1 NUMERICAL SIMULATION OF SAND CASTING PROCESS

A Thesis Presented to The Faculty of the College of Engineering and Technology Ohio University

In Partial Fulfilment of the Requirements for the Degree Master of Science

by

Kuah Teng Hock , .. - November 1987 ACKNOWLEDGEMENT

I take this opportunity to express my sincere thanks to Dr. Khariul Alam for the advice and guidance he provided during this research undertaking. I also wish to thank him for being such an understanding and helpful teacher during my four years stay at Ohio University.

I would like to thank Dr. Jay Gunasekara for guiding the project, specially during its infancy. Dr. Gary Graham was extremely helpful throughout my research, and helped me with numerical computations, specially in the area of finite elements.

In addition, I would like to thank Dr. Albert R. Squibb who was very helpful during the conduction of the experiment.

I owe a special word of thanks to the following people who have been a great source of moral support and help : Chong Hee, Wai Kuen, Lay Tin, Kim Fong, Tsu Cheng, and Amy Allen. A special word of thanks to Daniel Allen my good friend and colleague who has helped greatly during the conduction of the experiment.

Finally, I am grateful to my mother and family for being patience and understanding during my studies at Ohio University.

This project was funded by Ohio University Research Committee, and by Edison Material Technology Center (EMTEC). TABLE

Chapter 1 : Overview of Metal Casting 1.1 Introduction 1.2 History of Metal Casting 1.3 Modem Metal Casting Industry 1.4 History of Computer-Aided Casting

Chapter 2 : Literature Survey 2.1 Literature Review 2.2 Summary of Available References for Computer Simulation of Casting

Chapter 3 : Numerical Modeling and Solidification 3.1 Introduction 3.2 Heat Transfer During Solidification 3.3 Finite Element Technique 3.4 Numerical Modeling 3.5 Taylor Series Expansion of Space and Time Derivatives 3.6 Alternating Direction Implicit (ADI) Method 3.6.1 Derivation of AD1 Method 3.7 Tridiagonal Matrix System Chapter 4 : Finite Difference Equations for An Axisymmetric Case 4.1 Introduction 4.2 Derivation of Difference Equation of General Points 4.3 Derivation of Difference Equation at Points Lying at Centerline 4.4 Derivation of Difference Equation at Grid Points at the Boundaries 4.5 Derivation of Difference Equation for Grid Points at Interface Between Different Media 4.5.1 Grid Points (i,j)on Interface Parallel to z-axis Separating Medium A (below) and Medium B (above) 4.5.2 Grid Points (i,j) on Interface Parallel to r-axis Separating Medium A (left) and Medium B (right) 77 4.5.3 Centerline Point (i,l) on Interface of Constant z 82 4.6 Computer Implementation of AD1 Method 87

Chapter 5 : Experiment and Simulation 5.1 Experimental Model 5.2 Computer Model 5.3 Discussion of Results iii

Chapter Conclusions 126

References

Appendix A Finite Element Technique

Appendix B Computer Program Listing LIST OF FIGURES & TABLES

Fiaures :

1-1 Cast Bronze dating from China's Shang Dynasty 5 3-1 Flow of heat due temperature Gradient 3 1 3-2 Heat flux to and from a differential element 33 3-3 Interface between metal and sand 35 Temperature variations in a semi-infinite medium Grids pattern for different physical region AD1 calculation procedure Cylindrical Bar Left and right hand boundary grid points for lSt At/2 Top boundary grid points Schematic diagram of interface Schematic diagram of interface Schematic diagram of interface r Schematic diagram on constant z General flowchart for program SAND Drawing of pattern used in casting Drawing of pattern used in casting Locations of thermocouple 5-3 Schematic of experiment arrangement for temperature recording 5-4 Profile of the casting in the experiment v

Computer simulation profile 97 Physical dimension of cylindrical bar for simulation 98 Specific heat of pure aluminum 100 Thermal conductivity of pure aluminum 101

Specific heat of aluminum alloy used 102 Thermal conductivity curve of aluminum alloy used 103 Functional relationships for thermal conductivity of molding sand 104 Functional relationships for the thermal diffustivity molding sand 104 Cooling curves from experiment 108 Cooling curves from computer simulation 109 Cooling curves for location #2 near freezing range 111

Cooling curves for location #4 near freezing range 112

Temperature at location #6 (interface) 113 Temperature profile at location #2 (centerline) obtained by experiment, and by simulation at different values of thermal conductivity (k) 114 Temperature profile at location #4 (centerline) obtained by experiment, and by simulation at different values of thermal conductivity (k) 115 Temperature profile at location #3 : Effect of of variation of convective heat transfer

coefficient 117

Temperature profile at location #4 : Effect of of variation of convective heat transfer

coefficient 118 Temperature profile at location #1 : Effect of variation of convective heat transfer coefficient Effect of variation of latent heat on the solidification process Enthaply curve for the metal. Position of calculated freezing front at various time after pour Position of calculated freezing front at various time after pour Position of calculated freezing front at various time after pour 2-D Domain for the heat conduction

Tables .:

1 Computer software applicable to metal casting simulation 23 2 Computer Software not applicable to metal casting simulation 24 3 Computer software with pending status. 24 Nomenclature

temperature at begin and end of a time step ( OF ) temperature at first half of a time step ( OF ) indexes for column and row respectively radial direction radial spacing ( ft ) axial direction axial spacing ( ft ) time ( sec. ) index for time heat transfer coefficient ( Btu/ sec . it2 OF ) thermal dif fusivity ( ft2/sec ) Chapter 1 Overview of Metal Casting

1.1 Introduction

The purpose of this work is to develop a computer algorithm to simulate the solidification /freezing of liquid metal in a sand casting of a cylinder bar. This work represents a first step towards a research effort at the Mechanical Enginerring Department, of Ohio University, to develop a computer model to study the entire casting process.

A complete computer casting simulation should encompass some of the following capabilites : 1. Formulate an accurate physical description of casting and solidification process in concise mathematical form. 2. Accurate data of thermal properties involved i.e. metal and mold should be made available. 3. A suitable numerical method for computation. 4. Geometric modeling capabilities. 5. Graphic capabilities to show the results of simulation.

This thesis is organized into six chapters. The first chapter will trace the history of metal casting till 2 modern time and describe available casting process in practice. Chapter 2 is a literature review and survey of available publications and reports in the area of Computer - Aided Casting. Chapter 3 describes the heat transfer modeling of that occur during solidification in metal casting. Chapter 4 gives the details of mathematical modeling used for this work and the computer implementation of this research to simulate the cylindrical bar in sand casting process. The analysis used in this model follows the approach developed by Pehlke [21]. A discussion of experimental work, the computer simulation and results obtained are included in chapter 5. ~onclusionsare drawn in chapter 6. 1.2 Historv of Metal Casting

Metal casting is one of the oldest of all industries on record. Its historical development can be traced back as far as 4000 years B.C.. The art of metal casting is fundamental to civilization and has been practiced throughout the ancient world in Europe, Central and South ~merica,India, China, and Northern Africa. Metal casting is a process of manufacturing or fabricating a desired metal shape by pouring molten metal into a cavity called a mold, and allowing the molten metal to solidify. After solidification and cooling of the molten metal, the desired object can be removed from the mold and used or processed further.

The discovery and first use of metal has been traced with some accuracy to the area north of the Black Sea in the Carpathian Mountains of Russia. Gold was the first metal used by early man to make utensils for daily use. This is because gold is malleable enough to be shaped without splitting at ambient temperatures. Later, early man discovered copper and found that copper could best be shaped by heating and hammering. With further heating it was discovered that these metals melted into liquid which would then resolidify upon cooling. As a result, gold and copper became the first metals to be cast, and early man now had a method to obtain more complex shapes in metal. Due to the migratory nature of the people in earlier times, casting spread towards the Orient, and later, westwards into the Near East, the Mediterranean basin, and the rest of Europe. The early Chinese foundrymen were able to produce intricate and delicate metal objects by using stone molds, sectional loam molds and, later, lost wax techniques. The Chinese people had apparently discovered bronze, and it is believed they had advanced knowledge of metallurgy for their time period. Figure 1-1 shows a cast bronze dating from China's Shang Dynasty (1726 to 1122 B.C.) which was produced by the lost wax process. In the Mediterranean basin, the ~gyptiansput effort into improving the casting techniques used by the Chinese. In fact, credit should be given to these early Egyptians for the discovery of the lost wax process of casting and also the use of a cope and drag, and of core molding.

Around the year 3500 B.C. people in Europe began doping (mixing) copper with other substances and discovered that copper mixed with tin became much harder than copper alone. This led to the discovery of bronze. During the bronze age (3500 B.C. - 1000 B.C.), early man used the newly discovered metal to cast objects such as spearheads, axheads, and agricultural implements. Initial molds used were made of sand, and were open-half molds. Gradually, stone and fire clay were used to make these molds. In this period of time, melting was intially done on clay-lined ~igure1-1 : Cast bronze dating from China's Shang Dynasty (1766-1122 B.C.). The delicate filigree was acheived by the lost-wax process of casting, still used today in casting many of the finest art bronze. ref. [2 J holes in the ground which subsequently led to the inventions of permanent melting media by artisans which eventually evolved into furnaces.

According to history, iron was discovered as far back as 2000 B.C.. In the year 600 B.C., the Chinese used iron for casting, while in India, cast crucible steel was first produced about 500 A.D., but the process faded and was rediscovered by an Englishman Benjamin Huntsman in 1750 A.D.. During the fourteen century, iron began to gain popularity as a castable metal in Europe, but prior to that iron was only shaped by forging. The Europeans used furnaces that resembled a small blast furnace to achieve the high temperature needed to melt the iron ore. An excellent air supply, necessary for developing the high melting temperature, was delivered by large bellows operated by hand, foot, or water power. It was during this period that cast iron began to appear in cannon slots (barrels) and grave slabs.

In the time known as the Iron Age (1000 B.C.- 1000 A.D.), metal casting was considered as an art rather than a science and it was not until the beginning of sixteenth century when a man named Vannoccio Birringuccio (1480-1539) began to put down in writing detailed accounts of foundry practices. Birringuccio can be called the father of the foundry industry ". Today, three of the important foundry practices written by Barringuccio can go unchallenged : 7 (i) making and arranging the molds well, (ii) smelting and liquefying the materials of the metals well, and (iii) making the composition of their associations according to the results you wish to have. In the same century, more scientists began to explore into the science of foundry. An impressive contribution was made by Reaumur (1638 - 1757). He produced malleable iron and showed a clear understanding of the range of cast iron structure, and of the factors influencing the production of white, grey, and mottled irons.

The widespread use of cast iron did not come until 1730 when an Englishman, Abraham Darby, initiated the use of coke as a fuel. This allowed the production cost of cast iron to be two-thirds the earlier cost and subsequently, coke became the principle fuel of the iron foundry industry. The demand for iron casting greatly expanded when steam engine was invented by James Watt in 1765. In fact, the steam engine was used by John Wilkinson of England to provide the air blast needed to produce.the first iron metal-clad cupola.

America's first foundry industry was established in 1642 near Lynn, Massachusetts on the Saugus River. It was known as the Saugus Iron Works. The iron ore in Saugus area proved suitable for the start of an industry which eventually triggered the establishment of more than five thousand plants in the United States. Eventually, the 8 casting industry began to spread along the eastern sea coast as far south as Virginia.

Today, foundries are America's fifth largest industry, producing 15 to 20 million tons of castings each year. Hence, metal casting is still a keystone industry in today's society. 1.3 Modern Metal Castins Industrv

In spite of the early interest of several scientists, metal casting was still looked upon as an art well until the present century. During this modern era, casting process has become more of science where more control is exerted over the whole of casting production beginning from raw materials to finished product.

Traditionally, the foundryman was on his own. He determined the type of casting method to be used after recieving information on the product to be manufactured. From here, the foundryman would carry out his own mold material preparation, molding, pouring and sometime to the extend of melting his own metal and fettling the casting. In certain foundries the operator was virtually a sub-contractor selling his product to the company. Hence, the only technical control was done by the craftsman. This form of practice has changed in modern foundry industry.

