pcr. The pressure at liquidus is simply the sum of the ambient pressure of the
system and the local head pressure. The critical pressure is determined by considering the
mechanical equilibrium necessary for a stable pore to exist. This equilibrium is given by
the Young–Laplace equation as Pcr = Pp – Pσ, where Pp is the pressure inside the pore and Pσ
is the capillary pressure.
The capillary pressure is given by Pσ = 2σ / r0 , where σ is the surface tension between the
pore and the surrounding liquid and r0 is the initial radius of curvature at pore formation.
For pure shrinkage, in the absence of dissolved gases in the melt, the pressure inside the
pores is negligibly small (due only to the vapor pressure of the elements in the melt). With
this, the Young–Laplace equation simplifies to
Note that eqn (2.5) implies that the critical pressure is a negative number; this is
reasonable, because the surface tension must be overcome before porosity can form.
The point in space at which shrinkage porosity begins to form can be determined by
integrating eqn. (2.4) over the mushy zone from the critical point to the liquidus, assuming
that the viscosity, temperature gradient, and cooling rate are constant over the interval
being considered:
Where xcr, Tcr, and gl,cr are the position, temperature, and liquid fraction respectively, at
which the melt pressure drops to Pcr and porosity begins to form.
......................................................2,
0
1
=
liql ll l
cr cr
lll
ll cr
Tg dT g dxdx dTKG dt KG dTx T
dT g dT dgG dt Kdg dtg
μ β μ β
μ β
⎡ ⎤ ⎛ ⎞= ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
∫ ∫
∫ ................................................... .(2.6)eqn
26
22
( , )...................................................................................... .(2.7)Δ ⎡ ⎤Δ = ⎢ ⎥⎣ ⎦l f
cr l crT dTP I g eqn
dtμ βλ
cr 22*,
l fl f
cr
P ( ).......................................................... .(2.9)TdTT
Pdt
Δ= = =
Δ⎛ ⎞Δ ⎜ ⎟ Δ⎝ ⎠
yy l cr
G NN I g eqnλ λμ β
μ β
Eqn. (2.6) is essentially the same expression that was derived by Niyama et al. Lee et al.,
Sigworth and Wang, and Dantzig and Rappaz. Subsequent differences in their resulting
criteria (as well as in the present criterion) stem from assumptions made regarding the
solid fraction-temperature curve and the permeability in the integral in eqn. (2.6), as well
as the manner in which the result is cast and used. In order to non-dimensionalize the final
integral in eqn. (2.6), one can introduce a dimensionless temperature, sol fT T Tθ = − Δ ,
where ∆Tf = Tliq - Tsol is the freezing range, Tliq is the liquidus temperature, and Tsol is
the temperature at which the alloy is 100 pct solidified. Introducing this expression along
with eqn.(2.2) into eqn. (2.6) yields
Where
In the present study, however, this integral is evaluated numerically, using available alloy
solid fraction-temperature curve data. The use of realistic solid fraction-temperature curve
data makes the present criterion more general and accurate than previous criteria. This is
especially true for industrially relevant multi component alloys for which analytical
expressions for θ(gl) cannot be obtained.
Introducing the Niyama criterion and rearranging eqn. (2.8) provides an expression for the
present dimensionless Niyama criterion, Ny*:
The dimensionless Niyama criterion given by eqn. (2.9) accounts not only for the local
thermal conditions ( dT/dt ,G) considered by the original Niyama criterion, but also for the
properties and solidification characteristics of the alloy (µl, β, ∆Tf, and λ2) and the critical
pressure drop across the mushy zone (∆Pcr). The most important feature of eqn.(2.9) is
21
, 2
,
(1 )( ) 180 ............................................................................. .(2.8)⎡ ⎤−
= ⎢ ⎥⎣ ⎦
∫ ll cr l
lg ll cr
g dI g dg eqng dg
θ
27
[ ]1 / 3
*5 / 6 ................................ .(2.11)cr cr
y yffl l
G P dT PN C C N eqndT dt T dt Tλλ μ β μ β
−Δ Δ⎡ ⎤= = ⎢ ⎥Δ Δ⎣ ⎦
that Ny* can be expressed as a function of only the solid fraction-temperature curve, the
SDAS, the critical pressure drop, and the other parameters in the denominator of the third
term in eqn.(2.9). It will be demonstrated later that this feature generalizes the predictive
capability of this new criterion. The SDAS can be determined as a function of the cooling
rate from the relation
Where Cλ is constant which depends on material. Using eqn.(2.10), the following alternate
form can be written
Eqn (2.11) indicates that the dimensionless Niyama criterion is proportional to G(dT/dt)5/6,
rather than to G(dT/dt)1/2, as in the original Niyama criterion.
The difference from the original Niyama criterion in the dependence on thermal conditions
is due solely to the SDAS and the effect that this arm spacing has on the permeability in
the mushy zone. The increased dependence on the cooling rate can be expected to make
the dimensionless Niyama criterion more generally applicable for widely varying section
thicknesses and different mold materials (sand, steel, copper, etc.).
D.2 Shrinkage porosity prediction
The dimensionless Niyama criterion developed in previous section is now used to predict
shrinkage location and pore volume fraction once the critical liquid fraction has been
determined; it is possible to use the continuity equation to approximate the final pore
volume fraction, gp. By assuming that local feeding flow ceases once shrinkage porosity
forms (i.e., the remaining shrinkage is fed by porosity formation only), the continuity
equation can be simplified and integrated to give the relation
1 / 3
2 ..................................................................................................... .(2.10)dTC eqndtλλ
−⎡ ⎤= ⎢ ⎥⎣ ⎦
,' .............................................................................................................. .(2.12)p l crg g eqnβ=
28
2 / 3
........................................................................................................... .(2.13)=fGtLCC eqn
Vs
Where ' /( 1) s l sβ β β ρ ρ ρ= + = −
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.
2 ................................................................................................ .(2.14)f
sL
n TFRN eqnGV Dμ
ρ βΔ
=
30
dTVs G cooling rate / temp gradient at end of solidificationdt
⎡ ⎤= =⎢ ⎥⎣ ⎦
1 36 2P 2 3 9 9 6 x1 0 F R N 9 1x1 0 F R N eq n 2 1 5..................(% ) . . ( ) . ( ) .( . )−−= + −
Where n is a constant, µ is the viscosity of the melt, ∆T is the freezing range of the alloy,
ρL is the density of liquid, β is the shrinkage ratio and D is the characteristic size of solid
particles. The values of n and D depend on casting macrostructure. n = 16 for columnar
dendrites, and 216 for equiaxed dendrites and eutectic phases. D represents the
characteristic length scale of the solid phase; the value of D may be approximated to an
estimated value of dendrite/grain size.
A high value of FRN indicates higher resistance to feeding and hence higher potential for
pore formation. FRN is useful in casting simulations which address solidification kinetics
during freezing. Suri et al. used the FRN criteria to predict measured porosity in plate
casting by correlating the FRN with percent porosity using a second order polynomial and
it is given by
They also attempted to apply the FRN criteria to more complex aluminum casting with
varying degrees of success. A Drawback with this approach is that it requires
quantification of casting variables such as the liquid metal viscosity and the primary
dendrite arm spacing, and this limits the model’s applicability to specific alloy system for
which this information is available. Nevertheless, the FRN criteria by Suri et al. seems be
quite effective in predicting porosity.
G. Franco criterion
Chiesea et al. (1998) proposed the following criteria for porosity evaluation in aluminum,
commonly known as the Franco criteria.
Where P is probability of local porosity, K is Melt quality factor, tsl is local solidification
time (time from liquidus to solidus), Vs is solidus velocity and m & n are constants. K, m
and n are constants determined from pouring test bars and measuring porosity as a
function of tsl and Vs. K is 0.241; m is 0.7 and n is 0.12 experimentally determined factors
* * ................................................................................................... .(2.16)m nP K t V eqnssl=
31
for Aluminum casting. The formula was proposed for aluminum and was derived
assuming directional solidification. If the criteria are to be applied to graphitic iron, it must
be established that the nature of solidification is sufficiently directional. Alternatively, the
extension of the criteria to mushy solidification of eutectic graphitic irons needs to be
validated.
The existing thermal parameter criteria proposed in literature so far, including temperature
gradient G, are tabulated in Table 2.3. Some of these criteria can be reduced to the form
of Gx/V
s
y (x varies over the range 0~2 and y varies over the range of 0.25~1), among
which the Niyama criterion that can be reduced to G/Vs is a representative one.
2.6 Summary
Regarding to the shrinkage porosity defect prediction for castings, following aspects have
been clarified with this literature review.
It is usually difficult to eliminate shrinkage porosity completely from castings,
while reducing it or moving it to an unimportant area can be a choice. So there is
requirement of shrinkage porosity prediction with the help of simulation software
because it eliminates shop floor trials.
Classification of porosity is well defined in literature.
Porosity formation and mathematical model is also available in literature. The
summary of approaches and modeling of porosity formation is already given in
Table 2.1.
There are three main approaches, modulus and equisolidification method, criterion
function method, and direct numerical simulation method, with which shrinkage
and porosity can be predicted.
Modulus method can be used as a quick guide at the initial stage of mold designing
for simple geometry. It is very difficult to find modulus of last freezing region for
complex geometry. This method is not taking into account variation of mold
temperature and material physical properties.
Equi-solidification method is reliable in predicting gross shrinkage and porosity at
a final solidification area.
32
Table 2.3: Thermal parameters based criteria for porosity prediction (Research Thesis- Minami Rin, 2005)
Author Year Criterion Metal Threshold value
Bishop et al. 1951 G
Cast bar
Cast steel plate
1.3 – 2.6 0C/cm 0.2 - 0.4 0C/cm
Davies
1975 G/Vs NA NA
Khan
1980 1/Vsn NA NA
Niyama et al.
1982 G/(dT/dt)1/2 Steel 1
Lecomte – Beckers
1988 G/Vs NA NA
Lee et al. (LCC)
1990 Gts2/3/ Vs Al alloy 1-3
S T Kao et al
1994 G0.38 / Vs 1.62 NA NA
Suri et al. (FRN) 1994
2sL
n TFRNGV Dμ
ρ βΔ
=
Al alloy % porosity =
23.9 + 9.6x10-6(FRN) + 9.1x10-13(FRN)2
F. Chisea (FCC)
1998 1/tsm Vsn Al alloy 1.52
Carlson et al. (Dimensionless Niyama)
2009
cr2*
l f
PdTTdt
yGN λ
μ β
Δ=
⎛ ⎞Δ ⎜ ⎟⎝ ⎠
WCB steel Al A356 Mg alloy A Z91D.
610 (0.01 pct) 137(0.1 pct) 211 (0.01 pct) 23 (0.1 pct)
776(0.o1 pct) 99 (0.1 pct)
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
parameters during casting solidification.
Scope
Ferrour sand casting – Ductile iron, plain carbon steel, stainless steel
Objectives
Study and comparison of various models for prediction of shrinkage.
To apply the different criterion function to L shape casting, available in literature,
using finite element method and vector element method.
To decide the benchmark shape and check it for shrinkage porosity using FEM and
VEM.
To perform experiments using benchmark shape for the development of empirical
model to predict the size of shrinkage porosity.
To develop the empirical model considering effects of geometric and thermal
parameters in T junction.
3.3 Approach to Project
This is an attempt to develop some systematic approach for shrinkage porosity prediction
using various criterion functions. As discussed in section 2.9, shrinkage porosity can be
predicted by various techniques but they are limited to particular metal- process combination.
36
This project includes comparison of the various criterion functions applied to L shape casting,
available in literature, using finite element method and vector element method. Shrinkage
porosity prediction can be made with the help of finite element method and compared with
VEM. The attempt is also made to predict size of shrinkage porosity considering geometric
parameters along with thermal parameters for T junction. Various experiments will be carried
out and sufficient data will be generated for further analysis using casting simulation
software. The empirical model will be developed for prediction of porosity size for ductile
iron, plain carbon steel and stainless steel.
So, the aim of this project is to study existing method for prediction of location and develop
some empirical model which will predict size of porosity considering geometric parameters
along with thermal parameters.
37
Chapter 4
Shrinkage Defect Location
To predict the locations where porosity would occur, a judging criterion is needed. The
popular thermal-parameter based criteria from the literature review are summarized in
Table 2.2 (Chapter 2). There has been so far no agreement on which criterion is the best in
predicting shrinkage porosity, as a matter of fact, however, the Niyama criterion, G/R1/2,
has been widely implemented in current well-used commercial casting simulation
software. The reasons of the popularity of this criterion were also discussed in chapter 2.
Therefore it is preferable to use Niyama criterion to predict the location of shrinkage
porosity. Threshold value of other criterion functions for the cast steel is not availabe as
they are limited to particular metal-process combination. The attempt is made to establish
the threshold value for cast steel.
