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Quantification of the heat transfer during the plasma arc remelting of titanium alloys Ji, Shiwei 2016

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    Quantification of the Heat Transfer during the Plasma Arc Remelting of Titanium Alloys  by  Shiwei Ji  B.A.Sc., Tsinghua University, 2013  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  In  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (Materials Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)    August 2016  ©  Shiwei Ji, 2016   ii  Abstract  Plasma-arc cold hearth melting (PAM) is an important technology used in the melting process for titanium alloys. Compared to the more common, electron beam cold hearth re-melting process, PAM allows an inert gas environment which significantly reduces the evaporation rate of alloying elements. To develop a better understanding of the effects of the plasma torch in the PAM process, a numerical model is being developed. However, this model requires an accurate description of the torch heat flux distribution. This research presented in this thesis focused on developing and verifying an inverse heat transfer analysis methodology to characterize the heat flux distribution from a plasma torch. A test block trial was conducted with in an industrial scale plasma arc furnace to measure the temperature history in a test block during heating and cooling. Following the trial, the test block was sectioned to get the liquid pool profile. The distribution of heat flux calculated from the inverse analysis assumed a Gaussian-like distribution, decreasing radially from the centerline to the edge of the block. Predictions for temperature history and liquid pool profile are in good agreement with the measured results from the experiment. Sensitivity analysis was performed to find some key factors that influence the prediction.   iii  Preface The forward finite element heat transfer model was developed using ABAQUS by the author. The inverse code in Python was based on previous work by Dr. Jianglan Duan. The author made modifications to the code and algorithm, and made verifications. The temperature data was measured by engineers at an industrial partner’s site in Morgantown PA. Throughout the program, my supervisors, Professors Steve Cockcroft and Daan Maijer, provided all kinds of support on research direction and tutorial of the knowledge. Dr. Lu Yao also gave many helful suggestions and comments on the details of the model.   iv  Table of Contents  Abstract ...................................................................................................................... ii Preface ....................................................................................................................... iii Table of Contents ...................................................................................................... iv List of Tables .............................................................................................................. vi List of Figures ............................................................................................................ vii Acknowledgments ..................................................................................................... xi Chapter 1 Introduction .............................................................................................. 1 1.1 Titanium Alloys and Applications ..................................................................... 1 1.2 Defects in Titanium Alloys ................................................................................ 3 1.3 Melting and Casting Technologies for Titanium Alloys .................................... 4 Chapter 2 Literature Review .................................................................................... 10 2.1 Plasma Arc Melting Process Characterization ................................................ 10 2.2 Plasma Arc Melting Process Models ............................................................... 13 2.3 Inverse Heat Conduction Analysis .................................................................. 17 Chapter 3 Scope and Objectives .............................................................................. 23 Chapter 4 Methodology ........................................................................................... 25 4.1 Experimental Method – Furnace and Torch ................................................... 25 4.2 Development of an Inverse Heat Conduction Analysis .................................. 29 Chapter 5 Verification of Inverse Heat Conduction Analysis .................................. 42 5.1 Case 1 - Time Dependent Heat Flux ............................................................... 43 v  5.2 Case 2 - Time and Location Dependent Heat Flux .......................................... 50 5.3 Case 3 - Gaussian Heat Flux Distribution ........................................................ 53 Chapter 6 Results and Discussion ............................................................................ 58 6.1 Experimental Results ...................................................................................... 58 6.2 IHCP Analysis of Plasma Torch Heat Flux ........................................................ 65 6.3 Sensitivity Analysis .......................................................................................... 77 Chapter 7 Conclusions and Future work ................................................................. 81 Bibliography ................................................................................................................. 84 Appendix A: .................................................................................................................. 88   vi  List of Tables  Table 1.1 Chemical composition of Ti64 (wt%) ....................................................... 1 Table 1.2 Mass percentage of materials used on different fighter aircraft  .......... 2 Table 2.1 One of the operating conditions for experiment. ................................. 12 Table 2.2 Percentage change in pool depth at r=0 mm caused by percentage change in parameters. ................................................................................................ 17 Table 4.1 Chemical composition of Ti64 (wt%) ..................................................... 26 Table 5.1 Determination of future time for Case 1 using the 2nd position .......... 47 Table 5.2 Determination of future time for the 3rd position ................................ 49 Table 5.3 Result comparison between validations for time dependent ............... 50 Table 5.4 Comparison of the No. of iterations for the 1st time step for different initial heat flux ......................................................................................................... 56 Table 6.1 Anisotropic thermal conductivity data. ................................................. 70  vii  List of Figures  Figure 1.1 VAR furnace used for manufacturing of titanium alloy ......................... 5 Figure 1.2 Schematic of the electron beam cold hearth re-melting process (EBCHR). .......................................................................................................................... 7 Figure 1.3 Schematic of the plasma cold hearth re-melting process (PAM)........... 7 Figure 2.1 Schematic of a plasma furnace installed at the IRC in the University of Birmingham. ................................................................................................... 11 Figure 2.2 Schematic of experimental set up. ...................................................... 12 Figure 2.3 Schematic diagram of the melt and gas flow conditions on a cut-through of a PAM refining hearth. ............................................................................... 14 Figure 2.4 Sensitivity matrix used in the inverse model of Gadala et al. .............. 18 Figure 2.5 Assumption of constant heat flux over the future time. ..................... 20 Figure 2.6 Idealized interface heat flux variation with time. ................................ 21 Figure 2.7 Validation results, (a) temperature data obtained at 2 mm away from the interface (b) temperature data obtained at 5 mm away from the interface. 21 Figure 4.1(a) Test block sitting at the top of the mould in the plasma arc furnace and (b) initial position of the plasma torch relative to surface of test block for initial horizontal alignment. ..................................................................................... 27 Figure 4.2(a) Thermocouple positions from the top view; (b) No.1-No.10 thermocouple positions from the cross section view; (c) No.11-No.15 thermocouple positions from the cross section view. Note: dimensions are in inches. (Unit: in) ............................................................................................. 28 Figure 4.3 Finite elements and dimension of the domain. ................................... 30 Figure 4.4 Nomenclature of the surfaces in the calculation domain. ................... 31 Figure 4.5 Density of Ti64 as a function of temperature. ..................................... 33 Figure 4.6 Specific heat capacity as a function of temperature. .......................... 34 viii  Figure 4.7 Thermal conductivity as a function of temperature ............................ 34 Figure 4.8 Sensitivity matrix .................................................................................. 37 Figure 4.9 Flow chart of inverse heat conduction analysis methodology ............ 38 Figure 4.10 Heat flux approximated as uniform in each segment. ....................... 40 Figure 4.11 Schematic diagram of possible heat flux profile calculated by inverse model. ............................................................................................................ 41 Figure 5.1 Flow chart of verification procedure.................................................... 42 Figure 5.2 Thermocouple locations and positions used for model verification. .. 43 Figure 5.3 Specified time dependent heat flux. .................................................... 44 Figure 5.4 Comparison of the specified and calculated heat fluxes for Case 1 at the 1st thermocouple position ............................................................................. 45 Figure 5.5 Comparison of specified input and calculated heat flux using the inverse model for Case 1, the 2nd position, future time = 5s. ................................... 47 Figure 5.6 Comparison of specified input and calculated heat flux using the inverse model for Case 1, the 2nd position, future time = 4.4s. ................................ 48 Figure 5.7 Comparison of specified input and calculated heat flux using the inverse model for Case 1 using the 3rd thermocouple positions. .............................. 49 Figure 5.8 Specified time and location dependent heat flux. ............................... 