In more recent times, casting processes to be used were determined by the foundry shop floor supervisor. Further advancement of the foundry industry, and the mass production of a variety of castings has brought about greater specialization. Now, operators have limited tasks and their supervisors may even be removed from the direct responsiblities of casting production. Casting production 10 is now left to a technical staff whose role is to establish and record methods and casting techniques for a particular product. This is supported by metallugrical control using modern laboratory facilities to determine such things as metal composition, metallographic structure and properties. This scientific approach is used from the beginning when the raw materials are processed to the finish product, resulting a better quality castings of a more predictable form and structure.

The Modern foundry industry can be broadly classified into Jobbing and Captive (Mass Production) foundries. Jobbing Foundry is directly related to traditional foundry industry where a small number of castings of each design is made, and therefore, a large variety of patterns must be handled. Jobbing foundry is not a part of a manufacturing plant. Captive foundries, on the other hand, are part of a manufacturing plant, and castings from these foundries are used in the plant as parts of a finished product.

The numerous processes for the casting of metals can be divided into two main groups : casting with expendable molds and casting with permanent molds. Some of the expendable molds are as follows : 1. Sand Casting - Metal castings produced in sand molds. 2. Investment Casting (Lost Wax or Precision Casting) - A casting process in which a wax or thermoplastic pattern is used. 3. Full-Mold Casting - A casting technique based on a

foam plastic ( a polystrene type ) pattern placed in a one-piece molding box. It is then gasified by the molten metal. 4. Plaster-Mold Casting - A process of pouring molten metal into plaster molds or plaster bonded molds. Some of the permanent mold casting types are : 1. Die-Casting - Casting resulting from die-casting process i.e. pouring molten metals under pressure into metals molds. 2. Permanent Casting - Metals castings produced in metal mol ds . 3. Ceramics-Mold Casting - A reusable pattern is used for this process, with a refractory slurry mold coat that is allowed to gel before the pattern is removed. After the mold is fired molten metal is poured into the heated mold. 4. Centrifugual casting - The casting of metal in rotating molds. The metal is forced from the center of the mold to periphery by centrifugal action.

Finally, the various types of foundry based on the type of metal cast are the following : 1. Nonferrous foundries a) Brass and bronze, producing alloys with copper as the base metal, and other alloying elements. b) ~agnesium,producing a variety of alloys, with magnesium as the base metal (Dowmetal is an example). c) Aluminium, producing a variety of alloys, with aluminum as the base metal, and other alloying elements. 2. Iron Foundries a) Gray iron, producing high-carbon ferrous alloy. b) White iron, producing medium-carbon ferrous alloy. c) Ductile iron, producing a spherodial graphite ferrous alloy. d) Alloy gray iron, producing a variety of irons containing special-purpose alloying elements. 3. Steel Foundries a) Carbon steel, which is a relatively low-carbon ferrous alloy. b) Alloy steel, which is any steel with appreciable amounts of special alloying elements. 4. Malleable-Iron ~oundries a) Malleable-iron producing an annealed white iron with graphite in nodular or temper iron. 1.4 History of Com~uter-Aided Czstinas

The use of the computer as a tool to simulate metal casting began in the 19401s. It was first initiated by the American Foundrymenls Society (AFS) Heat Transfer Committee. Early research sponsored by the society was conducted by Parchies (761 from Columbia University. Parchies was the first to use a large analog computer to study the solidification patterns, heat losses from the laddles, runner design, heat flow in molding materials and problems of freezing pattern prediction in the casting process. Results obtained were helpful to the foundry industry.

With advancements in computer technology, digital computer - based simulation subsequently replaced or/supplemented the analog computer. With this, a numerical approach - the Finite Difference method was used to solve problems in casting simulation. The method was pioneered by Dunsinhere [78] and others for various engineering applications. In 1962, Forsund 1751 appeared to be the first to use this method to solve a heat transfer problem of the foundry on a digital computer. This new tool was used by Henzel and Keverian [77] of the General Electric Company in 1965 to predict freezing patterns in large castings. Their work was a success and was cited by sully (741 in a recent conference : " it contained all the elements of a good solidification and put the current status of modeling into perspectivem. A year later, guidelines for research work in the computer simulation of metal casting were drawn up by the AFS. Some of these directions were :(i) simulation of two dimensional solidification patterns, and (ii) three dimensional simulation of a complex combination of shaped-molding materials and riser insulating compounds.

A number of universities in U.S.A. have been conducting research in computer modeling of casting. The Unversity of Michigan has conducted research by using finite difference approximations using both explicit and implicit methods. Some of their early research was the calculation of solidification patterns in sand castings by matching the computed results with exi;ting experimental data reported in published literatures. The shapes used

were "TI1 and I1Lw sections in steel, and flanged barrel shape in 85-5-5-5 bronze. Some of the recent works in University of Michigan include : a) Evaluation of overall and relative costs of various computer-based methods of simulation, including finite element versus finite difference techniques. b) Investigating and quantifying the nature of heat transfer across interfaces, including gaps which may form between a casting and the mold medium or chill. c) Extending knowledge of how thermal expansion- - contraction phenomena within the mold or the casting interaction with gating and feeding in affecting unsoundness distribution.

Research projects at Georgia Insitute of Technology, include : a) Evaluating the current state of the art of computer-aided geometric modeling, in particular, the interface with computation schemes. b) Building appropriate models of thermal transport in the molding medium using both theoretical and experimental approaches. c) Determining the extent of thermal convention in the liquid metal within the mold cavity, using computational and experimental means. A summary of studies of other works in the area of computer simulation of heat transfer and solidification using experimental data can be found in the.paper by Lewis, Liou and Shin [73]. Cha~ter2 Literature Survey

2.1 Literature Review

Stoehr [25] has suggested that the following factors should be considered when developing a good computer code to simulate solidification in sand casting : a. Simplicity from user's view point. b. Realistic representation of the casting and mold design. c. Flexibility for representing a wide variety of designs . d. Accuracy of results as indicated by : i. Agreement with analytical solutions. ii. Agreement with experimental results. e. Stability (i.e. computational method used). f. Easily interpreted results. g. Economically justifiable cost.

It is also important to maximize the use of any symmetry that exists in the design of the cast part in order to save computer storage; and proper representation of physical properties (such as specific heat, thermal conductivity, etc.). Any unrealistic representations will result in incorrect simulation and also influence the computer time requirement. Stoehr 1251 suggested some ways 17 to overcome these problems. For instance, the specific heat capacity of materials can be given an artifically high value by taking into account the latent heat of fusion. Alternatively, the program can be designed to accept segmented linear temperature representations of material properties and a latent heat of fusion distributed over the solidification range. Accuracy of computer simulation is influenced by the computational method used and the proper representation of physical and geometrical factors. The experimental results are influenced by the amount of data known about the system to be modeled, and examples of such factors are heat transfer coefficients and transformation temperature. Stability criterion is dependent on the method used i.e. either finite difference (FDM) or finite element method (FEM). If explicit FDM is used for a 2-D problem, the time step to employ is (At2 0.25(Ax)2/a) and for a 3-D problem, the time step is (At2 0.33 (Ax)2/a) , and for FEM, the time step should be ( AtZ 0.25 (Ax)2/a) , regardless of geometric dimension. Results from the computer analysis should be in informative format, such as graphical output. Finally simulation cost must be justifiable if only it can result in better quality of casting, design and reduces the number of trials in casting process.

Grant (263 studied the permanent mold casting of an automotive aluminum brake drum . He was able to predict the temperatures and heat flow of operating molds. Grant used a finite difference approach model. The model was solved using a commerical heat transfer computer code named TRUMP developed by Lawerence Livermore Laboratory, University of California. Some of the simulations TRUMP can perform are constant temperature heat rejection, temperature variant material properties and three dimensional shapes.

Rigger 1271 modeled the die-casting process using the computer code developed by the company Gatecliff, which uses the finite element method . Three complex geometries were used in the study. They were small engine piston, air-cooled engine crankcase and connecting rod. All simulation models were checked by actual experimental work.

Solidification sequence, total time to solidify, die temperature response and thermal cycle of die casting machine were investigated. The results indicate that simple computer model can be applied to simulate solidification process in die casting. The computer model was able to predict actual location where porosity occurred in the cast part. Solidification time is influenced by changes in die temperatures and the temperature response of the die is affected by the initial die temperatures. The results of the complete operating thermal cycle of the die casting machine compare favorably with the actual process. The results show that spraying time, die close and open 19 times are important factor that can made the difference in quality of casting.

Riegger recommended that : 1. It may be possible to eliminate or move porosity by examining the application of temperature gradients, insulating cores, relocation gates and changes in section size. 2. Faster solidification produces better mechanical properties. Therefore microstructual refinement through localized removal of heat can improve mechanical properties in critical areas. 3. Control of appropriate die temperature should be investigated.

Comini (231 presented a general approach to the solution of transient heat conduction problems with non-linear physical properties and boundary conditions. The quasilinear parabolic heat conduction equation with the appropriate boundary conditions was formulated using the Galerkin Method. A three time level difference scheme was utilized for the solution of the resulting matrix differential equations. This allows a direct evaluation of thermophysical properties and heat transfer coefficients at the intermediate time level, thus eliminating the need for iterations within each time step. The temperature dependent properties of specific heat (c) is calculated by assuming that the enthalpy (H) follows the same distribution function as the temperature (T! and c is given by :

The formulation was tested on four problems which were i)Solidification of infinite slabs of liquid, ii) Solidification of a corner region, iii) Slabs with radiation boundary conditions and, iv) Ground freezing. Results of these simulation were compared to available exact solution and was shown to be accurate.

Morgan et al. [29] suggested that the specific heat capacity (c) calculated by Comini et al. [23] should be evaluated from the temperature and enthaply changes at previous time levels m and m+l. This new method calculated c as :

This new method was suggested as Hibbitt 1291 found that the averaging process by Comini can lead to erroneous results in certain circumstances.

Pham [28] evaluated the specific heat using the 2 1 lumped-capacitance principle- It is based on the explicit enthaply method and Pham's three-level enthaply method. By lumping the thermal capacitance of the material at a node, a diagonal capacitance matrix is obtained- Thus the singular behaviour of specific heat near the phase-change point is overcome. The method can be described as follows: First, the heat flow into a node is calculated. The enthalpy change between two time level at a node is then evaluated. Using this change of enthalpy a new nodal enthalpy is estimated. With this new enthalpy an estimation of a new temperature is estimated, using a function relating enthalpy to temperature. Finally the specific heat (c) at the middle time level is evaluated. Taking this c value the lumped C-matrix is calculated and the finite element equation for heat conduction is then solved.

The properties of sand used in the mold has significant influence on the casting process. Hartley and Babcock 1591 did some experimental work to determine the thermal property of mold medium used for metal casting . They concluded that, the amount of moisture present during mixing of bonded sands has a significant influence on the thermal conductivity (k) of dry mixture. An optimum moisture content exists which results in the most significant increase in thermal contact area between sand grains for a given binder content. Secondly,. there exist an optimum content of binder to be added for-anincrease in thermal conductivity. Thereafter, a further increase results in a decrease in k. Thirdly, an increase in dry density increases the thermal conductivity of bounded and unbonded sands. The behaviour is linear even at elevated temperature. Lastly, specific heat of bentonite-bonded sands increases uniformly with temperature.

Com~uterCodes for simulation of Castinq

Jechura et al. [53] suweycd 25 different computer codes that used FDM and FEM to simulate the transient heat transfer occuring during the solidification of metal casting. The software were evaluated on the basis of whether the following phenomenon were simulated : 1. The effects of natural convection. 2. Various boundary conditions, including both convective and radiative heat transfer, and also a mold-metal interface resistance.

. 3. Problems in three spatial dimensions.

