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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: Place:

Signature Name: Amit V. Sata

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).

i

Table of Contents

Abstract Table of Contents List of Figures List of Tables Nomenclatures

i ii iv vi viii

1

INTRODUCTION 1.1 Porosity in Metal Casting 1.2 Need of Defect Prediction 1.3 Organization of Report

1 1 3 3

2

LITERATURE REVIEW 2.1 Classification and Formation of Porosity 2.2 Factors Affecting Shrinkage Porosity 2.3 Modeling of shrinkage porosity 2.4 Casting Solidification Simulation

4 4 8 10 12 15 16 17 17 19 31

2.5

2.4.1. Finite element method 2.4.2. Vector element method Shrinkage Porosity Prediction

2.6

2.5.1. Modulus and equi-solidification time method 2.5.2.Criterion function method Summary

3

PROBLEM DEFINITION 3.1 Motivation 3.2 Goal, Scope and Objectives 3.3 Approach to Project

34 34 35 35

4

SHRINAKGE DEFECT LOCATION 4.1 Approach to Predict Location of Shrinkage Porosity

37 37 38 44 46

4.2

4.1.1. Solidification simulation using FEM 4.1.2. Solidification simulation using VEM Summary

ii

5

SHRINAKGE DEFECT SIZE PREDICTION 5.1 Benchmark shape 5.2 Solidification Simulation of Benchmark Shape

5.4.3 Plain carbon steel

47 47 49 51 53 55 57 57 63 67 71 71 76 79

5.4.4 Stainless steel Summary

82 87

5.3

5.2.1 Solidification simulation : Ductile iron 5.2.2 Solidification simulation : Plain carbon steel 5.2.3 Solidification simulation : Stainless steel Casting Experiements and Results

5.4

5.3.1 Ductile iron 5.3.2 Plain carbon steel 5.3.3 Stainless steel Empirical Model Development 5.4.1 Approach 5.4.2 Ductile iron

5.5 6

89 89 91

SUMMARY AND FUTURE WORK 8.1 Summary 8.2 Future work Annexure I Annexure II Annexure III Annexure IV

: Comparison of Casting Simulation Software : Data for Regression Analysis – Ductile iron : Data for Regression Analysis - Plain Carbon Steel (AISI 1005) : Data for Regression Analysis – SS 410

93 95 97 99

101 105

References Acknowledgement

iii

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

iv

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

Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, 2 and 3 - Plain Carbon Steel

80

5.25 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

v

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

vi

Table

Description

Page

5.19

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

vii

Nomenclatures   CMI V/A D m c0

Casting/mold interface Modulus of casting Diffusion co-efficient Liquidus slop

L x T K σ r0 µl ρs ρl λ2

Length of the mushy zone Spatial co ordinate Temperature Permeability Surface tension it is between pore and surrounding liquid Initial radius of curvature at pore formation Liquidus viscosity Solidus density Liquidus density Secondary dendrite arm spacing (SDAS)

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)

viii

Pliq Pcr ∆Pcr Xcr, Tcr, gl,cr, , R1 R2

Melt pressure Critical pressure Critical pressure drop = Pliq - Pcr Position at which the melt pressure drops to Pcr and porosity begins to form Temperature at which the melt pressure drops to Pcr and porosity begins to form Liquid fraction at which the melt pressure drops to Pcr and porosity begins to form Thickness ratio = t/T Length ratio = l/T

ix

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: ¾ Improper closure: flash, mismatch ¾ Incomplete filling: cold shut, misrun ¾ Gaseous entrapments: blow holes, gas porosity ¾ Solid inclusions: sand inclusions, slag inclusions ¾ Solidification shrinkage: cavity, porosity, centerline, sink ¾ 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 1   

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 liquidto-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) 2   

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. ¾ Chapter 1 gives introduction of casting process and need of defect prediction ¾ Chapter 2 gives detail literature review of shrinakge porosity formation, modeling and various prediction methods. ¾ Chapter 3 introduces the problem definition. ¾ Chapter 4 gives information about location based predication method and comparison of various criterion functions. ¾ 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. ¾ Chapter 6 includes summary and future work.   

3   

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.

4   

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 ¾ External defects: pipe shrinkage and caved surfaces; ¾ 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.

5   

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)

6   

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.

7   

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).

8   

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.

9   

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).

10   

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.

11   

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) ¾ Predict the pattern of solidification, indicating where shrinkage cavities and associated defects may arise. ¾ Simulate solidification with the casting in various positions, so that the optimum position may be selected. ¾ Calculate the volumes and weights of all the different materials in the solid model. ¾ Provide a choice of quality levels, allowing, for example, the highlighting or ignoring of micro-porosity. ¾ Perform over a range of metals, including steel, white iron, grey iron and ductile iron and non- ferrous metals.

12   

Table 2.1: Categories of approach on the basis of literature

Sr. No.

1

2

3

4

Category

Analytical solutions.

Criterion function models, based on empirical functions.

Numerical solutions of Stokes flow (Darcy's law), coupled with energy and mass conservation, and continuity equations.

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.

Author

Walther et al. Piwonka and Flemings

Pellini, Niyama et al.

Kubo& Pehlke Combeau et al. Rousset et al. Zhu et al. Huang et al. A. Reis et al

Lee et al. Viswanathan et al. Pequet et al. Carlson et al.

13   

Modeling Approach

Focused on shrinkage driven pore growth, developing models that range from exact mathematical solutions to approximate asymptotic analytical solutions using 1D Darcy’s law.

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).

Presents 2D numerical model by coupling Darcy’s law to equations of continuity estimating the fluid flow.

Presents 3D models that included the concept of pore nucleation and growth.

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

14   

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. 15 

 

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, q = − k Where,

ΔT ΔS

ΔT is thermal gradient (G) ΔS

⎛ −1⎞ Hence, G = ⎜ ⎟q ⎝ k ⎠

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.

16   

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. 2

tf = B (V/A)  

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.

 

17   

 

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.  

18   

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:  

¾ Flow modeling: velocity vectors, pressures, and surface tracking ¾ Heat Transfer modeling: temperature, temperature gradients (of filled metal and mold both), and heat transfer process (conduction, convection, and radiation) ¾ Solidification: change in physical properties (density, viscosity, coefficient of conductivity, etc.).   A combination of these parameters can be easily obtained from numerical solutions.

19   

   

 

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 1/2

G / (dT/dt) 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:

20   

Table 2.2: Proposed and calculated critical values of several solidification parameters for centerline porosity prediction (S. Minakawa et al., 1985)

50 mm

25 mm

12.5 mm

5 mm

G

Proposed Critical Values 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

Parameters

¾

Calculated critical values for plate thickness listed.

The criterion itself simple and only requires data obtainable from temperature measurements for verification.

¾

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, c0 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. 

¾

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). 

¾

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) 21   

 

   

Fig.2.7: The relation between the experimentally determined critical Niyama criterion value and the calculated tf(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

22   

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   dP K  glul = −( μ ) dx  .....................................................................................................eqn .(2.1) l

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 K= Ko

gl3 ...........................................................................................................eqn .(2.2) (1 − gl ) 2

Where, K0= λ2/180; in which λ2 is the secondary dendrite arm spacing (SDAS).

23   

 

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 u 1 = − β R = − β [dT / dt ] / G ......................................................................................eqn .(2.3)

Substituting this expression into eqn.(2.1) yields dP dx

=

μ l β gl ⎡ dT ⎤ K g

⎢⎣ d t ⎥⎦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .e q n . ( 2 . 4 )

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 24   

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 ⎡ 2σ ⎤ Pcr = - Pσ = - ⎢ .....................................................................................................eqn .(2.5) ⎣ r0 ⎥⎦  

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:

0

∫ x

cr

T liq

μl β gl ⎡ dT ⎤

μ β g ⎛ dx ⎞ dx = ∫ l l ⎜ ⎟ dT ⎢ ⎥ KG ⎣ dt ⎦ T cr KG ⎝ dT ⎠

=

μl β ⎡ dT ⎤

1

gl ⎡ dT ∫ ⎢ ⎥ G ⎣ dt ⎦ gl ,cr Kdgl ⎢⎣ dt 2

⎤ ⎥⎦dgl .........................................................................................................eqn .(2.6)

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.