51 Figure 5.9 Comparison of specified and calculated heat fluxes for Case 2 at two times ............................................................................................................... 52 Figure 5.10 Comparison of the specified and calculated heat fluxes for Case 2 at three locations ............................................................................................... 52 Figure 5.11 Specified heat flux in the form of Gaussian distribution. .................. 53 Figure 5.12 Comparison of specified and calculated heat flux for Case 3 at the 3rd thermocouple location. .................................................................................. 54 Figure 5.13 Schematic of specified input and 3 initial heat fluxes. ...................... 55 Figure 5.14 Comparison of specified input and calculated heat flux using the inverse ix  analysis for Case 3 with different initial heat fluxes. ..................................... 56 Figure 6.1 Photo of the cooled test block before being sectioned. ...................... 58 Figure 6.2 Photo of the cross-sectioned test block showing the solidified liquid pool profile; a) original cross-sectional image and b) cross-sectional image with liquid pool outlined. ................................................................................................. 59 Figure 6.3 Raw temperature data measured by thermocouples installed in the test block. .............................................................................................................. 60 Figure 6.4 Effect of smoothing on temperature data measured at location 7. .... 61 Figure 6.5 Smoothed temperature data for each thermocouple location. .......... 61 Figure 6.6 Measured temperature profile at 278s ................................................ 63 Figure 6.7 Comparison of temperature data assessing symmetry of heating in the test block. ....................................................................................................... 64 Figure 6.8 Revised positions of thermocouples from the above view of the block.65 Figure 6.9 Calculated average heat flux distribution based on the inverse analysis. 66 Figure 6.10 Heat flux results from the inverse analysis at different times using isotropic thermal conductivity. ...................................................................... 67 Figure 6.11 Comparisons of temperature curves using isotropic thermal conductivity. ........................................................................................................................ 68 Figure 6.12 Comparisons of the measured and calculated pool profiles using isotropic thermal conductivity. ...................................................................... 69 Figure 6.13 Heat flux results from the inverse analysis at different times using anisotropic thermal conductivity. .................................................................. 71 Figure 6.14 Predicted heat flux distribution from the inverse model using anisotropic thermal conductivity. ..................................................................................... 72 Figure 6.15 Comparisons of measured temperature and predicted temperatures using anisotropic thermal conductivity. ......................................................... 73 Figure 6.16 Comparisons of the measured and calculated pool profiles using the x  anisotropic thermal conductivity. .................................................................. 74 Figure 6.17 Components of the predicted heat flux distribution from the inverse model using anisotropic thermal conductivity. ............................................. 76 Figure 6.18 Three different heat flux distributions. .............................................. 78 Figure 6.19 Effect of heat flux distribution on predicted temperature curves ..... 78 Figure 6.20 Effect of torch on time on predicted temperature curves ................. 79 xi  Acknowledgments First of all, I would like to appreciate my supervisors, Dr. Steve Cockcroft and Dr. Daan Maijer. Sometimes they gave me comments to let me know that the target was still far away, and sometimes they gave me encouragements to make me going further. I think, after many many years, maybe I would not remember the results got from the models here in Canada, but I would never forget the extremely rigorous attitude towards knowledge that I saw from both of them. I would like to express my sincere thanks to Dr. Lu Yao and Dr. Jianglan Duan. Both of them have taught me a lot about many details of the research. Although they are not my supervisors, I consider them as important teachers, as well as good friends.      1  Chapter 1 Introduction 1.1 Titanium Alloys and Applications Titanium (Atomic number = 22) is a transition metal. Pure titanium has a density of 4.52 g/cm3, equal to 57% of iron. The strength of high purity titanium is moderate, but the strength of some titanium alloys is up to 800 to 900 MPa, which is comparable to that of quenched and tempered structural steels. Alloy additions like manganese, aluminum, chromium, vanadium, and molybdenum have been shown to increase the strength of titanium alloys [1]. Ti-6wt%Al-4wt%V (Ti64), with the nominal composition shown in Table 1.1, is the most widely used structural titanium alloy. Table 1.1 Chemical composition of Ti64 (wt%) [2] Ti Al V Fe C N H O Other Balance 5.5-6.8 3.5-4.5 0.3 0.08 0.05 0.015 0.2 0.5  In recent years, the problem of energy conservation has become a prominent societal focus. One strategy being pursued in the transportation sector is to reduce weight. The use of titanium and its alloys in the aerospace sector due to its relatively high strength and low density is well established in components for both aero-engines and airframes [3-4]. Broader use of titanium in transportation, for example in the automotive applications, has been very limited due to its relatively high cost. Additionally, titanium alloys offer good corrosion resistance to sea water, and sulfuric and nitric acid environments resulting in its 2  use in offshore applications, in the chemical industry, in food processing equipment and in bio-medical applications. Titanium is also used in sports equipment. For example, golf clubs and bike frames. [5-6] Focusing on its application in the aerospace industry, in the early 1950’s, the United States first successfully used titanium alloys on aircraft. Looking at the last two commercial Boeing aircraft to be introduced, roughly 7% of Boeing 777 airframe weight is titanium and 15% of the Boeing 787 is titanium. In comparison, the percent of aluminum by weight has dropped from 70% in the 777 to 20% in the 787. Table 1.2 shows the increase in percentage of titanium alloy usage in fighter aircraft since the 1960’s. In aircraft engines, titanium alloys are commonly used in parts that experience relatively low temperature service conditions (<500℃). Modern jet engines, like the GE90, employ titanium alloys to achieve a weight savings of more than 150 kg [7-10]. Table 1.2 Mass percentage of materials used on different fighter aircraft [11] Fighter Year of Design Ti Steel Composite F-14 1969 24 17 1 F-15 1972 27 6 2 F-17 1983 25 5 10 F-22 1989 41 5 25   3  1.2 Defects in Titanium Alloys The use of titanium alloys in jet engines follows strict quality control guidelines and the manufacturers have established frozen-practice procedures for the processes used to produce these alloys. In titanium alloys, defects caused by inclusions or compositional in-homogeneities can significantly influence mechanical properties such as fracture toughness and fatigue strength. Inclusions are classified according to their composition and density. Titanium and its alloys have a high solubility of oxygen and nitrogen, both of which act as potent interstitial hardeners. During titanium processing, especially during sponge production, titanium can be exposed to oxygen and nitrogen at elevated temperatures resulting in local regions with elevated hardness. These so-called High Interstitial Defects (HID’s), or Type I inclusions, can degrade fatigue performance if not removed during subsequent melt-consolidation processing. When classified according to density, there are low density inclusions (LDIs) and high-density inclusions (HDIs). LDIs are mainly titanium-nitrogen and titanium-oxygen enriched regions, while HDIs are commonly tantalum, molybdenum, tungsten and tungsten carbides. The later may be introduced to the process with the scrap (revert) metal that often comprises a significant portion of process feedstock in addition to virgin sponge. Sources include fractured pieces of cutting bits used in machining operations. The need to consolidate both sponge and revert material into primary ingot and to ensure the elimination of these defects from premium quality titanium intended for aerospace applications has resulted in the development of a number of sophisticated 4  melt-consolidation technologies. Two of the more recent technologies, Electron Beam Cold Hearth Re-melting (EBCHR) and Plasma Arc Cold Hearth Re-melting (PAM), exploit density separation and extended liquid metal residence times to facilitate removal of these inclusions.  1.3 Melting and Casting Technologies for Titanium Alloys 1.3.1 Vacuum Arc Remelting (VAR) The VAR process, depicted schematically in Figure 1.1, is the dominant commercial method used to melt specialty metals such as nickel and titanium alloys for critical applications. During the VAR process, an electrode of the alloy to be melted is sequentially drip melted by passing a high current (in the range of kA) between the electrode and ingot being formed to create an arc. The liquid metal drops vertically from the electrode into the liquid pool that forms in the ingot within the crucible gradually consuming the electrode. Adjustment is needed to maintain the distance between the electrode and liquid pool to produce a stable arc and melt pool. In order to encourage homogeneity, a magnetic field is applied to stir the liquid pool. The Lorentz forces caused by this magnetic field leads to a circumferential flow, which must be carefully controlled to optimize the operation of the process for the production of high quality ingots [12].  The VAR process operates in a vacuum environment to eliminate the pick-up of nitrogen and oxygen. The loss of alloying elements through evaporation is low because the surface area of the liquid metal exposed to the vacuum environment is small. A major 5  disadvantage of the VAR process is that it is difficult to remove HDI’s because they drop from the electrode directly into the liquid pool of the ingot. Moreover, the residence time in the liquid pool is relatively short and temperature exposure is lower compared to hearth melting processes limiting the dissolution and homogenization (refining) that can occur. According to SAE-AMS2380, the VAR processing must be preceded by a hearth melting process to produce high quality titanium alloys (i.e. premium-quality titanium) when recycled materials are being included [13].  