4. Variable thermal properties due to temperature and/or time effects.

5. Latent heat effects due to solidification, . either directly or indirectly.

Using these criteria they were able to catagorize these programs into three different types of status; applicable, non -applicable and pending status (see Tables 1,2 and 3). In addition, all computer simulations were Table 1 : Computer Software Applicable to Metal-Casting Simulation. ref. [53 1

Pronram Organization and Cantact ANSY S Swanson Analysis System, Lac. FEII - Frontal solution method Boustoa, PA CM deal with a wavefront of David Dietrich war 1000 degrees of free- (4 12) 746-3304 dom. 11

HEATING5 Oak Ridge National Laborarory FDM-Geometry is approximated Oak BAdge, TN u a set of rectangular Betty XcCill parallelipipeds whose faces (615) 574-6176 are parallel to the coordin- are planes. 12

MARC Analysis Research Corp. FEM-Originally developed for Palo Alto, CA welding problems. Mchael B. Hsu (415) 326-7511

ISC/NASTRAN HaCNeal-Schwendler Corp. PM-Recently issued Version Lo8 Angeles, CA 60 can deal with temperacure- Jerry A. Joseph dependent rpecific heats.9 (213) 254-3456 SINDA COSMIC Library FDX - Solves the general dif- University of Georgia fusion-type equations. Athens, CA Stephen J. Horton (404) 542-3265

Institut fur Statik und Large FLPl package, available Dynamik der Luft- und on CDC, IBM and UNIVAC equip- ilaumfahrtkonstruktionen ment. University of Stuttgart Pf af f enwaldring 27 7000 Stuttgart 80 West Germany Professor J. H. Argyrls 0711-7841

Oak Ridge National Laboratory FDM-Treats the latent heat Osk Ridge, TN effect as a heat release at ~ettyncciil constant temperature, using (6U)574-6176 a modified specific heat or using a combination of both?3

UECUS Yes tinghouse ?El4 - Originally developed and Pittsburgh, PA used in-house, now being S. E. Gabrielse marketed on the (412) 256-5040 Table 2 : Computer Software Not Applicable to Metal-Casting simulation. ref.[53]

Program Organization AD INAT K. Bathe VPI - Ccmstant specific heat, Hassachusetts Institute of no other provtsio for latent Technology but affects -14 915

AY ER Los Alamos Scientific Lab.

AZLAS VPW-lo tr nsienc heat trmfer .rd FETE Poster Vheelar Corp. Pa- Two dinensions withauc capabilities for latent heat effect.l7 10 CT STRUDL CM: Strudl PM-No heat transfer.

XWO STRUDL HcAuto Pen-No heat transfer. 10

SAPIV University of Southern PM-No heat transfer. 10 California at Berkeley 10 STARDYNE CDC FPW-No heat transfer.

SUPERB SDRC FEH-No transient heat Cincinnati, OH transfer .I6 10 SAP6 University of Southern FM-No heat transfer. California

Table 3 : Computer Software with Pending Status. ref.[53].

-. Program Organization Coamen t s

ASAS Atkins Research and FW Developenc

BETA Booing FDU

CINDA-3G Center for Information and FDH-Presumably superseded Nwrical Data Analysis by HITAS and SINDA and Synthesis Purdue University

FESS University of Wales 8t FEM FINESSE Svansea

HITAS

NISA Engineering Mechanics FEM Research Corporation done using three standard shapes which were close to industrial castings and results of the run were verified against experiment conducted under identical conditions. Comparisions were made of the number of arithmetric operations needed for solutions.

Thornton and Wieting [24] discussed an algorithm for solving steady state thermal analysis using finite elements with temperature dependent thermal parameters. The numerical procedure recommended was a modified Newton-Raphson method. With this method, the conductance matrix [K] was held constant for sewal iterations and was recalculated only when the convergence rate deteriorates. It should be noted that solving the heat equation with temperature dependent parameters require solution of a set of nonlinear, algebraic equations,

[X(T)]{T) = (Q) ------(1) where [K(T)] is the temperature dependent system conductance matrix, (T) is the nodal temperature vector and {Q) is the system nodal heat load vector. The Newton-Raphson iterative solution alogrithm for solving equation 1 is : where [JIn = Jacobian matrix

{R), = Residual nodal heat load vector

A component term of Jacobian matrix and Residual vector is given by :

* N where (Kij = C (a(Kil)e/aTj)(Tl)e ------(5) i=1

The above procedure is based on two assumptions. Firstly, thermal parameters are constant within an element and secondly an element thermal parameter depends only on the average element temperature. However, these assumptions are valid only for lower order finite elements with simple temperature variations. If a higher-order element is used, then the two assumptions are not valid as the thermal parameters should be allowed to vary within the element.

The researchers applied the solution procedure to a convectively cooled structured with significant varying thermal parameters and the solution converged in four iterations.

A general description of Finite Element Modeling of Heat Transfer problems is given in Appendix A.

Kim and Desai [36] developed a novel finite element method to analyze transient thermal contact problems. This computation methodology is called the fictitious finite element layer. In casting ,thermal contact problems are encountered at mold and liquid metal interfaces. With this method, a fictitious layer of finite elements is introduced at the interface between the mold and liquid metal flowing in a runner channel. The layer is then assigned fictitious thermal properties.of very high thermal conductivity and a very low thernal capacity. By such action an accurate interface temperature can be evaluated immediately following the thermal contact on pouring of liquid metal.

The actual values of the thermal properties are chosen by numerical experimentation which also includes a variation of grid size, Ax and computational time step, ~t to render an optimum fourier number, aAt/(Ax)2, of the fictitious layer. The researchers tested this method with various fourier number on a 1-D problem and obtained results that agreed with analytical results.. Various numerical schemes such as the crank-Nicoloson central difference, the Galerkin and the fully implicit backward 28 difference formulation were used to solve this finite element formulation. It was found that for thermal problem the fully implicit backward-difference numerical scheme is the most appropriate method. This is because the solution obtained is free from computional oscillation. The method was then extended to solve a two dimensional transient forced convection heat transfer in channel. The results obtained in this solution proved accurate in 2-D case. It was noted that the results obtained by the fictitious finite element layer was consistent in form to those obtained by applying the so called method of vectorial dimensional analysis. As this method can incorporate interface resistances in metal casting, it was used by Berry et al. [53] to simulate the forced and natural convection during filling of a casting and thermal performance of gating systems sprues in sand casting system.

Mazzantini and Zorizi [63] reported the solidification model developed for the process design of continous casting machines. In this paper description of the CAD system architecture is provided. The system consist of : i) The process program TEF3D which performs the thermal analysis. ii) The pre-processor COCOl which prepares the input data for TEF3D and iii) The post processors COC02 and FEMDS which aggregrates and presents the results in a more meaningful form for interpretation by casting engineer. The finite element method is used for 2 9 the solidification. The finite element mesh used are 4 to 8 nodes 2-D elements in conjunction with 2 to 3 nodes 1-D boundary elements. It was found that the computing time is shorter if higher order elements are used. The TEF3D finite elements program takes into account the temperature dependent properties of the materials such as conductivity, mass density and heat capacity. The boundary condition used for modeling of a test problem are heat removal mechanisms such as water cooling of mold, convection in a spray cooling zones and radiation of cast product. The results obtained with the computer simulation was verified with experimentation and good agreement was observed. 2.2 Summary of available References for Com~uter Simulation of Castins.

As there are many reported publications on the subject of computer simulation of casting, it is difficult to list them all. However, a short list of references within the reach of the researcher is given at the end of this thesis.

Some mathematical and geometric models of casting are contained in references [23] to [34]. They deal with the subject of numerical methods used to solve various casting process and possible mathematical formulation. References [40] to [53] deal with the subject of computer software that is available for use in casting simulation or have been used by researchers of these papers.

Defects that occur in casting can be found in references 1541 to [62]. These papers deal with subjects such as porosity both at micro and marco level. Some mathematics of this phenomena is also discussed. Finally reference [63] to [71] give the publications on the subject of mold and metal interface phenomena. Chapter 3 Numerical Modeling and Solidification

3.1 Introduction

In this chapter, the heat transfer in solidification will be analyzed, and nuiaerical solutions will be described. The analysis will be carried out for an axisymmetric (2-dimensional) case. The numerical model will be later compared with experimental results.

3.2 Heat Transfer Durins Solidification

The governing equateionsfor heat transfer in casting will be first established. The derived equations are applicable to solidifying metal, in the surrounding mold, or at the interface between the two.

T (TEMPERATURE)

qx (RESULTING HEAT FLOW)

I X I .

Figure 3- 1 : Flow of Heat due to Temperature Gradient. Our attention will be centered on the temperature which will, in general, be a function of position co-ordinates and time. Thus, temperature is given by :

T = f(x,y,z,t) during any period of time. The temperatures vary with position co-ordinate as shown in figure 3-1.

This variation will result in the flow of heat towards the direction of decreasing temperature. Using the ~ourier'sLaw of Heat conduction it is expressed mathematically as :

aT qi = -ki------ax: (1)

where i = co-ordinate direction ( x,y,z ) q = heat flux density in the i direction T = temperature Xi = position co-ordinate

ki = thermal conductivity of the medium in the xi direction

It should be noted that the partial derivative is used here as temperature can be a variation in any of the position co-ordinate or in all three directions. The thermal conductivities ki in three co-ordinate directions will not be equal in anisotropic materials but in the case that is considered here isotropy is assumed i.e. kx = ky = k, = k. The two dixensional case of the energy equation which governs the temperature variations with space and time in the conduction medium will be formulated. The derivation is based on the Law of Conservation of Energy, which states that the amount of heat which enters a region plus what is generated inside is equal to the amount of heat which leaves plus the amount stored. Referring to figure 3-2 we have :

Figure 3-2 : Heat flux to and from a differential element. 34 After algebric manipulation equation (2) becomes :

Applying Fourier Law of Heat Conduction equation (1) to equation (3) we have :

Thus, equation (4) becomes the general equation that describes the heat transfer during casting solidification for a two dimensional case.

If the heat transfer were to occur in a cylindrical co-ordinate system with symmetry condition the temperature will then be a function of :

T = f( r,z,t ) where r is the radial distance, z the axial distance and t the time. If the thermal energy balance is performed on a differential cylindrical annulus the resulting equation will be : It is important to find the derivations of the two more equations that can describe the interface between two media, for instance between metal and mold. Such an interface is shown in figure 3-3.

A Ti

Metal Sand

Figure 3-3 : Interface between metal and sand.

The first equation simply states that the heat flux is continous at the interface. Using equation (1) we have :

where the subscripted m and s denotes metal and sand respectively. xi and xf refer to directions immediately to the left and to the right of the interface (locate at x=x~) respectively. As the thermal conductivity of metal (km) is greater than sand (k,); equation (6) implies that in the vicinity of the interface, temperature gradient will be much steeper on the sand than in the metal.

The interfacial temperature Ti, immediately after the pouring of the molten metal T-, can be modeled by considering a semi-infinite medium with a single face at x=O, extending indefinitely in the positive x direction.

At time t = 0, the initial interface temperature is To, it is then suddenly brought to Ti, at time greater than zero (see figure 3-4).

Ti h-Fixed interface temperature Uor to)

I Initial tsmperature at t-0

Figure 3-4 : Temperature variations in a semi-infinite medium.

The subsequent temperature inside a semi-infinite medium is known to obey the relation :

X T(x,t) = To + (Ti - To)erfc ------(7) 2w in which the complementary error function is defined by

eric z = -1 e-82 d~

From equation (I), the heat flux density into the semi-infinite medium is :

Since both the metal and sand act as a semi-infinite media for a short period of time after they have been in contact, equating their heat flux equation (6) and using equation (9) we then have :

km(Tmn - Ti) - ks(Ti - Tso)

which simplify into :

where Ti is the interfacial temperature. 3.3 Finite Element Technique

The present analysis uses Finite Difference method. Finite element methods have become popular in recent years. The finite element technique solution has been summarized in Appendix A. Numerical Modelinq of Castinq 3.4 Numerical Modelinq

The finite difference method (FDM) is a numerical method or technique that can be used to solve the three classes of partial differential equations (PDE) encountered in engineering. The three types of PDE's are :

I. raraDollc e.g. near Lquarlon : s+--rar+S="- at

a2u a2u 2. Elliptic e.g. Laplace Equation : s+ay~= f (X,Y) a2u a2u 3. Hyperbolic e.g. Wave Equation : z=c2s

The basis of FDM is that derivatives in a governing partial differential equation are substituted by difference expression, which can be obtained by the Taylor series expansion. The first step towards obtaining a solution using FDM is to divide the physical region into a grid of nodes. The grid shape used is governed by the special nature of the physical problem being solved. Figure 3-5 shows some of these grid patterns.

Secondly, the governing partial differential equations that described the problem can be transformed into a corresponding co-ordinate system that can best fit the problem where solution is sought. Once this has been determined, the difference equation is then applied at each node in a given region and the functional relationships 40 between a node and its surrounding nodes are established. As a result, a set of linear algebraic equations is formed. With the proper boundary conditions, these simultaneous equations can be solved using an appropriate algorithm.