25   

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, θ = T −T sol ΔT f , 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 ΔPcr =

μ l βΔT f ⎡ dT 2 ⎢⎣ dt λ2

⎤ ⎥⎦ I ( gl , cr )......................................................................................eqn .(2.7)

Where 2

⎡ (1 − gl ) ⎤ d θ 180 ⎢ 2 ⎥ dg dgl .............................................................................eqn .(2.8) ∫ ⎣ gl ⎦ l gl ,cr 1

I ( gl , cr ) =

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*:

N y* =

G λ 2 ΔPcr ⎛ dT ⎞ ⎟ ⎝ dt ⎠

μ lβ ΔTf ⎜

=

N yλ 2 = I ( gl , cr )..........................................................eqn .(2.9) μ lβ ΔTf ΔPcr

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

26   

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 ⎡ dT ⎤ ⎣ dt ⎥⎦

λ2 = C λ ⎢

−1/ 3

.....................................................................................................eqn .(2.10)

Where Cλ is constant which depends on material. Using eqn.(2.10), the following alternate form can be written G N = Cλ [dT dt ]5 / 6 * y

ΔPcr ⎡ dT ⎤ = Cλ ⎢ μ l β ΔT f ⎣ dt ⎥⎦

−1/ 3

Ny

ΔPcr ................................eqn .(2.11) μ l β ΔT f

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 gp = β ' gl , cr ..............................................................................................................eqn .(2.12)

27   

Where β ' = β /( β + 1) = ρ s − ρ l ρ s

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 2/ 3

LCC =

Gtf ...........................................................................................................eqn .(2.13) Vs

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. 28   

(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. FRN =

n μΔT f 2 ................................................................................................eqn .(2.14) ρL GVs β D

29   

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

P (% ) = 2 3 . 9 + 9 . 6 x1 0 − 6 ( F R N ) − 9 . 1x1 0 − 1 3 ( F R N )2 .................. e q n .( 2 . 1 5 ) 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. P = K * tslm * Vs n ...................................................................................................eqn .(2.16)

⎡ dT ⎤ Vs = ⎢ ⎥ G = cooling rate / temp gradient at end of solidification ⎣ dt ⎦ 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

30   

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 x

y

of G /Vs (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.   ¾ 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. ¾ Classification of porosity is well defined in literature. ¾ 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. ¾ 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. ¾ 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. ¾ Equi-solidification method is reliable in predicting gross shrinkage and porosity at

a final solidification area. 31   

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

Al alloy

% porosity = 23.9 + 9.6x10-6(FRN) + 9.1x10-13(FRN)2

F. Chisea (FCC)

1998

Al alloy

1.52

WCB steel Al A356 Mg alloy A Z91D.

610 (0.01 pct) 137(0.1 pct) 211 (0.01 pct) 23 (0.1 pct)

Carlson et al. (Dimensionless Niyama)

2009

FRN =

n μΔT 2 ρL GVs β D

1/tsm Vsn

N y* =

G λ 2 ΔPcr ⎛ dT ⎞ ⎟ ⎝ dt ⎠

μlβΔTf ⎜

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

32   

¾ Both modulus and equi-solidification method not consider the effect of inter

dendrite and gas evolution. ¾ 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. ¾ Foundries use the Niyama criterion primarily in a qualitative fashion, to identify

regions in a casting that are likely to contain shrinkage porosity. ¾ 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).

¾ 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. ¾ 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. ¾ 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. 

33   

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 34

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    ¾ Study and comparison of various models for prediction of shrinkage. ¾ To apply the different criterion function to L shape casting, available in literature, using finite element method and vector element method. ¾ To decide the benchmark shape and check it for shrinkage porosity using FEM and VEM.   ¾ To perform experiments using benchmark shape for the development of empirical model to predict the size of shrinkage porosity.  ¾ 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. 35

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.

36

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

37

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.   38

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.

fs =

Tl −T ......................................................................e q n .( 4 .1) Tl − T s

Where fs is solid fraction, Tl and Ts are liquidus and solidus temperature respectively. 39

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.

40

Fig. 4.2: Geometric parameters of L shape casting

Table 4.1: Nomenclature for L junction 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

Table 4.2: Properties of Cast steel and Sand   Cast Steel Temperature (K)

Conductivity

Density 3

(W/m K)

(kg/ m )

Specific heat

Interface Heat

(KJ/kg K)

Transfer Co efficient (W/m2K)

255

29.9

7500

0.507

1723.70

32

0.804

1783.15

25.3

0.837

1852.59

25.3

570 7000

0.837

1 490

1.17

Sand 300

0.519

41

11.25

Table 4.3: Input parameters for Cast steel 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

Fig. 4.3: Modelling and meshing: cast steel L shape casting

42

Solidification Simulation

Probable locations of Shrinkage Porosity

Observed Location of Porosity

Fig. 4.4: Solidification simulation of L junction using FEM

43

Table 4.4: Comparison of various Criteria for case I   Criterion Function Davis LCC

Node

Temp

G

r

Vs

fs

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

Niyama

Bishop

  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).

44

Location of hot spot using VEM

Feed path of L junction using VEM

Observed Location of Porosity

Fig. 4.5: Solidification simulation using VEM   45

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. ¾ Solidification time is 267 sec for the L junction as per simualtion using FEM. ¾ 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. ¾ 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. ¾ Solidification time is 226.2 sec for the L junction as per simulation using VEM. ¾ 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.

46

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.

47

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

STEM ARM

Fig. 5.1 Benchmark shape

Table 5.1 Variations in benchmark shape Stem thickness t mm 5

Stem length l1 mm 40

Stem length l2 mm 60

Stem length l3 mm 80

Total length L mm 240

1

20-5-40-60-80-240-40

Arm thickness T mm 20

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

Case #

Nomenclature

48

Depth d mm 40

Case I

Case II

Case III

Case IV

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.

49

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 Element element behavior Material mode 1 Material model 2 Material model 3

Full Transient Thermal Plane 55 Plane thickness Casting Mould Atmosphere Area 1- for Metal Area 2- for sand mould Area 3 – for Atmosphere 2.5 mm 5 mm

Mesh attributes Mesh size (casting) Mesh size (Mould + Atmosphere) Load condition Ref. Temp Initial condition of Metal (Pouring temperature)

 

Initial condition of sand Load step Time Time step

 

50

300 1953 (AISI 1005) 1973 (SS 410) 1667 (SG Iron 500/7) 300 1 500 sec 0.5 sec

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 51

Case

FEM

VE EM

1.

2.

3.

4.

ification sim mulation usiing FEM an nd VEM: Duuctile iron Figg. 5.4 Solidif

52

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

53

Case C

FEM

V VEM

1.

2.

3.

4.

  Fig. 5.6 5 Solidificaation simula ation using FEM and V VEM: Plainn carbon steeel 54

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

55

  Case

FEM

VE EM

1.

2.

3.

4.

Fig.. 5.8 Solidiffication simulation usinng FEM andd VEM: Stainless steel 56

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

57

has been increaasing intereest in appliication of SG iron annd it comess as alternaate as steell prodduct. A littlee work has been b done on o the preddiction of shhrinkage porrosity for SG iron so itt is deecided to caast the SG irron as a initiative workk for predicction of shriinkage poro osity for SG G iron.. The chemiical compossition is shoown in tablee 5.3

A. Exxperiments for ductile iron  S. Industriess - Ichalkarranji (Dist.: Kolhapur), The experimenttation workk was conduucted at S.S m is heatted to 14000 0C in induuction furnaace and then n poured inn Mahharashtra, Inndia. The metal gravvity. Filter was w not put in i feeding channels. c Thhe benchmaark castings were propeerly cleaned d withh the help off shot blastiing process. The layouut of cavity in moldingg box, spruee and gating g systeem is as shoown in fig. 5.10. The setup s of bennchmark cassting is as sshown in figg. 5.11. Thee bencchmark casting is as shown in fig. 5.12. Table 5.4 shows thee details off set up off expeeriment for SG S Iron.

Case I

Case II

Case III

Case IV

Fig. 5.9 Wooden W pattterns for caasting

58

Table 5.3 Chemical Composition - Ductile iron (Source: S.S. Industries, Ichalkaranji, India)          

Sr No. 1. 2. 3. 4. 5. 6.

Element Carbon Manganese Silicon Phosphorus and sulphur Magnesium Ferrous

Wt. % 3.67 1.3 1.65 0.04 0.03 93

  Table 5.4 Experimental details - Ductile iron 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

Fig. 5.10: Layout, runner, gating and cavity of casting – Ductile iron       59

.

F Fig.5.11: S Setup of cassting – Ducttile iron

Case I

Case III

Case III

Case IV V

Fiig. 5.12: Beenchmark caasting – Ductile iron 60

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

Sink + Porosity (cm3)

Case # 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

61

Case 1

Case 2

Case 3

Case 4

Fig. 5.13 Porosity in benchmark casting – Ductile iron

62

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. 1. 2. 3. 4. 5.

Element Carbon Silicon Manganese Phosphorus and Sulphur Ferrous

63

Wt. % 0.045 0.48 0.85 0.09 98

Table 5.7 Experimental details for 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

Gating

Runner

Fig. 5.14: Layout, runner, gating and cavity of casting – Plain carbon steel

Fig. 5.15: Setup of casting – Plain carbon steel

64

Case I

Case II

Case IIII

Case IIV

Fig. 5.16: bench hmark castinng – Plain carbon c steeel    B. Reesults    As expected, e f fair amountt of porosiity can be observed in the diffferent sectiions of thee bencchmark castting. Strong g centerlinee porosity is i present in n 1st case nnear junctio on 1. Largee amou unt of poro osity can bee observed at junction ns of case 2, 2 3 and 4. In the folllowing, thee resullts are presented for each e of the case in Tabble 5.7. Porosity distriibution for benchmark k castiing is shown n in 5.17.