4.1 Approach to Predict Location of Shrinkage Porosity
The location of shrinkage porosity can be predicted with the help of various criterion
functions which further depends on thermal parameters like temperature gradient, cooling
rate, solidification velocity etc. These thermal parameters can be found out using
solidification simulation. There are various numeric techniques are used for solidification
simulation like FEM, FDM, FVM etc. As discussed in chapter 2, finite element method
(FEM) and vector element method (VEM) will be used to predict the location of the
porosity in casting with the help of solidification simulation.
4.1.1. Solidification simulation using FEM
A FEM based commercial software; ANSYS® is used for solidification simulation. The
output of the analysis is thermal gradient and temperature at each node. Various criteria
38
are calculated with the help of these outputs. The analysis is carried out in three steps as
given below:
Pre-processing is used to define geometry, material property, and element type for the
analysis.
Processing phase defines analysis type like transient or steady state, apply loads and solve
the problem. It can also be referred as solution phase
Post processing is to review the result in the form of graphs or tables. The general
postprocessor is used to review results at one sub step (time step) over the entire model.
The time-history postprocessor is used to review results at specific points in the model
over all time steps.
The following assumptions are made for the analysis:
• Contact resistance between the mold and cooling material is negligible.
• In practice the temperature difference between the mould surface and surrounding air
is not substantial hence radiation transfer can be ignored.
• Mould cavity is instantaneously filled with molten metal.
• Outer surface of the mould is initially assumed to be at ambient temperature.
• The bottom surfaces of the casting are always in contact with the mould.
• The vertical surfaces of casting are in contact with the mould i.e. no air gap in between
Approach to locate shrinkage porosity is already discussed in previous section. Various
steps for thermal analysis are given below.
A. Modelling
Modelling can be done with the help of any available CAD software and can be inserted in
FEM based software using geometry transformation. Model can also be generated using
FEM based software. In FEM based software (ANSYS®), model generation means
generation of nodes that represents the spatial volume. Modelling consists of defining two
parts one is sand mould and other is the castings with proportional dimensions.
39
...................................................................... .( 4 .1)T Tlf eq ns T T sl−
=−
B. Meshing
The main aim of the analysis is to get temperature distribution with respect to time.
Element, PLANE 55 is chosen in ANSYS® which has capability of transient heat transfer
analysis. The element has four nodes with a single degree of freedom, temperature, at each
node. PLANE 55 is a four node quadrilateral element with linear shape functions.
C. Input parameters
In present work, analysis is carried out for the time from pouring temperature to
solidification temperature. Input parameters required for model are as follows:
Initial boundary conditions for sand mould, casting and atmosphere air.
Thermal boundary condition is convective heat transfer co efficient.
Material specifications like density, specific heat and thermal conductivity is required for
sand, cast and atmospheric air.
D. Processing and post processing
Processing is carried out by applying proper boundary conditions and material
specifications for the solidification time by selecting proper time steps. Post processor
stage includes collection of results from processing stage. A little consideration is required
while collecting results from processor stage. This consideration is that the resultant
thermal gradient and temperature should be taken few steps before the solidification time
because shrinkage porosity criteria are evaluated near the end of solidification, when
solidification forms. This is important to note, as the choice of evaluation temperature can
significantly influence the resulting criteria values (Carlson et al., 2008). Solid fraction at
that time step should be in between 0.9 to 1. Solid fraction is approximately calculated
using equation given below.
Where fs is solid fraction, Tl and Ts are liquidus and solidus temperature respectively.
40
Fig. 4.1: Approach to locate shrinkage porosity
The approach to predict the location of shrinkage porisity can also be explained by fig.
4.1. L junction of cast steel is analysed for checking the possibility of shrinkage porosity
in casting. Experimental results are availabe from literature (Joshi et al., 2010).
E. Cast Steel L Junction ‐ (L‐90‐30‐30‐240‐240‐0‐0)
Solidification simulation of ‘L’ junction with equal elements (arm length and thickness)
was carried out as shown in fig. 4.2. The nomenclature of L junction can be explained by
table 4.1. These casting was produced in cast steel (0.2% C) with conventional green sand
casting process at pouring temperature of 1680 °C. The filter was not used in molten metal
channel. The temperature dependent properties of cast steel taken into consideration are as
shown in table 4.2. The results are shown in fig. 4.3 and 4.4 The Comparsion of various
criterion functions is shown in table 4.4.
41
Fig. 4.2: Geometric parameters of L shape casting
Table 4.1: Nomenclature for L junction
Table 4.2: Properties of Cast steel and Sand
Case #
Nomenclature for L Junction
Angle θ
Thick t
(mm)
Height h
(mm)
Length L
(mm)
Inner radius
r1 (mm)
Outer radius
r2 (mm)
Time Sec
1 L-90-30-30-240-240-0-0 90 30 30 240 0 0 267
Cast Steel
Temperature (K)
Conductivity
(W/m K) Density
(kg/ m3) Specific heat
(KJ/kg K) Interface Heat
Transfer Co
efficient (W/m2K)
255 29.9 7500 0.507
570 1723.70 32 0.804
1783.15 25.3 0.837
1852.59 25.3 7000 0.837
Sand
300 0.519 1 490 1.17 11.25
42
Table 4.3: Input parameters for Cast steel
Fig. 4.3: Modelling and meshing: cast steel L shape casting
Analysis Full Transient Thermal
Element Plane 55
Element behavior Plane thickness
Material mode 1 / 2 / 3 Casting / Mould / Atmosphere
Mesh attributes 1/ 2 / 3 Metal / Mould / Atmosphere
Mesh size (casting) 3 and refined by 1
Mesh size (Mould + Atmosphere) 3
Load condition
Ref. Temp 300 Initial condition of Metal (Pouring temperature) 1953
Initial condition of sand 300
Load step 1
Time 267
Time step 1
Model Meshing
43
Solidification Simulation
Probable locations of Shrinkage Porosity
Observed Location of Porosity
Fig. 4.4: Solidification simulation of L junction using FEM
44
Table 4.4: Comparison of various Criteria for case I
Shrinakge porosity is likely occurring at a place where threshold value of Niyama is less
than unity. It can be observed from table 4.4 that the value of Niyama which is nearer to
unity on node number 289 and 1467. These nodes are illustrated in fig. 4.4 by circling on
to it. The threshold values of other criterion functions are also calculated on those nodes.
4.1.2. Solidification simulation using VEM
As discussed in chapter 2, casting simulation is carried out using Vector Element Method
which identifies the feed metal paths. It based on the principle that the direction of the
highest temperature gradeint (feed path of metal) at any point inside the casting is given by
vector sum of individual thermal flux in all directions. The starting point may be any point
of casting geometry. After taking resultant of the all vector are indicating the location of
hot spot.
Solidification simulation of L-junctions with equal elements (arm length and thickness)
was carried out using VEM based casting simulation software AutoCAST®. The results of
progressive solidification simulation using VEM are shown in Figure 4.5. The location os
hot spot with maximum temperature can also been seen in fig. 4.5. The feed paths are also
simulated and it is shown in fig. 4.6.
In simulation images, outer dark regions indicate solidified portion of the casting; inner
bright regions are those that will solidify subsequently. The white regions represent the hot
spot. Molten metal in hot spot region feeds surrounding regions as they solidify. If hot spot
is not fed by liquid metal from a feeder, it will eventually manifest into porosity defect. In
L junction, a significant hot spot is observed. The location of hot spot predicted by
solidfication simulation software is matching with the observed location of defect. It can
be easily seen in radiographical image of L junction (fig. 4.4).
Node Temp G r Vs fs Criterion Function
Niyama Davis LCC Bishop
289.0 1731.8 0.8 0.8 1.01 0.9 0.9 0.8 32.1 0.8
1467.0 1731.5 1.0 0.8 0.81 0.9 1.1 1.3 50.8 1.0
45
Location of hot spot using VEM
Feed path of L junction using VEM
Observed Location of Porosity
Fig. 4.5: Solidification simulation using VEM
46
4.2 Summary
Solidification simulation of L junction is carried out using FEM and VEM. The results can
be seen in fig. 4.4 and 4.5. Following points can be observed from simulation using FEM
and VEM.
Solidification time is 267 sec for the L junction as per simualtion using FEM.
It can be seen for the case Cast Steel L junction that solidification simulation is
predicting the shrinkage porosity location with the help of available criteria. It is
reasonably matches location which has been shown in experimental results.
Comparison of various criterion functions has also been prepared. By comparing
threshold value of LCC criterion and Davis criterion can be taken as 36.5 and 0.9
respectively for the case of Cast Steel.
Solidification time is 226.2 sec for the L junction as per simulation using VEM.
The VEM result of location of shrinkage porosity is matching with actual result of
shrinkage defect location.
As seen, solidifacation simulation using FEM and application of criterion function giving
the location of shrinakge defect. The methodology is well established and understood. It
is also required to predict the size of the shrinakge defect. It is equally important to
understant the dependency of the geometric parameters along with thermal parameter to
size of defect. This problem can be solved by developing empirical model which can
predict size of shrinakge defect.
47
Chapter 5
Shrinkage Defect Size Prediction
The locations of shrinkage porosity can be predicted with the help of solidfication simulation
using FEM or VEM. There is requirement of criterion functions (CFs) to predict it using
FEM. The methodology is well defined in chapter 4. The purpose of the present investigation
is to develop an empirical model that can be used to predict the size of shrinkage that forms.
The ability to predict actual shrinkage pore volume fractions (or percentage) completely
avoids the need to know threshold value.
A benchmark shape, a combination of three T junction is used for the understanding the
dependency of geometric parameters along with thermal parameter to the size of the shrinkage
defect. Various experiements were carried out for different ferrous metals. The experimental
data were used to set the limiting value of gradient in commerical casting simulation software
- AutoCAST. Further, simulation experiements were carried out using it. Finally, the results
from experiments and simulation were used as input to regression analysis to evolve an
empirical model for each metal. The empirical model is also validated for the stainless steel.
5.1 Benchmark Shape
As a need of development of empirical model for shrinkage porosity prediction, it is required
to decide benchmark shape which is having higher probability of porosity. It also required that
variation in the shape also leads to change the location as well as size of the shrinkage
porosity. To develop the model and illustrate the feature of the developed model, a simple
geometry has been simulated. Benchmark shape is as shown in fig.5.1 and table 5.1 shows
different variations in benchmark shape.
48
Benchmark shape is a combination T-junction J1, J2 & J3. Four benchmark casting will be
cast with variations as shown in table 5.1. The arm thickness (T), depth (d) and total length
(L) will remain constant. Stem thickness (t) will have four variations and stem length will
have three variations (Table 5.1). Ductile iron, plain carbon steel (AISI 1005) and stainless
steel 410 are taken for experiements as they are widely used in industry. 3D model of the
different variations are shown in fig.5.2
Fig. 5.1 Benchmark shape
Table 5.1 Variations in benchmark shape
Case #
Nomenclature
Arm thickness
T mm
Stem thickness
t mm
Stem length
l1
mm
Stem length
l2
mm
Stem length
l3
mm
Total length
L mm
Depthd
mm
1 20-5-40-60-80-240-40 20 5 40 60 80 240 40 2 20-10-40-60-80-240-40 20 10 40 60 80 240 40 3 20-20-40-60-80-240-40 20 20 40 60 80 240 40 4 20-30-40-60-80-240-40 20 30 40 60 80 240 40
ARM STEM
49
Fig. 5.2 3D model of Benchmark Shape
5.2 Solidification Simulation of Benchmark Shape
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
Mesh size (casting) 2.5 mmMesh size (Mould + Atmosphere) 5 mm
Load conditionRef. Temp 300
Initial condition of Metal (Pouring temperature) 1953 (AISI 1005)1973 (SS 410)1667 (SG Iron 500/7)
Initial condition of sand 300 Load step 1 Time 500 sec Time step 0.5 sec
51
B. Solidification simulation using VEM
The hot spot can be found out using vector element method (VEM). Mesh size for
solidification simulation using VEM is 0.5 mm and kept constant for each metal. The
locations of hot spot are marked with red colour. The hot spot of the bench mark casting for
different metals are shown in following section.
5.2.1 Solidification simulation ‐ Dutile iron
Temperature dependent properties are shown in fig. 5.3 (a), (b) and (c). The solidification
simulations are shown in fig. 5.4 (a), (b) and (c). IHTC for metal to sand mould and thermal
conductiviy are taken as 700 W/m2 K and 36 W/m K respectively and assumed to be constant
throughout the solidification.
(a)
(b)
Fig. 5.3 Temperature dependent (a) Specific heat (b) Density: Ductile iron
Fig
Case
1.
2.
3.