Figure 1.1 VAR furnace used for manufacturing of titanium alloy [14] 1.3.2 Cold Hearth Melting (CHM) CHM technologies can be divided into two variants based on the heat source –Electron Beam Cold Hearth Melting (EBCHR) and Plasma-Arc Cold Hearth Melting (PAM). A schematic of a typical layout for an Electron Beam (EB) Cold Hearth 6  Re-melting furnace (EBCHR) is shown in Figure 1.2. The equivalent layout for a Plasma Cold Hearth Re-melting Process (PAM) is shown in Figure 1.3. These processes may be divided into three operations: i) melting, ii) refining and iii) casting. Melting and refining may be carried out in separate hearths (water-cooled copper containment vessels) or in a single hearth. A solid metal skull forms in the hearth to contain the liquid titanium (titanium cannot be contained in conventional metallurgical refractories owing to it high reactivity). In the melting zone/hearth, an electron beam or plasma arc heats and melts the alloy. The use of high intensity and high density heat sources, such as electron beams and plasma torches, results in higher temperatures in the liquid metal compared to the VAR method [15]. The liquid metal flows from the melting zone/hearth into the refining zone/hearth, where additional heat is applied (often by additional sources such as other plasma torches) to ensure the metal remains liquid at an elevated temperature. Inclusions and impurities are removed by different mechanisms such as evaporation, dissolution and density separation – e.g. density separation occurs when the HDIs sink to the bottom of the melt and become trapped in the refining hearth, whereas LDIs melt or dissolve as they transit the refining hearth. In the casting area, liquid metal flows slowly into a mold to form an ingot that is continuously withdrawn.  7   Figure 1.2 Schematic of the electron beam cold hearth re-melting process (EBCHR). [16]  Figure 1.3 Schematic of the plasma cold hearth re-melting process (PAM). [16] Electron beam furnaces are more commonly used in the specialty metals industry and account for the largest percentage of CHM production. During the process, electron 8  beams are guided by electromagnetic coils allowing rapid movement of the beam and significant flexibility in depositing heat to the surface of the metal. The disadvantage of the process is that the vacuum environment needed to enable the generation and transmission of the electron beam and high liquid metal surface area in the hearth results in high evaporation rates of the liquid metal and its alloying elements. In addition to the preferential loss of alloying elements, such as aluminum and chromium, the build-up of condensate on the furnace walls and mould can lead to solid material of a composition high in the volatile alloy constituent occasionally breaking away and entering the liquid pool. In the case of Ti64 this can lead to the so-called Type-II, aluminum rich, alpha stabilized defect. In the PAM process, an inert gas environment is used to avoid oxygen and nitrogen pickup and to also generate the plasma used to provide the process heat. For the production of titanium alloys, argon and helium are used. The commercial processes typically use inert gas pressures at or slightly below atmospheric pressure, which significantly reduces the evaporation rate of alloying elements compared with the electron beam melting process. Thus, for alloys containing high concentrations of elements with high vapor pressures, such as aluminum, chromium and manganese, the PAM process has a distinct advantage over EBCHR. The focus of the work in this thesis is to better understand the heat transfer (quantity and distribution of heat) from a plasma torch to titanium under typical conditions existing in a commercial PAM process. This understanding is critical to fully exploit the 9  advantages of this technology and to improve the industrial-scale process. The data resulting from this work may be directly used in thermal-fluid models of the process that can be applied to aid in the design of the layout of commercial furnaces and to optimize their operation – e.g. torch positioning, outputs and patterns.    10  Chapter 2 Literature Review There have been a limited number of studies focus on considering the refining capabilities / performance of the plasma arc cold hearth melting process. In this chapter, the experimental studies on characterizing plasma arc melting will be introduced first, and then models developed to predict the thermal-fluid flow conditions that exist in this process will be reviewed. Finally, numerical techniques to estimate the heat flux from a distributed spatially dependent heat source will be reviewed. 2.1 Plasma Arc Melting Process Characterization Figure 2.1 shows a schematic of the facility that Ward et al. used to investigate the plasma arc remelting process for a nickel-based superalloy [17]. Experiments were performed using this facility in the Interdisciplinary Research Centre (IRC) at the University of Birmingham (Figure 2.1). The plasma torches are rated at 150 kW in the IRC furnace. Melting is performed in an argon atmosphere to reduce oxidation and evaporation.  11   Figure 2.1 Schematic of a plasma furnace installed at the IRC in the University of Birmingham. [17] Huang et al. investigated the behavior of inclusions in the refining hearth of a PAM furnace [18]. For the experimental setup, the plasma torch was positioned with a 6'' standoff above the center of a block. The orientation of the torch can be controlled by changing the tilt-angle from 0 to 10 degrees (shown in Figure 2.2). In this work, a fixed tilt-angle of 10 degree was used. The torch pattern (motion) was found to have some effect on the shape of molten pool. However, it was determined the motion does not influence the overall mass and energy balances. An example of one set of operating 12  conditions is given in Table 2.1. The profile of the melt pool was determined by etching sections with a dilute solution of hydrofluoric and nitric acids. In addition, thermocouples were placed at a series of radial locations at approximately mid-height.  Figure 2.2 Schematic of experimental set up. [18]  Table 2.1 One of the operating conditions for experiment. [18] Material Ti-6-4 Geometry 2-D Cylinder Furnace Chamber Pressure (atm) 1.1 Torch Status Stationary Tilt-angle (degree) 10 Torch Electrical Current (A) 750 Torch Electrical Voltage (V) 200 Torch Power (kW) 150  13  2.2 Plasma Arc Melting Process Models Huang et al. developed one of the first models of the PAM process [18]. Their model considered the heat transfer and flow occurring in the PAM refining hearth and was used to investigate the behavior of inclusions introduced with the feed. In this work, Huang used a solidification model containing fluid flow for 2-D axisymmetric PAM hearth to study the geometry of liquid pool.  The heat flux from the plasma torch was assumed to exhibit a Gaussian distribution and torch gas shear was not included. Huang performed a sensitivity analysis on the effects of process parameters on the liquid pool depth. Results showed that, the torch power, Marangoni coefficient and heat transfer coefficient play very important roles, while the torch thermal focus was negligible. The emissivity of the top surface was shown to affect a small portion of heat flux near the top surface where the temperature is high. 14   Figure 2.3 Schematic diagram of the melt and gas flow conditions on a cut-through of a PAM refining hearth. [19] Huang et al. developed another model incorporating the effects of the Lorentz forces generated by the interaction between the melt and the transferred plasma arc. Lorentz forces are the forces occurring on charged objects when they are moving in a magnetic field. In the PAM refining hearth, the magnetic field is produced by the plasma arc. They also incorporated a surface shear force to describe the effects of the high velocity gas exiting the plasma torch. In this model, the Lorentz and surface shear forces act as driving forces on the surface of the liquid pool. Figure 2.3 shows a vector plot of flow pattern in the melt pool and gas plume.  The boundary conditions applied on the skull/hearth interfaces and melt free surface are: 15  For the skull/hearth interfaces:    (2.1) where    = 5.669x      W/(  2   4) is the Stefan-Boltzmann constant, ε is the emissivity and a constant value of 0.6 was used and hc is a contact heat transfer coefficient, meant to describe contact conduction heat transfer between the skull and the water-cooled copper hearth, equal to 100 W/  2 K, and 𝑇𝑤  is the temperature of cooling water. For the free surface of the melt:    (2.2) where qn is the net heat loss or gain due to radiation from the top surface and heat input from the torch. The heat flux from the torch was assumed to follow a Gaussian distribution:    (2.3) where 𝑟   is the distance from the center of plasma torch axis, 𝑟 𝑎  is the characteristic radius, V is torch electrical voltage (V), and I is the torch electrical current (A). The model was used to predict the trajectory of inclusion particles in the hearth. The model predictions were shown to be sensitive to the torch thermal parameters such as heat flux, gas shear stress and electromagnetic forces. The heat flux relationship describing the plasma torch in Huang's model was highly concentrated, which produced large temperature gradients and strong surface tension flows. Additionally, the electromagnetic force effects incorporated into the model where shown to affect the pool 16  profile. The predictions also showed that the Joule heating effect caused by the electromagnetic field is very small and can be neglected [19]. Lothian et al. developed a numerical model of the remelting process for a nickel-based superalloy disk, which accounts for heat transfer, buoyancy, fluid flow and electromagnetic effects. Their results showed that the average Lorentz force is in general much smaller than buoyancy, but, in a small region beneath the torch, large Lorentz forces exist. The heat flux applied in the model to represent the torch was a Gaussian distribution, and the torch was held stationary above the center of the pool. The distribution followed the equation:   (2.4) Where R is the distance from the center of plasma torch, 𝑟    is the characteristic radius, and 𝑄0  is the peak heat flux. The model was used to conduct a sensitivity study. The results shown in Table 2.2 indicate that torch power and surface tension gradient are the main parameters that dominate the shape of the pool. [20]    17  Table 2.2 Percentage change in pool depth at r=0 mm caused by percentage change in parameters. [20] Parameter Change Percentage Pool depth change Sensitivity Specific heat capacity +25% +0% Negligible Surface Tension Gradient +25% +8% Has Effect Latent heat +25% +0% Negligible Viscosity -50% +2% Negligible Torch Power -25% -5% Has Effect  In the models describe above, the heat flux distribution from the plasma arc was assumed to be Gaussian. The literature reviewed to date suggests that there is a need to validate the Gaussian distribution assumption with measurements in a plasma furnace. Inverse heat conduction analysis could be used to calculate the heat flux distribution, given a set of temperatures measured in a plasma furnace. The inverse methodology is briefly introduced in the next section.  