65

Case 1

Case 2

Case 3

Case 4

Fig. 5.17:: Porosity in n benchmarrk shape- Pllain carbonn steel   66

Table 5.8: Porosity distribution – Plain carbon steel  

Case #

 

1 2 3 4

   

Junction 1 0.1 0.2 1.6 2.5

Porosity (cm3) Junction 2 Junction 3 0 0 2.2 0 2.3 0.1 2.5 0.45

Total 0.1 2.4 4 5.45

  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)        

S. No. 1. 2. 3. 4. 5. 7.

Element Carbon Manganese & silicon Phosphorus and sulphur Chromium Nickel Ferrous

67

Wt. % 0.15 2 0.07 12.5 0.75 85

Table 5.10 Experimental details for SS 410 (275 mm x 325 mm x 50 mm) x 2

Size of molding box

Φ20 mm x 50 mm

Size of Sprue

Φ20 mm

Size of pouring basin Size of gating

Two gates of 25 mm x 15 mm

No. of cavity

Single

Pouring Temp.

1700 0C

Feeder

Not used

Runner

Gating

Fig. 5.18: Layout, runner, gating and cavity of casting – Stainless steel – Stainless steel

Fig. 5.19: Setup of casting – Stainless steel

68

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.  

69

Case 1

Case 2

Case 3

Case 4

Fig. 5.21: Poroosity in bencchmark shappe - SS 4100

70

Table 5.11: Porosity distribution - SS 410 Case # 1 2 3 4

Junction 1 0 0 0.8 3.1

Porosity (cm3) Junction 2 Junction 3 0 0 2.1 0 2.3 0.4 2.6 0.8

Total 0 2.1 3.5 6.5

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®. 71

¾ Benchmark casting is divided into three T – junction (refer fig. 5.1). ¾ 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 ¾ 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) ¾ Measurements were made of pore volume of shrinkage porosity for each T- junction of each benchmark casting. ¾ Prepare 3D model and .stl file of each variation of T-junction. ¾ 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. ¾ 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. ¾ Find out maximum gradient for each T – junction with the help of simulation. It can be shown in fig. 5.22 by red ellipse. ¾ 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. ¾ 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) ¾ Find out solidification time and Find out cooling rate (r) Cooling rate (r) = (pouring temp. – solidus temp.) / Solidification time (sec).

72

Fiig. 5.22: Maaximum graadient

Fig. 5.23: Addjustment off percent lim miting valuee of gradiennt in AutoCA AST®

73

¾ 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®. ¾ Find out the relationship between thickness ratio (R1) with the limiting value of gradient (G) using curve fitting. ¾ 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. ¾ Limiting value of gradient can be found out with the help of relationship between limiting gradient and thickness ratio ¾ The % limiting value of gradient can be found out using maximum gradient and limiting value of gradient. ¾ The value of %limiting value of gradient will be inserted to casting simulation software for each variation ¾ Find out volume of porosity using casting simulation. This will generate sufficient data of each variable to carry out further analysis.

74

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.

y= m1x1 + m2x2 + m3x3 + m4x4 + ............................+ mnxn + b ………………….…........ eqn. (5.1) Where, the dependent y-value is a function of the independent x-values. The m-values are coefficient 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. ¾ 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. ¾ 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. ¾ 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. ¾ 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. ¾ 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.

75

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

AutoCAST Junction t/T

1

2

3

0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5

l/T

Measured surface sink cm3

2 2 2 2 2 2 2 2 2 2 2 2

0.25 0.3 0.4 0.75 0.25 0.45 0.9 2.2 0.5 0.75 1.6 2.5

Shrinkage Porosity, (cm3) 0.25 0.3 0.42 0.75 0.25 0.455 0.89 2.17 0.49 0.74 1.58 2.55

76

Limiting Max. Limiting value of Gradient Value of gradient 0 C/ mm Gradient % 0 C/ mm 9.57 9.47 8.61 7.96 9.42 9.32 8.49 7.88 9.28 9.18 8.38 7.8

2.89 4.1 7.15 9.32 2.88 4.88 9 12.5 4.35 6.65 11.2 12.05

0.28 0.39 0.62 0.74 0.27 0.45 0.76 0.99 0.40 0.61 0.94 0.94

 

1.20

 

1.00

     

Junction 1

Junction 2

Junction 3

0.80 0.60

 

0.40

 

0.20

 

0.00 0

 

0.5

1

1.5

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. 3

2

G = -0.170R1 + 0.308R1 + 0.29R1 + 0.187 (Junction 1)………………….………....eqn. (5.2) 3

2

G = -0.018R1 - 0.122R1 + 0.833R1 + 0.070 (Junction 2)………….………...……....eqn. (5.3) 3

2

G = -0.340R1 + 0.367R1 + 0.700R1 + 0.210 (Junction 3)…………………...……....eqn. (5.4)   The data is generated for regression in same manner. (Refer Annexure II)

77

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. ¾ R Square is 0.931 which is acceptable and Adjusted R square 0.912 which is acceptable. ¾ 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. ¾ 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.

%P =

G 2 .3 5 xR 20 .0 1 ………………………………………………..……..…….....eqn. (5.6) r 0 .91 xR 11.3 6 Table 5.13 Regression Statistics – Ductile iron

R Square

0.83

Adjusted R Square

0.81

Standard Error

0.28

Observations

78

78

Table 5.14 Regression analysis – Ductile iron

Intercept ln (R1) ln(R2) ln(G) ln(r)

Coefficients

t Stat

P-value

0.00 -1.36 0.01 2.35 -0.91

#N/A -3.267 0.070 7.991 -2.471

#N/A 0.002 0.945 0.000 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. 3

2

G = -1.523R1 + 3.534R1 - 1.370R1 + 0.288 (Junction 1)…………………..………..eqn. (5.7) 3

2

G = 3.215R1 - 10.21R1 + 9.900R1 - 1.872 (Junction 2)………………………....…..eqn. (5.8) 3

2

G = -0.373R1 + 1.088R1 - 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).

79

Table 5.15: Limiting value of gradient for junction 1, 2 and 3 – Plain Carbon Steel

2

3

Shrinkage Porosity, (cm3)

0.25 0.5 1 1.5

2 2 2 2

0.1 0.2 1.6 2.5

0.1 0.20 1.59 2.51

12.44 12.3 11.18 10.34

1.15 2.41 8.31 10.1

0.14 0.30 0.93 1.04

0.25 0.5 1 1.5 0.25 0.5

3 3 3 3 4 4

0 2.2 2.3 2.5 0 0

0 2.2 2.3 2.53 0 0

12.24 12.1 11.03 10.24 12.05 11.92

0.12 7.65 9.3 8.2 0.37 0.3

0.01 0.93 1.03 0.84 0.04 0.04

1 1.5

4 4

0.1 0.45

0.1 0.455

10.89 10.13

1.66 3.05

0.18 0.31

1.40

Junction 1

Max. Limiting Gradient Value of 0 C/ mm Gradient %

Limiting value of gradient 0 C/ mm

l/T

Junction t/T

1

AutoCAST

Measured shrinkage porosity cm3

Junction 2

Junction 3

1.20 1.00 0.80 0.60 0.40 0.20 0.00 0

0.5

1

1.5

Fig. 5.25: Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, 2 and 3 - Plain Carbon Steel  

80

Step 3: Regression analysis  Following points should be observed from regression analysis. ¾ R Square is 0.955 which is acceptable and Adjusted R square 0.938 which is also acceptable. ¾ 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. ¾ 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

R Square

0.955

Adjusted R Square

0.938

Standard Error

0.392 70

Observations

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

81

So, equation can be written as.

%P =

G 1 .9 1 3 x R 20.21 5 ……………………………………………………….…....eqn. (5.11) r 1 .6 7 5 x R11.2 8 2

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

1

2

3

0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5

AutoCAST

l/T

Measured shrinkage porosity cm3

Shrinkage Porosity, (cm3)

2 2 2 2 2 2 2 2 2 2 2 2

0 0 0.8 3.1 0 2.1 2.3 2.6 0 0 0.4 0.8

0 0 0.801 3.1 0 2.1 2.3 2.6 0 0 0.4 0.8 82

Limiting Max. Limiting value of Gradient Value of gradient 0 C/ mm Gradient % 0 C/ mm 12.34 12.2 11.09 10.25 12.14 12.01 10.94 10.15 11.96 11.82 10.8 10.45

0.1 0.1 5.31 10.55 0.17 6.75 7.4 8 0.3 0.3 2.82 3.79

0.01 0.01 0.59 1.08 0.02 0.81 0.81 0.81 0.04 0.04 0.30 0.40

3

2

G = -1.365R1 + 3.928R1 - 2.349R1 + 0.375 (Junction 1)………………....……….eqn. (5.12) 3

2

3

2

G = 3.378R1 - 10.13R1 + 9.279R1 - 1.718 (Junction 2)……………………..…….eqn. (5.13) G = -0.86R1 + 2.224R1 - 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. ¾ R Square is 0.931 which is acceptable and Adjusted R square 0.912 which is also acceptable. ¾ The P – value of each variable are acceptable. The P- value of length ration is slightly on higher side. ¾ 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)

1.30

Junction 1

1.10

Junction 2

Junction 3

0.90 0.70 0.50 0.30 0.10 -0.10

0

0.5

1

1.5

Fig. 5.26: Relationship between thickness ratio (R1) and limiting value of gradient (G) for Junction 1, Junction 2 and Junction 3 – SS 410 83

Table 5.19 Regression Statistics – SS 410

R Square Adjusted R Square Standard Error Observations

0.931 0.912 0.552 69.000

Table 5.20 Regression analysis – SS 410 Coefficients

t Stat

P-value

0 -0.821 0.693 1.983 -0.899

#N/A -1.837 3.722 21.713 -1.603

#N/A 0.0707707 0.0004146 1.7E-31 0.1137873

Intercept ln (R1) ln(R2) ln(G) ln(r)

So, equation can be written as.