4.
g. 5.4 Solidifification sim
FEM
52
mulation usiing FEM an
nd VEM: Du
VE
uctile iron
EM
53
5.2.2 Solidification simulation ‐ Plain carbon steel
Temperature dependent properties are shown in fig. 5.5 (a), (b) and (c). The solidification
simulations are shown in fig. 5.6 (a), (b) and (c). IHTC for metal to sand mould is taken as
570 W/m2 K and assumed to be constant throughout solidification.
(a)
(b)
(c)
Fig. 5.5 Temperature dependent (a) Thermal conductivity (b) Specific heat (c) Density: Plain carbon steel
C
Fig. 5
Case
1.
2.
3.
4.
5.6 Solidificaation simula
FEM
54
ation using FEM and V
VEM: Plain
V
n carbon ste
VEM
eel
55
5.2.3 Solidification simulation ‐ Stainless steel
Temperature dependent properties are shown in fig. 5.7 (a), (b) and (c). The solidification
simulations are shown in fig. 5.8 (a), (b) and (c). IHTC for metal to sand mould is taken as
600 W/m2 K and assumed to be constant throughout solidification.
(a)
(b)
(c)
Fig. 5.7 Temperature dependent (a) Thermal conductivity (b) Specific heat (c) Density: stainless steel
Fig.
Case
1.
2.
3.
4.
. 5.8 Solidiffication sim
FEM
56
ulation usinng FEM and
d VEM: Sta
VE
inless steel
EM
57
Analysis for benchmark shape can be done with the help of FEM and VEM. It can be
observed that both approaches can be used for locating the hotspot region and both giving
comparatively good results. As vector element method (VEM) directly gives the location of
hotspot. Also, thermal gradient, solidfication time etc. can be obtained very easily using it. So,
it is preferrable to use VEM for further solidification analysis.
5.3 Casting Experiments and results
The various experiments were carried out to develop empirical model to predict size of
shrinkage defect considering geometric parameters alongwith thermal parameters. As
discussed in previous section, benchmark shape consist a combination of three T junctions.
The various ferrous metals are choosen for the experimentation. The concept behind the
selection of ferrous metals is that they are widely used in industry. Also, an attempt is made to
cover as distinct as possible types of ferrous metals like plain carbon steel, alloy steel and
high carbon steel.
The experiements are carried out for ductile iron (high carbon), plain carbon steel (low
carbon) and stainless steel (alloy steel). The experimentation work is kept restricted to sand
casting process. The details of various metals are discussed below.
Experiments were carried out for each variation in benchmark shape. Each variation of
benchmark shape has been cast twice for the same condition to accommodate the any
uncertainty. The patterns are made by solely wood (fig. 5.9). Allowances in making of
patterns are neglected due to simple shape. The following section will be based on
experiements of benchmark casting and its results.
5.3.1. Ductile iron
Ductile iron frequently referred to as nodular or SG iron is a recent member of the family of
cast irons. It contains spheroid graphite in the as cast condition, through the addition of
nucleating agents such as cerium or magnesium to the liquid iron. In the recent years, there
has
prod
is de
iron.
A. Ex
The
Mah
grav
with
syste
benc
expe
been increa
duct. A little
ecided to ca
. The chemi
xperiments
experiment
harashtra, In
vity. Filter w
h the help of
em is as sho
chmark cas
eriment for S
asing intere
e work has b
ast the SG ir
ical compos
for ductile
tation work
ndia. The m
was not put i
f shot blasti
own in fig.
ting is as
SG Iron.
Case I
Case III
est in appli
been done o
ron as a ini
sition is sho
iron
k was condu
metal is heat
in feeding c
ing process
5.10. The s
shown in
Fig. 5.9 W
58
ication of
on the pred
tiative work
own in table
ucted at S.S
ted to 1400
channels. Th
. The layou
setup of ben
fig. 5.12.
Wooden pat
SG iron an
diction of sh
k for predic
e 5.3
S. Industries
0 0C in indu
he benchma
ut of cavity
nchmark cas
Table 5.4
tterns for ca
nd it comes
hrinkage por
ction of shri
s - Ichalkar
uction furna
ark castings
in molding
sting is as s
shows the
Case II
Case IV
asting
s as alterna
rosity for S
inkage poro
ranji (Dist.:
ace and then
were prope
g box, sprue
shown in fig
e details of
ate as steel
G iron so it
osity for SG
Kolhapur)
n poured in
erly cleaned
e and gating
g. 5.11. The
f set up of
l
t
G
,
n
d
g
e
f
59
Table 5.3 Chemical Composition - Ductile iron (Source: S.S. Industries, Ichalkaranji, India)
Table 5.4 Experimental details - Ductile iron
Fig. 5.10: Layout, runner, gating and cavity of casting – Ductile iron
Sr No. Element Wt. % 1. Carbon 3.67 2. Manganese 1.3 3. Silicon 1.65 4. Phosphorus and sulphur 0.04 5. Magnesium 0.03 6. Ferrous 93
Size of molding box (350 mm x 350 mm x 100 mm) x 2
Size of Sprue Φ25 mm x 90 mm
Size of pouring basin Square c/s 50 mm
Size of gating Two gates of 20 mm x 5 mm
No. of cavity 2
Pouring Temp. 1394 0C
Feeder Not used
Gating
Runner
.
F
Case I
Case III
Fi
Fig.5.11: S
ig. 5.12: Be
60
Setup of cas
enchmark ca
sting – Duct
asting – Du
tile iron
Case II
Case IV
ctile iron
I
V
61
B. Results
In order to individuate the shrinkage porosity distribution, the benchmark castings were cut 20
mm from top surface parallel to length (L). The benchmark castings were properly cleaned
after cutting. Porosity locations can be visualized after cutting of benchmark casting. Porosity
size measured by filling measured quantity of water into porosity cavity and volume of water
was taken as volume of shrinkage porosity. To reduce the complexity in calculations and for
the sake of simplicity in measurement of shrinkage porosity, benchmark shape is sub divided
into three junctions J1, J2 and J3 (refer Fig. 5.1). Shrinkage porosity measurements were
taken separately at each junction for every metal. The procedure for measurement of porosity
will be same for each metal.
Small amount of shrinkage porosity is found in the case of SG iron due to surface sink. Strong
surface sink with the small amount of shrinkage porosity can be observed for the benchmark
casting (refer fig. 5.12). Fortunately, they are at same location where shrinkage porosity was
expected. Both surface sink and shrinkage porosity is taken into consideration. The
Summation of both will be considered as shrinkage porosity for the sake of development of
model. Strong centerline surface sink with shrinkage porosity is present in 1st and 2nd case.
Large amount of surface sink with small amount of shrinkage porosity can be observed at
junctions of case 3 and 4. In the following, the results are presented for each of the case in
Table 5.5. Porosity distribution for benchmark casting is shown in 5.13.
Table 5.5: Surface sink and Shrinkage porosity distribution - Ductile iron
Case # Sink + Porosity (cm3)
Junction 1 Junction 2 Junction 3 Total
1 0.25 0.25 0.5 1
2 0.3 0.35 + 0.1 0.65 +0.1 1.5
3 0.3 + 0.1 0.6 + 0.2 1.4 + 0.2 2.9
4 0.5 + 0.2 1.9 + 0.3 2.3 + 0.2 5.4
62
Case 1
Case 2
Case 3
Case 4
Fig. 5.13 Porosity in benchmark casting – Ductile iron
63
5.3.2 Plain carbon steel (AISI 1005)
Plain Carbon Steel has been selected as the cast metal because of it is widely used in industry
as large scale cast-steel products. Large scale cast steel products have been used as
fundamentals components of structures in broad fields of industries such as power generation,
construction, vessels and automobiles. We know that steel is very difficult metal for sand
casting process because of its low fluidity. Investment casting increases the castability of
plain carbon steel but it increases the cost of the product and also limits of size of product. So,
it is required to manufacture sound casting to avoid any solidification defects. The chemical
composition of plain carbon steel is given in table 5.6.
A. Experiments for plain carbon steel
The experiments were conducted at Manek Casting Pvt. Ltd. - Rajkot, Gujarat, India. Casting
is formed in green sand mould and silica is used as sand. The metal is heated to 1675 0C in
induction furnace and then poured in gravity. Filter was not put in feeding channels. The
benchmark castings were properly cleaned with the help of shot blasting process. The layout
of cavity in molding box, sprue and gating system is as shown in fig. 5.14. The setup of
benchmark casting is as shown in fig. 5.15. The benchmark casting is as shown in fig. 5.16.
Table 5.7 shows the details of set up of experiment for plain carbon steel.
Table 5.6 Chemical Composition - Plain Carbon Steel (Source: Manek Casting Pvt. Ltd.: Rajkot, India)
S. No. Element Wt. % 1. Carbon 0.045 2. Silicon 0.48 3. Manganese 0.85 4. Phosphorus and Sulphur 0.09 5. Ferrous 98
64
Table 5.7 Experimental details for Plain carbon steel
Fig. 5.14: Layout, runner, gating and cavity of casting – Plain carbon steel
Fig. 5.15: Setup of casting – Plain carbon steel
Size of molding box (340 mm x 340 mm x 125 mm) x 2
Size of Sprue Φ25 mm x 125 mm
Size of pouring basin Square c/s 50 mm
Size of gating Two gates of 25 mm x 10 mm
No. of cavity Double
Pouring Temp. 1657 0C
Feeder Not used
Runner
Gating
B. Re
As e
benc
amou
resul
casti
esults
expected, f
chmark cast
unt of poro
lts are pres
ing is shown
Case I
Case II
Fig.
fair amount
ting. Strong
osity can be
ented for e
n in 5.17.
I
II
5.16: bench
t of porosi
g centerline
e observed
each of the
65
hmark castin
ity can be
e porosity i
at junction
case in Tab
ng – Plain c
observed
is present in
ns of case 2
ble 5.7. Po
Case
Case I
carbon stee
in the diff
n 1st case n
2, 3 and 4.
rosity distri
II
IV
el
ferent secti
near junctio
In the fol
ibution for
ions of the
on 1. Large
llowing, the
benchmark
e
e
e
k
Fig. 5.17:: Porosity in
66
Case 1
Case 2
Case 3
Case 4
n benchmarrk shape- Pllain carbonn steel
67
Table 5.8: Porosity distribution – Plain carbon steel
5.3.3 Stainless steel (SS 410)
Stainless steel is an ideal base material for a host of commercial applications because of its
resistance to corrosion and staining, low maintenance, and familiar luster make it. SS 410
casting product is widely used in manufacturing of impeller of water pump. Sand casting of
SS 410 is very difficult process like other types of steel. Investment casting is better option to
cast SS 410 but it increases the cost of the product. Table 5.9 shows chemical composition of
SS 410.
A. Experiments for stainless steel
The experimentation work was conducted at Maruti Metals - Rajkot, Gujarat, India. The metal
is heated to 1750 0C in induction furnace and then poured in gravity. Filter was not put in
feeding channels. The benchmark castings were properly cleaned with the help of shot
blasting process. The layout of cavity in molding box, sprue and gating system is as shown in
fig. 5.18. The setup of benchmark casting is as shown in fig. 5.19. The benchmark casting is
as shown in fig. 5.20. Table 5.10 shows the details of set up of experiment for SS 410.
Table 5.9 Chemical Composition of SS 410 (Source: Maruti Metals: Rajkot, India)
Case # Porosity (cm3) Junction 1 Junction 2 Junction 3 Total
1 0.1 0 0 0.1 2 0.2 2.2 0 2.4 3 1.6 2.3 0.1 4 4 2.5 2.5 0.45 5.45
S. No. Element Wt. % 1. Carbon 0.15 2. Manganese & silicon 2 3. Phosphorus and sulphur 0.07 4. Chromium 12.5 5. Nickel 0.75 7. Ferrous 85
68
Table 5.10 Experimental details for SS 410
Fig. 5.18: Layout, runner, gating and cavity of casting – Stainless steel – Stainless steel
Fig. 5.19: Setup of casting – Stainless steel
Size of molding box (275 mm x 325 mm x 50 mm) x 2
Size of Sprue Φ20 mm x 50 mm
Size of pouring basin Φ20 mm
Size of gating Two gates of 25 mm x 15 mm
No. of cavity Single
Pouring Temp. 1700 0C
Feeder Not used
Gating
Runner
69
Case I Case II
Case III Case IV
Fig. 5.20: Benchmark casting – SS 410
B. Results
As expected, fair amount of porosity can be observed in the different sections of the
benchmark casting. There is no porosity observed for the case 1. Porosity present at only
junction 2 for the case 2. Large amount of porosity can be observed at junctions of case 3 and
4. In the following, the results are presented for each of the case in Table 5.10. Porosity
distribution for benchmark casting is shown in 5.21.