(4 (dl

Figure 3-5 : Grid Pattern for Different Physical Regions. ref.[l9] 41 3.5 Taylor Series Expansions of Space and Time Derivatives.

In order to solve the PDE by the FDM, each derivative term in the PDE will have to be replaced by appropriate difference expression. The Taylor series expansion is one such method to obtain these difference expressions. In general, the Taylor series expansion is given by :

I h2 f (x t h) = f (x) hf (x) + - f'I(x) k - + 2 ! 3 !

To obtain the finite difference approximation of the second-order space derivative of the terms in the axisymmetric heat equation :

are evaluated around Ti,j,n for the Ti,j+l,n 1 Titj-l,n first term in (1) and Ti+l,j , Ti-l,j,n around Tiljlnfor the third term.

a2T Evaluation of 7ar :

(~r)~a2~ (hr13 a3~(hr14 a4~ Tij-ln Tijn - Ar -aT + -

ar 2! ar" 3! ar.' + 4! ar4 Adding equations (3) and (4), we have :

------(5) After simplification of equation (5)

where the last term of equation (5) is neglected for practical purposes, 0 [ (Ar)2] indicates the order of the term that is neglected.

aT Evaluation of -ar : Using the same a2proach, equation (4) is subtracted from equation (3) ,

simplifying yields,

a2~ Evaluation of 7 : az Applying the same technique to obtain the second order-space derivative in z direction : ------(9) Adding equations (8) and (9) and simplifying, one obtains :

Evaluation of WA -at : Similarly for the first-order time derivative term, we have :

where T*ijn+l/2 = functional value at the end of At/2 half time steps TABLE 1 - FDM APPROXIMATION

Ti+lin - 2Tiin + Ti-lin 2.[a2T]1 = - -. - ijn (Az) 3.6 Alternatins - Clirection Implicit (AD11 Method.

The AD1 method was developed by Peacemen and Rachford (1955) and Douglas (1955). As the name implies it is an implicit method. The finite difference approximation of the space derivatives is in terms of values yet to be computed. Thus the solution will involve solving a system of linear simultaneous equations, arranged in a matrix form to give a tridiagonal system of linear algebraic equations. The solution for heat conduction equation in the cylinder co-ordinate system will be solved by the AD1 approximation. The heat equation in cylindrical co-ordinate is :

In the AD1 method, each time step of At is split into two half-time steps, each of duration At/2. During the first half-time step (lst At/2) the space derivative of equation (1) are approximated implicitly in the z-direction and explicitly in the r-direction. This produces a tridiagonal matrix for each j row of grid points. With the completion of the first-half time step, the second-half time step is solved with the approximation made explicitly in the z-direction and implicitly in the r-direction. (Note that the approxomations in the first half-time step is reversed of that made in the second half-time step; thus the name Alternating Direction Implicit Method.) This 46 results in solving tridiagonal matrix for each i column of grid points.

The AD1 method is a second-order accurate with a truncation error of O[ (At)2, (Ar]2, (Az)*] and is known to be unconditionally stable. It is also a very efficient method due to the availability of algorithm for the rapid solution of tridiagonal systems.

EXPLICIT IN Z-DIRECTION 2*~t/2 n+ 1

EXPLICIT IN

IMPLICIT IN 2-DIRECTION lStat/2 n+m

n b Z iAZ

Figure 3-6 : AD1 CalculaUon Procedure. 3.6.1 - Derivation of AD1 Method.

To illustrate the AD1 method just described, consider a general grid point (i,j), one which is neither on a boundary nor on an interface surface. During a time step At the revelant AD1 approximations to equation (1) are

First At/2 :

Second At/2 :

where aijnt the thermal diffusivity at point (i,) is evaluated at the prevailing temperature Tijn at the beginning of the time step, for either the sand or metal, whichever is appropriate.

Expanding equation (2) by the substitution of the appropriate FDA from Table 1 in section 3.5 we have after algebra : aijAt aijnht aijnAt where q = = , P2 = P3 = P1 P2 2 (AZ) 2 2 (Ar)2 4rjAr - P4 = P1 + P2 and 2 2 i 2 imax-1.

From equation (4) it becomes clear that there are 3 knowns variables, essentially T*i-lj, T*ij, and T*i+lj. Hence it is logical to consider a whole row at once in order to develop the linear algebraic simultaneous equations into a tridiagonal matrix system which looks

like :

------(5) Note: subcript n+1/2 and n are dropped for convenience in the T and d vectors respectively. where di=P3Tij-ln + (1-2Pl)Ti jn + P4Tij+ln

Similarly expanding equation (3) for the next second ~t/2,(with the appropriate FDA substitution from Table 1 of section 3.5) : Now consider a whole column at once in order to develop a tridiagonal matrix, which looks like :

Note: i). subscripts n+l and n+1/2 are dropped for convenience in vectors T and d respectively. . ii). Values of d2, dimax-1, and djmax-1 are in general

given as d2 + d2 + Ts, dimax-1 + dimax-1 + Ts,

and djmax-1 + djmax-1 + Ts respectively.

The solution to the two tridiagonal matrices will be discussed in the next section. Tridiasonal Matrix Systems

In this section, the solution techniques to a tridiagonal matrix system will be discussed. A tridiagonal matrix system can be generalized as the following form :

A technique for solving a tridiagonal system of linear algebraic equations such as equation (1) is due to Thomas (1949) and is called the Thomas algorithm. In this algorithm, the system of equations is put into the upper triangular matrix form by replacing the diagonal elements bi with :

bi + bi - ci-1 for i=2,3 ...... n bi-1 and the di's with

d + d - d for i=2,3 ...... n bi-1 The unknowns are then computed using back substitution starting with .: and continuing with

Consider the tridiagonal matrix formed by the set of following equations :

blTl + ClT2 . = dl a2T2 + b2T2 + c2T3 = d2 a3T2 + b3T3 + c3T4 = d3 . . - . IT,,^ + bn-lTn-l + cn-lTn = dpl

. anTn-l + bnTn = dn

Now, instead of storing an n x n matrix one need only to store the vectors {a), {b) and {c) with dimensions n-l,n, and n-1 respectively. Choosing the diagonal element d as a pivot for the first derived system, one needs to eliminate T1 from the second equation only, and all other equations will remain the same. The first derived system is therefore the following : blTl + ClT2 . = dl b2 (1)~2 + c2T3 . = d2(l)

a3T2 + b3T3 + c3T4 . = d3 . . - where b2 (I) = b2. - 9 c1 bl and d2(l) = d2 - %dl bl By choosing the diagonal b2 for computing the second derived system, one needs to eliminate T2 from the third equation only. Now the second derived system has the form :

By continuing in this fashion we proceed from one equation to the next we have a generalized formulae of :

for i=2,3,....n. By this procedure, one obtains the upper triangular system of the form :- By back substitution from the last equation of the above system, we have :

and the preceeding equations imply that Cha~ter4 Finite Difference Equations for an Axisymmetric Case

4.1 Introduction

Consider the axisymmetric cylinder shown in figure 4-1. The boundary equations will be developed for all points including the axis of symmetry, and the surfaces on the three other sides.

_.----

(a)

z I I I I I I I I axisof L- ,-,-,,-,---,------I symmetry

Figure 4- 1 : Cylindrical Bar.

4.2 Derivation of Difference Emations at General Points

This section deals with the derivation of equations to be used at grid points not including points at boundaries, centerline, and interfaces. The governing partial differential equations for the physical problem (axisymmetric) to be studied is represented in the cylindrical co-ordinate system as :-

Since the AD1 method is employed as a solution solver to the above heat equation (I), it is necessary to derive the equation for a grid point for two half time steps. The first half time step is implicit in the z-direction and explicit in the r-direction. While the second half time step is implicit in the r-direction and explicit in the z-direction. With this in mind, the derivative of equation (1) is replaced by the appropriate FDA from Table 1 in section 3.5 First (Implicit z-direction)

Note : subscripts n and n+1/2 are dropped for convenience in derivation. Using the following assignments :

Substituting equation (4) into equation (3) we have :

After simplification of equation (5) we have : The next, second half-time step of At/2 is implicit in r-direction. Again terms in equation (1) are replaced by the difference expressions from Table 1 in section 3.5.

Thus :

Second At/2 ( Implicit in r-direction )

Note : subscripts n+1/2 and n+l are dropped for convenience in derivation.

Substituting equation (5) into (8), we have :

after algebra, equation (9) becomes : summarizing the results of the derivation we have from the tridiagonal-matrix coefficients i.e. a's (subdiagonal), bls (main diagonal) and cls (super-diagonal) and dls (right hand side vector) as :

First At/2 :

ai ' Ci = -q ------(lla) bi = (1+2q) ------(lib) di = P3Tij-ln + (1-2P1)Tijn + P4Tij+ln ------(11~) for i= 2,3, ...... M-1

Second At/2 :

* dj = @i-ljn+l/2 + (l-~q)~fjn+l/2+ @i:ljn+l/~ ---- (12d) for j = 2,3 ,.....N-1 4.3 Derivations of Djfference Emation at Point Lyinq on the Centerline.

At the centerline, which is the axis of symmetry (see figure 4-l), special treatment is required for the term 1 aT of the heat equation -r -ar :

aT because at centerline r=O and 0. By applying a -ar = limiting'technique such as the Lf Hopital rule we have :

ar lim r*Or-

lim = r+oa -03ar

ar" = lim r+0- 1

- a2T - ar"

lim therefore we have o -r = 9 60 By applying the result just obtained to equation (I), we have :

Equation (3) will be the equation to be used for grid points at the centerline of the physical region to be studied. To obtain an algebraic difference expression it should be noted that Tij-l 5 Tij+l from the condition of symmetry. Thus, the difference expression is :

The first-half time step is :

at the centerline j-1 Note : subscripts n and n+1/2 are dropped for convenience in derivation.

Let p1= 2ildLTj 2 (Ar)

after simplification of equation (8)

Similarly, the subsequent second-half time step will have the difference expression :

Substituting equation (7) into equation (10) we have (after algebra) :

Summarizing the derivation, we have the coefficients of the tridiagonal matrix coefficient :

First At/2 :

Second At/2 :

It should be noted that the equations (12) and (13) are different from equations (11) and (12) only in (derivation in section 4.2) their bi,ci and di terms. 4.4 Derivations of difference equations for srids ~ointsat

the boundaries (of ~ointsadjacent to to^, left, and risht-hand boundaries).

The equations (6).and (10) from section 4.2 for the general grid points will be used, for top, left and right -hand boundaries. For convenience equations (6) and (10) are rewritten here :

First At/2 :

Second At/2 :

The locations of these grid points are shown in figures 4-2 and 4-3. i= 1 i-m j=n ' j-n (2.n-1 ) ( m-1, n-1) d2 = di + qTip-i,~ d=-5= &-I+ ~T~EA-1,~'-

(2.1) (m-1 *jI,, (Id2= di + q~i,~ drl= dm-1 + qTm.j.~

i= 1 - (2.1 ) (m-1.1) - i =m j=l d2 = di + qTi.1.~ dm-1 = d&-I+ qTm.1.~ j= 1

Figure 4-2 : Left and Right hand boundary grid points for 1'' AV2.

For the first-half time step i.e. lSt At/2, if i=2 and j P 1, equation (1) becomes :

Since, the left hand edge i=1 is a fixed boundary * and Tljn+l/2 ' Ts = Tljnt where Ts is the surface temperature, equation (3) is then :

Let d2 = P3T2 jmI + (1-2P1) T2j + P4T2 j To generalize equation (5) for i=2 and m-1 and j=2...... n-1, we have :

where bi = 1+2q ------(7a) Ci = -q ------(7b)

and i=2 or m-1, j=2,3, ...n-1.

If i=2 or m-1 and j=l, we have to use equation (9) from section 4.3 which is written here for convenience,

* -qTi-ln+l/2 + (1+29)~1,+1/2 - q~i:ln+l/~ = cl-4pl)Ti1n

+ 4P1Ti2, ------(8) and noting that Tr1 = Ts = T11, Tm-11* = Ts = Tm-11 we then have :

I Let di = (1-4P1)Til + 4P1Ti2 where bi = 1+2q

Ci ' '9

I dl = di + mi-lj --..------(llc) and i=2 or m-1, j=l

i-m 1=n

Figure 4-3 : Top Boundary grid poinb.