%P =

G 1.9 83 xR 20.693 …………………………………………………………....eqn. (5.16) r 0.9 xR 10.821

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.

84

 

Wooden pattern

Casting  

Location of shrinkage porosity

Fig. 5.27: T junction casting for validation – SS 410

85

Fig. 5.28: Feed F path : validation casting

Fig. 5.29::Hotspot: Validation V caasting

86

Limiting value of gradient = G = R11.19 / R20.73 (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.

87

5.5 Summary  This section can be summarized as given below: ¾ Analysis for benchmark shape can be done with the help of FEM and VEM. ¾ Vector element method (VEM) directly gives the location of hotspot. Also, thermal gradient, solidfication time etc. can be obtained very easily using it. ¾ 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. ¾ 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 ¾ The empirical model is developed for ductile iron, plain carbon steel and stainless steel (SS 410). ¾ The empirical model can be written in general form as per followings % Porosity = Thermal parameters * Geometric Parameters ¾ The empirical model can be written in generalized form. a

% P =

G x R d2 rbx R

c 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .e q n .( 5 .1 7 )

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 Plain carbon Steel Stainless Steel SG Iron

a

b

C

d

1.913 1.983 2.35

1.675 0.9 0.91

1.282 0.821 1.36

0.215 0.693 0.01

88

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. ¾ It can be summarised from literature review that there are various methods are available for prediction of shrinkage porosity. ¾ It is very convenient to predict shrinkage porosity with the help of criterion function due to their advantages. ¾ 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. ¾ Niyama, LCC, Davis and Bishop Criterion are considered for comparison for sand casting. ¾ Niyama, LCC, Davis and Bishop Criterion are compared for steel sand casting of LJunction. ¾ The simulation results are also compared with AutoCAST software for L-junction. ¾ 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 89 

major metal process combination then comparison can also be made between the other criterion functions. ¾ 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

Criterion Function

Metal

Threshold Value

LCC

Cast Steel

36.5

Davis

Cast Steel

0.9

Bishop

Cast Steel

0.9

¾ 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. ¾ The experimentation is carried out to develop some empirical model which will provide quantitative prediction of shrinkage porosity. ¾ 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. ¾ 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

90 

% Porosity = Thermal Parameters * Geometric Parameters a

%P =

G xR2d c

r b xR 1

Where a, b

=

Thermal Constant

c, d

=

Geometric Constant

The values of above constants are given in table 5.17. ¾ 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. ¾ 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. ¾ 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. ¾ 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.

91 

¾ It will be required to perform some experiments to validate the prediction made by empirical model. It can also be validated by industrial casting.  ¾ 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.       

92 

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.

93   

Procast www.esigroup.com

FEM

2. Magma - www.magma.com

FDM

3. Flow 3d - www.flow3d.com

FDM & FVM

Cast tech www.avenue.castech.fi Power cast 5. www.technalysis.com Mavis flow - www.mavis6. flow.co.uk 4.

7. Cap cast - www.ekkinc.com

SOLIDCAST www.castingsimulation.com NOVA FLOW 9. www.novacast.se SIMtech - www.simtec10. inc.com BKK metal casting 11. simulation software 8.

12. Cast view

13. Camcast simulator

FEM FEM

PC unix based system Sun/ Win NT

















PC windows







-

Windows /Unix Unix workstations







-







-

FEM

PC

-



-

-

FEM

PC DOS / Windows









FDM

NA

‐ 

√ 

‐ 



FDM

Windows





-

-

FEM

Windows/ Unix









FEM

IRIX

-



-

-

FEM

Workstations SG1 Sun









Navier stoke & heat transfer

Unix workstations









   

94   

Platform

stress

1.

Approach

Micro structure

Software

Solidification

Sr. no.

flow

( http://www.thefreelibrary.com/1999+casting+simulation+software+survey.-a054773076)

Annexure II Data for Regression Analysis – SG Iron (500/7)    Sr no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

G τs, r V Glimit % R1 lnR1 R2 ln(R2) 0 max lnG ln(r) C/mm cm3 0C/mm Glimit min. 0C/sec 0.25 -1.39 2 0.69 9.57 72.00 0.276 -1.29 2.885 3.81 1.067 0.07 0.30 -1.20 2 0.69 9.57 73.60 0.297 -1.21 3.105 3.81 1.067 0.07 0.35 -1.05 2 0.69 9.57 75.20 0.319 -1.14 3.333 3.81 1.067 0.07 0.40 -0.92 2 0.69 9.57 76.80 0.341 -1.07 3.567 3.81 1.067 0.07 0.45 -0.80 2 0.69 9.57 78.40 0.364 -1.01 3.808 3.81 1.067 0.07 0.50 -0.69 2 0.69 9.47 80.00 0.388 -0.95 4.095 4.12 0.987 -0.01 0.55 -0.60 2 0.69 9.39 81.60 0.411 -0.89 4.381 4.42 0.920 -0.08 0.60 -0.51 2 0.69 9.34 83.20 0.435 -0.83 4.659 4.73 0.860 -0.15 0.65 -0.43 2 0.69 9.3 84.80 0.459 -0.78 4.935 5.04 0.807 -0.21 0.70 -0.36 2 0.69 9.29 86.40 0.483 -0.73 5.195 5.36 0.759 -0.28 0.75 -0.29 2 0.69 9.28 88.00 0.506 -0.68 5.453 5.68 0.716 -0.33 0.80 -0.22 2 0.69 9.27 89.60 0.529 -0.64 5.707 5.99 0.679 -0.39 0.85 -0.16 2 0.69 9.22 91.20 0.552 -0.59 5.983 6.62 0.614 -0.49 0.90 -0.11 2 0.69 9 92.80 0.574 -0.56 6.373 6.89 0.590 -0.53 0.95 -0.05 2 0.69 8.79 94.40 0.595 -0.52 6.766 7.17 0.567 -0.57 1.00 0.00 2 0.69 8.61 96.00 0.615 -0.49 7.143 7.45 0.546 -0.61 1.05 0.05 2 0.69 8.44 97.60 0.634 -0.46 7.515 7.74 0.525 -0.64 1.10 0.10 2 0.69 8.28 99.20 0.652 -0.43 7.879 8.02 0.507 -0.68 1.15 0.14 2 0.69 8.16 100.80 0.669 -0.40 8.202 8.3 0.490 -0.71 1.20 0.18 2 0.69 8.09 102.40 0.685 -0.38 8.464 8.57 0.475 -0.75 1.25 0.22 2 0.69 8.02 104.00 0.699 -0.36 8.712 8.85 0.460 -0.78 1.30 0.26 2 0.69 7.96 105.60 0.711 -0.34 8.933 9.11 0.446 -0.81 1.35 0.30 2 0.69 8.12 107.20 0.722 -0.33 8.886 9.37 0.434 -0.83 1.40 0.34 2 0.69 8.04 108.80 0.730 -0.31 9.082 9.63 0.422 -0.86 1.45 0.37 2 0.69 7.98 110.40 0.737 -0.31 9.233 9.87 0.412 -0.89 1.50 0.41 2 0.69 7.96 112.00 0.741 -0.30 9.312 10.11 0.402 -0.91 0.25 -1.39 3 1.10 9.42 76.00 0.270 -1.31 2.870 3.67 1.11 0.10 0.30 -1.20 3 1.10 9.42 78.40 0.308 -1.18 3.274 3.67 1.11 0.10 0.35 -1.05 3 1.10 9.42 80.80 0.346 -1.06 3.671 3.67 1.11 0.10 0.40 -0.92 3 1.10 9.42 83.20 0.383 -0.96 4.061 3.67 1.11 0.10 0.45 -0.80 3 1.10 9.42 85.60 0.419 -0.87 4.443 3.67 1.11 0.10 0.50 -0.69 3 1.10 9.32 88.00 0.454 -0.79 4.869 3.97 1.02 0.02 0.55 -0.60 3 1.10 9.24 90.40 0.488 -0.72 5.284 4.27 0.95 -0.05 0.60 -0.51 3 1.10 9.18 92.80 0.522 -0.65 5.686 4.58 0.89 -0.12 0.65 -0.43 3 1.10 9.15 95.20 0.555 -0.59 6.065 4.9 0.83 -0.19 0.70 -0.36 3 1.10 9.13 97.60 0.587 -0.53 6.431 5.22 0.78 -0.25 0.75 -0.29 3 1.10 9.12 100.00 0.619 -0.48 6.782 5.53 0.74 -0.31 0.80 -0.22 3 1.10 9.12 102.40 0.649 -0.43 7.117 5.84 0.70 -0.36 0.85 -0.16 3 1.10 9.13 104.80 0.679 -0.39 7.435 6.21 0.65 -0.42 0.90 -0.11 3 1.10 8.87 107.20 0.708 -0.35 7.979 6.49 0.63 -0.47 0.95 -0.05 3 1.10 8.67 109.60 0.736 -0.31 8.487 6.78 0.60 -0.51 1.00 0.00 3 1.10 8.49 112.00 0.763 -0.27 8.987 7.07 0.58 -0.55 1.05 0.05 3 1.10 8.44 114.40 0.789 -0.24 9.352 7.74 0.53 -0.64 1.10 0.10 3 1.10 8.17 116.80 0.815 -0.20 9.972 7.65 0.53 -0.63 1.15 0.14 3 1.10 8.06 119.20 0.839 -0.18 10.412 7.94 0.51 -0.67 1.20 0.18 3 1.10 7.98 121.60 0.863 -0.15 10.812 8.22 0.49 -0.70 1.25 0.22 3 1.10 7.92 124.00 0.885 -0.12 11.180 8.51 0.48 -0.74