Fig. 5.21: Poro
70
Case 1
Case 2
Case 3
Case 4
osity in benc
chmark shappe - SS 4100
71
Table 5.11: Porosity distribution - SS 410
5.4 Empirical Model Development
From the previous section, it can be observed that large amount of shrinkage porosity were
observed in the case of plain carbon steel and SS 410 while in the case of ductile iron
combination of surface sink and shrinkage porosity found. In this section, the analysis will be
carried out for the development of the model which will predict the size of shrinkage porosity.
The following approach can be followed to analyse results.
5.4.1 Approach
Pore volume (volume of shrinkage porosity) of benchmark casting from different metals is
already available from experimental data. We assume that there is no effect of one junction to
other junction during solidification in benchmark casting for the sake of simplicity in
calculation. Proper approach is required which will develop a empirical model. This model
will further used for quantitative prediction of shrinkage porosity. The model can be
developed using both experimental data and simulation software. It is required that an
approach should create a bridge between experimental data and AutoCAST®. It is also
required to generate sufficient data using simulation for further analysis and finally, proper
forecasting technique should be used to develop the empirical model.
Step 1: To create bridge between experimental data and simulation software
Following approach is adopted to develop bridge between experimental data and simulation
software AutoCAST®.
Case # Porosity (cm3) Junction 1 Junction 2 Junction 3 Total
1 0 0 0 0 2 0 2.1 0 2.1 3 0.8 2.3 0.4 3.5 4 3.1 2.6 0.8 6.5
72
Benchmark casting is divided into three T – junction (refer fig. 5.1).
Followings have been taken as variables for experimentation of benchmark casting
(Refer Table 5.1)
Arm thickness, T = 20 mm (remains constant for each benchmark casting)
Stem thickness, t = 5 mm, 10 mm, 20 mm & 30 mm
Stem length, l = 40 mm, 60 mm, 80 mm
Thickness ratio (R1) and Length ratio (R2) have been taken as geometric parameters
for development of empirical model.
Thickness ratio (R1) = t/T (R1 = 0.25, 0.5, 1 and 1.5)
Length ration (R2) = l/T (R2 = 2, 3 and 4)
Measurements were made of pore volume of shrinkage porosity for each T- junction
of each benchmark casting.
Prepare 3D model and .stl file of each variation of T-junction.
The simulations were made with the commercial software AutoCAST®. The program
is based on Vector Element Method, calculates the equations of thermal transmissions
and shows outputs such as thermal variables (maximum gradient, maximum
temperature, solidification time) through .stl images. We utilize the program to
calculate the maximum gradient, % limiting value of gradient and solidification time.
The thermal properties of the metals used were entered into the software program to
run simulation (if metal is not available in material library of AutoCAST®). Mesh size
is kept 0.5 mm for each case and feedpath is adjusted to dense.
Find out maximum gradient for each T – junction with the help of simulation. It can be
shown in fig. 5.22 by red ellipse.
Adjust the casting simulation software AutoCAST® for volume of shrinkage porosity
(Macro porosity) with available experimental data of porosity volume by adjusting %
limiting value of gradient. It can be shown in fig. 5.23.
Find out the limiting value of gradient (G) with the help of maximum gradient and %
limiting value of gradient.
Limiting value of gradient (G) = Maximum gradient * (% limiting value of gradient)
Find out solidification time and Find out cooling rate (r)
Cooling rate (r) = (pouring temp. – solidus temp.) / Solidification time (sec).
FFig. 5.23: Ad
Fi
djustment of
73
ig. 5.22: Ma
f percent lim
aximum gra
miting value
adient
e of gradiennt in AutoCAAST®
74
Find out limiting value of gradient for each variation of thickness ratio (R1) and length
ratio (R2) of T- junction in same manner by simulating with AutoCAST®.
Find out the relationship between thickness ratio (R1) with the limiting value of
gradient (G) using curve fitting.
Find out relation between thickness ratio (R1) and limiting value of gradient (G) which
has best curve fitting. Follow the same procedure for each variation of length ratio
(R2).
Step 2: Generation of data
It is required to generate sufficient data for further analysis after creating bridge between
simulation software and experimental data. There should be enough data required for the
analysis. Thermal gradient (G), cooling rate (r), thickness ratio (R1), length ratio (R2) will be
variables for our analysis.
For the generation of data, we increase the thickness ratio (R1) by 0.05 (Stem thickness is
increased by 1 mm) for every length ratio (R2). These variations in thickness ratio (R1) are
from minimum thickness ratio (minimum thickness ratio = 0.25) to maximum thickness ratio
(maximum thickness ratio = 1.5) at the difference of 0.05. Number of variations in thickness
ratio (R1) will be 26 for every length ratio (R2). Total number of variations is 78 for each
metal and their simulations have been carried out to generate data using AutoCAST®.
The sufficient amount of data can be generated by following method.
Limiting value of gradient can be found out with the help of relationship between
limiting gradient and thickness ratio
The % limiting value of gradient can be found out using maximum gradient and
limiting value of gradient.
The value of %limiting value of gradient will be inserted to casting simulation
software for each variation
Find out volume of porosity using casting simulation.
This will generate sufficient data of each variable to carry out further analysis.
75
Step 3: Regression analysis
Regression analysis is used to investigate and model the relationship between a response
variable and one or more predictors. It is well defined function. It is based upon least squares
method and calculates equation of straight line (in the form of equation 7.1) that best fits data.
+ + + ............................+ + n1 2 2 3 4 n1 3 4y= m x + m x m x m x m x b ………………….…........ eqn. (5.1)
Where, the dependent y-value is a function of the independent x-values. The m-values are co-
efficient corresponding to each x-value and b is a constant value.
Regression can be carried out using either Minitab® or Microsoft Excel®. The interpretation of
results is also very important task. The results of regression analysis are interpreted in
following manner.
R Square is measure of the explanatory power of the model. In theory, R square
compares the amount of the error explained by the model as compared to the amount
of error explained by averages. The higher the R-Square the better. An R-Square
above .5 is generally considered quite well.
Adjusted R Square is a modified version of R Square, and has the same meaning, but
includes computations that prevent a high volume of data points from artificially
driving up the measure of explanatory power. An Adjust R Square above .20 is
generally considered quite well.
The t-statistic is a measure of how strongly a particular independent variable explains
variations in the dependent variable. The larger the t-statistic is the good for model.
The P-value is the probability that the independent variable in question has nothing to
do with the dependent variable. It should be less than 0.1.
F is similar to the t-stat, but F looks at the quality of the entire model, meaning with all
independent variables included. The larger the F is better.
76
In our case, we have four independent variables to predict the size of porosity. These variables
are thermal gradient (G) & cooling rate (r) as thermal parameters and thickness ratio (R1) &
length ratio (R2) are as geometric parameters. Results with zero porosity will be neglected for
the purpose of achieving good accuracy in regression analysis and all the values from data are
converted to natural logarithmic scale to avoid the linear relationship. We are taking into
account percentage porosity instead of porosity to accommodate the changes in volume. The
percentage porosity is defined by porosity volume divided by volume of casting.
The analysis will be carried out as explained earlier with the help of regression analysis. An
effort is made to develop the empirical model to predict the size of shrinkage defect. The
analyses for different metals are as follows.
5.4.2 Ductile iron
Step 1: To create bridge between experimental data and simulation software
Table 5.12: Limiting value of gradient for junction 1, 2 and 3 – Ductile iron
Junction t/T l/T Measured surface sink cm3
AutoCAST Max.
Gradient 0C/ mm
Limiting Value of
Gradient %
Limiting value of gradient 0C/ mm
Shrinkage Porosity, (cm3)
1
0.25 2 0.25 0.25 9.57 2.89 0.28 0.5 2 0.3 0.3 9.47 4.1 0.39 1 2 0.4 0.42 8.61 7.15 0.62
1.5 2 0.75 0.75 7.96 9.32 0.74
2
0.25 2 0.25 0.25 9.42 2.88 0.27 0.5 2 0.45 0.455 9.32 4.88 0.45 1 2 0.9 0.89 8.49 9 0.76
1.5 2 2.2 2.17 7.88 12.5 0.99
3
0.25 2 0.5 0.49 9.28 4.35 0.40 0.5 2 0.75 0.74 9.18 6.65 0.61 1 2 1.6 1.58 8.38 11.2 0.94
1.5 2 2.5 2.55 7.8 12.05 0.94
77
Fig. 5.24: Relationship between thickness ratio (R1) and limiting value of gradient (G) for
Junction 1, 2 and 3 – Ductile iron
Step 2: Generation of data
Maximum value of gradient can be found by casting simulation and limiting value of gradient
can be found out with the help of above equation 5.2 to 5.4. The % limiting value of gradient
can be easily calculated using maximum gradient and limiting value of gradient. The %
limiting value of gradient will be inserted to casting simulation software to get volume of
porosity. Solidification time can also be found out using simulation software. Cooling rate can
also be calculated using solidification time.
G = -0.170R1
3 + 0.308R1
2 + 0.29R1 + 0.187 (Junction 1)………………….………....eqn. (5.2)
G = -0.018R1
3 - 0.122R1
2 + 0.833R1 + 0.070 (Junction 2)………….………...……....eqn. (5.3)
G = -0.340R1
3 + 0.367R1
2 + 0.700R1 + 0.210 (Junction 3)…………………...……....eqn. (5.4)
The data is generated for regression in same manner. (Refer Annexure II)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 0.5 1 1.5
Junction 1 Junction 2 Junction 3
78
7.6.3 Step 3: Regression analysis
Regression analysis is carried out to develop the empircal model which will provide
quantitative prediction of shrinkage porosity using Microsoft Excel®. The regression statistics
are as given in table 5.13. Results from regression analysis are shown in table 5.14.
Following points should be observed from regression analysis.
R Square is 0.931 which is acceptable and Adjusted R square 0.912 which is
acceptable.
The P – value of each variable are acceptable. The P- value of length ration is slightly
on higher side but it is due to less no. of variation in length ratio.
Gradient is having highest t-stat value and lowest P- value. It means Gradient highly
affects on porosity volume.
Regression model can be given as
ln(%P) = -1.36*ln(R1) + 0.01*ln(R2) + 2.35*ln(G) –0.91*ln(r) ………...……..……..eqn. (5.5)
So, equation can be written as.
2.35 010.
291 1.360.
1
G xR% Pr xR
= ………………………………………………..……..…….....eqn. (5.6)
Table 5.13 Regression Statistics – Ductile iron
R Square 0.83
Adjusted R Square 0.81
Standard Error 0.28
Observations 78
79
Table 5.14 Regression analysis – Ductile iron
Coefficients t Stat P-value
Intercept 0.00 #N/A #N/Aln (R1) -1.36 -3.267 0.002ln(R2) 0.01 0.070 0.945ln(G) 2.35 7.991 0.000ln(r) -0.91 -2.471 0.016
5.4.3 Plain Carbon Steel (AISI 1005)
We will follow the same approach to derive the empirical model for plain carbon steel that we
have discussed in previous section.
Step 1: To create bridge between experimental data and simulation software
Pore volume of shrinkage porosity and limiting value of gradient for each variation of
junction J1, J2 and J3 is as following.
Step 2: Generation of data
The relationship between thickness ratio (R1) and limiting value (G) of gradient can be given
as per followings.
G = -1.523R1
3 + 3.534R1
2 - 1.370R1 + 0.288 (Junction 1)…………………..………..eqn. (5.7)
G = 3.215R1
3 - 10.21R1
2 + 9.900R1 - 1.872 (Junction 2)………………………....…..eqn. (5.8)
G = -0.373R1
3 + 1.088R1
2 - 0.687R1 + 0.154 (Junction 3)…………………..………..eqn. (5.9)
Now, we have limiting value of gradient, cooling rate, thickness ratio (R1) and length ratio
(R2). The data can be generated in same manner. (Ref. Annexure III).
80
Table 5.15: Limiting value of gradient for junction 1, 2 and 3 – Plain Carbon Steel
Junction t/T l/T Measured
shrinkage porosity cm3
AutoCAST Max.
Gradient 0C/ mm
Limiting Value of
Gradient %
Limiting value of gradient 0C/ mm
Shrinkage Porosity, (cm3)
1
0.25 2 0.1 0.1 12.44 1.15 0.14 0.5 2 0.2 0.20 12.3 2.41 0.30 1 2 1.6 1.59 11.18 8.31 0.93
1.5 2 2.5 2.51 10.34 10.1 1.04
2
0.25 3 0 0 12.24 0.12 0.01 0.5 3 2.2 2.2 12.1 7.65 0.93 1 3 2.3 2.3 11.03 9.3 1.03
1.5 3 2.5 2.53 10.24 8.2 0.84
3
0.25 4 0 0 12.05 0.37 0.04 0.5 4 0 0 11.92 0.3 0.04 1 4 0.1 0.1 10.89 1.66 0.18
1.5 4 0.45 0.455 10.13 3.05 0.31
Fig. 5.25: Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, 2 and 3 - Plain Carbon Steel
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 0.5 1 1.5
Junction 1 Junction 2 Junction 3
81
Step 3: Regression analysis
Following points should be observed from regression analysis.