2.3 Inverse Heat Conduction Analysis Beck published the original work on analyzing an inverse heat conduction problem (IHCP) [21]. A classical IHCP may be defined as the calculation of the heat flux at the surface of a solid body from the transient temperature data measured some distance 18  below the surface. As there is no guaranteed unique, stable solution, IHCP is considered to be an ill-posed problem. In practice, the heat flux solution is found to be highly sensitive to small variations in temperature data. Beck also introduced a sequential procedure to calculate a time-dependent heat flux q(𝑡𝑛  ), where the heat flux at the nth time step is estimated iteratively before moving on to the next time step q(𝑡𝑛+1  ). A sum of the square of the differences between the measured and predicted temperatures is used to evaluate the error of the solution. When the error is below a criterion, the solution of q(𝑡𝑛  ) is accepted and n is increased by one and the procedure is repeated. Gadala et al. applied an inverse heat transfer analysis procedure for the heat transfer problem of cooling water impinging on steel plates using a two-dimensional finite element model [22]. Figure 2.4 shows the form of sensitivity matrix Gadala used in the inverse model.   Figure 2.4 Sensitivity matrix used in the inverse model of Gadala et al.  In Figure 2.4, i is the number of the time step, r is the evaluation location in body and spans values from 1 to L, where L is the number of thermocouples, and s is the heat flux 19  segment and spans values from 1 to J, where J is the number of heat flux components on the surface. J must be less than or equal to L. In Gadala et. al.’s model, they used the same value of L and J.  They found that the heat flux calculated by the inverse analysis was sensitive to random measurement errors in the input data and that therefore the uniqueness and stability of inverse heat conduction problem solution could not be guaranteed. [22] Hamasaiid et al. developed an inverse model to calculate the heat transfer coefficient (HTC) at a casting-mold interface in die coating process of aluminum alloys. They studied how the HTC is affected by coatings applied to the die. The peak HTC was shown to decrease with increasing coating thickness- i.e. 50% decrease with coating thickness increases from 10 to 100 μm. Sensitivity of the interfacial HTC was also shown to decrease as coating thickness increases. [23] In situations where there is a significant lag time in the response of the thermocouple due to the material having a low thermal diffusivity or the placement of the thermocouple (far from the source or sink of heat), the future time step method has been shown to be effective. Referring to Figure 2.5, starting at current time t, the approach adopted is to solve the forward heat conduction equation for R time steps into the future while holding the estimated heat flux constant. At the end of RΔθ seconds, where Δθ is the time-step size, an assessment of the error in ith iteration is made. The error is determined by comparing the predicted temperature, at the location of the thermocouple(s), with an objective function that utilizes the measured temperature (described below). Depending 20  on the magnitude of the error the estimated heat flux is either updated (refined) and the process rerun or the estimated heat flux q1 becomes the predicted heat flux a time t. In the event that the second condition holds, the time-step counter is augmented to t+t and the process is restarted. The process is repeated until the desired analysis time tend is achieved.   Figure 2.5 Assumption of constant heat flux over the future time. Zhang et al. developed and applied this method to calculate the HTC at the mold/casting interface during the directional solidification of a taper cylinder A356 sample, based on data obtained from thermocouples cast in to the metal undergoing solidification. In development of the model, an idealized time dependent interfacial heat flux was used to generate simulated thermocouple data to verify the numerical technique. This idealized heat flux is shown in Figure 2.6. 21   Figure 2.6 Idealized interface heat flux variation with time. [24] Zhang et al. applied this idealized heat flux in 2 verification cases with different simulated thermocouple positions. Case 1 was for thermocouples placed 2 mm from the interface and Case 2 was for thermocouples placed 5 mm away from the interface.   Figure 2.7 Validation results, (a) temperature data obtained at 2 mm away from the interface (b) temperature data obtained at 5 mm away from the interface. [24] The results of this verification analysis show good agreement for the 2 mm case, but reveal solution instability for the 5 mm case. A key finding of the result was that placing 22  the thermocouples within a material undergoing a phase change (with the associated release of heat) represents a particularly challenging problem as latent heat increases the thermal inertia of the material surrounding the sensors. To overcome this challenge, the thermocouple needed to be placed within 2mm of the interface. Additionally, this necessitated a large value of R (the number of future time steps) in the inverse algorithm than would normally be recommended in the literature. Results from the inverse model calculation also showed that the interface HTC value varies strongly with time when solidification happens. [24]    23  Chapter 3 Scope and Objectives The objectives of this study are to: 1) develop and verify a method to characterize the heat flux distribution from a plasma torch used for plasma arc melting; and 2) validate this method using data obtained from a commercial PAM torch. In order to accomplish this objective, the following sub-tasks were undertaken: i) To instrument a test block with a number of thermocouple sensors, to obtain data suitable for input to an inverse model. ii) To perform an instrumented trial to measure the transient temperature response of a Ti64 test block in an industrial plasma arc furnace. iii) To section the test block to determine the liquid pool profile as a secondary means of IHC model validation. The instrumented thermocouple trial was conducted on an industrial scale plasma arc furnace located at an industrial partner’s site, in Morgantown PA. Planning and setup for the trial was performed by engineers from the industrial partner, in consultation with UBC. During the trial, the temperature of an instrumented test block was measured while heat was applied with a plasma torch. The inverse heat transfer model was developed using ABAQUS as the forward conduction code, which is a commercial finite element analysis package. The inverse code that estimated the surface heat flux distribution and coordinated operation of the forward conduction model was coded in Python and based on previous work by Dr. 24  Jianglan Duan of Casting Group in the Department of Materials Engineering at the University British Columbia. Modifications were made to the code and algorithm to make it suitable for the test block trial.      25  Chapter 4 Methodology In this Chapter, the experimental methodology used to generate data for calculating the heat flux obtained from an industrial scale plasma torch will be introduced, including the geometry of the test block and the locations of thermocouples. Following this, the development of a thermal model, capable of predicting the temperature evolution of the test block, and the implementation of this model within an inverse heat transfer analysis will be presented.   4.1 Experimental Method – Furnace and Torch A trial was conducted to measure the temperature history in a test block during heating with a plasma torch. For this trial, the power of the plasma torch was set at 300kW (Current: 1000 A, Voltage: 300 V), and was used to heat the block for 5 minutes. The average pressure of the chamber was 425 Torr (0.56atm), torch standoff distance is 0.3048m (12 in), and the gas used in the furnace was helium. The measured temperatures were subsequently used as input data for an inverse heat transfer analysis to calculate the heat flux distribution from the plasma torch. The trial was conducted in an industrial-scale plasma-arc furnace operated by TIMET at their Morgantown, PA facility. Plant technical personnel conducted the plant trial with oversight provided by UBC. The trial setup will be presented in this section and the measured temperatures will be presented in Chapter 6.  26  4.1.1 Test Block A cylindrical sample of Ti64 was used as the test block for this trial. The composition of Ti64 is shown in Table 4.1. Table 4.1 Chemical composition of Ti64 (wt%) [2] Ti Al V Fe C N H O Other Balance 5.5-6.8 3.5-4.5 0.3 0.08 0.05 0.015 0.2 0.5  The test block has a diameter of 0.4191m (16.5 in) and a thickness of 0.0762m (3 in). The test block was placed on top of copper standoff blocks that were resting on the starter block in the mould of the plasma furnace. The standoff blocks were used to ensure that the surface of the test block was at a height consistent with the metal surface during casting operations and to provide space to run the thermocouple wires from the test block to a pass-through at the furnace wall. The plasma torch was located directly above the top surface of the block. Figure 4.1 shows a photo of the test block and its setup in the mould cavity.  27  (a)   (b)  Figure 4.1(a) Test block sitting at the top of the mould in the plasma arc furnace and (b) initial position of the plasma torch relative to surface of test block for initial horizontal alignment.  4.1.2 Thermocouple Positions The test block was instrumented with 15 K-type thermocouples. The thermocouples were installed in holes that were 0.0127m (0.5 in) deep and were drilled from the bottom surface of the test block. Thus, the thermocouple tips were 0.0635m (2.5 in) below the top surface of the test block. The thermocouples were positioned along two lines oriented 90 from each other as shown in Figure 4.2. This layout was designed to assess the symmetry of temperature distribution in the block. If the heating conditions were symmetric, the trial will provide 10 independent temperature-time curves that can be used in the inverse calculation. 28  (a)   (b)   (c)  Figure 4.2(a) Thermocouple positions from the top view; (b) No.1-No.10 thermocouple positions from the cross section view; (c) No.11-No.15 thermocouple positions from the cross section view. Note: dimensions are in inches. (Unit: in)  29  4.2 Development of an Inverse Heat Conduction Analysis An inverse heat conduction methodology is used to estimate the heat flux experienced by the surface of a test block based on the measured temperature data. A forward heat conduction model was first developed to predict the temperature distribution in the test block. This model was then incorporated into an inverse heat transfer analysis to calculate the plasma torch heat flux using the measured temperature data as input.   