95   

P cm3 0.250 0.269 0.258 0.269 0.275 0.300 0.305 0.281 0.251 0.233 0.232 0.252 0.239 0.208 0.281 0.420 0.497 0.575 0.604 0.592 0.714 0.762 0.736 0.783 0.793 0.750 0.250 0.313 0.344 0.377 0.397 0.455 0.497 0.464 0.466 0.436 0.417 0.391 0.594 0.681 0.815 0.890 0.773 1.260 1.390 1.540 1.910

%P ln(%P) 0.347 0.365 0.343 0.350 0.351 0.375 0.374 0.338 0.296 0.270 0.264 0.281 0.262 0.224 0.298 0.438 0.509 0.580 0.599 0.578 0.687 0.722 0.687 0.720 0.718 0.670 0.329 0.399 0.426 0.453 0.464 0.517 0.550 0.500 0.489 0.447 0.417 0.382 0.567 0.635 0.744 0.795 0.676 1.079 1.166 1.266 1.540

-1.06 -1.01 -1.07 -1.05 -1.05 -0.98 -0.98 -1.09 -1.22 -1.31 -1.33 -1.27 -1.34 -1.50 -1.21 -0.83 -0.67 -0.55 -0.51 -0.55 -0.38 -0.33 -0.38 -0.33 -0.33 -0.40 -1.11 -0.92 -0.85 -0.79 -0.77 -0.66 -0.60 -0.69 -0.71 -0.81 -0.87 -0.96 -0.57 -0.45 -0.30 -0.23 -0.39 0.08 0.15 0.24 0.43

Sr no. 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

V Glimit % G R1 lnR1 R2 ln(R2) 0 max lnG C/mm cm3 0C/mm Glimit 1.30 0.26 3 1.10 7.86 126.40 0.907 -0.10 11.542 1.35 0.30 3 1.10 8.04 128.80 0.928 -0.07 11.541 1.40 0.34 3 1.10 7.97 131.20 0.948 -0.05 11.891 1.45 0.37 3 1.10 7.91 133.60 0.966 -0.03 12.218 1.50 0.41 3 1.10 7.88 136.00 0.984 -0.02 12.490 0.25 -1.39 4 1.39 9.28 80.00 0.403 -0.91 4.339 0.30 -1.20 4 1.39 9.28 83.20 0.444 -0.81 4.783 0.35 -1.05 4 1.39 9.28 86.40 0.485 -0.72 5.230 0.40 -0.92 4 1.39 9.28 89.60 0.527 -0.64 5.678 0.45 -0.80 4 1.39 9.28 92.80 0.568 -0.57 6.124 0.50 -0.69 4 1.39 9.18 96.00 0.609 -0.50 6.637 0.55 -0.60 4 1.39 9.09 99.20 0.649 -0.43 7.145 0.60 -0.51 4 1.39 9.04 102.40 0.689 -0.37 7.618 0.65 -0.43 4 1.39 9.01 105.60 0.727 -0.32 8.065 0.70 -0.36 4 1.39 8.99 108.80 0.763 -0.27 8.490 0.75 -0.29 4 1.39 8.98 112.00 0.798 -0.23 8.886 0.80 -0.22 4 1.39 8.98 115.20 0.831 -0.19 9.252 0.85 -0.16 4 1.39 8.97 118.40 0.861 -0.15 9.603 0.90 -0.11 4 1.39 8.76 121.60 0.889 -0.12 10.153 0.95 -0.05 4 1.39 8.56 124.80 0.915 -0.09 10.686 1.00 0.00 4 1.39 8.38 128.00 0.937 -0.07 11.181 1.05 0.05 4 1.39 8.22 131.20 0.956 -0.04 11.630 1.10 0.10 4 1.39 8.07 134.40 0.972 -0.03 12.039 1.15 0.14 4 1.39 7.06 137.60 0.983 -0.02 13.927 1.20 0.18 4 1.39 7.89 140.80 0.991 -0.01 12.560 1.25 0.22 4 1.39 8.07 144.00 0.994 -0.01 12.322 1.30 0.26 4 1.39 8 147.20 0.993 -0.01 12.416 1.35 0.30 4 1.39 7.92 150.40 0.987 -0.01 12.466 1.40 0.34 4 1.39 7.86 153.60 0.976 -0.02 12.422 1.45 0.37 4 1.39 7.85 156.80 0.960 -0.04 12.230 1.50 0.41 4 1.39 7.8 160.00 0.938 -0.06 12.029

 

96   

τs, min. 8.79 9.06 9.33 9.59 9.84 3.55 3.55 3.55 3.55 3.55 3.83 4.14 4.44 4.76 5.07 5.37 5.68 5.69 6.13 6.42 6.71 7 7.29 7.58 7.87 8.15 8.43 8.7 8.97 9.23 9.49

r ln(r) C/sec 0.46 -0.77 0.45 -0.80 0.44 -0.83 0.42 -0.86 0.41 -0.88 1.146 0.14 1.146 0.14 1.146 0.14 1.146 0.14 1.146 0.14 1.062 0.06 0.982 -0.02 0.916 -0.09 0.854 -0.16 0.802 -0.22 0.757 -0.28 0.716 -0.33 0.715 -0.34 0.663 -0.41 0.633 -0.46 0.606 -0.50 0.581 -0.54 0.558 -0.58 0.536 -0.62 0.517 -0.66 0.499 -0.70 0.482 -0.73 0.467 -0.76 0.453 -0.79 0.441 -0.82 0.429 -0.85

0

P cm3 2.010 2.030 2.040 2.160 2.170 0.500 0.572 0.587 0.616 0.698 0.750 0.850 0.869 0.844 0.779 0.743 0.844 0.919 1.200 1.390 1.600 1.870 2.120 2.570 2.590 2.520 2.600 2.550 2.640 2.540 2.500

%P ln(%P) 1.590 1.576 1.555 1.617 1.596 0.625 0.688 0.679 0.688 0.752 0.781 0.857 0.849 0.799 0.716 0.663 0.733 0.776 0.987 1.114 1.250 1.425 1.577 1.868 1.839 1.750 1.766 1.695 1.719 1.620 1.563

0.46 0.45 0.44 0.48 0.47 -0.47 -0.37 -0.39 -0.37 -0.28 -0.25 -0.15 -0.16 -0.22 -0.33 -0.41 -0.31 -0.25 -0.01 0.11 0.22 0.35 0.46 0.62 0.61 0.56 0.57 0.53 0.54 0.48 0.45

Annexure III Data for Regression Analysis ‐ Plain Carbon Steel (AISI 1005)    Sr no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

R1 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

lnR1 R2 ln(R2) -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.37 0.41 -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10

0

Gmax V Glimit C/mm cm3 0C/mm 12.44 72.00 0.14 12.44 73.60 0.15 12.44 75.20 0.18 12.44 76.80 0.21 12.44 78.40 0.25 12.30 80.00 0.30 12.20 81.60 0.35 12.13 83.20 0.41 12.09 84.80 0.47 12.06 86.40 0.54 12.05 88.00 0.61 12.05 89.60 0.67 11.97 91.20 0.74 11.69 92.80 0.81 11.42 94.40 0.87 11.18 96.00 0.93 10.96 97.60 0.98 10.76 99.20 1.03 10.61 100.80 1.07 10.51 102.40 1.10 10.42 104.00 1.12 10.34 105.60 1.13 10.54 107.20 1.13 10.45 108.80 1.12 10.37 110.40 1.09 10.34 112.00 1.04 12.24 76.00 0.02 12.24 78.40 0.27 12.24 80.80 0.48 12.24 83.20 0.66 12.24 85.60 0.81 12.10 88.00 0.93 12.00 90.40 1.02 11.93 92.80 1.09 11.89 95.20 1.13 11.86 97.60 1.16 11.85 100.00 1.17 11.85 102.40 1.16 11.81 104.80 1.14 11.53 107.20 1.11 11.27 109.60 1.07 11.03 112.00 1.03 10.81 114.40 0.99 10.62 116.80 0.94 10.47 119.20 0.90 10.37 121.60 0.86 10.28 124.00 0.83

lnG -1.95 -1.87 -1.74 -1.57 -1.39 -1.22 -1.05 -0.89 -0.75 -0.62 -0.50 -0.39 -0.30 -0.21 -0.14 -0.07 -0.02 0.03 0.07 0.10 0.12 0.13 0.12 0.11 0.08 0.04 -4.19 -1.32 -0.73 -0.42 -0.21 -0.08 0.02 0.08 0.12 0.15 0.15 0.15 0.13 0.11 0.07 0.03 -0.01 -0.06 -0.11 -0.15 -0.19