R Square is 0.955 which is acceptable and Adjusted R square 0.938 which is also
acceptable.
The P – value of each variable are acceptable. The P- value of length ration is slightly
on higher side but it is due to less number of variations in length ratio.
Gradient is having highest t-stat value and lowest P- value. It means gradient is having
largest effect on porosity volume.
Regression model can be given as
ln(%P) = -1.282*ln(R1) + 0.215*ln(R2) + 1.913*ln(G) – 1.675*ln(r) …….…….......eqn. (5.10)
Table 5.16: Regression Statistics – plain carbon steel
Table 5.17: Regression analysis – plain carbon steel
Coefficients t Stat P-value
Intercept #N/A #N/A #N/A
ln (R1) -1.381 -5.882 1.47783 e-07
ln(R2) 0.215 1.580 0.118796482
ln(G) 1.913 28.482 8.49336 e-39
ln(r) -1.675 -4.697 1.38352 e-05
R Square 0.955
Adjusted R Square 0.938
Standard Error 0.392
Observations 70
82
So, equation can be written as.
1.913 0.215
21.2821.6751
G xR% Pr xR
= ……………………………………………………….…....eqn. (5.11)
5.4.4 Stainless Steel (SS 410)
Step 1: To create bridge between experimental data and simulation software
Pore volume of shrinkage porosity and limiting value of gradient for each variation of
junction J1, J2 and J3 is as following.
Step 2: Generation of data
The relationship between thickness ratio (R1) and limiting value (G) of gradient can be given
as per followings.
Table 5.18: Limiting value of gradient for junction 1, 2 and 3 – SS 410
Junction t/T l/T Measured
shrinkage porosity cm3
AutoCAST Max.
Gradient 0C/ mm
Limiting Value of
Gradient %
Limiting value of gradient 0C/ mm
Shrinkage Porosity, (cm3)
1
0.25 2 0 0 12.34 0.1 0.01 0.5 2 0 0 12.2 0.1 0.01 1 2 0.8 0.801 11.09 5.31 0.59
1.5 2 3.1 3.1 10.25 10.55 1.08
2
0.25 2 0 0 12.14 0.17 0.02 0.5 2 2.1 2.1 12.01 6.75 0.81 1 2 2.3 2.3 10.94 7.4 0.81
1.5 2 2.6 2.6 10.15 8 0.81
3
0.25 2 0 0 11.96 0.3 0.04 0.5 2 0 0 11.82 0.3 0.04 1 2 0.4 0.4 10.8 2.82 0.30
1.5 2 0.8 0.8 10.45 3.79 0.40
83
G = -1.365R1
3 + 3.928R1
2 - 2.349R1 + 0.375 (Junction 1)………………....……….eqn. (5.12)
G = 3.378R1
3 - 10.13R1
2 + 9.279R1 - 1.718 (Junction 2)……………………..…….eqn. (5.13)
G = -0.86R1
3 + 2.224R1
2 - 1.294R1 + 0.233 (Junction 3)………………………..….eqn. (5.14)
The data is generated for regression in same manner. (Refer Annexure IV)
Step 3: Regression analysis
Following points should be observed from regression analysis.
R Square is 0.931 which is acceptable and Adjusted R square 0.912 which is also
acceptable.
The P – value of each variable are acceptable. The P- value of length ration is slightly
on higher side.
Gradient is having highest t-stat value and lowest P- value. It means Gradient is
having largest effect on porosity volume.
Regression model can be given as
ln(%P) = -0.821*ln(R1) + 0.693*ln(R2) + 1.983*ln(G) – 0.9*ln(r) ………………..eqn. (5.15)
Fig. 5.26: Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, Junction 2 and Junction 3 – SS 410
-0.10
0.10
0.30
0.50
0.70
0.90
1.10
1.30
0 0.5 1 1.5
Junction 1 Junction 2 Junction 3
84
Table 5.19 Regression Statistics – SS 410
Table 5.20 Regression analysis – SS 410
Coefficients t Stat P-value Intercept 0 #N/A #N/Aln (R1) -0.821 -1.837 0.0707707 ln(R2) 0.693 3.722 0.0004146 ln(G) 1.983 21.713 1.7E-31ln(r) -0.899 -1.603 0.1137873
So, equation can be written as.
6931.9 83 0.
29 0.8210.
1
G xR% Pr xR
= …………………………………………………………....eqn. (5.16)
Step 4: Validation
To validate the above result of SS 410, casting was made of T junction having thickness ratio
(R1) and length ratio (R2) is 1.75 and 5 respectively. The patterns made of soley wood and and
pattern making allowances are neglected due to simple geometry.
Pattern and casting of T junction is as shown in fig. 5.27. Solidification simulation is carried
out using VEM. It includes feed path and location of hot spot. It can be seen in fig. 5.28 and
5.29. To measure the shrinkage defect, T junction is cut into two halves. Volume of cavity is
measured by filling water into the shrinkage cavity. The equation of limiting value of gradient
can be found out using data generated for regression analysis.
R Square 0.931Adjusted R Square 0.912Standard Error 0.552Observations 69.000
85
Wooden pattern Casting
Location of shrinkage porosity
Fig. 5.27: T junction casting for validation – SS 410
Fig. 5.28: F
Fig. 5.29:
86
Feed path :
:Hotspot: V
validation
Validation ca
casting
asting
87
Limiting value of gradient = G = R11.19 / R2
0.73 (from regression analysis of G, R1 and R2)
By putting R1 = 1.75 and R2 = 5, limiting value of gradient, G = 0.61
Now, % limiting value of gradient = limiting value of gradient x 100 / maximum gradient
Solidification simulation gives following results:
Maximum gradient= 10.26 0C/mm
(Solidification simulation using Vector Element Method - Refer fig. 5.28)
Solidification time = 7.31 min
(Solidification simulation using Vector Element Method - Refer fig. 5.29)
Cooling rate (r) = (pouring temp. – solidus temp.) / Solidification time (sec) = 0.5 0C/sec
Where Pouring temp. = 1700 0C,
Solidus temp. = 1482 0C
By putting the values of limiting value of gradient, cooling rate, thickness ratio and length
ratio in eqn.(5.16).
%P = 1.44
Porosity volume = (Volume of casting) x (% Porosity) / 100
= 204 x 1.44 / 100
= 2.94 cm3
Observed porosity volume = 2.5 cm3
It can be seen that predicted shrinakge volume and observed value is 2.94 cm3 and 2.5 cm3
reseptively.
88
a d2
cb1
e q n . ( 5 . 1 7 )G x R
% Pr x R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .=
5.5 Summary
This section can be summarized as given below:
Analysis for benchmark shape can be done with the help of FEM and VEM.
Vector element method (VEM) directly gives the location of hotspot. Also, thermal
gradient, solidfication time etc. can be obtained very easily using it.
From experimental results, it can be observed that large amount of shrinkage porosity
were observed in the case of plain carbon steel and SS 410 while in the case of ductile
iron combination of surface sink and shrinkage porosity found.
The model development includes three steps:
1. To create bridge between experimental data and software
2. To generate data for further analysis
3. Regression analysis
The empirical model is developed for ductile iron, plain carbon steel and stainless
steel (SS 410).
The empirical model can be written in general form as per followings
% Porosity = Thermal parameters * Geometric Parameters
The empirical model can be written in generalized form.
a, b, c and d are constants and they are nothing but co efficient of regression analysis for each
metal. They are as shown in table 5.21
Table 5.21 Co efficient of empirical model
Metal a b C d
Plain carbon Steel 1.913 1.675 1.282 0.215 Stainless Steel 1.983 0.9 0.821 0.693 SG Iron 2.35 0.91 1.36 0.01
89
Chapter 6
Summary and Future Work
6.1 Summary
Shrinkage porosity in castings contributes directly to customer concerns about reliability and
quality. Controlling shrinkage porosity depends on understanding its sources and causes.
Significant improvements in product quality, component performance, and design reliability
can be achieved if shrinkage porosity in castings can be predicted, controlled or eliminated.
Regarding to the shrinkage porosity prediction for castings, following aspects have been
summarised.
It can be summarised from literature review that there are various methods are
available for prediction of shrinkage porosity.
It is very convenient to predict shrinkage porosity with the help of criterion function
due to their advantages.
Threshold value of Niyama and LCC criterion is readily available for steel casting
and Al alloys respectively in literature. A threshold value is the value below which
shrinkage porosity likely to occur.
Niyama, LCC, Davis and Bishop Criterion are considered for comparison for sand
casting.
Niyama, LCC, Davis and Bishop Criterion are compared for steel sand casting of L-
Junction.
The simulation results are also compared with AutoCAST software for L-junction.
If properties like Secondary Dendrite Arm Spacing (SDAS), critical pressure drop
during solidification, dendrite size of metal, melt quality factor are known to us for
90
major metal process combination then comparison can also be made between the
other criterion functions.
The comparison of various criterion functions is clarified the threshold values of LCC
for steel, Bishop criterion for Al alloys, Niyama criterion for Al alloys & ductile iron
and Davis criterion for steel as well as Al alloys. It is also possible to find threshold
value of Suri et al. criterion and FCC for steel if above mentioned metallurgical
values are available.
Table 6.1: Threshold Value of Various Criterion Function
Bench mark shape is decided to establish the relationship of geometric parameters
and thermal parameters with quantity of shrinkage porosity in T junction of various
metals like ductile iron, plain carbon steel and SS 410.
The experimentation is carried out to develop some empirical model which will
provide quantitative prediction of shrinkage porosity.
The regression analysis is carried out to develop empirical relationship for various
metals to find out proportion of porosity with the help of experimental data and
casting simulation.
Finally, empirical relationship is found out which will predict size of porosity using
thermal gradient, cooling rate, thickness ratio and length ratio of T junction. It can be
written as follows
Criterion
Function Metal
Threshold
Value
LCC Cast Steel 36.5
Davis Cast Steel 0.9
Bishop Cast Steel 0.9
91
% Porosity = Thermal Parameters * Geometric Parameters
Where a, b = Thermal Constant
c, d = Geometric Constant
The values of above constants are given in table 5.17.
The experiement was carried out for validation of empirical model of T junction for
SS 410. The T junction is having thickness ratio and lenght ratio of 1.75 and 5
respectively. The calculated volume of shirnkage defect is 2.94 cm3 and
experiemented result is 2.5 cm3. So, it is approximately matched with available
results.
6.2 Future Work
Following aspects on which work can be done extended in future after the end of project
work.
The comparison of various criteria is already shown in chapter 4. It can be observed
that some criterion is well accepted for steel or aluminum alloy and they are also
limited to sand casting process.
There is no criterion available which will predict the shrinkage porosity for aluminum
alloy combine with gravity die casting. So, there is also required to work on this
metal- process combination. It can also been done for other non ferrous metals like
copper, zinc, brass, bronze etc.
There is very limited literature is available on prediction of shrinkage porosity for
investment casting process and also there is no such criterion is available to predict
shrinkage porosity. So, it is also required to check validity of available criterion for
investment casting process.
a d2
cb1
G xR% P =
r xR
92
It will be required to perform some experiments to validate the prediction made by
empirical model. It can also be validated by industrial casting.
Correction factor can be developed to accommodate the variation in prediction and
actual results of shrinkage porosity for different metal-process combination. Then it
will be useful to predict shrinkage porosity accurately for different metal-process
combination. Empirical model can also be developed for various junctions.
93
Annexure I
Comparison of Casting Simulation Software
There are three basic types of computer simulation tools: empirical programs based on
experimental results and experience; semi-empirical programs based on experimental results
in addition to basic physics; and physics-based programs that require complex mathematics
and accurate material thermo-physical data. Empirical and semi-empirical programs use
tables of experimental results, rules & guidelines and physical & algebraic equations to
model a physical process. Typically, the problem is broken down into many small
calculations via a finite difference method or finite element method.
Simple empirical programs only can model simple, repeatable processes where the process
variables only change slightly. While feeding distance (low carbon steel in green sand) could
be accurately predicted, defect prediction from a simple empirical analysis would likely be in
error. Semi-empirical programs can effectively model relatively simple casting processes.
Process variables must have limited, known ranges, and the physics must be simple and
straightforward. An example of such a situation is steel solidification characteristics for
simple shapes, and in this case, semi-empirical programs can provide accurate defect
predictions.
Physics-based programs can model simple or complex processes. The physics can be
complex, the process variables can change greatly and new processes can be analyzed. This
approach is well-suited for solidification predictions for all metal types.