4.2.1 Forward Heat Conduction Model The forward heat conduction model is used to calculate the transient temperature response of the test block. The governing heat conduction equation that is solved in this model is:   (4.1) where  is the density (kg/  3), 𝑐   is the specific heat capacity (J/kg·K), k is the thermal conductivity (W/m·K), T is the temperature (K), and Q is a heat source associated with the latent heat of phase transformations (W/  3). After applying boundary conditions consistent with the conditions in the plasma torch trial, the governing equations may be solved using the finite element (FE) method. In this study, the commercial FE software, ABAQUS (v16.4), was used to perform the FE calculations. The inverse heat transfer model was coded in Python, and the forward heat transfer calculation was implemented to run in ABAQUS by a trigger. Figure 4.3 shows the domain of the model. By assuming axisymmetric conditions 30  exist, the test block can be represented by a rectangular domain with an area of 0.0762m x 0.21m (8.25 in x 3 in) and the rotational axis (r = 0) aligned with line AB (refer to Figure 4.3). A uniform element edge length of 4 mm has been used and the total number of nodes and elements is 3166 and 1007, respectively.  Figure 4.3 Finite elements and dimension of the domain.  4.2.1.1 Boundary Conditions Boundary conditions must be selected to represent the heat transfer conditions existing during the heating of the test block. A zero heat flux condition exists at the symmetry boundary along the centerline. The boundary conditions applied to the top, side and bottom surfaces will now be defined (refer to Figure 4.4). 31   Figure 4.4 Nomenclature of the surfaces in the calculation domain.  Top Surface The boundary condition applied to the top surface includes heat input from the plasma torch and heat losses due to radiation and convection. The expression representing the net heat flux to the top is:  𝑞 𝑛      = 𝑞       + 𝑞 𝑎     + 𝑞  𝑛     (4.2) In Eq. 4.2, 𝑞       is the spatially dependent heat flux (W/m2) from the plasma torch (to be calculated through the inverse calculation), 𝑞 𝑎     is the heat loss due to radiation to the inside of the plasma furnace and 𝑞  𝑛     is the heat loss (or gain) due to convection to the gases inside the furnace. The radiative losses from the top are estimated using Eq. 4.3, which assumes a small grey body in a large enclosure. 32    (4.3) In Eq. 4.3,    = 5.669x      W/(  2   4) is the Stefan-Boltzmann constant,   is the emissivity, and 𝑇   𝑛𝑎    is the temperature of the furnace wall above the test block and is assumed to be 200˚C. The emissivity is equal to 0.6 [19]. The convective contribution to the heat balance is estimated using Eq. 4.4.   (4.4) In Eq. 4.4, 𝑕  𝑛   is the convective heat transfer coefficient, assumed to be 50 W/(  2    ). Although this value has been assumed, its sensitivity was assessed with the model and was shown to have very little effect compared to the heat flux from the plasma torch. 𝑇 𝑎   is the temperature of the gas above the test block, assumed to be a constant 500˚C. Side surface The side surface of the test block is cooled via radiation to the inside wall of the mould. The radiation heat flux on this edge is calculated using:   (4.5) where 𝑇𝑤𝑎    is the temperature of the water cooled mold, assumed to be a constant equal to 25 C. The emissivity of the top surface is also applied to the side surface. Bottom surface The thermal effect of the copper standoffs used to raise the test block must be accounted for in the boundary conditions applied to the bottom edge of the test block. The outer portion (~0.0508m) of the bottom edge of the block was assumed to be in contact with the standoffs. The boundary condition applied in this region is calculated 𝑞𝑟𝑎𝑑𝑡𝑜𝑝  =  𝑠  (𝑇 4- 𝑇𝑓𝑢𝑟𝑛𝑎𝑐𝑒4 )   𝑞𝑐𝑜𝑛𝑣𝑡𝑜𝑝 = 𝑕𝑐𝑜𝑛𝑣  (𝑇  - 𝑇𝑔𝑎𝑠 )     𝑞𝑟𝑎𝑑𝑠𝑖𝑑𝑒  =  𝑠  (𝑇 4- 𝑇𝑤𝑎𝑙𝑙4 ) 33  using the following expression:   (4.6) where h (W/  2 K) is the contact heat transfer coefficient, assumed to be 1000 W/  2 K and 𝑇        is the temperature of the copper support, which is assumed to be a constant equal to 25 °C. Heat transfer from the rest of the bottom surface was assumed to be due to radiation. The radiation heat loss is calculated using:    (4.7) where 𝑇𝑎     is the temperature of the ambient environment below the test block and is assumed to be equal to 25 °C. The emissivity is 0.6. 4.2.1.2 Material Properties Thermophysical properties for Ti64 used in the model are based on values compiled by Mills [25]. Figures 4.5 – 4.7 show the variation of density, specific heat capacity and thermal conductivity, respectively, as a function of temperature applied in the model.   Figure 4.5 Density of Ti64 as a function of temperature. 𝑞𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑏𝑜𝑡𝑡𝑜  = 𝑕   (𝑇  - 𝑇𝑐𝑜𝑝𝑝𝑒𝑟 )   𝑞𝑟𝑎𝑑𝑏𝑜𝑡𝑡𝑜  =  𝑠  (𝑇 4- 𝑇𝑎 𝑏𝑖4 )  34   Figure 4.6 Specific heat capacity as a function of temperature.  Figure 4.7 Thermal conductivity as a function of temperature [25].    The density was held constant at temperatures below the solidus temperature. This is necessary to conserve mass because a fixed grid analysis is used. To incorporate the heat consumed (generated) during the solid to liquid (liquid to solid) phase transformation, the latent heat option was used in ABAQUS which incorporates a specified amount of latent heat linearly between two temperatures. A latent heat of 286 kJ/kg was defined for the solid/liquid phase transformation, and the solidus and liquidus temperatures were 1595C and 1625C, respectively [26]. The thermophysical property data can be found in 35  Appendix A in a tabulated format.  4.2.2 Inverse Heat Conduction Analysis Methodology    The inverse heat conduction analysis will be used to estimate the heat flux from the measured temperature-time data from the test block trial. The heat flux applied to the top surface of the test block by the plasma torch is both time and location dependent. To determine the dependence of the heat flux with time, the heat flux is updated every time step. The spatial dependence in the heat flux is accounted for by discretizing the heat flux into a series of m constant heat fluxes, qn, where m is the number of thermocouples. The time step size of the inverse analysis is specified as 1s in this work. The heat fluxes used for the first time step are estimated and input by the user. Subsequently, the heat fluxes are updated based on the difference between the predicted and measured temperatures and using a relationship that incorporates the sensitivity of the temperature at each thermocouple position to a small change to the applied heat fluxes.  Since the thermocouples (TC) were located 0.0635m (2.5 in) below the top surface of the test block where heat from the plasma torch was applied, there is a delay in the temperature response due to changes in the heat flux at the surface. This distance was chosen so as to avoid the thermocouples contacting molten titanium, which would cause Type-K thermocouples to fail within a short time.  To address the challenge associated with the thermocouples being placed relatively distant form the top surface, the future time-step version of the inverse method was 36  adopted [24]. In the future time-step method, for each inverse time step, the forward model must be run to some “future time” that enables the effects of changes in the heat flux conditions to diffuse through the material and be of a reasonable magnitude. This future time is referred to as 𝑡       . The error is defined as the sum of the squared differences between the measured and predicted temperatures and is calculated as:   (4.8) where 𝑇𝑛  𝑡  𝑡         is temperature predicted by the forward model,  𝑛  𝑡  𝑡         is the measured temperature, 𝑡  𝑡        is the current time being considered in the inverse analysis plus the future time, and m is the number of heat fluxes that are being calculated in the inverse analysis. In this work, m is equal to 10.     If the error calculated in Eq. 4.8 is greater than a set criterion, the heat fluxes applied at time t are not acceptable. The heat fluxes, 𝑞𝑛 , are then updated to (𝑞𝑛  +△𝑞𝑛  ) until the error calculated from Eq. 4.8 is below the set criterion. For each iteration, △qn is calculated as:  △𝑞𝑛   =    1 △𝑇𝑛  +         (4.9) where S is a sensitivity coefficient matrix calculated by determining the temperature change, T(q+q), resulting from a change, q, (where q is equal to 1000 W/m2) to the current heat flux in each segment separately. The forward model is used to evaluate the sensitivity matrix. △qn and △Tn in Eq. 4.9 are vectors and    1 is the inverse of the sensitivity matrix. The sensitivity matrix, S, takes the form shown in Figure 4.8. Error =  (𝑇𝑛 (𝑡 + 𝑡𝑓𝑢𝑡𝑢𝑟𝑒 ) −  𝑛 (𝑡 + 𝑡𝑓𝑢𝑡𝑢𝑟𝑒 )) 2  𝑛=1  37   Figure 4.8 Sensitivity matrix [27] Iterations are continued until the error evaluation function reaches the criterion shown below:  1    𝑇𝑛  𝑡  𝑡        −  𝑛  𝑡  𝑡          2  𝑛 1 < Crit (4.10) Crit is the value that the average of the sum of the squared temperature differences must be below to proceed to the next time step. The choice of Crit depends on the required accuracy of the calculated heat flux and the robustness for achieving convergence. Smaller values require that the difference between calculated and measured temperature are smaller. However, more iterations and thus more computation time is needed to achieve convergence. In this work, the Crit was assessed by several test calculations during the verification process, and finally 0.001 was used.    Once the temperatures predicted with the updated (qn+△qn) satisfy the requirement in Eq. 4.10 (if qn satisfies the requirement itself, no changes will be needed), the heat flux for the current time step has converged and the inverse analysis will continue to the next time step.A flow chart of the inverse analysis methodology applied in this study is given in Figure 4.9.           38   Figure 4.9 Flow chart of inverse heat conduction analysis methodology 39  For some types of inverse problems, for example when the heat flux changes greatly with location, estimates of △qn exhibit significant numerical variability which causes instability in the predicted heat flux with time. In these cases, a factor, f, may be introduced to dampen the update of q:   (4.11) When f is less than 1, additional inverse iterations will be required to achieve a viable solution and the effects of instabilities can be reduced. In this study, an f of 0.1 has been used. To determine the location dependent heat flux distribution, the top surface of the test block has been divided into discrete segments. Since the temperature was measured at ten radial locations, the top surface was divided into 10 segments. In this manner, ten heat flux (qn) values can be calculated directly using the inverse methodology. Often, the heat flux is assumed to be uniform within each segment [21], and the segments are centered radially on the thermocouple locations, as shown in Figure 4.10. However, this results in discontinuities between each segment leading to inaccuracies in the predicted temperatures and calculated heat fluxes.     