Next, to obtain the equation for the second At/2, we have from equation (2) for i=2131.....m-ll j=n-1

-P3Tin-2 n+l + (1+2P1)Tin-ln+l - P4Tinn+~' mif ln-ln+l/2

+ (1-2q) ~ln-ln+l/~

+ ~~itln-ln+l/~--- (12) again with Tin TS = ~f~~ it simplifies to : * * * where d2" qTi-ln-ln+1/2 + (1-2q)Tin-ln+~/2 + qTi+ln-ln+l/2 and the coefficients of the tridiagonal matrix are : 4.5 Derivation of Difference Equation for srid points at interface between different media.

In this section, the derivation will be on the interface temperature between metal and sand or vice versa. To account for the interface temperature between two media during solidification process we have to make some assumptions like good thermal contact between the media so that an interface temperature is applicable to both media. Therefore, there is continuity of heat flux at the interfaces.

For the case to be discussed here will be two such occurences as : 1. Point (i,j) on interface parallel to z-axis. 2. Point (i,j) on interface parallel to r-axis.

Before proceeding with the derivation of the above cases, it is appropriate to generally state the Taylor series expansion of a second order derivative at an interface.

The Taylor's series of the interfacial temperature of

TijWl and Tij+l which is given as : (1, j+l )

Medium B

Intarf ace (1-1,j) (1.j) f (i+i,j) 2

Medium A

I (Lj-1) Figure 4-4 : Schematic Diagram of Interface.

Note :

Rearranging equations (1) and (2), The condition of continuity at interface is mathematically expressed as : 4.5.1 Grid Point (i,j) on Interface Parallel to z-axis separatinq Medium A (below) and Medium B (abovel.

I (i,j+l)

Medium B

Intsrface

(1-1,j 1 , 1 (i+l,j1 f z

Medium A

U,j-11 Figure 4-5 : Schematic Diagram of Interface.

The heat transfer equation in medium A for the first half time-step is given as :

and for medium B is :

By replacing the second-order derivative of 1 and 2 with the equations (3) and (4) from section 4.5 we have : - ST aT where 6$~;jn+l/2 - 3 and AtT = -at

Next, the continuity assumption of heat flux at the interface is :

K~Tr~= K~Tr~ ------(5)

The value of TrA and TrB is given by equation (3) and ------(7) substituting equations (6) and (7) into (5) we have:

Let C5j = A rj -.-----_(gal

substituting equation (9) into (8) and after some algebra we have :

Substituting equation (13) into (12) we have :

Applying the same procedure to the second half-time step to interface A and B, the following equation is obtained :

Summarizing the results for the tridiagonal matrix coefficients, we have :

First At/2 :

di = PATij-ln + (1-PA-PB)Tijn + PBTijn ----- (16C)

Second At/2 :

aj = -PA

4.5.2 Grids Points (i,i) on interface Parallel to r-axis separatina Medium A (Left) and Medium B (Riaht).

r 4

Medium A , Medium B

o-- Interface

(1-1,j) (i,j 1 (i+l,j1 - b- -7

Figure 4-6 : Schematic Diagram on Interface R

The heat transfer equation in medium A for the first half time-step is given as :

and for medium B is :

~eplacingthe second-order derivative of equations (1) and (2) in similar manner to equations (3) and (4) from section 4.5, we have : Note : subscripts n and n+1/2 are dropped for convenience in derivation.

where TZA = - and TZB = - azaT I A azaT I B

The condition of continuity of heat flux at interface is given as :

Substituting equation (5) into (3) and multipling by

KA, we have : Next, multiplying equation (4) by Kg, we have :

Adding equations (6) and (7), and replacing AtT by its difference expression,

KA KB Let C4 = - + - aA aB

Put equation (9) into (8) : At - 2Ti + Tij-l)' TZj - Tij ------+2C4 (Ar) (Tij+l (10)

P5 At (KA + Kg) ------= 2C4 (Ar)1 (llc)

Finally substituting equation (11) into (10) :

Applying the same procedure to the second half-time step to the interface A and B, we will arrive at the following equation : Summarizing the results for the tridiagonal matrix coefficients, we have :

First At/2 :

Second At/2 : 4.5.3 Centerline Point (i.1) on interface of constant z.

Medium A ' ( 2 Medium B

Ar -Inbrface

Figure 4-7 :-schematic Diagram on constant z

The heat transfer equation in medium A for the first half time-step is given as :

and for medium B is

Next, replacing the second-order derivative of equations (1) and (2) in a similar manner to the equation (3) and (4) from section 4.5, we have : Note : subscripts n and n+1/2 are dropped for convenience in derivation.

where TZA = - and TZB = - aTaz I A aTaz I B

The condition of continuity of heat flux at interface z is given as :

Substituting equation (5) into (3) and multiplying by KAl we have :

- Multiplying equation (4) by Kg, Adding equations (6) and (7). and replacing AtT by its difference expression,

~et~4 = 5 + 5 -----____(9) a~ "B combine equations (89) and (9) and simplify : At P5 = [KA + Kg) ------2C4 (Ar) (11~)

Finally, substituting equation (11) into (lo), and after simplication we have :

Applying the same procedure to the second half-time step to the interface A and B, the following equation is obtained :

Summarizing the results for the tridiagonal matrix coefficients, we have :

First At/2 : dl = (1-4P5)Tiln + 4P5TiZn

Second At/2 :

bl = (1+4P5)

cl = -4P5 4.6 Computer Implementation of the AD1 Method.

The general flowchart of the computer implementation of the AD1 method for simulating of the sand casting of the cylinder bar (see figure 5-1) is shown in figure 4.8. The main program is made up of four main subroutines : 1. INPUT - This subprogram will assign all input parameters like axial spacing, radial spacing, time step etc, initializes all grid points temperatures and numbering of grid points with an identification number to facilitate the channelling of subsequent calls to the correct physical property. It also does the computation of initial interface temperatures and constant parameters require in the evaluating of the tridiagonal cofficient matrix. Finally, all input parameters are printed out in this subroutine. 2. PRINT - This subprogram will print all grid points temperature as required. Due to the huge output, the printing is done in three parts. 3. ROW - This subprogram will evaluate the coefficent matrix vector ai,bi,ci,di of the tridiagonal system. It will solve for temperatures using the TDMA solver for the first half time step. Within this subroutine it will call MATPRO and PROMS for physical properties of either sand or metallor both. It will pass to this secondary subprogram with a type number for the purpose of identifying whether the grid points is an interface or general points and a START+ ASSIG- Of ASSmUUm OF CALL INPUT * ALLurPm . TYPE MRIBEB VALUES FOR EVEBl[ GRID PO^ L 1 r

I I EVALUATE I ROW DO COEFFICIEMI CALL IOU CALL BOW LOOP 1 TO TRXDIAGONAL u-1 ai,bi,ci,di SOLVER FOR ROV -

~0~ I UTPRO I CALL PROUS

I I EVALUATE CALL TDnA COLUMN DO COEFFICIEM CALL COB LOOP 2 TO -+ TBIDIAGONAL aj,bj,cj,dj SOLVER x-1 FOB CoLm I I

CALL PROUS

Figure 4-6 : 9- FLOWCHART EQB klumMam temperature to evaluate thermal properties. The results from this subprogram are conductivity and thermal diffusivity. 4. COLS - This subprogram is similar in structure to ROW except that the grid points temperature is evaluated at the second half time step.

An important point to note in the computation is the use of two matrices T and TSTAR. These matrices are used to store temperature compu-Led,where TSTAR is for the first At/2 and T is for the beginning and second At/2. With these matrices they can be used repeatly over successive time steps thereby avoid the triple subscripted array. But temperatures must be outputted before the next time step if required, otherwise their values will be lost. Chapter 5 Experiment and Simulation

The mathematical model for the solidification simulation of the cylinder casting in this study is based on previously derived equations in chapter 4. The results from the computer simulation will be compared to the experimental results obtained.

5.1 Experimental Model

An experiment was carried out to sand cast the cylinder bar geometry for this computer simulation resezrch. The wood pattern was designed (see figures 5-la

C 5-lb) with the intention of having a high probability of cavity in the casting. The pattern was made at the Industrial Technology Casting workshop at Ohio University. With this wood pattern the sand mold was hand rammed. No risers were used because it was desired to produce a defective casting for simulation. It can be expected that the cylinder would develop marco-shrinkage defects. Before the pouring of the liquid metal, thermocouples were placed at various locations in the sand mold (see figure 5-2). The thermocouple setup was then connected to an IBM PC (see figure 5-3) with a data acquisition hardware (MetroByte Corporation DASH-8). The data acquisition package records temperature measurement according to specified time

...... c ...... 1 ...... * ...... 7 ...... e...... SAND:::::::::::::::::::::::::::::::::::::...... SAND iiiiiiii ...... * 5 ...... " ...... METAL 3- -3 ...... r - location of thermocouple

Figure 5-2 : Locations of Thermocouple. Thermocouple at location 1 was inserted after pouring. All other thermocouples were fasertsd before pouring the metal.

95 intervals. A five second time interval was used for this experiment. The main advantage of this data acquisition package is that the recording can be viewed from the screen as the temperature changes. In addition the data recorded can be used to plot cooling curves without any additional processing.

Com~uterModel

The actual casting was three dimensional because of gating and sprue (see figure 5-4). In addition, a pool of metal solidifies on the top of sand over the gating. However, the analytical model is a two dimensional axisymmetric model. The gating system was turned 90° and modified so that the casting became axisymmetric (see figure 5-5) .

The computer simulation was performed with certain assumptions. Firstly, it is assumed that the liquid metal filled the mold instantaneously at the pouring temperature of 1200 OF. In addition, once the sand mold is filled, the liquid metal is assumed stagnant, i.e. no convective mass and energy transport in the liquid phase will be taken into account. Radiation effects are neglected. Convective heat losses to the air are considered only for the hot metal pool formed above the sprue. A perfect thermal contact is assumed to exist at all metal-sand interfaces, meaning that thermal resistance at the interface caused by air gap is Metal pool formed after pouring of molten metal

Cylinder Sprue

Figure 5-4 : Profile of the casting in the experiment...... SAND ......

Figure 5-5a : Computer Simulation Profile. Figure 5-5b :Physical Dimension of Cylindrical Bar for Simulation. 99 not considered. Thus, the condition of continuity of heat flux exits at interface. The thermal properties of metal and sand a function of temperature. However, accurate data are not easily available. The metal used was assumed to be pure aluminium but the freezing point of this metal was observed to be approximately 1120 OF, which is substantially different from the freezing point of pure aluminum (1240~~).The metal used was actually aluminum mixed with other compounds, such as silicates, during repeated sand casting.

overcome this, the thermal properties of pure aluminium were modified as follows. The specific heat and thermal-conductivity curves for pure aluminum (as shown in

Figure 5-6) were shifted by 115 OF so that the freezing point could be matched. The thermal properties used in this simulation are in Figures 5-7 and 5-8. The sand properties used in this simulation are shown in Figures 5-9 and 5-10.

Because of axisymmetry only one half of the geometry is modeled. The boundary condition of UI 0 is applied at -ar = centerline. Another two boundary conditions are constant wall temperature at the sand surface and convective boundary for the pool of liquid metal which solidified on the top. The convective boundary condition, h[Ti - Tm] = aT k- was used on this surface The heat transfer ar , . c, 0 ooolo7s (t-940)40.27 028-

-0 \

o*

6 0- I I 1 I 1 I 1 0 18- I 200 400 600 800 KXX) 1200 I400 16OO It 1EMPERATURE. *F

Figure 5-6a : Specific of Aluminum ref.12 11. Figure 5-6b : Therrna conducUvit?7 of Aluminum ref.[2 11. Figure 5-7 : Specific Heat of 'Aluminum Alloyg used in the casting. The latent heat is incorporated into the model by adding a sharp peak to the specific heat curve at the freezing point of 1 125 degrees farenheit. The metal is now assumed over the range of 10 degrees farenheit. Figure 5-8 :Thermal Conductivity of 'Aluminum AUoy- used. - 0 8 Ref. r 8 RI(. 0 *Rd 8 Aai. 0 8 Aof * Rd. .8 0 Rot. * Rol. v 8 Rai

Figure 5-9 : Functfonal Relationships for Thermal Conductivity of molding Sand ref .[2 1 I. Equation (a) is used in this simulation.