Comparison of various simulation software based on survey of 1999 is shown in table.
94
( http://www.thefreelibrary.com/1999+casting+simulation+software+survey.-a054773076)
Sr. no. Software Approach Platform flo
w
Solid
ifica
tion
Mic
ro st
ruct
ure
stre
ss
1. Procast www.esigroup.com FEM PC unix
based system √ √ √ √
2. Magma - www.magma.com FDM Sun/ Win NT √ √ √ √
3. Flow 3d - www.flow3d.com FDM & FVM PC windows √ √ √ -
4. Cast tech - www.avenue.castech.fi FEM Windows
/Unix √ √ √ -
5. Power cast - www.technalysis.com FEM Unix
workstations √ √ √ -
6. Mavis flow - www.mavis-flow.co.uk FEM PC - √ - -
7. Cap cast - www.ekkinc.com FEM PC DOS / Windows √ √ √ √
8. SOLIDCAST www.castingsimulation.com FDM NA ‐ √ ‐ -
9. NOVA FLOW - www.novacast.se FDM Windows √ √ - -
10. SIMtech - www.simtec-inc.com FEM Windows/
Unix √ √ √ √
11. BKK metal casting simulation software FEM IRIX - √ - -
12. Cast view FEM WorkstationsSG1 Sun √ √ √ √
13. Camcast simulator
Navier stoke &
heat transfer
Unix workstations √ √ √ √
95
Annexure II
Data for Regression Analysis – SG Iron (500/7)
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
1 0.25 -1.39 2 0.69 9.57 72.00 0.276 -1.29 2.885 3.81 1.067 0.07 0.250 0.347 -1.062 0.30 -1.20 2 0.69 9.57 73.60 0.297 -1.21 3.105 3.81 1.067 0.07 0.269 0.365 -1.013 0.35 -1.05 2 0.69 9.57 75.20 0.319 -1.14 3.333 3.81 1.067 0.07 0.258 0.343 -1.074 0.40 -0.92 2 0.69 9.57 76.80 0.341 -1.07 3.567 3.81 1.067 0.07 0.269 0.350 -1.055 0.45 -0.80 2 0.69 9.57 78.40 0.364 -1.01 3.808 3.81 1.067 0.07 0.275 0.351 -1.056 0.50 -0.69 2 0.69 9.47 80.00 0.388 -0.95 4.095 4.12 0.987 -0.01 0.300 0.375 -0.987 0.55 -0.60 2 0.69 9.39 81.60 0.411 -0.89 4.381 4.42 0.920 -0.08 0.305 0.374 -0.988 0.60 -0.51 2 0.69 9.34 83.20 0.435 -0.83 4.659 4.73 0.860 -0.15 0.281 0.338 -1.099 0.65 -0.43 2 0.69 9.3 84.80 0.459 -0.78 4.935 5.04 0.807 -0.21 0.251 0.296 -1.22
10 0.70 -0.36 2 0.69 9.29 86.40 0.483 -0.73 5.195 5.36 0.759 -0.28 0.233 0.270 -1.3111 0.75 -0.29 2 0.69 9.28 88.00 0.506 -0.68 5.453 5.68 0.716 -0.33 0.232 0.264 -1.3312 0.80 -0.22 2 0.69 9.27 89.60 0.529 -0.64 5.707 5.99 0.679 -0.39 0.252 0.281 -1.2713 0.85 -0.16 2 0.69 9.22 91.20 0.552 -0.59 5.983 6.62 0.614 -0.49 0.239 0.262 -1.3414 0.90 -0.11 2 0.69 9 92.80 0.574 -0.56 6.373 6.89 0.590 -0.53 0.208 0.224 -1.5015 0.95 -0.05 2 0.69 8.79 94.40 0.595 -0.52 6.766 7.17 0.567 -0.57 0.281 0.298 -1.2116 1.00 0.00 2 0.69 8.61 96.00 0.615 -0.49 7.143 7.45 0.546 -0.61 0.420 0.438 -0.8317 1.05 0.05 2 0.69 8.44 97.60 0.634 -0.46 7.515 7.74 0.525 -0.64 0.497 0.509 -0.6718 1.10 0.10 2 0.69 8.28 99.20 0.652 -0.43 7.879 8.02 0.507 -0.68 0.575 0.580 -0.5519 1.15 0.14 2 0.69 8.16 100.80 0.669 -0.40 8.202 8.3 0.490 -0.71 0.604 0.599 -0.5120 1.20 0.18 2 0.69 8.09 102.40 0.685 -0.38 8.464 8.57 0.475 -0.75 0.592 0.578 -0.5521 1.25 0.22 2 0.69 8.02 104.00 0.699 -0.36 8.712 8.85 0.460 -0.78 0.714 0.687 -0.3822 1.30 0.26 2 0.69 7.96 105.60 0.711 -0.34 8.933 9.11 0.446 -0.81 0.762 0.722 -0.3323 1.35 0.30 2 0.69 8.12 107.20 0.722 -0.33 8.886 9.37 0.434 -0.83 0.736 0.687 -0.3824 1.40 0.34 2 0.69 8.04 108.80 0.730 -0.31 9.082 9.63 0.422 -0.86 0.783 0.720 -0.3325 1.45 0.37 2 0.69 7.98 110.40 0.737 -0.31 9.233 9.87 0.412 -0.89 0.793 0.718 -0.3326 1.50 0.41 2 0.69 7.96 112.00 0.741 -0.30 9.312 10.11 0.402 -0.91 0.750 0.670 -0.4027 0.25 -1.39 3 1.10 9.42 76.00 0.270 -1.31 2.870 3.67 1.11 0.10 0.250 0.329 -1.1128 0.30 -1.20 3 1.10 9.42 78.40 0.308 -1.18 3.274 3.67 1.11 0.10 0.313 0.399 -0.9229 0.35 -1.05 3 1.10 9.42 80.80 0.346 -1.06 3.671 3.67 1.11 0.10 0.344 0.426 -0.8530 0.40 -0.92 3 1.10 9.42 83.20 0.383 -0.96 4.061 3.67 1.11 0.10 0.377 0.453 -0.7931 0.45 -0.80 3 1.10 9.42 85.60 0.419 -0.87 4.443 3.67 1.11 0.10 0.397 0.464 -0.7732 0.50 -0.69 3 1.10 9.32 88.00 0.454 -0.79 4.869 3.97 1.02 0.02 0.455 0.517 -0.6633 0.55 -0.60 3 1.10 9.24 90.40 0.488 -0.72 5.284 4.27 0.95 -0.05 0.497 0.550 -0.6034 0.60 -0.51 3 1.10 9.18 92.80 0.522 -0.65 5.686 4.58 0.89 -0.12 0.464 0.500 -0.6935 0.65 -0.43 3 1.10 9.15 95.20 0.555 -0.59 6.065 4.9 0.83 -0.19 0.466 0.489 -0.7136 0.70 -0.36 3 1.10 9.13 97.60 0.587 -0.53 6.431 5.22 0.78 -0.25 0.436 0.447 -0.8137 0.75 -0.29 3 1.10 9.12 100.00 0.619 -0.48 6.782 5.53 0.74 -0.31 0.417 0.417 -0.8738 0.80 -0.22 3 1.10 9.12 102.40 0.649 -0.43 7.117 5.84 0.70 -0.36 0.391 0.382 -0.9639 0.85 -0.16 3 1.10 9.13 104.80 0.679 -0.39 7.435 6.21 0.65 -0.42 0.594 0.567 -0.5740 0.90 -0.11 3 1.10 8.87 107.20 0.708 -0.35 7.979 6.49 0.63 -0.47 0.681 0.635 -0.4541 0.95 -0.05 3 1.10 8.67 109.60 0.736 -0.31 8.487 6.78 0.60 -0.51 0.815 0.744 -0.3042 1.00 0.00 3 1.10 8.49 112.00 0.763 -0.27 8.987 7.07 0.58 -0.55 0.890 0.795 -0.2343 1.05 0.05 3 1.10 8.44 114.40 0.789 -0.24 9.352 7.74 0.53 -0.64 0.773 0.676 -0.3944 1.10 0.10 3 1.10 8.17 116.80 0.815 -0.20 9.972 7.65 0.53 -0.63 1.260 1.079 0.0845 1.15 0.14 3 1.10 8.06 119.20 0.839 -0.18 10.412 7.94 0.51 -0.67 1.390 1.166 0.1546 1.20 0.18 3 1.10 7.98 121.60 0.863 -0.15 10.812 8.22 0.49 -0.70 1.540 1.266 0.2447 1.25 0.22 3 1.10 7.92 124.00 0.885 -0.12 11.180 8.51 0.48 -0.74 1.910 1.540 0.43
96
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
48 1.30 0.26 3 1.10 7.86 126.40 0.907 -0.10 11.542 8.79 0.46 -0.77 2.010 1.590 0.4649 1.35 0.30 3 1.10 8.04 128.80 0.928 -0.07 11.541 9.06 0.45 -0.80 2.030 1.576 0.4550 1.40 0.34 3 1.10 7.97 131.20 0.948 -0.05 11.891 9.33 0.44 -0.83 2.040 1.555 0.4451 1.45 0.37 3 1.10 7.91 133.60 0.966 -0.03 12.218 9.59 0.42 -0.86 2.160 1.617 0.4852 1.50 0.41 3 1.10 7.88 136.00 0.984 -0.02 12.490 9.84 0.41 -0.88 2.170 1.596 0.4753 0.25 -1.39 4 1.39 9.28 80.00 0.403 -0.91 4.339 3.55 1.146 0.14 0.500 0.625 -0.4754 0.30 -1.20 4 1.39 9.28 83.20 0.444 -0.81 4.783 3.55 1.146 0.14 0.572 0.688 -0.3755 0.35 -1.05 4 1.39 9.28 86.40 0.485 -0.72 5.230 3.55 1.146 0.14 0.587 0.679 -0.3956 0.40 -0.92 4 1.39 9.28 89.60 0.527 -0.64 5.678 3.55 1.146 0.14 0.616 0.688 -0.3757 0.45 -0.80 4 1.39 9.28 92.80 0.568 -0.57 6.124 3.55 1.146 0.14 0.698 0.752 -0.2858 0.50 -0.69 4 1.39 9.18 96.00 0.609 -0.50 6.637 3.83 1.062 0.06 0.750 0.781 -0.2559 0.55 -0.60 4 1.39 9.09 99.20 0.649 -0.43 7.145 4.14 0.982 -0.02 0.850 0.857 -0.1560 0.60 -0.51 4 1.39 9.04 102.40 0.689 -0.37 7.618 4.44 0.916 -0.09 0.869 0.849 -0.1661 0.65 -0.43 4 1.39 9.01 105.60 0.727 -0.32 8.065 4.76 0.854 -0.16 0.844 0.799 -0.2262 0.70 -0.36 4 1.39 8.99 108.80 0.763 -0.27 8.490 5.07 0.802 -0.22 0.779 0.716 -0.3363 0.75 -0.29 4 1.39 8.98 112.00 0.798 -0.23 8.886 5.37 0.757 -0.28 0.743 0.663 -0.4164 0.80 -0.22 4 1.39 8.98 115.20 0.831 -0.19 9.252 5.68 0.716 -0.33 0.844 0.733 -0.3165 0.85 -0.16 4 1.39 8.97 118.40 0.861 -0.15 9.603 5.69 0.715 -0.34 0.919 0.776 -0.2566 0.90 -0.11 4 1.39 8.76 121.60 0.889 -0.12 10.153 6.13 0.663 -0.41 1.200 0.987 -0.0167 0.95 -0.05 4 1.39 8.56 124.80 0.915 -0.09 10.686 6.42 0.633 -0.46 1.390 1.114 0.1168 1.00 0.00 4 1.39 8.38 128.00 0.937 -0.07 11.181 6.71 0.606 -0.50 1.600 1.250 0.2269 1.05 0.05 4 1.39 8.22 131.20 0.956 -0.04 11.630 7 0.581 -0.54 1.870 1.425 0.3570 1.10 0.10 4 1.39 8.07 134.40 0.972 -0.03 12.039 7.29 0.558 -0.58 2.120 1.577 0.4671 1.15 0.14 4 1.39 7.06 137.60 0.983 -0.02 13.927 7.58 0.536 -0.62 2.570 1.868 0.6272 1.20 0.18 4 1.39 7.89 140.80 0.991 -0.01 12.560 7.87 0.517 -0.66 2.590 1.839 0.6173 1.25 0.22 4 1.39 8.07 144.00 0.994 -0.01 12.322 8.15 0.499 -0.70 2.520 1.750 0.5674 1.30 0.26 4 1.39 8 147.20 0.993 -0.01 12.416 8.43 0.482 -0.73 2.600 1.766 0.5775 1.35 0.30 4 1.39 7.92 150.40 0.987 -0.01 12.466 8.7 0.467 -0.76 2.550 1.695 0.5376 1.40 0.34 4 1.39 7.86 153.60 0.976 -0.02 12.422 8.97 0.453 -0.79 2.640 1.719 0.5477 1.45 0.37 4 1.39 7.85 156.80 0.960 -0.04 12.230 9.23 0.441 -0.82 2.540 1.620 0.4878 1.50 0.41 4 1.39 7.8 160.00 0.938 -0.06 12.029 9.49 0.429 -0.85 2.500 1.563 0.45