𝑞𝑢𝑝𝑑𝑎𝑡𝑒  = q + f△q           40   Figure 4.10 Heat flux approximated as uniform in each segment.  In this work, the heat flux distribution was varied linearly across each segment. Heat flux values on the surface of the block, aligned radially with each thermocouple, were calculated using the inverse methodology. The heat flux was linearly interpolated between these 10 locations. Beyond the bounding points (i.e. between the centerline and position 1 and between the outer diameter and position 10), the variation of the heat flux was assumed so as to to provide a reasonable profile. Two weighting factors were used: the first, factor1, was applied to estimate the flux at r = 0; and the second, factor2, was applied to estimate the flux at r = 0.21m (8.25in).  q(r =0) = q(@ 1st thermocouple) * factor1 q(r =8.25 in) = q(@ 10th thermocouple) * factor2      (4.12) The resulting continuous heat flux distribution is shown schematically in Figure 4.11. 41   Figure 4.11 Schematic diagram of possible heat flux profile calculated by inverse model.   42  Chapter 5 Verification of Inverse Heat Conduction Analysis A multi-stage verification process has been used to during the development and tuning of the methodology used for the inverse heat conduction analysis in this study. To begin the verification process, the forward model was run using ABAQUS to calculate the temperature history based on an assumed representative specified heat flux. The calculated temperature results were then adopted as “hypothetical” thermocouple data and used as input data into the inverse analysis. The resulting estimated heat fluxes were then compared with those originally input for each case to assess the accuracy of the inverse methodology. These steps are shown below in a flow chart highlighting the flow of data through this process:  Figure 5.1 Flow chart of verification procedure. 43  The verification was performed using 3 cases: 1) a uniform heat flux changing with time; 2) a spatially varying heat flux changing with time; and 3) a specified heat flux with a Gaussian distribution.  The temperature output from the forward model (hypothetical thermocouple data) corresponded to the radial positions of the thermocouples mounted in the test block, shown in Figure 5.2. The vertical position was varied between three positions, 1st, 2nd and 3rd depending on the conditions under evaluation.   Figure 5.2 Thermocouple locations and positions used for model verification.  5.1 Case 1 - Time Dependent Heat Flux A time dependent heat flux, shown in Figure 5.3, was applied to the top surface of the test block in Case 1. The heat flux varies sinusoidally with time during the first 300s and then 44  becomes constant after 300s as shown in Eq. 5.1. For Case 1, an initial guess of 80000 W/m2 (equal to 80% of the initial specified heat flux) was used.    q={4     ∗ 𝑠𝑖𝑛  .   47 ∗ 𝑡𝑖 𝑒          𝑡𝑖 𝑒 < 3  𝑠                                                                         𝑡𝑖 𝑒 ≥ 3  𝑠       (W/  2)        (5.1)   Figure 5.3 Specified time dependent heat flux. Previous studies reported that the accuracy of inverse analysis predictions can be influenced by distance between the thermocouple location and the surface where the heat flux is applied [24]. For Case 1, where the heat flux is uniform over the surface but changes with time, temperature data at three distances away from the top surface was extracted from the forward model and used to test the capability of the inverse heat transfer analysis.   01000002000003000004000005000006000000 100 200 300 400 500 600 700 800 900Heat Flux (W/m2)Time (s)45   5.1.1 Verification at the 1st Position (Top Surface) As discussed in Chapter 4.2.2, a future time has been introduced in the analysis to account for the time required for the changes in the heat flux to influence the temperature at the thermocouple locations. For the 1st position on the top surface of the test block, no future time was required to calculate accurate heat fluxes.   Figure 5.4 Comparison of the specified and calculated heat fluxes for Case 1 at the 1st thermocouple position Figure 5.4 shows a comparison of the calculated and specified heat fluxes versus time at the location of 0.0127m (0.5 in) away from center line on the top surface. The results indicate that the heat fluxes calculated with the inverse analysis agree with the 46  specified heat fluxes. The average error between the specified and calculated heat fluxes is 1.07%.  5.1.2 Verification at the 2nd Position (0.5 in Below the Top Surface) As the second position is 0.0127m (0.5 in) below the top surface, a future time was used with the inverse analysis to account for the time required for changes in the heat flux condition to reach the thermocouple locations. The value of future time was determined by trial and error by running the inverse analysis with multiple future times. If the value of future time is too small, the temperature change at the thermocouple position is zero and results in a zero elements in the sensitivity matrix. If any row of the sensitivity matrix is filled with zeros, the matrix inversion will generate a “singular matrix” error and the will inverse analysis crash because △q =   1•△   +         cannot be evaluated. When the value of the future time is too large, the inverse analysis result is not accurate because the temperature at time t + t future is also influenced by the heat flux after time t that changes with time.    An appropriate future time was determined by calculating the heat flux using various future times. The results are summarized in Table 5.1.    47  Table 5.1 Determination of future time for Case 1 using the 2nd position  Initially, future time steps based on integer values (i.e. 1, 2, 3, 4 and 5) were used to map the response of the inverse analysis. Singular matrix errors resulted from 1s to 4s. At 5s, the inverse analysis finished. The future time was then refined to show that 4.4s was the smallest functional future time that provided a stable solution.  Figure 5.5 Comparison of specified input and calculated heat flux using the inverse model for Case 1, the 2nd position, future time = 5s. 01000002000003000004000005000006000000 100 200 300 400 500 600 700 800 900Heat Flux (W/m2)Time (s)Specified Inverse resultFuture Time Model status 1s Singular Matrix 2s Singular Matrix 3s Singular Matrix 4s Singular Matrix 4.3s Singular Matrix 4.4s Finished 4.5s Finished 5s Finished 48   Figure 5.6 Comparison of specified input and calculated heat flux using the inverse model for Case 1, the 2nd position, future time = 4.4s. The inverse analysis results for future time equals to 4.4s and 5.0s are shown in Figure 5.5 and Figure 5.6. When the future time was equal to 4.4s, the average error between inverse analysis result and the specified heat flux was 3.10%, whereas it was 5.15% for 5.0s case. A more accurate result was achieved with the smaller future time.   5.1.3 Verification at the 3rd Position (2.5 in below the top surface) Using the same method discussed in 5.1.2, a series of future times were used to determine a suitable value for use in the inverse analysis at the 3rd position (0.0635m (2.5 in) below the surface). As shown in Table 5.2, the future time had to be increased to 51s to provide enough time for changes in the surface heat transfer conditions to affect 01000002000003000004000005000006000000 100 200 300 400 500 600 700 800 900Heat Flux (W/m2)Time (s)Specified Inverse result49  the temperatures at these sub-surface positions. Figure 5.7 shows the comparison of the calculated and specified heat fluxes at the 3rd position. The average error between the calculated and specified heat fluxes is 4.64%.  Table 5.2 Determination of future time for the 3rd position Value of Future Time Model status 30s Singular Matrix 35s Singular Matrix 40s Singular Matrix 45s Singular Matrix 50s Singular Matrix 51s Finished 53s Finished 55s Finished   Figure 5.7 Comparison of specified input and calculated heat flux using the inverse model for Case 1 using the 3rd thermocouple positions. 01000002000003000004000005000006000000 100 200 300 400 500 600 700 800 900Heat Flux (W/m2)Time (s)Specified Inverse Result50  Table 5.3 compares the error when the temperature data at different vertical thermocouple locations are used in the inverse analysis. The calculation error increases as the distance to the top surface becomes larger. This is caused by the drop in sensitivity of temperature variation to changes in heat flux at the top surface as the distance to the surface increases.  Table 5.3 Result comparison between validations for time dependent  heat flux at different positions Distance to the top surface Error  0 in  1.07% 0.0127m (0.5 in) 3.10% 0.0635m (2.5 in) 4.64%   5.2 Case 2 - Time and Location Dependent Heat Flux In Case 2, the temperature data from 0.0127m (2.5 in) below the top surface was used to assess the capability of the inverse analysis methodology applied in this study to calculate a time and location dependent heat flux. Figure 5.8 shows the spatially dependent heat flux used in Case 2 at two times (t = 0s and 40s). The heat flux was assumed to vary linearly with time between 0 and 40 s. Based on the results shown in Section 5.1.3, a future time of 51s was applied. The initial guess for the heat flux to start the inverse analysis was equal to 90% of specified input heat flux.  51   Figure 5.8 Specified time and location dependent heat flux.  Figure 5.9 shows the comparison of calculated and specified heat flux at different times, and Figure 5.10 shows the calculated heat flux varying with time at 3 locations.  52   Figure 5.9 Comparison of specified and calculated heat fluxes for Case 2 at two times  Figure 5.10 Comparison of the specified and calculated heat fluxes for Case 2 at three locations 53  The results for Case 2 show that, although at some points the inverse result is higher or lower than the specified heat flux, the model will correct itself in the next time step or at the region nearby, ensuring that the error between the specified and calculated heat flux is small.   5.3 Case 3 - Gaussian Heat Flux Distribution A Gaussian heat flux distribution, shown in Figure 5.11, was applied to the top surface of the test block in Case 3. The heat flux was location dependent, but time independent. Eq. 5.2. shows the equation used to define the heat flux.   (5.2) In Eq. 5.2, r is the distance from the center line and σ is the standard deviation, which was equal to 0.065 m.  Figure 5.11 Specified heat flux in the form of Gaussian distribution. q = 2.26x10 6·e − r  22σ  2   (W/m 2) 54  As described in the methodology, the distribution was built from linear segments between each heat flux point associated with a thermocouple location. Two weighting factors were used to increase the heat flux at the centerline of the test block and to decrease the heat flux radially towards the outer diameter. The weighting factors used were: q(r =0) = q(@ 1st thermocouple) * 1.019 q(r =8.25 in) = q(@ 10th thermocouple) * 0.219        (5.3) For the inverse analysis, a future time of 51s and an initial guess equal to 90% of the specified input heat flux was employed. The result of the inverse analysis is shown in Figure 5.12. The inverse analysis has accurately determined the time independent, Gaussian heat flux distribution.  