Figure 5- 10 : Functional relationships for the thermal diffusttvity molding sand. coefficient h is taken to vary with temperature and the function used is h = 1.12[~]1'4(from ref. [ll]). Finally, unequal axial and radial spacing are used because finer detail is required in the axial direction, as more heat is expected to traverse in this direction. 5.3 Discl~ssionof Results.

The experimental data are plotted in Figure 5-11. The channel numbers correspond to the thermocouple location numbers shown in Figure 5-2. The results of the simulation are plotted, and compared with the experimental results in Figures 5-11 to 5-21.

From the experimental curves it can be observed that the freezing point of this metal (which was aluminum doped with impurities due to repeated casting processes) during this run is approximately 1120 OF. This temperature is lower than the freezing point of pure aluminum (1242 OF). Location #1 and #6 are always below freezing point. This was due to the fact that the thermocouple at location #1 was inserted after pouring the metal, and there is a lag in the temperature history until the thermocouple warms up. The thermocouple at location #6 is at the interface, and it is expected to measure the interface temperature which'is greatly affected by air-gap formation.

Thermocouples at location #I, #2, #3 and #4 show behavior typical of freezing metal. The metal cools rapidly until the freezing point is reached. The cooling rate then slows down until the latent heat has been released; and when the latent heat has been dissipated, the cooling rate goes up again. These results are more obvious 107 when the cooling curve is examined closely around the freezing point, as shown in Figure 5-11 and 5-12. The last two figures show that the solidification process is completed within the first 2-3 minutes of pouring.

The sand mold does not heat up appreciably at points which are more then one inch away from the metal. For example, the location #7 (2.1 inches from the metal surface) is below 100 OF for the duration of the simulation. It appears, therefore, the assumption of constant temperature (80 OF) boundary conditon at the sand surface is very reasonable, particularly during the initial period when freezing takes place.

Simulation Results:

The results of the simulation are shown in Figure 5-12. One of the important differences that can be seen by comparing this figure with the experimental curves in Figure 5-11 is that the cooling rate is much faster during the final stages of the simulation. The reasons for this difference is quite well known. As the metal cools down, it shrinks and separates from the sand mold, and an interfacial layer of insulating air gap is formed. The present simulation has not considered this effect, which results in slower cooling rates in the experimental curves.

The results during the solidification process which

occcurs during the first few minutes are much more comparable, as can be seen from Figure 5-13, and Figure 5-14. In both these cases, the experimental curve shows lower temperatures which are expected to be due to cooling during the filling of the mold.

The interface temperatures are compared in Figure 5-15. There is a large discrepancy between the theoretical cooling rates and the rates obtained from the experiment. This is not surprising since it is well known that an air gap is formed at the interface during cooling. This slow down the cooling rate substantially, as can be seen from the experimental results. Simulation was also carried out to determine the effect of changing the thermal conductivity of the metal. ( The parameter which controls the cooling rate is actually the thermal diffusivity. Since the latent heat is incorporated into the specific heat curve, changing the thermal diffusivity would also change the latent heat). From Figure 5-15, it can be seen that when the conductivity is reduced to 70% of the original value, the cooling rate slow down appreciably.

In Figures 5-16 and 5-17 cooling rates are shown at the centerline of the casting. These curves indicate that the simulation gives answers which are close to the experimental results during the solidification process. The simulations have been carried out with various values of thermal conductivity. For higher values of thermal

.. -'a' A COOLING CURVE FOR BLUJdliI SAND CASTING

Figure 5- 15 : Temperature at location '6 (interface).

conductivity, the cooling progresses at a faster rate, as can be expected.

As was mentioned earlier, excess of metal on top of the sprue formed a fin-like structure on the sand, The convective heat loss of metal surface (which was quite hot) to the air was incorporated into'the model by using a convective boundary condition. The effect of the convective heat transfer from this surface was examined by changing the values of heat transfer coefficient (H) from the value obtained from theory. The results are shown in Figures 5-18 to 5-20. It can be seen that the convective heat transfer from.the surface of the metal was rather small. It can thus be concluded that convective heat

transfer was not a dominant heat transfer mechanism. .

Figure 5-20 shows that the location #1 never showed any phase change in the experiment. This was due to the fact that the thermocouple was inserted after pouring, and there was a time delay before the correct temperature was reached. During the later part of cooling process, this thermocouple reported a much higher heat transfer rate.

Figure 5-21 shows the effect of variation of latent heat value on the cooling curve at loaction #2. This figure shows that the temperature history depends strongly on the latent heat produced during the solidification process. If the latent heat has a lower value, the cooling COOLING CURE FOR ALUmNUM SAM) CASTING ~OCO~Al' CKMU" # 3 f.3

Figure 5- 18 : Temperature profile at location *3 : Effect of variation of convective heat transfer coefficient. COOLING CURVE FOR ALUMINUM SAND CASTING

Figure 5- 19 : Temperature profile at location *4 : Effect of variation of convective heat transfer coefficient. COOLING CURVE FOR ALUMNUM SAND CASTING

Figure 5-20 : Temperature profile at location 1 : Effect of variation of convective heat transfer coefficient. COOLING CURUES FOR, DIFFERENT SPECIFIC HEIT UCILUES 1300 I

A EXPERIMENTAL A CWPUTER SIHUL&IION LtlOOX

e COMPUTER SIWIATIOW L=5W 0 CWPUTER SIMULllTION L=lSW'X

N.B. FOR LOCITIOH @2

Figure 5-2 1 : Effect of variation of latent heat on the solidification process. proceeds at a faster rate.

The overall cooling rate can be examined by considering the energy lost by the metal piece. The casting weighed 8.5 pounds. The dominant energy term is the latent heat generated in freezing; this would be approximately 1200 BTU. Almost all of this heat is dissipated to the surrounding sand in approximately 3 to 10 minutes. The convective heat losses are estimated to be

8.5 BTU/ min- it2 which is small.

The freezing fronts for this casting process are shown in Figures 5-23 (a), (b), and (c). It is obvious that the main body of the casting freezes last; and therefore, one would expect to produce a shrinkage cavity. In simulation axisymmetry has been used. However, it was observed that a very large shrinkage cavity was formed on the top surface of the cylinder. This indicates that cavity was formed while the surrounding metal was still liquid, and buoyancy forces pushed the cavity to the top. The position of this cavity did not change in repeated experiments. It is possible that some gas bubbles were trapped and also became part of the cavity on the top of the cylinder. This would be a poor design for casting. As mentioned earlier, the casting was designed to produce a shrinkage cavity.

At time - 60 seconds

Y ""

At time - 75 seconds I

At time 90 seconds

~EZ: @ Completely Frozen 0 Liquid and Partially Frozen

Figure 5-23b : Position of calculated freezing front at various time after pour. At time = 105 seconds

At time = 12 0 seconds

Y

At time = 135 seconds

Kh]L : Completsly Frozen

0 Liquid and Partially Frozen

Figure 5-2 3c : Position of calculated freezing front at various time after pour. Chapter 6 Conclusions

An experimental and analytical study has been carried out to determine the solidification process in casting. The experimental studies were conducted on sand casting of aluminum. A data acquisition system was used to obtain the time-temperature history of the casting after the metal has been poured into the mold. The piece to cast was axisymmetric, which made the analysis two dimensional.

The analytical study was based on the work done by Pehkle [21]. A two dimensional analysis was carried out in which the gating and sprue were included. The algorithm developed for simulation of the casting is based on a finite difference, AD1 method. Comparision of the experiment show that results are reasonable for the initial phase of cooling when the solidification takes place.

During the later period of cooling, typically after 5 minutes, the simulation and experimental results do not compare well. This is probably due to the fact that the present simulation has neglected the formation of air-gap at the metal-sand interface, which results in much slower cooling rates in the experiment. The cooling rates are primarily affected by thermal properties of the metal and sand, and the geometry of the casting design. The dynamics of the cooling process is strongly influenced by latent heat of fusion of the metal.

For more accurate simulation results, the following improvements are recommended : (i) Use of accurate thermal properties of metal and sand. (ii) Simulation of the air-gap formation in the casting. (iii) Simulstion of fluid-flow and convection in the liquid metal.

A number of commercial programs are available which incorporate some of the above improvements. A discussion of these programs are included in the review chapters of this thesis. The simulation of fluid flow is rather complex, and has not been done satisfactorily to date. New computer programs are be.ing developed which are expected to improve the simulation capability at reasonable cost.

Some of the commercial packages for solidification simulation are : Solidification Waves, Nova Cast, and Spider. In addition new packages such as AFSOLID (available from American Foundrymenls Society), and CAST3 (developed for the U.S. Air Force) are now available for the casting industry. These programs should make significant improvements in the design of castings in the industry. The analysis presented in this thesis can be used to predict marco-shrinkage cavities. At present, a great deal of effort is being directed towards the prediction of other defects, such as : (i) Gas porosity (ii) Interdendritic Porosity, (iii) Subsurface Porosity, (iv) Centerline Shrinkage. These defects cannot be predicted by the traditional solidification analysis. It can be expected that advanced commercial software will, in future, incorporate the prediction of some or all of these defects. REFERENCES

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78. Dusinberre, G.M., "Heat-Transfer Calculation by Finite Differencew, International Textbook Co., Scranton, PA, 1961. This section contains the finite element solution to non-linear heat conduction problems with phase change. Finite Elenent Solution of Non-Linear Heat Conduction Problems With Special Reference to Phase Chanae.

The governing differential equation of transient heat conduction is a quasilinear parabolic equation :-

subject to the following boundary conditions :- Initial Boundary Condition T = To(x,y,O) in D, t=O -- (2) Fixed Boundary Condition T = Tw(x,y,t) on rl t>O -- (3) and the Non-Homogenous Boundary is

+ q + qh = o on r2 t>~------(4)

Figure A-1 : 2 -1) Domain for the heat conduction. 139

Nomenclature :

p = Material density (kg/m3)

c = Specific heat (J/kg K) or ( ~/rn~K) T = Temperature (K)

t = Time (s)

X,Y = Spatial co-ordinates (m) Kx,Ky = Thermal Conductivities (W/m K) Q = Rate of internal heat generation per unit

volume (w/m3 )

%,Ly = Direction cosines of the outward normal to the boundary surface.

q = Heat flux density through boundary (w/m3) qh = Heat loss at boundary due to convection (w/m3) i.e. h(T - Ta)

h = Heat transfer coefficient (w/m2 K) Ta = Ambient temperature (K)

To = Initial temperature of system. Tw = Wall temperature at the boundary at t>O

Galerkin's Method

In general the Galerkin's criterion is given by :

I [Le- f(e)]NidD= 0 for i=1,2,3,.... r ---- (5) ~(e) where L (@ ) = Differential equation. f = Forcing function defined over an element. 140 r = Number of unknown parameters assigned to an element

Ti = Discrete nodal temperature.

Green's Theorem (Intesration bv Parts in 2-Dl

where D = Domain.

Z = Boundary or surface.

u = Weighting function.

+ V.v = Derivative in the differential operator 2

Finite Element Formulation of Heat Transfer Eauation

The differential equation governing two-dimensional

transient heat conduction as given by equation 1 is : 141 Applying GalerkintsMethod (equation 6) to an element of the above equation we have : I dxdy = 0

Next, integrating the first two terms by parts using Green's Theorem (eqn.5), where we can avoid second derivative in integrals imposing unnecessary continuity conditions between elements. We have the following :

+ u(V.v) d$ = u(v.n) dC (Vu)do ------(9) i 'A D Z D

where u r ~i

thus, equation 7 becomes : A A A A N.B. i.i = j.j Simplifying equation 10 we have :

Applying the natural boundary condition (non-homogenous) equation 4 we have : and also using the interpolation function, given as :

substituting equations (12) and (13) into (11) we have :

NiQdxdy - dXdy = 0 ----- (14 +I I 1 Nip C&[I NI{T} ~(e) ~(e) which simplies into : Equation 15 is an elemental equation where i=1,2, ...r. Putting (15) into a global matrix form we have :

where :

aN- aN aN. aN. Kij = I/ [K -1.2+ K 2.2 dxdy ------(17) Xa~ax yay a~I Secondly the enthaply change between levels m and m+2 time step at node i is then given as :

Thus, the estimated new nodal enthaply is

the estimated new temperature is

T; = fT( HTf ) ------(28) and the specific heat at the middle time level is

The values of Ci(m+l)Can then be used to calculate the lumped C-matrix of equation 23. After the finite-element equation 23 has been solved, the following temperature correction step is applied

T~(~+~)(corrected) = fT[ fH( Tf ) + ci (m+l)(T~ (m+2) - ~f)1 ------(30) Putting equation 16 in more compact form we have :

where {F} = {q} - {Q} - {qT2 ------(24)

and the value of Cij and terms in equation 24 are given by the previous equation 18,19,20,and 22.