97   

% Glimit 1.15 1.24 1.42 1.67 2.00 2.41 2.87 3.37 3.91 4.46 5.03 5.59 6.19 6.91 7.62 8.31 8.97 9.57 10.08 10.48 10.78 10.96 10.74 10.69 10.50 10.10 0.12 2.17 3.92 5.39 6.60 7.66 8.49 9.11 9.52 9.76 9.84 9.79 9.66 9.64 9.54 9.37 9.14 8.88 8.59 8.30 8.07

τ s, min. 2.06 2.06 2.06 2.06 2.06 2.23 2.39 2.56 2.73 2.90 3.07 3.24 3.57 3.72 3.88 4.03 4.18 4.33 4.33 4.63 4.78 4.92 5.06 5.20 5.33 5.46 1.99 1.99 1.99 1.99 1.99 2.14 2.31 2.48 2.65 2.82 2.99 3.15 3.35 3.51 3.66 3.82 3.98 4.13 4.29 4.44 4.60

r C/sec 1.32 1.32 1.32 1.32 1.32 1.22 1.14 1.06 1.00 0.94 0.88 0.84 0.76 0.73 0.70 0.67 0.65 0.63 0.63 0.59 0.57 0.55 0.54 0.52 0.51 0.50 1.37 1.37 1.37 1.37 1.37 1.27 1.18 1.10 1.03 0.96 0.91 0.86 0.81 0.77 0.74 0.71 0.68 0.66 0.63 0.61 0.59

0

ln(r) 0.28 0.28 0.28 0.28 0.28 0.20 0.13 0.06 0.00 -0.07 -0.12 -0.18 -0.27 -0.31 -0.36 -0.39 -0.43 -0.47 -0.47 -0.53 -0.57 -0.59 -0.62 -0.65 -0.67 -0.70 0.31 0.31 0.31 0.31 0.31 0.24 0.16 0.09 0.02 -0.04 -0.10 -0.15 -0.21 -0.26 -0.30 -0.34 -0.38 -0.42 -0.46 -0.49 -0.53

P cm3 0.10 0.10 0.12 0.11 0.12 0.21 0.27 0.36 0.43 0.49 0.56 0.57 0.70 0.85 1.18 1.59 1.81 2.11 2.36 2.46 2.89 2.98 2.83 2.93 2.76 2.51 0.00 0.19 0.89 1.38 1.73 2.20 2.40 2.60 2.67 2.63 2.50 2.24 1.97 2.01 2.14 2.30 2.30 2.50 2.50 2.45 2.59

%P ln(%P) 0.14 -1.97 0.14 -1.98 0.15 -1.87 0.15 -1.92 0.15 -1.89 0.26 -1.36 0.33 -1.11 0.43 -0.85 0.51 -0.67 0.57 -0.57 0.64 -0.45 0.63 -0.46 0.77 -0.26 0.92 -0.09 1.25 0.22 1.66 0.50 1.85 0.62 2.13 0.75 2.34 0.85 2.40 0.88 2.78 1.02 2.82 1.04 2.64 0.97 2.69 0.99 2.50 0.92 2.24 0.81 0.00 #NUM! 0.24 -1.43 1.10 0.09 1.66 0.51 2.02 0.70 2.50 0.92 2.65 0.98 2.80 1.03 2.80 1.03 2.69 0.99 2.50 0.92 2.19 0.78 1.88 0.63 1.88 0.63 1.95 0.67 2.05 0.72 2.01 0.70 2.14 0.76 2.10 0.74 2.01 0.70 2.09 0.74

Sr no. 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

R1 1.30 1.35 1.40 1.45 1.50 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50

lnR1 R2 ln(R2) 0.26 0.30 0.34 0.37 0.41 -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.37 0.41

3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1.10 1.10 1.10 1.10 1.10 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39

0

V Glimit Gmax C/mm cm3 0C/mm 10.21 126.40 0.81 10.45 128.80 0.80 10.35 131.20 0.80 10.28 133.60 0.82 10.24 136.00 0.86 12.05 80.00 0.04 12.05 83.20 0.04 12.05 86.40 0.03 12.05 89.60 0.03 12.05 92.80 0.03 11.92 96.00 0.04 11.81 99.20 0.04 11.75 102.40 0.05 11.71 105.60 0.06 11.68 108.80 0.08 11.67 112.00 0.09 11.66 115.20 0.11 11.66 118.40 0.13 11.38 121.60 0.15 11.13 124.80 0.16 11.03 128.00 0.18 10.68 131.20 0.20 10.48 134.40 0.22 10.34 137.60 0.24 10.25 140.80 0.25 10.49 144.00 0.27 10.39 147.20 0.28 10.29 150.40 0.29 10.21 153.60 0.30 10.19 156.80 0.31 10.24 160.00 0.31

lnG -0.22 -0.23 -0.23 -0.20 -0.16 -3.11 -3.33 -3.48 -3.53 -3.47 -3.33 -3.14 -2.94 -2.74 -2.55 -2.37 -2.21 -2.06 -1.93 -1.81 -1.70 -1.61 -1.52 -1.45 -1.38 -1.32 -1.27 -1.23 -1.20 -1.18 -1.16

 

98   

% Glimit 7.90 7.61 7.71 7.96 8.36 0.37 0.30 0.26 0.24 0.26 0.30 0.37 0.45 0.55 0.67 0.80 0.94 1.09 1.27 1.47 1.65 1.88 2.08 2.28 2.46 2.54 2.70 2.83 2.95 3.02 3.05

τ s, min. 4.75 4.90 5.04 5.18 5.32 1.92 1.92 1.92 1.92 1.92 2.07 2.23 2.40 2.57 2.74 2.90 3.07 3.07 3.31 3.47 3.62 3.78 3.94 4.10 4.25 4.40 4.56 4.70 4.85 4.99 5.13

r C/sec 0.57 0.55 0.54 0.52 0.51 1.41 1.41 1.41 1.41 1.41 1.31 1.22 1.13 1.06 0.99 0.94 0.88 0.88 0.82 0.78 0.75 0.72 0.69 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.53

0

ln(r) -0.56 -0.59 -0.62 -0.65 -0.67 0.35 0.35 0.35 0.35 0.35 0.27 0.20 0.12 0.06 -0.01 -0.07 -0.12 -0.12 -0.20 -0.24 -0.29 -0.33 -0.37 -0.41 -0.45 -0.48 -0.52 -0.55 -0.58 -0.61 -0.64

P cm3 2.49 2.32 2.37 2.46 2.53 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.09 0.10 0.11 0.23 0.36 0.34 0.40 0.44 0.47 0.48 0.48 0.46

%P ln(%P) 1.97 1.80 1.81 1.84 1.86 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.07 0.08 0.09 0.17 0.26 0.24 0.28 0.30 0.31 0.31 0.31 0.28

0.68 0.59 0.59 0.61 0.62 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! -4.60 -4.41 -4.44 -4.51 -4.58 -4.70 #NUM! -4.63 -2.67 -2.55 -2.45 -1.77 -1.34 -1.41 -1.29 -1.21 -1.17 -1.16 -1.18 -1.26

Annexure IV Data for Regression Analysis – Stainless Steel (SS 410)    Sr no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

R1 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

lnR1 R2 ln(R2) -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.37 0.41 -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 0.69 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10

0

Gmax V Glimit lnG C/mm cm3 0C/mm 12.34 72.00 0.01 -4.43 12.34 73.60 0.01 -4.43 12.34 75.20 0.01 -4.43 12.34 76.80 0.01 -4.43 12.34 78.40 0.01 -4.43 12.20 80.00 0.01 -4.43 12.10 81.60 0.04 -3.12 12.03 83.20 0.08 -2.47 11.99 84.80 0.13 -2.02 11.97 86.40 0.19 -1.68 11.95 88.00 0.25 -1.40 11.95 89.60 0.31 -1.17 11.88 91.20 0.38 -0.97 11.59 92.80 0.45 -0.80 11.33 94.40 0.52 -0.66 11.09 96.00 0.59 -0.53 10.87 97.60 0.66 -0.42 10.67 99.20 0.73 -0.32 10.52 100.80 0.79 -0.23 10.43 102.40 0.85 -0.16 10.33 104.00 0.91 -0.09 10.26 105.60 0.96 -0.04 10.46 107.20 1.00 0.00 10.36 108.80 1.04 0.04 10.29 110.40 1.07 0.06 10.25 112.00 1.08 0.08 12.14 76.00 0.02 -3.84 12.14 78.40 0.25 -1.41 12.14 80.80 0.43 -0.84 12.14 83.20 0.59 -0.53 12.14 85.60 0.71 -0.34 12.01 88.00 0.81 -0.21 11.90 90.40 0.88 -0.12 11.84 92.80 0.93 -0.07 11.79 95.20 0.96 -0.04 11.77 97.60 0.97 -0.03 11.76 100.00 0.97 -0.03 11.75 102.40 0.95 -0.05 11.71 104.80 0.92 -0.08 11.71 107.20 0.89 -0.12 11.18 109.60 0.85 -0.16 10.94 112.00 0.81 -0.21 10.73 114.40 0.77 -0.27 10.53 116.80 0.73 -0.32 10.38 119.20 0.69 -0.37 10.29 121.60 0.67 -0.41 10.20 124.00 0.65 -0.43