97
Annexure III
Data for Regression Analysis ‐ Plain Carbon Steel (AISI 1005)
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
1 0.25 -1.39 2 0.69 12.44 72.00 0.14 -1.95 1.15 2.06 1.32 0.28 0.10 0.14 -1.972 0.30 -1.20 2 0.69 12.44 73.60 0.15 -1.87 1.24 2.06 1.32 0.28 0.10 0.14 -1.983 0.35 -1.05 2 0.69 12.44 75.20 0.18 -1.74 1.42 2.06 1.32 0.28 0.12 0.15 -1.874 0.40 -0.92 2 0.69 12.44 76.80 0.21 -1.57 1.67 2.06 1.32 0.28 0.11 0.15 -1.925 0.45 -0.80 2 0.69 12.44 78.40 0.25 -1.39 2.00 2.06 1.32 0.28 0.12 0.15 -1.896 0.50 -0.69 2 0.69 12.30 80.00 0.30 -1.22 2.41 2.23 1.22 0.20 0.21 0.26 -1.367 0.55 -0.60 2 0.69 12.20 81.60 0.35 -1.05 2.87 2.39 1.14 0.13 0.27 0.33 -1.118 0.60 -0.51 2 0.69 12.13 83.20 0.41 -0.89 3.37 2.56 1.06 0.06 0.36 0.43 -0.859 0.65 -0.43 2 0.69 12.09 84.80 0.47 -0.75 3.91 2.73 1.00 0.00 0.43 0.51 -0.67
10 0.70 -0.36 2 0.69 12.06 86.40 0.54 -0.62 4.46 2.90 0.94 -0.07 0.49 0.57 -0.5711 0.75 -0.29 2 0.69 12.05 88.00 0.61 -0.50 5.03 3.07 0.88 -0.12 0.56 0.64 -0.4512 0.80 -0.22 2 0.69 12.05 89.60 0.67 -0.39 5.59 3.24 0.84 -0.18 0.57 0.63 -0.4613 0.85 -0.16 2 0.69 11.97 91.20 0.74 -0.30 6.19 3.57 0.76 -0.27 0.70 0.77 -0.2614 0.90 -0.11 2 0.69 11.69 92.80 0.81 -0.21 6.91 3.72 0.73 -0.31 0.85 0.92 -0.0915 0.95 -0.05 2 0.69 11.42 94.40 0.87 -0.14 7.62 3.88 0.70 -0.36 1.18 1.25 0.2216 1.00 0.00 2 0.69 11.18 96.00 0.93 -0.07 8.31 4.03 0.67 -0.39 1.59 1.66 0.5017 1.05 0.05 2 0.69 10.96 97.60 0.98 -0.02 8.97 4.18 0.65 -0.43 1.81 1.85 0.6218 1.10 0.10 2 0.69 10.76 99.20 1.03 0.03 9.57 4.33 0.63 -0.47 2.11 2.13 0.7519 1.15 0.14 2 0.69 10.61 100.80 1.07 0.07 10.08 4.33 0.63 -0.47 2.36 2.34 0.8520 1.20 0.18 2 0.69 10.51 102.40 1.10 0.10 10.48 4.63 0.59 -0.53 2.46 2.40 0.8821 1.25 0.22 2 0.69 10.42 104.00 1.12 0.12 10.78 4.78 0.57 -0.57 2.89 2.78 1.0222 1.30 0.26 2 0.69 10.34 105.60 1.13 0.13 10.96 4.92 0.55 -0.59 2.98 2.82 1.0423 1.35 0.30 2 0.69 10.54 107.20 1.13 0.12 10.74 5.06 0.54 -0.62 2.83 2.64 0.9724 1.40 0.34 2 0.69 10.45 108.80 1.12 0.11 10.69 5.20 0.52 -0.65 2.93 2.69 0.9925 1.45 0.37 2 0.69 10.37 110.40 1.09 0.08 10.50 5.33 0.51 -0.67 2.76 2.50 0.9226 1.50 0.41 2 0.69 10.34 112.00 1.04 0.04 10.10 5.46 0.50 -0.70 2.51 2.24 0.8127 0.25 -1.39 3 1.10 12.24 76.00 0.02 -4.19 0.12 1.99 1.37 0.31 0.00 0.00 #NUM!28 0.30 -1.20 3 1.10 12.24 78.40 0.27 -1.32 2.17 1.99 1.37 0.31 0.19 0.24 -1.4329 0.35 -1.05 3 1.10 12.24 80.80 0.48 -0.73 3.92 1.99 1.37 0.31 0.89 1.10 0.0930 0.40 -0.92 3 1.10 12.24 83.20 0.66 -0.42 5.39 1.99 1.37 0.31 1.38 1.66 0.5131 0.45 -0.80 3 1.10 12.24 85.60 0.81 -0.21 6.60 1.99 1.37 0.31 1.73 2.02 0.7032 0.50 -0.69 3 1.10 12.10 88.00 0.93 -0.08 7.66 2.14 1.27 0.24 2.20 2.50 0.9233 0.55 -0.60 3 1.10 12.00 90.40 1.02 0.02 8.49 2.31 1.18 0.16 2.40 2.65 0.9834 0.60 -0.51 3 1.10 11.93 92.80 1.09 0.08 9.11 2.48 1.10 0.09 2.60 2.80 1.0335 0.65 -0.43 3 1.10 11.89 95.20 1.13 0.12 9.52 2.65 1.03 0.02 2.67 2.80 1.0336 0.70 -0.36 3 1.10 11.86 97.60 1.16 0.15 9.76 2.82 0.96 -0.04 2.63 2.69 0.9937 0.75 -0.29 3 1.10 11.85 100.00 1.17 0.15 9.84 2.99 0.91 -0.10 2.50 2.50 0.9238 0.80 -0.22 3 1.10 11.85 102.40 1.16 0.15 9.79 3.15 0.86 -0.15 2.24 2.19 0.7839 0.85 -0.16 3 1.10 11.81 104.80 1.14 0.13 9.66 3.35 0.81 -0.21 1.97 1.88 0.6340 0.90 -0.11 3 1.10 11.53 107.20 1.11 0.11 9.64 3.51 0.77 -0.26 2.01 1.88 0.6341 0.95 -0.05 3 1.10 11.27 109.60 1.07 0.07 9.54 3.66 0.74 -0.30 2.14 1.95 0.6742 1.00 0.00 3 1.10 11.03 112.00 1.03 0.03 9.37 3.82 0.71 -0.34 2.30 2.05 0.7243 1.05 0.05 3 1.10 10.81 114.40 0.99 -0.01 9.14 3.98 0.68 -0.38 2.30 2.01 0.7044 1.10 0.10 3 1.10 10.62 116.80 0.94 -0.06 8.88 4.13 0.66 -0.42 2.50 2.14 0.7645 1.15 0.14 3 1.10 10.47 119.20 0.90 -0.11 8.59 4.29 0.63 -0.46 2.50 2.10 0.7446 1.20 0.18 3 1.10 10.37 121.60 0.86 -0.15 8.30 4.44 0.61 -0.49 2.45 2.01 0.7047 1.25 0.22 3 1.10 10.28 124.00 0.83 -0.19 8.07 4.60 0.59 -0.53 2.59 2.09 0.74
98
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
48 1.30 0.26 3 1.10 10.21 126.40 0.81 -0.22 7.90 4.75 0.57 -0.56 2.49 1.97 0.6849 1.35 0.30 3 1.10 10.45 128.80 0.80 -0.23 7.61 4.90 0.55 -0.59 2.32 1.80 0.5950 1.40 0.34 3 1.10 10.35 131.20 0.80 -0.23 7.71 5.04 0.54 -0.62 2.37 1.81 0.5951 1.45 0.37 3 1.10 10.28 133.60 0.82 -0.20 7.96 5.18 0.52 -0.65 2.46 1.84 0.6152 1.50 0.41 3 1.10 10.24 136.00 0.86 -0.16 8.36 5.32 0.51 -0.67 2.53 1.86 0.6253 0.25 -1.39 4 1.39 12.05 80.00 0.04 -3.11 0.37 1.92 1.41 0.35 0.00 0.00 #NUM!54 0.30 -1.20 4 1.39 12.05 83.20 0.04 -3.33 0.30 1.92 1.41 0.35 0.00 0.00 #NUM!55 0.35 -1.05 4 1.39 12.05 86.40 0.03 -3.48 0.26 1.92 1.41 0.35 0.00 0.00 #NUM!56 0.40 -0.92 4 1.39 12.05 89.60 0.03 -3.53 0.24 1.92 1.41 0.35 0.00 0.00 #NUM!57 0.45 -0.80 4 1.39 12.05 92.80 0.03 -3.47 0.26 1.92 1.41 0.35 0.00 0.00 #NUM!58 0.50 -0.69 4 1.39 11.92 96.00 0.04 -3.33 0.30 2.07 1.31 0.27 0.00 0.00 #NUM!59 0.55 -0.60 4 1.39 11.81 99.20 0.04 -3.14 0.37 2.23 1.22 0.20 0.01 0.01 -4.6060 0.60 -0.51 4 1.39 11.75 102.40 0.05 -2.94 0.45 2.40 1.13 0.12 0.01 0.01 -4.4161 0.65 -0.43 4 1.39 11.71 105.60 0.06 -2.74 0.55 2.57 1.06 0.06 0.01 0.01 -4.4462 0.70 -0.36 4 1.39 11.68 108.80 0.08 -2.55 0.67 2.74 0.99 -0.01 0.01 0.01 -4.5163 0.75 -0.29 4 1.39 11.67 112.00 0.09 -2.37 0.80 2.90 0.94 -0.07 0.01 0.01 -4.5864 0.80 -0.22 4 1.39 11.66 115.20 0.11 -2.21 0.94 3.07 0.88 -0.12 0.01 0.01 -4.7065 0.85 -0.16 4 1.39 11.66 118.40 0.13 -2.06 1.09 3.07 0.88 -0.12 0.00 0.00 #NUM!66 0.90 -0.11 4 1.39 11.38 121.60 0.15 -1.93 1.27 3.31 0.82 -0.20 0.01 0.01 -4.6367 0.95 -0.05 4 1.39 11.13 124.80 0.16 -1.81 1.47 3.47 0.78 -0.24 0.09 0.07 -2.6768 1.00 0.00 4 1.39 11.03 128.00 0.18 -1.70 1.65 3.62 0.75 -0.29 0.10 0.08 -2.5569 1.05 0.05 4 1.39 10.68 131.20 0.20 -1.61 1.88 3.78 0.72 -0.33 0.11 0.09 -2.4570 1.10 0.10 4 1.39 10.48 134.40 0.22 -1.52 2.08 3.94 0.69 -0.37 0.23 0.17 -1.7771 1.15 0.14 4 1.39 10.34 137.60 0.24 -1.45 2.28 4.10 0.66 -0.41 0.36 0.26 -1.3472 1.20 0.18 4 1.39 10.25 140.80 0.25 -1.38 2.46 4.25 0.64 -0.45 0.34 0.24 -1.4173 1.25 0.22 4 1.39 10.49 144.00 0.27 -1.32 2.54 4.40 0.62 -0.48 0.40 0.28 -1.2974 1.30 0.26 4 1.39 10.39 147.20 0.28 -1.27 2.70 4.56 0.60 -0.52 0.44 0.30 -1.2175 1.35 0.30 4 1.39 10.29 150.40 0.29 -1.23 2.83 4.70 0.58 -0.55 0.47 0.31 -1.1776 1.40 0.34 4 1.39 10.21 153.60 0.30 -1.20 2.95 4.85 0.56 -0.58 0.48 0.31 -1.1677 1.45 0.37 4 1.39 10.19 156.80 0.31 -1.18 3.02 4.99 0.54 -0.61 0.48 0.31 -1.1878 1.50 0.41 4 1.39 10.24 160.00 0.31 -1.16 3.05 5.13 0.53 -0.64 0.46 0.28 -1.26
99
Annexure IV
Data for Regression Analysis – Stainless Steel (SS 410)