To account for the variation of heat capacity during the phase change, it is suggested by Pham [28] that the Cp can be estimated as follows :-

The first step is to use equation 23 at the middle time level m+l to estimate the rate of heat gain Awwendix B

This section contains the listing for the program SAND. The code is based on a program developed by Pehlke r211 ...... * THIS COMPUTER PROGRAM DOES THE SIMULATION PROCESS * * OF SOLIDIFICATION IN A SAND CASTING OF A CYLINDER * * BAR. THE NUMERICAL DISCRETIZATION USED IS THE * * FINITE DIFFERENCE METHOD WHICH IS SOLVED BY THE * * ALTERNATING DIRECTION IMPLICIT METHOD. THE MAIN * * PROGRAM WILL CALL FOUR SUBROUTINES. THE FOLLOWING * * ARE THE SUBPROGRAMS USED : * * 1. INPUT * * 2. PRINT * * 3. ROWS * * 4. COLS * * WITHIN THE SUBROUTINES, 2 SUBPROGRAMS (TDMA AND * * PROMS) AND 1 FUNCTION CALL (MATPRO) ARE USED. * * NOTE : THIS COMPUTER CODE IS A MODIFICATION OF THE * * PROGRAM FROM REF.[21] FOR THIS RESEARCH. * ...... * COMMON BLOCK VARIABLES USED ARE : * * ...... * * A,B,C, : COEFFICIENTS OF THE TRIDIAGONAL MATRIX. * * RHS,D : RIGHT HAND SIDE VECTOR OF THE TRIDIAGONAL* * MATRIX * * T : TEMPERATURE AT THE BEGINING AND END OF * * SOLUTION PROCEDURE. (OF) * * TSTAR : TEMPERATURE AT THE ~ALFTIME STEP OF * * SOLUTION. (OF). * * TYPE : IDENTIFICATION NUMBER FOR METAL-SAND * * INTERFACE. * * C1-C6 : CONSTANT REQUIRE IN SOLUTION PROCEDURE. * * 12-16 : POSITION OF METAL-SAND INTERFACE IN THE * * 2-DIRECTION. * * J1-35 : POSITION OF METAL-SAND INTERFACE IN THE * * R-DIRECTION. * * NCOL : NUMBER OF COLUMN GRID IN THE Z-DIRECTION.* * NROW : NUMBER OF ROW GRID IN THE R-DIRECTION. * * IFREQ : THE INTERVAL TO PRINT THE SOLUTION. * * TZERO : INITIAL TEMPERATURE OF LIQUID METAL.(OF) * * TW : PRESCRIBED TEMPERATURE B. C. (OF) * * DZ : AXIAL SPACING. (FEET) * * DR : RADIAL SPACING. (FEET) * * DT : TIME STEP. (SEC) * * TMAX : TOTAL TIME OF SIMULATION REQUIRED.(SEC) * ......

MAIN PROGRAM INTEGER TYPE .OO) NCOL 1 ,NROW,IFREQ,TZERO,TWIDZ,DR,DT,TMAX CALL INPUT TIME=O .0 K=O CALL PRINT (TIME,T,NCOL,NROW,) TIME= TIME + DT K=K+l CALL ROWS CALL COLS IF(K/IFREQl*IFREQl.EQ.K) CALL PRINT(TIME,T,NCOL,NROW) IF (TIME .LE. TMAX+DT/2.0) GO TO 99 STOP END SUBROUTINE INPUT C ...... C * FUNCTION : 1.ASSIGN ALL GRIDS WITH A TEMPERATURE * C * 2.ASSIGN ALL GRIDS WITH A IDENTFICATION* C * NUMBER. * C * 3.GIVE LOCATIONS OF INTERFACE IN THE * C * RADIAL AND AXIAL DIRECTION. * C * 4.COMPUTE CONSTANT REQUIRED FOR FUTURE * C * COMPUTATION. * C * 5. COMPUTE THE INITIAL METAL-SAND * C * INTERFACIAL TEMPERAm. * C * * C * CALL SEQUENCE : CALL INPUT * C ...... C BEGIN INTEGER TYPE REAL KA, KB COMMON/COM2/T(lOO,lOO)ITSTAR(1OOt1OO)tTYPE(1OOllOO) COMM0N/COM3/Clt C2,C3 (100) C5 (100) ,C6 (100) COMMON/DATA/I2,I3,I4,I5,I6II7IJ1IJ2,J3,J4,J5tJ6,NCOL 1 ,NROW,IFREQtTZERO,TWIDZtDRIDTtTMAX I2=17 13~53 I4=59 I5=68 I6=73 I7=75 NCOL=75 Jl=l J2=5 J3=9 54-17 55-32 J6=45 NROW=61 IFREQ=24 TZERO=1200.0 TW=80.0 DZ=0.25/12.0 DR=0.1/12.0 DT-2.500 ?mAX=1800.00 C COMPUTE CONSTANT REQUIRE IN SUBROUTINE ROWS AND COLS Cl=DT/ (2.0*DR*DR) C2=DT/ (2.0*DZ*DZ) DO 100 N1=2 ,NROW R= (Nl-1)*DR C3 (Nl)=DT/ (4.0*R*DR) C5 (Nl)=-2.0/DR+l.O/R C6 (Nl)-2.O/DR+l. O/R 100 CONTINUE C COMPUTE INITIAL INTERFACIAL TEMPERATURE. TMEAN= (TW+TZERO)/2.0 DO 200 N2=1,5 CALL MS(TMEAN,ALPHAA,ALPHABIKAIKB) F=KB/KA*SQRT ( ALPHAA/ALPHAB) TMEAN=TW+ (TZERO-TW)/ (l.O+F) 200 CONTINUE C ASSGIN ID. NUMBER TO ALL GRID POINTS. DO 1 N1-1,NROW DO 2 N2-1,NCOL TYPE (N2,Nl) =l T (N2,N1) =TW TSTAR (N2, N1) =TW CONTINUE CONTINUE DO 4 N4=J1, J5 TYPE (I7, N4) =3 T (I7,N4) =TMEAN CONTINUE DO 5 N5=12,13 TYPE (N5,J4) =4 T (N5I J4)=TMEAN CONTINUE DO 6 N6=13,14 TYPE (N6,J2) =4 T (N6,J2) =TMEAN CONTINUE DO 7 N7=14,15 TYPE (N7,J3) =4 T (N7,J3 ) =TMEAN CONTINUE DO 8 N8=15,16 . TYPE(N8, J2)=4 T (N8,J2 ) =TMEAN CONTINUE DO 9 N9=16,17 TYPE (N9,J5) =4 T(NgIJ5)=TMEAN CONTINUE JJ4.44-1 DO 10 N1O=Jlt JJ4 TYPE (I2,NlO) =3 T (12,N10) =TMEAN CONTINUE JJ2=J2+1 JJ444-1

CONTINUE JJ3=J3-1 DO 12 N12=JJ2, JJ3 TYPE (I4,N12) =3 T(I4,N12)=TMEAN TYPE (I5,N12)=3 T (15,N12) =TMEAN CONTINUE JJ5=J5-1 DO 13 N13=JJ2,JJ5 TYPE(I6,N13)=3 T(I6,N13)=TMEAN CONTINUE L1=12+1 L2=13-1 L341 L4=J4-1 DO 14 N14=L3 ,L4 DO 15 N15=L1, L2 TYPE (N15,N14) =2 T (N15,N14) =TZERO CONTINUE CONTINUE Ll=Jl L2*2-1 DO 16 N16=LltL2 DO 17 N17=13,14 TYPE (N17,N16) =2 T (N17,N16) =TZERO CONTINUE CONTINUE L1=14+1 L2=15-1 L3=J1 L4*3-1 DO 18 N18=L3 ,L4 DO 19 N19=L1, L2 TYPE (N19,N18) =2 T (N19,N18) =TZERO CONTINUE CONTINUE Ll==Jl L2=J2-1 DO 20 N20=LltL2 DO 21 N21=15,16 TYPE(N21fN20)=2 T (N21, N2 0)=TZERO CONTINUE CONTINUE LlrJl L245-1 L3=16+1 L4=16+1 DO 22 N22=LltL2 DO 23 N23=L3,L4 TYPE (N23,N22) =2 T(N23,N22)=TZERO' CONTINUE CONTINUE JJ=J5-1 DO 24 N24=Jl1JJ TYPE (I7,N24)=2 T (I7,N24) =TZERO TSTAR (17,N24) =TZERO CONTINUE Ll=NCOL/2 L2=Ll+l PRINT ALL INITIAL INPUT. WRITE(6,211) 12,13,14,15,16,M WRITE(6,213) J2,J3,J4,55,N WRITE(6,214) WRITE(6,215) TZERO,TW,DZ,DR,DT WRITE(6,201) WRITE(6,202) (I,I=l, Ll) WRITE(6,203) DO 25 N25=1,NROW J=NROW+l-N25 WRITE(6,204) J1 (TYPE(1,J),I=l,Ll) CONTINUE