Figure 5.12 Comparison of specified and calculated heat flux for Case 3 at the 3rd thermocouple location. 55  Additional calculations were performed to examine the influence of the initial heat flux assumed in the model. In addition to 90%, the inverse analysis was run with initial heat fluxes of 70% and 50% of the specified Gaussian heat flux distribution for comparison. The 3 initial heat fluxes and the specified heat flux for Case 3 are shown in Figure 5.13.  Figure 5.13 Schematic of specified input and 3 initial heat fluxes. The heat fluxes calculated from the 3 difference initial heat flux distributions are shown in Figure 5.14. Although the initial heat fluxes were different, the results of the inverse analysis are nearly identical. The only difference experienced was the number of iterations required to complete the 1st time step in the analysis, shown in Table 5.4. When the difference between initial heat flux and the specified heat flux was larger, more iterations were needed for the 1st time step to reach the convergence. This results in 56  longer computational times. All the 3 cases in Table 5.4 were completed using the same damping factor, if computational time was an issue, a variable damping factor or limiting the maximum heat flux change per iteration could be implemented to reduce the number of iterations required to find a solution.   Figure 5.14 Comparison of calculated heat flux using the inverse analysis for Case 3 with different initial heat fluxes.  Table 5.4 Comparison of the No. of iterations for the 1st time step for different initial heat flux Initial heat flux  (% of the specified) No. of iterations for the 1st time step 90% 4 70% 17 50% 33  57  This verification exercise has demonstrated that the inverse analysis technique developed for this study has the ability to the predict time and location dependent heat fluxes based on the temperature measured 0.0635m (2.5 in) below the top surface of the test block. In the next chapter the verified inverse analysis procedures will be used to calculate the time and location dependent heat flux from a plasma torch based on the temperature data measured during the test block trial.   58  Chapter 6 Results and Discussion 6.1 Experimental Results Following the methodology described in Chapter 4, an instrumented test block was installed in an industrial-scale PAM furnace at TIMET’s Morgantown, PA operations. The plasma torch was set to 300kW (Current: 1000 A, Voltage: 300 V). The torch was used to heat the center of the test block for 278s. After the block had cooled, it was removed from the furnace and sectioned for examination. Figure 6.1 shows the test block after removal from the furnace prior to sectioning.  Figure 6.1 Photo of the cooled test block before being sectioned. The top surface of the test block, shown in Figure 6.1, indicates that the heating from the plasma torch was not completely centered on the top surface. The shape of the area 59  that melted and re-solidified is circular in the lower left quadrant of the image, but transitions to an irregular shape in the upper right quadrant.  Figure 6.2 shows a cross-section from the test block heated during the trial. The cross-section was made along the line indicated in Figure 6.1. In Figure 6.2a, the liquid pool that formed during heating can be seen due to the visible microstructure change. The profile of the liquid pool has been highlighted with an orange line in Figure 6.2b.  a)  b)  Figure 6.2 Photo of the cross-sectioned test block showing the solidified liquid pool profile; a) original cross-sectional image and b) cross-sectional image with liquid pool outlined. In Figure 6.2, the profile of the liquid pool is nearly symmetric about the center line of the block. The left side of the pool is 17 mm larger than than the right half. This asymmetry is consistent with the asymmetry observed on the top surface of the test block. Figure 6.3 shows the raw temperature data measured during the test block trial. Prior to ~300s when the torch was on, there is significant noise in the measured temperatures because of the electrical field created by the torch.  60   Figure 6.3 Raw temperature data measured by thermocouples installed in the test block. IHCP analysis is particularly sensitive to noise in the input temperature data [21]. Thus, the raw temperature data was smoothed using a 4th-order polynomial smoothing algorithm implemented in VBA in Excel. An example of the effects of the smoothing process is shown in Figure 6.4, where the blue dots are the raw temperatures measured at location 7 and the red line is the smoothed data. Figure 6.5 shows the smoothed temperature data for all thermocouple locations (i.e. 1-10). 61   Figure 6.4 Effect of smoothing on temperature data measured at location 7.   Figure 6.5 Smoothed temperature data for each thermocouple location. 62  Figure 6.5 shows that, the rate of temperature increase gradually increases to a constant value after the torch is turned on, and then following a time delay begins to decrease after the torch is turned off at 278s. The temperature at each thermocouple continues to increase after the torch is turned off as the gradients in the test block   dissipate and heat is transported throughout the test block. The time to reach the peak temperature at each thermocouple location will be influenced by the release of latent heat as the liquid metal solidifies. The rate of heating at each location is reflected in the slopes of the curves, which decrease as the distance to the center of the block increases. When the torch was turned off at 278s, a significant temperature difference is evident from the center to the outer thermocouple as shown in Figure 6.6. The temperature data in Figure 6.6 was obtained by extracting the temperatures from the ten thermocouples at 278s. In the chapter on methodology, it was assumed that the heat flux would follow a Gaussian distribution and to some degree, Figure 6.6 supports this assumption.  63   Figure 6.6 Measured temperature profile at 278s During the test block trial, temperature data was collected from 15 thermocouples. The temperatures measured with thermocouples 11-15 was intended to assess whether the heating applied to the test block is symmetric. Figure 6.7 shows a comparison of temperature data at 3 different distances from the center of the block.  64   Figure 6.7 Comparison of temperature data assessing symmetry of heating in the test block. The thermocouples at 0.0127m (0.5 in) from the centerline (TC1 and TC11) show very similar temperature history. However, as the distance from the centerline increases, the difference between the temperatures measured at TC5 and TC13 and between TC7 and TC14 increases. This comparison supports the conclusions made based on the picture of the top surface of the test block that heating was not symmetric about the centerline. The scale of the asymmetry was assessed based on the image of the test block top surface and suggests that thermocouple locations corresponding to TC13 and TC14 are ~0.01778m (0.7 in) further from the centerline than TC5 and TC7. The revised positions of thermocouples can be seen in Figure 6.8. 65   Figure 6.8 Revised positions of thermocouples from the above view of the block. Additionally, measurement errors occurred near the end of the trial in thermocouples TC11 and TC14, which resulted in discontinuities in the temperature data. Since the data from thermocouples TC1-10 was continuous, this data was used for the inverse model.  6.2 IHCP Analysis of Plasma Torch Heat Flux The IHCP analysis procedure described in Chapter 4 was used to analyze the test block temperature data and calculate the heat flux output by the plasma torch during the test block trial.  6.2.1 Calculated heat flux (isotropic thermal conductivity) The calculated heat flux from the IHCP analysis is shown in Figure 6.9. The heat flux presented in Figure 6.9 is the heat flux averaged over the time when the plasma torch 66  was heating the test block, because it did not change much with time. Additional heat flux results from the inverse analysis at different times are shown in Figure 6.10. The difference between these results is below 5%.  Figure 6.9 Calculated average heat flux distribution based on the inverse analysis. 67   Figure 6.10 Heat flux results from the inverse analysis at different times using isotropic thermal conductivity. The predicted and measured temperatures are compared in Figure 6.11. The results indicate that when the test block was being heated and the temperature increased, the predicted temperatures agree well with those measured. The inverse analysis was only used to calculate the heat flux when the plasma torch was on. After 278s, when the plasma torch was turned off, heat loss was calculated based on the radiation and convection boundary conditions defined in the model, which are approximations. Thus, the predicted temperatures do not match the measured data as well for the time when the torch was off. 68   Figure 6.11 Comparisons of temperature curves using isotropic thermal conductivity. A contour plot of the temperature distribution at 278 s is compared with an optical image of the test block cross-section in Figure 6.12. The liquidus isotherm has been highlighted with a black line in the contour image to delineate the liquid pool shape. Comparing the predicted liquid pool shape with the shape observed on the cross-section of the test block indicates that while the thermocouple data is accurately reproduced with the inverse model the liquid pool profile is not. Based on parallel work by Dr. Lu Yao in the Department of Materials Engineering on modeling the plasma hearth, it was postulated that the disagreement in the pool profile is mainly caused by advective transport of heat in the liquid pool resulting in differences in the radial heat transfer in the liquid pool. When a plasma torch is directed towards a liquid, heat transfer within the liquid is affected by a number of fluid flow phenomena that are not described accurately in the conduction-only model used as for the forward model in the IHCP analysis. These 69  phenomena include flow due to thermal buoyancy, thermal Marangoni and surface shear caused by the plasma plume and have been shown to enhance heat transport both vertically and radially in the liquid pool.   Figure 6.12 Comparisons of the measured and calculated pool profiles using isotropic thermal conductivity. To approximate enhanced radial heat transport, an anisotropic thermal conductivity was implemented in the forward thermal model when the metal temperature was above the liquidus. 