99   

% Glimit 0.10 0.10 0.10 0.10 0.10 0.10 0.37 0.71 1.11 1.56 2.07 2.60 3.18 3.86 4.57 5.31 6.06 6.82 7.53 8.19 8.81 9.36 9.60 10.04 10.36 10.56 0.18 2.02 3.57 4.85 5.88 6.75 7.42 7.87 8.15 8.26 8.23 8.10 7.90 7.60 7.61 7.39 7.15 6.91 6.68 6.48 6.38

τs, min. 2.70 2.70 2.70 2.70 2.70 2.92 3.13 3.35 3.58 3.80 4.03 4.24 4.68 4.89 5.08 5.28 5.48 5.68 5.88 6.08 6.27 6.46 6.64 6.82 7.00 7.17 2.6 2.6 2.6 2.6 2.6 2.81 3.03 3.25 3.47 3.7 3.92 4.14 4.4 4.4 4.8 5.01 5.21 5.42 5.63 5.83 6.03

r C/sec 1.35 1.35 1.35 1.35 1.35 1.24 1.16 1.08 1.01 0.96 0.90 0.86 0.78 0.74 0.72 0.69 0.66 0.64 0.62 0.60 0.58 0.56 0.55 0.53 0.52 0.51 1.40 1.40 1.40 1.40 1.40 1.29 1.20 1.12 1.05 0.98 0.93 0.88 0.83 0.83 0.76 0.73 0.70 0.67 0.65 0.62 0.60

0

ln(r) 0.30 0.30 0.30 0.30 0.30 0.22 0.15 0.08 0.01 -0.04 -0.10 -0.15 -0.25 -0.30 -0.34 -0.37 -0.41 -0.45 -0.48 -0.51 -0.55 -0.58 -0.60 -0.63 -0.66 -0.68 0.33 0.33 0.33 0.33 0.33 0.26 0.18 0.11 0.05 -0.02 -0.08 -0.13 -0.19 -0.19 -0.28 -0.32 -0.36 -0.40 -0.44 -0.47 -0.51

P cm3 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.03 0.04 0.06 0.08 0.31 0.57 0.80 0.99 1.28 1.77 2.05 2.53 2.61 2.70 2.93 3.07 3.10 0 0.166 0.831 1.34 1.77 2.1 2.48 2.59 2.63 2.5 2.17 1.86 1.73 1.57 2.16 2.3 2.28 2.21 2.1 1.89 1.94

%P

ln(%P)

0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.03 0.03 0.03 0.05 0.07 0.09 0.34 0.61 0.83 1.01 1.29 1.76 2.00 2.43 2.47 2.52 2.69 2.78 2.77 0.00 0.21 1.03 1.61 2.07 2.39 2.74 2.79 2.76 2.56 2.17 1.82 1.65 1.46 1.97 2.05 1.99 1.89 1.76 1.55 1.56

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! -4.07 -3.68 -3.50 -3.47 -3.02 -2.68 -2.46 -1.09 -0.50 -0.18 0.01 0.25 0.56 0.69 0.89 0.90 0.92 0.99 1.02 1.02 #NUM! -1.55 0.03 0.48 0.73 0.87 1.01 1.03 1.02 0.94 0.77 0.60 0.50 0.38 0.68 0.72 0.69 0.64 0.57 0.44 0.45

Sr no. 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

R1 1.30 1.35 1.40 1.45 1.50 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50

lnR1 R2 ln(R2) 0.26 0.30 0.34 0.37 0.41 -1.39 -1.20 -1.05 -0.92 -0.80 -0.69 -0.60 -0.51 -0.43 -0.36 -0.29 -0.22 -0.16 -0.11 -0.05 0.00 0.05 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.37 0.41

3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

1.10 1.10 1.10 1.10 1.10 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39 1.39

0

V Glimit Gmax lnG C/mm cm3 0C/mm 10.13 126.40 0.65 -0.44 10.36 128.80 0.66 -0.42 10.27 131.20 0.69 -0.38 10.19 133.60 0.74 -0.31 10.15 136.00 0.81 -0.21 11.96 80.00 0.04 -3.22 11.96 83.20 0.04 -3.22 11.96 86.40 0.04 -3.22 11.96 89.60 0.04 -3.22 11.96 92.80 0.04 -3.22 11.82 96.00 0.04 -3.22 11.72 99.20 0.05 -2.98 11.65 102.40 0.07 -2.64 11.61 105.60 0.10 -2.35 11.59 108.80 0.12 -2.10 11.58 112.00 0.15 -1.89 11.57 115.20 0.18 -1.71 11.56 118.40 0.21 -1.55 11.29 121.60 0.24 -1.42 11.04 124.80 0.27 -1.30 10.8 128.00 0.30 -1.19 10.59 131.20 0.33 -1.11 10.4 134.40 0.36 -1.03 10.26 137.60 0.38 -0.97 10.17 140.80 0.40 -0.92 10.41 144.00 0.41 -0.89 10.3 147.20 0.42 -0.87 10.21 150.40 0.42 -0.86 10.13 153.60 0.42 -0.87 10.11 156.80 0.41 -0.89 10.45 160.00 0.39 -0.93

 

100   

% Glimit 6.38 6.35 6.69 7.23 7.97 0.33 0.33 0.33 0.33 0.33 0.34 0.43 0.61 0.82 1.05 1.30 1.56 1.83 2.15 2.48 2.81 3.12 3.42 3.69 3.90 3.95 4.08 4.15 4.15 4.06 3.77

τs, min. 6.23 6.42 6.61 6.8 6.98 2.52 2.52 2.52 2.52 2.52 2.72 2.93 3.15 3.37 3.6 3.8 4.02 4.03 4.35 4.55 4.75 4.96 5.15 5.37 5.58 5.78 5.98 6.17 6.36 6.54 6.72

r C/sec 0.58 0.57 0.55 0.53 0.52 1.44 1.44 1.44 1.44 1.44 1.34 1.24 1.15 1.08 1.01 0.96 0.90 0.90 0.84 0.80 0.76 0.73 0.71 0.68 0.65 0.63 0.61 0.59 0.57 0.56 0.54

0

ln(r) -0.54 -0.57 -0.60 -0.63 -0.65 0.37 0.37 0.37 0.37 0.37 0.29 0.22 0.14 0.08 0.01 -0.04 -0.10 -0.10 -0.18 -0.22 -0.27 -0.31 -0.35 -0.39 -0.43 -0.46 -0.50 -0.53 -0.56 -0.59 -0.61

P cm3 1.89 1.87 2.05 2.29 2.6 0.00 0.01 0.01 0.01 0.01 0.00 0.01 0.02 0.02 0.02 0.02 0.02 0.31 0.36 0.35 0.40 0.58 0.75 0.88 0.86 0.89 1.02 1.06 1.06 0.99 0.80

%P

ln(%P)

1.50 0.40 1.45 0.37 1.56 0.45 1.71 0.54 1.91 0.65 0.00 #NUM! 0.01 -4.53 0.01 -4.46 0.01 -4.31 0.01 -4.27 0.00 #NUM! 0.01 -4.26 0.02 -4.04 0.02 -4.02 0.02 -4.05 0.02 -4.08 0.01 -4.28 0.26 -1.35 0.29 -1.23 0.28 -1.27 0.31 -1.16 0.44 -0.81 0.56 -0.59 0.64 -0.44 0.61 -0.49 0.62 -0.48 0.69 -0.37 0.70 -0.35 0.69 -0.37 0.63 -0.46 0.50 -0.69

References  Journal articles  1. A. Tseng and J. Zou,"Numerical Modeling of Macro and Micro Behaviors of Materials in Processing" A Review," Journal of Compurational Physics,102(1992) 1-17. 2. A. Kermanpur,"Numerical simulation of metal flow and solidification in the multi-cavity casting moulds of automotive components," Journal of Materials Processing Technology, 2006(2008) 62–68. 3. A. Reis, Y. Houbaert, Zhian Xu, Rob Van Tol, A.D.Santos, J.F.Duarte, A.B. Magalhaes ,"Modeling of shrinkage defects during solidification of long and short freezing materials," Journal of Materials Processing Technology, 202(2008) 428–434. 4. A S Sabau and S Visvanathan,"Microporosity Prediction in Aluminum Alloy Castings" Metallurgical And Materials Trnascations B, 33B(2002) 243-255. 5. Barkhudarov M R: Advanced simulation of the flow and heat transfer processes in simultaneous engineering, Flow Science, Inc. 6. C. Bailey, P. Chow, M. Cross, Y. Fryer, K. Pericleous  ,"Multiphysics Modelling of the Metals Casting Process," Mathematical, Physical and Engineering Sciences, 452(1946)(1996) 459486. 7. Ch. Pequet et al.,"Modeling of Microporosity, Macroporosity and Pipe Shrinkage Formation during the Solidification of Alloys Using a Mushy-Zone Refinement Method: Applications to Aluminum Alloys," Metallurgical and Material Transactions A, 33A(2002) 2095-2106. 8. D. Joshi, B.Ravi, “Classification and Simulation Based Design of 3D Junctions in Casting”, AFS (2009). 9. D.R. Gunasegaram, D.J. Farnsworth,1, T.T. Nguyen,"Identification of critical factors affecting shrinkage porosity in permanent mold casting using numerical simulations based on design of experiments" Journal of Materials Processing Technology, 209(2009) 1209–1219. 10. D. R. Poirier, K. Yeum, and A. L. Maples, “A Thermodynamic Prediction for Macroporosity