WRITE(6,204) J, (TYPE(1,J),I=L2,NCOL) CONTINUE FORMAT('l','THE FIRST HALF TYPE MATRIX IS') FORMAT('l','THE SECOND HALF OF TYPE MATRIX IS') FORMAT('O1,'3 I=',5012) FORMAT ( ' 0 ' ) FORMAT(' ',I2,4Xf50I2) FORMAT(12X,'I2 = ',I2/12X,'I3 = ',I2/ 12X1114= ',I2/12X,'I5 = ',I2/ 12XItI6= ',12/12Xf'17 = ',I2/ 12XI1NO.OF COLUMN IN 2-DIRECTION = ',I2) FORMAT'(~~X,'J~= ',I2/12XI'J3 = ',I2/ 12XI1J4= ',I2/12X,'J5 = ',I2/ 12XItNO.OF ROW IN R-DIRECTION = ', 12) FORMAT(10',12X,'METAL - ',F6.1,'FARENHEIT '/ 12XttSAN- 'IF6.1/'01,9X,'3.AXIAL SPACING DZ = tIF7.4,'FEET1/'0',9Xlt4.RADIAL SPACING DR 'IF7e41'FEET'1/'0'19X,'S.TIME STEP = ',F7.4, ' SECOND ' ) FORMAT(101,'2.1NITIAL TEMPERATURE1) RETURN END ...... SUBROUTINE PRINT(TIME,T,NCOL,NROW) * FUNCTION : THIS SUBROUTINE PRINT ALL THE GRID * * TEMPERATURES ON REQUEST. * * * * CALL SEQUENCE : CALL PRINT(TIME,T,NCOL,NROW) * * PARAMETERS : * * TIME= SPECIFIC TIME TO OUTPUT SOLUTION. * * IT= INTEGER VALUES OF TEMPERATURES FOR * * PRINTING ONLY. * ...... INTEGER TYPE DIMENSION IT(100,100) ,T(100,100) DO 1 Nl=l,NCOL DO 2 N~=~;NROW IT(NlfN2)=T(N1,N2) 2 CONTINUE 1 CONTINUE IMID= (NCOL+l)/3 IMIDPl=IMID + 1 IMIDP2=IMID + IMIDPl - 1 IMIDP3=IMIDP2 + 1 NP1= NROW + 1 PTIME=TIME/GO.O WRITE (6,200) PTIME WRITE (6,201) (I ,I=l, IMID) WRITE (6,202) C PRINTING TEMPERATURE ROW-WISE . DO 3 N3= 1,NROW J=NPl-N3 WRITE(6,203) J, (IT(1,J) ,I=l,IMID) 3 CONTINUE WRITE(6,204) (I,I=IMIDPl,IMIDP2) WRITE (6,202) DO 4 N4=1,NROW J=NPl-N4 WRITE(6,203) J,(IT(I,J),I=IMIDP1,IMIDP2) CONTINUE WRITE (6,204) (I,I=IMIDP3, NCOL) WRITE(6,202) DO 5 N5=1,NROW J=NPl-N5 WRITE(6,203) J,(IT(I,J),I=IMIDP3,NCOL) CONTINUE ...... FORMAT FOR OUTPUT STATEMENTS ...... FORMAT('l', 'AT TIME =',F7.1, 'MIN1/'O','THE COMPLETE TEMPERATURE MATRIX, PRINTED IN THREE PARTS, IS') FORMAT('O', 'J 1=',12,2615) FORMAT('O1) FORMAT(' ',12,1Xt2615) FORMAT('ll,'J 1=',12,2615) RETURN END FUNCTIQN MATPRO (ID,TL, COND) C ...... C * FUNCTION : THIS SUBPROGRAM EVALUATE THE THERMAL * C * PROPERTIES OF SAND OR FITAL, DEPENDING * C * ON THE VALUE OF THE TYPE ID INPUTTED. * C * * C * CALL SEQUENCE : MATPRO (ID,TSTAR(Nl,N2) ,COND) OR * C * MATPRO (IDIT(N2, N1) ,COND) * C * PARAMETERS : * C * K,COND = THERMAL CONDUCTIVITY OF SAND OR METAL. * C * (BTU/SEC-FT OF) * C * CP = SPECIFIC HEAT OF WTAL (BTU/IB-OF) * C * ALPHA = THERMAL DIFFUSIVITY OF METAL OR SAND * C * (SQ.FT/SEC) * C * RHO = DENSITY OF METAL (IB/CU. FT) * C ...... C BEGIN INTEGER ID REAL KtCOND ,RHO IF(ID.EQ.l) GO TO 20 C THERMAL PROPERTIES OF METAL RH0=169.0 IF(TL.LT. 65.0) K=O.O3058*(TL+l15. )+134.5 IF(TL.GE.65.O.AND.TL.LT.495.O)K=0.01161*(65.O-TL) 1 + 140.0 IF(TL.GE.495.O.AND.TL.LT.1082.0) K=0.02263*(495.0-TL) 1 + 135.0 IF(TL.GE.1082.0.AND.TL.LT.1127.0) K=1.53*(1082.0-TL) 1 +121.7 IF(TL.GE. 1127.0) K=0.00774* (TL-1127.0) + 52.9 IF(TL.LT.825.0) CP=6.07E-05* (TL+115.0)+O. 213 IF(TL.GE.825.O.AND.TL.LT.1025.0) CP=1.075E-04*(TL-825 1 .0)+0.27 IF(TL.GE.1025.O.AND.TL.LT.1082.0) CP=2.02E-04*(TL-102 1 5.0)+0.2915 IF(TL.GE.1082.0.AND.TL.LE.1104.0) CP=0.336*(TL-1082.0 1 )+0.303 IF(TL.GT.llO4.O.AND.TL.LE.1127.0) CP=0.334*(1127.0-TL 1 )+0.313 IF(TL.GT. 1127.0) CP=0.280 ALPHA=( (l.O*K)/3600.)/ (RHO*CP) MATPRO=ALPHA COND=(K)/3600.0 GO TO 50 C THERMAL PROPERTIES OF SAND K=(0.1634-5.3969E-05*TL+2.4133E-08*(TL**2))/3600.0 ALPHA=(1.1315E-02-6.0427E-06*TL+3.5112E-O9*(TL**2)) 1 /3600.0 MATPRO=ALPHA COND=K 50 RETURN END ...... SUBROUTINE PROMS(TLIALPflAAIALPHABfKAIKB) * FUNCTION : THIS SUBROUTINE COMPUTES THE THERMAL * * PROPERTIES OF BOTH SAND AND METAL FOR * * COMPUTATIONS AT INTERFACE. * * * * CALL SEQUENCE : * * CALL PROMS (T (N2,N1) IALPHAA, ALPHAB ,KAI KB) OR * * CALL PROMS (TSTAR (N1, N2 ) ,ALPHAA ,ALPHAB ,KA ,KB) * * PARAMETERS: * * K,KA,KB = THERMAL CONDUCTIVITY OF SAND OR METAL * * (BTU/SEC-FT OF) * * ALPHAA/ALPHAB = THERMAL DIFFUSIVITY OF METAL OR * * SAND (SQ.FT/SEC) * * RHO = DENSITY OF METAL (IB/CU.FT.) * * CP = SPECIFIC OF METAL (BTU/IB OF) * ...... BEGIN REAL KA, KB,K THERMAL PROPERTIES OF METAL RH0=169.0 IF(TL.LT.65.0) K=0.03058*(TL+115.)+134.5 IF(TL.GE.65.0.AND.TL.LT.495.0) K=0.01161*(65.0-TL) 1 +140.0 IF(TL.GE.495.O.AND.TL.LT.1082.O)K=0.2263*(495.-TL) + 1 + 135.0 IF(TL.GE.lO82.O.AND.TL.LT.1127.0) K=1.53*(1082.0-TL)+ 1 +121.7 IF (TL.GE. 1127.0) K=0.00774* (TL-1127.0) +52.9 IF(TL.LT.825.0) CP=6.07E-05* (TL+115.0)+O. 213 IF(TL.GE.825.O.AND.TL.LT.1025.0) CP=1.075E-04*(TL-825 1 .0)+0.27 IF(TL.GE.1025.0.AND.TL.LT.1082.0) CP=2.20E-04*(TL-102 1 5.0)+.2915 IF(TL.GE.lO82.O.AND.TL.LE.1104.0) CP=0.336*(TL-1082.0 1 )+0.303 IF(TL.GT.llO4.O.AND.TL.LE.1127.0) CP=0.334*(1127.0-TL 1 )+0.313 IF(TL.GT.1127.0) CP=0.280 ALPHAA=(l. O*K/3600.0)/ (RHO*CP) KA=K/3600.0 THERMAL PROPERTIES OF SAND KB=(0.1634-5.3969E-05*TL+2.4133E-O8*(TL**2))/3600.0 ALPHAB=(1.1315E-02-6.0427E-06*TL+3.5112E-O9*(TL**2)) 1 /3600.0 RETURN END ...... SUBROUTINE TDMA (SNE, NE) * FUNCTION : THIS SUBROUTINE SOLVES A TRIDIAGONAL * * SYSTEMS OF EQUATION BY THE GAUSS * * ELIMINATION. FOR A BAND SYSTEM OF WIDTH* * TWO. * * * * CALL SEQUENCE : CALL TDMA(SNE,NE) * * PARAMETERS *: * * A(N)= SUBDIAGONAL COEFFICIENT OF THE SYSTEM. * * B(N)= DIAGONAL COEFFICIENT OF THE SYSTEM. * * C(N)= SUPERDIAGONAL COEFFICIENT OF THE SYSTEM * * D (N)= RIGHT SIDE VECTOR. * * DUM(N),TT = DUMMY ARRAY AND VARIABLE RESPECTIVELY.* * MS(N) = ARRAY TO STORE SOLUTION. * * SNE= STARTING NO. TO SOLVE THE SYSTEM. * *...... NE= NO. OF EQUATIONS TO SOLVE. * BEGIN COMMON/COMl/A(lOO) ,B(100) /C(lOO) D(100) ,S (100) INTEGER SNE,NE DIMENSION DUM (100) ASSIGNING B ARRAY TO A DUMMY ARRAY. DO 1 Nl=l,NE DUM (Nl)=B (Nl) CONTINUE JJ=SNE+l COMPUTE GAUSSIAN ELIMINATION FACTORS. DO 2 NZeJ,NE TT-A (N2)/DUM(N2-1) DUM(N2)=DUM(N2) -C(N2-1) *TT D(N2)=D(N2) -D(I-1) *TT CONTINUE COMPUTE SYSTEM SOLUTION AND STORE IN RHS VECTOR RHS (NE)=D (NE)/DUM (NE) II=NE-SNE Do 3 ~3=i,11 I-NE-N2 RHS (I)=(D(I) -C(I) *RHS (I+1))/DUM(I) CONTINUE RETURN END ...... SUBROUTINE ROWS * FUNCTION : THIS SUBROUTINE IS TO COMPUTE THE * * TEMPERATURE FOR THE FIRST HALF TIME * * STEP, IMPLICIT IN THE 2-DIRECTION. * * * * CALL SEQUENCE : CALL ROWS * ...... BEGIN INTEGER TYPE,ID REAL KA,KB COMMON/COMl/A(lOO) ,B(100) ,C(100) ,D(100) ,RHS (100) COMMON/COM2/T(lOO,lOO),TSTAR(1OOI1OO),TYPE(lOO~lOO) COMMON/COM3/Cl1 C2,C3 (100) I C5 (100) ,C6 (100) COMMON/DATA/I2,I3,I4,I5,I6II7IJ1IJ2,J3,J4,J5~J6~NCOL ,NROW,IFREQ,TZERO,TWIDZIDRfDTITMAX NC=NCOL-1 NR=NROW- 1 DO 1 Nl=l,NR DO 2 N2=2,NC ID=TYPE (N2, N1) IF(ID.GE.3) GO TO 3 COMPUTE VALUES FOR GRID POINTS OF TYPES 1 AND 2 ALPHA=MATPRO(ID,T(N2 ,N1) ,COND) Q=C2 *ALPHA Pl=Cl*ALPHA P2=C3 (Nl)*ALPHA P3=P1-P2 P4=Pl+P2 A (N2)=-Q B(N2)=1.0 + 2.0*Q C (N2)=-Q IF(Nl.EQ.1) GO TO 4 D(N~)=P~*T(N~ ,~1-1) + (1.0~2. o*P~) *T (N2 ,Nl) +P4 *T (N2,N1+1) GO TO 5 D(N2)=(l.0-4.0*P1)*T(N2f1)+440*P1*T(N2,2) IF(N2.EQ.2) D(2)=D(2) + Q*T(l,Nl) IF (N2. EQ. NC) D (NC)=D (NC) + Q*T (NCOL,N1) GO TO 2 IF(ID.EQ.3) GO TO 6 COMPUTE VALUE FOR GRID POINTS OF TYPES 4 ON CONSTANT R INTERFACES CALL PROMS (T(N2, N1) ,ALPHAA, ALPHAB, KA, KB) CAB=KA*CS (N1)/ALP=-KB*C6 (N1) /ALPHAB PA=2.O*Cl*KA*CS(Nl)/CAB PB=-2.0*Cl*KB*C6 (Nl)/CAB QAB=C2* (KA*C5 (Nl)-KB*C6 (Nl))/CAB A (N2) =-QAB B(N2)=1.0 + 2.0*QAB C (N2) =-QAB D (N2)=PA*T (N2,N1-1) +PB*T (N2, N1+1) + (1.0-PA-PB) *T (N2,N1) GO TO 2 C COMPUTE VALUES FOR GRID POINTS OF TYPES 3 ON C CONSTANT Z INTERFACES 6 IF (N2. EQ .I2 ) CALL PROMS (T(N2, N1) ,ALPHAB, ALPHAA, KB ,KA) IF (N2.EQ. 13) CALL PROMS (T (N2,N1) ,ALPHAA,ALPHAB, KA, KB) IF (N2,EQ. 14) CALL PROMS (T (N2,N1) ,ALPHAB, ALPHAA, KB,KA) IF (N2. EQ. 15) CALL PROMS (T (N2 ,N1) ,ALPHAA,ALPHAB,KA, KB) IF (N2. EQ. 16) CALL PROMS (T(N2, N1) ,ALPHAB, ALPHAA, KB,KA) C4=KA/ALPHAA+KB/ALPHAB QA=2.0*KA*C2/C4 QB=2.0*KB*C2/C4 P5=C1* (KA+KB)/C4 A (N2) =-QA C (N2) =-QB B(N2)=1.0 + QA +QB IF(Nl.EQ.1) GO TO 7 P6=C3 (Nl)* (KA+KB)/C4 D(N2)=(P5-P6) *T(N2 ,N1-1)+(1.0-2.0*P5) *T(N2 ,Nl)+ 1 (P5+P6) *T (N2,Nl+l) GO TO 2 D(N2)=T(N2,1)*(1,0-4,0*P5)+4.O*P5*T(N2,2) CONTINUE C CALLING EQUATION SOLVER, CALL TDMA (2,NC) DO 8 N8=2,NC TSTAR (N8,N1) =RHS (N8) 8 CONTINUE C CALCULATING TSTAR FOR Jl