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
1 0.25 -1.39 2 0.69 12.34 72.00 0.01 -4.43 0.10 2.70 1.35 0.30 0.00 0.00 #NUM!2 0.30 -1.20 2 0.69 12.34 73.60 0.01 -4.43 0.10 2.70 1.35 0.30 0.00 0.00 #NUM!3 0.35 -1.05 2 0.69 12.34 75.20 0.01 -4.43 0.10 2.70 1.35 0.30 0.00 0.00 #NUM!4 0.40 -0.92 2 0.69 12.34 76.80 0.01 -4.43 0.10 2.70 1.35 0.30 0.00 0.00 #NUM!5 0.45 -0.80 2 0.69 12.34 78.40 0.01 -4.43 0.10 2.70 1.35 0.30 0.00 0.00 #NUM!6 0.50 -0.69 2 0.69 12.20 80.00 0.01 -4.43 0.10 2.92 1.24 0.22 0.00 0.00 #NUM!7 0.55 -0.60 2 0.69 12.10 81.60 0.04 -3.12 0.37 3.13 1.16 0.15 0.01 0.02 -4.078 0.60 -0.51 2 0.69 12.03 83.20 0.08 -2.47 0.71 3.35 1.08 0.08 0.02 0.03 -3.689 0.65 -0.43 2 0.69 11.99 84.80 0.13 -2.02 1.11 3.58 1.01 0.01 0.03 0.03 -3.50
10 0.70 -0.36 2 0.69 11.97 86.40 0.19 -1.68 1.56 3.80 0.96 -0.04 0.03 0.03 -3.4711 0.75 -0.29 2 0.69 11.95 88.00 0.25 -1.40 2.07 4.03 0.90 -0.10 0.04 0.05 -3.0212 0.80 -0.22 2 0.69 11.95 89.60 0.31 -1.17 2.60 4.24 0.86 -0.15 0.06 0.07 -2.6813 0.85 -0.16 2 0.69 11.88 91.20 0.38 -0.97 3.18 4.68 0.78 -0.25 0.08 0.09 -2.4614 0.90 -0.11 2 0.69 11.59 92.80 0.45 -0.80 3.86 4.89 0.74 -0.30 0.31 0.34 -1.0915 0.95 -0.05 2 0.69 11.33 94.40 0.52 -0.66 4.57 5.08 0.72 -0.34 0.57 0.61 -0.5016 1.00 0.00 2 0.69 11.09 96.00 0.59 -0.53 5.31 5.28 0.69 -0.37 0.80 0.83 -0.1817 1.05 0.05 2 0.69 10.87 97.60 0.66 -0.42 6.06 5.48 0.66 -0.41 0.99 1.01 0.0118 1.10 0.10 2 0.69 10.67 99.20 0.73 -0.32 6.82 5.68 0.64 -0.45 1.28 1.29 0.2519 1.15 0.14 2 0.69 10.52 100.80 0.79 -0.23 7.53 5.88 0.62 -0.48 1.77 1.76 0.5620 1.20 0.18 2 0.69 10.43 102.40 0.85 -0.16 8.19 6.08 0.60 -0.51 2.05 2.00 0.6921 1.25 0.22 2 0.69 10.33 104.00 0.91 -0.09 8.81 6.27 0.58 -0.55 2.53 2.43 0.8922 1.30 0.26 2 0.69 10.26 105.60 0.96 -0.04 9.36 6.46 0.56 -0.58 2.61 2.47 0.9023 1.35 0.30 2 0.69 10.46 107.20 1.00 0.00 9.60 6.64 0.55 -0.60 2.70 2.52 0.9224 1.40 0.34 2 0.69 10.36 108.80 1.04 0.04 10.04 6.82 0.53 -0.63 2.93 2.69 0.9925 1.45 0.37 2 0.69 10.29 110.40 1.07 0.06 10.36 7.00 0.52 -0.66 3.07 2.78 1.0226 1.50 0.41 2 0.69 10.25 112.00 1.08 0.08 10.56 7.17 0.51 -0.68 3.10 2.77 1.0227 0.25 -1.39 3 1.10 12.14 76.00 0.02 -3.84 0.18 2.6 1.40 0.33 0 0.00 #NUM!28 0.30 -1.20 3 1.10 12.14 78.40 0.25 -1.41 2.02 2.6 1.40 0.33 0.166 0.21 -1.5529 0.35 -1.05 3 1.10 12.14 80.80 0.43 -0.84 3.57 2.6 1.40 0.33 0.831 1.03 0.0330 0.40 -0.92 3 1.10 12.14 83.20 0.59 -0.53 4.85 2.6 1.40 0.33 1.34 1.61 0.4831 0.45 -0.80 3 1.10 12.14 85.60 0.71 -0.34 5.88 2.6 1.40 0.33 1.77 2.07 0.7332 0.50 -0.69 3 1.10 12.01 88.00 0.81 -0.21 6.75 2.81 1.29 0.26 2.1 2.39 0.8733 0.55 -0.60 3 1.10 11.90 90.40 0.88 -0.12 7.42 3.03 1.20 0.18 2.48 2.74 1.0134 0.60 -0.51 3 1.10 11.84 92.80 0.93 -0.07 7.87 3.25 1.12 0.11 2.59 2.79 1.0335 0.65 -0.43 3 1.10 11.79 95.20 0.96 -0.04 8.15 3.47 1.05 0.05 2.63 2.76 1.0236 0.70 -0.36 3 1.10 11.77 97.60 0.97 -0.03 8.26 3.7 0.98 -0.02 2.5 2.56 0.9437 0.75 -0.29 3 1.10 11.76 100.00 0.97 -0.03 8.23 3.92 0.93 -0.08 2.17 2.17 0.7738 0.80 -0.22 3 1.10 11.75 102.40 0.95 -0.05 8.10 4.14 0.88 -0.13 1.86 1.82 0.6039 0.85 -0.16 3 1.10 11.71 104.80 0.92 -0.08 7.90 4.4 0.83 -0.19 1.73 1.65 0.5040 0.90 -0.11 3 1.10 11.71 107.20 0.89 -0.12 7.60 4.4 0.83 -0.19 1.57 1.46 0.3841 0.95 -0.05 3 1.10 11.18 109.60 0.85 -0.16 7.61 4.8 0.76 -0.28 2.16 1.97 0.6842 1.00 0.00 3 1.10 10.94 112.00 0.81 -0.21 7.39 5.01 0.73 -0.32 2.3 2.05 0.7243 1.05 0.05 3 1.10 10.73 114.40 0.77 -0.27 7.15 5.21 0.70 -0.36 2.28 1.99 0.6944 1.10 0.10 3 1.10 10.53 116.80 0.73 -0.32 6.91 5.42 0.67 -0.40 2.21 1.89 0.6445 1.15 0.14 3 1.10 10.38 119.20 0.69 -0.37 6.68 5.63 0.65 -0.44 2.1 1.76 0.5746 1.20 0.18 3 1.10 10.29 121.60 0.67 -0.41 6.48 5.83 0.62 -0.47 1.89 1.55 0.4447 1.25 0.22 3 1.10 10.20 124.00 0.65 -0.43 6.38 6.03 0.60 -0.51 1.94 1.56 0.45
100
Sr no. R1 lnR1 R2 ln(R2) Gmax
0C/mm V
cm3 Glimit
0C/mm lnG % Glimit
τs, min.
r 0C/sec ln(r) P
cm3 %P ln(%P)
48 1.30 0.26 3 1.10 10.13 126.40 0.65 -0.44 6.38 6.23 0.58 -0.54 1.89 1.50 0.4049 1.35 0.30 3 1.10 10.36 128.80 0.66 -0.42 6.35 6.42 0.57 -0.57 1.87 1.45 0.3750 1.40 0.34 3 1.10 10.27 131.20 0.69 -0.38 6.69 6.61 0.55 -0.60 2.05 1.56 0.4551 1.45 0.37 3 1.10 10.19 133.60 0.74 -0.31 7.23 6.8 0.53 -0.63 2.29 1.71 0.5452 1.50 0.41 3 1.10 10.15 136.00 0.81 -0.21 7.97 6.98 0.52 -0.65 2.6 1.91 0.6553 0.25 -1.39 4 1.39 11.96 80.00 0.04 -3.22 0.33 2.52 1.44 0.37 0.00 0.00 #NUM!54 0.30 -1.20 4 1.39 11.96 83.20 0.04 -3.22 0.33 2.52 1.44 0.37 0.01 0.01 -4.5355 0.35 -1.05 4 1.39 11.96 86.40 0.04 -3.22 0.33 2.52 1.44 0.37 0.01 0.01 -4.4656 0.40 -0.92 4 1.39 11.96 89.60 0.04 -3.22 0.33 2.52 1.44 0.37 0.01 0.01 -4.3157 0.45 -0.80 4 1.39 11.96 92.80 0.04 -3.22 0.33 2.52 1.44 0.37 0.01 0.01 -4.2758 0.50 -0.69 4 1.39 11.82 96.00 0.04 -3.22 0.34 2.72 1.34 0.29 0.00 0.00 #NUM!59 0.55 -0.60 4 1.39 11.72 99.20 0.05 -2.98 0.43 2.93 1.24 0.22 0.01 0.01 -4.2660 0.60 -0.51 4 1.39 11.65 102.40 0.07 -2.64 0.61 3.15 1.15 0.14 0.02 0.02 -4.0461 0.65 -0.43 4 1.39 11.61 105.60 0.10 -2.35 0.82 3.37 1.08 0.08 0.02 0.02 -4.0262 0.70 -0.36 4 1.39 11.59 108.80 0.12 -2.10 1.05 3.6 1.01 0.01 0.02 0.02 -4.0563 0.75 -0.29 4 1.39 11.58 112.00 0.15 -1.89 1.30 3.8 0.96 -0.04 0.02 0.02 -4.0864 0.80 -0.22 4 1.39 11.57 115.20 0.18 -1.71 1.56 4.02 0.90 -0.10 0.02 0.01 -4.2865 0.85 -0.16 4 1.39 11.56 118.40 0.21 -1.55 1.83 4.03 0.90 -0.10 0.31 0.26 -1.3566 0.90 -0.11 4 1.39 11.29 121.60 0.24 -1.42 2.15 4.35 0.84 -0.18 0.36 0.29 -1.2367 0.95 -0.05 4 1.39 11.04 124.80 0.27 -1.30 2.48 4.55 0.80 -0.22 0.35 0.28 -1.2768 1.00 0.00 4 1.39 10.8 128.00 0.30 -1.19 2.81 4.75 0.76 -0.27 0.40 0.31 -1.1669 1.05 0.05 4 1.39 10.59 131.20 0.33 -1.11 3.12 4.96 0.73 -0.31 0.58 0.44 -0.8170 1.10 0.10 4 1.39 10.4 134.40 0.36 -1.03 3.42 5.15 0.71 -0.35 0.75 0.56 -0.5971 1.15 0.14 4 1.39 10.26 137.60 0.38 -0.97 3.69 5.37 0.68 -0.39 0.88 0.64 -0.4472 1.20 0.18 4 1.39 10.17 140.80 0.40 -0.92 3.90 5.58 0.65 -0.43 0.86 0.61 -0.4973 1.25 0.22 4 1.39 10.41 144.00 0.41 -0.89 3.95 5.78 0.63 -0.46 0.89 0.62 -0.4874 1.30 0.26 4 1.39 10.3 147.20 0.42 -0.87 4.08 5.98 0.61 -0.50 1.02 0.69 -0.3775 1.35 0.30 4 1.39 10.21 150.40 0.42 -0.86 4.15 6.17 0.59 -0.53 1.06 0.70 -0.3576 1.40 0.34 4 1.39 10.13 153.60 0.42 -0.87 4.15 6.36 0.57 -0.56 1.06 0.69 -0.3777 1.45 0.37 4 1.39 10.11 156.80 0.41 -0.89 4.06 6.54 0.56 -0.59 0.99 0.63 -0.4678 1.50 0.41 4 1.39 10.45 160.00 0.39 -0.93 3.77 6.72 0.54 -0.61 0.80 0.50 -0.69
101
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Acknowledgements
I am very thankful to Prof. B. Ravi for providing me unidirectional focused guidance to
fulfill the understanding of the prediction of shrinakge poroisty using casting simualtion. I
am also thankful to Mr. Mayur Sutaria for providing me the understanding of the various
aspects of the project. I am also presenting my sincere gratitude to following persons for
allowing me to conduct experiemental work in their foundries.
Mr. Aseem Kulkarni – S.S. Industries, Ichalkaranji (Maharashtra)
Mr. Kaushikbhai Ghedia – Manek Casting Pvt. Ltd., Rajkot (Gujarat)
Mr. Nathabhai Patel –Maruti Metals, Rajkot (Gujarat)
Amit V. Sata
MTech
Manufacturing Engineering
Roll No. 08310301