Formation in Aluminum-Rich Al-Cu Alloys”, Met. Trans. 18A(1987) 1979-1987. 11. Dawei Sun and Suresh V. Garimella,"Numerical and Experimental Investigation of Solidification Shrinkage," Numerical Heat Transfer, Part A(52)(2007) 145–162. 12. Dieter Ott " Chaos in Casting: An Approach to Shrinkage Porosity," Gold bulletin 30(1)(1997) 13- 19. 13. E. Escobar de Obaldia, S.D. Felicelli,"Quantitative prediction of microporosity in aluminum alloys," Journal of Materials Processing Technology, 191 (2007) 265-269. 14. E. Niyama, T. Uchida, M. Morikawa, and S. Saito, “A Method of Shrinkage Prediction and its Application to Steel Casting Practice,” Am. Foundrymen’s Soc. Int. Cast Met. J., 7(3)(1982) 52-63. 101   

15. G. K. Sigworth and Chengming Wan, "Mechanisms of Porosity Formation during Solidification: A Theoretical Analysis" Metallurgical Transactions B 24B(1993) 349-364. 16. Gunasegarama, D.J. Farnsworthb, T.T. Nguyena, “Identification of critical factors affecting shrinkage porosity in permanent mold casting using numerical simulations based on design of experiments,” Journal of Materials Processing Technology 209(2009) 1209–1219. 17. Hallam, C., Griffiths, W., 2004. A model of the interfacial heat-transfer coefficient for the aluminum gravity die casting process. Metall. Mater. Trans. B 35, 721–733. 18. H. F. Bishop and W. S. Pellini, “The Contribution of Riser and Casting End Effects to Soundness of Cast Steel Bars”, AFS Trans., 59(1951)171. 19. H. Iwahori, K. Yonekura, Y. Sugiyama, Y. Ymamoto and M. Nakamura, “Behavior of Shrinkage Porosity Defects and Limiting Solid fraction of Feeding on Al-Si Alloys”, AFS Trans., 71(1985) 443-451. 20. I.H. Katzarov

Y.B. Arsov, P. Stoyanov, T. Zeuner, A. Buehrig-Polaczek, P.R. Sahm

,"Porosity formation in axi-symmetric castings produced by counter-pressure casting method," International Journal of Heat and Mass Transfer, 44 (2001) 111-119. 21. Imafuku I, Chijiiwa K 1983: A mathematical model for shrinkage cavity prediction in steel castings, AFS Transactions , 91:527–540. 22. J. A. Eady and D. M. Smith, “The Effect of Porosity on the Tensile Properties of Al-alloy Castings” Mat. Forum, 9(4)(1986) 217-223. 23. J.L.H. Green,"Porosity in Castings - An Everyday Reality," Materials and Design,5(1984) 1-5. 24. Jaroslav Mackerle,"Object-oriented programming in FEM and BEM: a bibliography (1990– 2003)" Advances in Engineering Software,35 (2004) 325–336. 25. John W. Gibbs and Patricio F. Mendez ,"Solid fraction measurement using equation-based cooling curve analysis," Scripta Materialia, 58 (2008) 699–702. 26. J.Y. Buffiere, S. Savelli, P.H. Jouneau , E. Maire, R. Fouge`res

“Experimental study of

porosity and its relation to fatigue mechanisms of model Al Si-7-Mg0.3 cast Al alloys” Materials Science and Engineering A316(2001)115-126. 27. K. Davey and S. Hinduja,"Modeling the transient thermal behaviour of the pressure die-casting process with the BEM," Appl. Math. Modelling, 14(1990) 395-409. 28. K.D. Carlson, S. Ou, R.A. Hardin, and C. Beckermann, “Development of New FeedingDistance Rules Using Casting Simulation: Part I. Methodology,” Metall. Mater. Trans. B, 33B(2002) 731-740. 29. Kent D. Carlson and Christoph Beckermann ,"Prediction of Shrinkage Pore Volume Fraction Using a Dimensionless Niyama Criterion," Metallurgical and Materials Transactions A,40A(2009) 163-175. 30. K. Kubo and R. Phelke, “Mathematical Modeling of Porosity Formation in Solidification”, Met. Trans B, June 16B (1985) 359-366. 102   

31. M. J. Couper, “Casting Defects and the Fatigue Behavior of an Al-alloy Casting”, Fatigue Fracture Engineering Material Structure, 13 (3)(1990) 213-227. 32. M. Xiong and A.V. Kuznetsov,"Comparison between Lever and Scheil Rules for Modeling of Microporosity Formation during Solidification," Flow, Turbulence and Combustion, 67(2001) 305–323. 33. Mark Jolly, " Casting Simulation: How well do Reality and Virtual Casting Match? State of the art review,"International Journal of Cast Metal,14(2002)303-313. 34. Nao- Aki Noda , "Predicting Locations of Defects in the Solidification Process for Large-Scale Cast Steel," Journal of Computational Science and Technology, 3(1)(2009) 1136-1143. 35. P.D.Lee, A.Chirazi, D. See. "Modeling microporosity in aluminium-silicon alloys: a review,"Journal of Light Metals 1(2001) 15-30. 36. Qiming Chen and C. Ravindran, "A Study of Thermal Parameters and Interdendritic Feeding in Lost Foam Casting," Journal of Materials Engineering and Performance, 9(4)(2000) 386395. 37. R. Monroe,"Porosity in Castings,"AFS Transactions,05-245(04)(2005) 1-28. 38. Rohallah Tavakoli,"Automatic optimal feeder design in steel casting process," Appl. Mech. Engrg. 197 (2008) 921–932. 39. S. J. Neises, J. J. Uicker and R. W. Heine “Geometric Modeling of Directional Solidification Based on Section Modulus”, AFS Trans., 61(1987) 25-29. 40. S.Minakawa, I V Samarasekera and F Weinberg ,"Centreline Porosity in Plate Casting", Metallurgical Transactions B, 16(B)(1985)823-829. 41. S. Shivkumar, D. Apelian and J. Zou, “Modeling of Microstructure Evolution and Microporosity Formation in Cast Aluminum Alloys”, AFS Trans.98 (1990) 897-904. 42. T.R. Vijayaram et al.,"Numerical simulation of casting solidification in permanent metallic molds," Journal of Materials Processing Technology, 178 (2006) 29–33. 43. Y.W. Lee, E. Chang and C.F. Chieu " Modeling of Feeding Behaviour of Solidifying AI-7Si0.3Mg Alloy Plate Casting," Metallurgical Transactions B, 21 B(1990) 715-722. 44. Yin-Henc S. Minakawa Chen et al. ,"Analysis of Solidification in Sand and Permanent Mold Casting Shrinkage Prediction," Int. J. Mach. Tools Manufact. 30(2)(1990) 175-189. 45. Yoshihiko Hangai et al.,"Quantitative evaluation of porosity in aluminum alloy die castings by fractal analysis of spatial distribution of area," Materials and Design 30 (2009) 1169–1173. 46. C. W. Hirt et al.," Casting Simulation: Mold filling and Solidification- Benchmark Calculations using Flow-3D® 47. C.W. Hirt, "Modeling Shrinkage Induced Micro-porosity,"FSI-03-TN66(2004). 48. E. Liotti, B. Previtali,"Study of the validity of the Niyama criteria function applied to the alloy AlSi7Mg," Memoria vincitrice del premio Aldo Daccò (2004).

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49. E. Niyama, T. Uchida, M. Morikawa and S. Saito, “A Method of Shrinkage Prediction and Its Application to Steel Casting Practice”, Inter. Foundry Congress 49 in Chicago, paper 10 (1982). 50. Kent D. Carlson et and Christoph Beckerman “ Use of the Niyama Criterion to predict shrinkage – related leaks in High Nickel Steel and Nickel-based Alloy Casting” Steel Founders' Society of America National T&O Conference – December 11-13, (2008).

Bibliography     51. B Cantor, K O'Reilly,"Solidification and casting," Institute of Physics Publishing Bristol and Philadelphia, 2003. 52. Ravi B “Metal casting- computer aided design and analysis,” PHI, 2008. 53. Doru Michael Stefanescu"Science and Engineering of Casting Solidification," Springer Science plus Business Media,2nd edition, 2008. 54. J. Campbell, "Castings", Butterworth – Heinemann Ltd. Oxford 1991. 55. Laurentiu Nastac,"Modeling and Simulation of Microstructure Evolution in Solidifying," Kluwer Academic Publisher, New York, 2004. 56. Minkowycs W J, Sparrow E M “Advances in numerical heat transfer,” Taylor & Francis, 1997

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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

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