6.2.2 Calculated heat flux (anisotropic thermal conductivity) A trial and error process determined the anisotropic thermal conductivity value for the liquid metal. In this process, the ABAQUS forward model was run using different conductivities in the radial and vertical directions until the predicted pool profile was comparable with the test block cross-section. Values of 200 W/m·k in the radial direction and 6 W/m·k in the vertical direction were used to achieve a liquid pool profile similar to 70  the cross-section. The anisotropic thermal conductivity applied in the model is shown in Table 6.1. It should be noted that an anisotropic thermal conductivity was only applied to the metal when it was liquid. Below the liquidus temperature, the thermal conductivity is isotropic.  Table 6.1 Anisotropic thermal conductivity data. T(°C) Radial kc, rad (W/m·k) Vertical  kc, vert (W/m·k) 35  7  7  91  7  7  233  9  9  521  13  13  795  18  18  907  19  19  1013  19  19  1215  23  23  1377  24  24  1595  27  27  1626  33  33  1646  200  6  1783  200  6  1894  200  6   Selected heat flux profiles at different times when the plasma torch is on are shown in Figure 6.13 obtained using the anisotropic thermal conductivity. 71   Figure 6.13 Heat flux results from the inverse analysis at different times using anisotropic thermal conductivity. The result plotted in Figure 6.14 is the average heat flux profile taken over the time that the plasma torch is on.  72   Figure 6.14 Predicted heat flux distribution from the inverse model using anisotropic thermal conductivity. The predicted and measured temperatures are compared in Figure 6.15. The curves agree well with each other with only some small differences observed at the time when the torch was turned off and beyond.  73   Figure 6.15 Comparisons of measured temperature and predicted temperatures using anisotropic thermal conductivity.  A contour plot of the temperature at 278 s is compared with an optical image of the test block cross-section in Figure 6.16. Figure 6.16 shows that, with anisotropic thermal conductivity or enhanced radial heat transport when the metal is liquid, the width of the pool profile more closely resembles that observed in the test block cross-section. Additionally, a shallower pool depth was obtained, which more closely represents what was observed from the test block cross-section.  Comparing the radial and vertical conductivities above the liquidus (1625C) presented in Table 6.1 with the original isotropic conductivities at just below the solidus (1595C), it is clear that a substantial increase in the radial component was required, whereas a reduction in the vertical component was need to better align the model with the experimentally derived pool profile. The increase in the radial component is consistent 74  with a strong radial component in the fluid flow along the surface of the melt pool associated with a strong shear force acting radially outward. This shear force stems from both the gas flow from the torch and the gradient in surface tension (Maragoni force) associated with the radial temperature gradient. The reduction in the vertical component is likely due to the recirculating upward flow in the middle of the pool, which results in the diffusional and advective contributions to heat transport opposing one another. This recirculating flow also likely gives rise to the pool profile seen in the center of the block, which is not reproduced in the diffusion only forward conduction model.  Figure 6.16 Comparisons of the measured and calculated pool profiles using the anisotropic thermal conductivity. The comparisons above show that the predicted heat flux profile obtained from the inverse analysis is reasonable provided and anisotropic thermal conductivity is also adopted in the liquid in the conduction model. By integrating the calculated heat flux over the area of the test block surface, the net power from the plasma torch applied to the 75  test block was calculated to be 84.9 kW. As the input power to the plasma torch was 300 kW, the yields a heat transfer efficiency of approximately 28%. This is consistent with other reported values [28]. This obviously indicates that, over all, the plasma-based heating process is relatively inefficient.  There are several potential reasons for this poor efficiency: 1) there is significant energy lost to the water used to cool the torch; and 2) significant energy is lost to heating up the helium gas. There potentially two mechanisms by which heat is transfer from the plasma to the work-piece: 1) convection and 2) radiation. To explore this, an attempt has been made to fit the total heat flux profile using two superimposed heat flux distribution, both of which are assumed to follow a Gaussian distribution. Figure 6.17 shows the convection and radiation components of predicted heat flux distribution from the plasma torch together with the sum of the two components and the predict profile. The parameters used are shown in Eq. 6.1 and 6.2. 76   Figure 6.17 Components of the predicted heat flux distribution from the inverse model using anisotropic thermal conductivity.    (6.1) In Eq. 6.1, r is the distance from the center line and σ is the standard deviation (=0.038 m).   (6.2) In Eq. 6.2, r is the distance from the center line and σ is the standard deviation (=0.12 m). The heat input due to convection from the plasma torch is more focused, while the radiation input is more spread out, as would be expected. This is consistent with other findings in the literature [28, 29]. The sum of the two contributions agrees very well with the heat flux calculated from inverse model. 01000000200000030000004000000500000060000000 2 4 6 8Heat Flux (W/m2)Distance to the center line (in)Inverse Result Convection+RadiationConvection Radiationqconv = 4.42x10 6·e − r  22σ  2   (W/m 2) qrad = 6.41x10 5·e − r  22σ  2   (W/m 2) 77   6.3 Sensitivity Analysis  A sensitivity analysis was conducted to examine the effects of different plasma torch heating parameters on the predicted temperature distribution. This analysis was conducted by systematically varying parameters in the forward model and evaluating the effects on the temperature in the test block at one location.  The heat flux distribution and the heating time (i.e. duration that the plasma torch is on) are two parameters that significantly influence the shape and peak value of the temperature variation at a given location beneath the torch. A sensitivity analysis was conducted to see how much influence these two parameters have.   6.3.1 Effect of Heat Flux Distribution Figure 6.18 shows three heat flux distributions used to assess the sensitivity. The blue curve (q2) is the baseline heat flux distribution, q1 and q3 represent an 22% increase and and 30% decrease, respectively. Figure 6.19 shows the heating rate in the test block increases as the heat flux increases.  78   Figure 6.18 Three different heat flux distributions.  Figure 6.19 Effect of heat flux distribution on predicted temperature curves  0200400600800100012000 200 400 600 800 1000Temperature(℃)Time (s)q1 q2 q379  6.3.2 Effect of Heating Duration The model was used to assess the impact of increasing and decreasing the heating duration by 40 s from the baseline duration of 250 s. The results in Figure 6.20 show that when the torch is turned off in each case, the temperature keeps increasing for another 50 s before decreasing. The three curves have the same slopes during the time when the torch is on because the heat flux distribution from the torch is the same. The analysis shows that the heating duration affects the maximum temperature achieved at a given location.   Figure 6.20 Effect of torch on time on predicted temperature curves 0200400600800100012000 200 400 600 800 1000Temperature(℃)Time (s)250s 290s 330s80  The results of the sensitivity analysis show that the predictions are sensitive to both the magnitude of the heat flux and duration over which it is applied as would be expected.  In this chapter, an inverse analysis was used to calculate the heat flux from measured temperature data. An anisotropic thermal conductivity was implemented in the conduction-only model used in this analysis to approximate the effects of fluid flow on the melt pool. The distribution of heat flux was observed to have a Gaussian-like distribution, decreasing radially from the centerline of the torch. The influence of heat flux distribution and the time to turn off the torch was studied with a sensitivity analysis.       81  Chapter 7 Conclusions and Future work 7.1 Summary and Conclusions An inverse heat transfer analysis methodology was developed to calculate the heat flux output from an industrial plasma torch based on measured thermocouple data. An instrumented thermocouple trial was conducted on an industrial scale plasma arc furnace to provide the input temperature data for this analysis. The temperature of an instrumented test block (15 K-type thermocouples placed 63.5 mm (2.5 in) below the top surface of the test block) was measured during heating with a plasma torch and after the torch was turned off.  The heat transfer model used in the inverse analysis was developed in ABAQUS, a commercial finite element analysis package. The inverse analysis methodology was verified using 3 numerical cases: 1) a uniform heat flux changing with time; 2) a spatially varying heat flux changing with time; and 3) a specified heat flux with a Gaussian distribution. The verification cases showed that the future time step version of the inverse conduction method would be required to solve the test block problem due to the fact that the thermocouples were located at a subsurface location 63.5 mm below the top surface subject to the heat flux from the torch. The future time step method accounts for the delay in the diffusional transport of heat from the surface to the thermocouple locations. For the test block, the value of the future time parameter was determined to be 51 s following a trial and error process. Additionally, the initial heat flux used for inverse analysis was 82  shown to have no influence on the calculated heat flux. However, it was shown to affect the number of iterations needed for the 1st time step to reach the convergence. The results from the inverse analysis on the measured temperature data showed that the heat flux distribution from plasma arc torch could be analytically approximated by the sum of two Gaussian distributions: one representing the convective contribution from the plasma gas; and the other, the radiative contribution from the plasma gas. The convective contribution was found to be the larger of the two and more narrowly focused (peak 4.42 MW/m2,  = 0.038), whereas the radiation contribution was smaller and more diffuse (peak 0.641 MW/m2,  = 0.12). The total power obtained from integrating the sum of the convective and radiative fluxes over the block surface was found to 84.9 kW. This result translates into an overall efficiency of heating for the plasma torch of 28.3%. Some suggestions were put forward for the low efficiency of the process; however, they were not verified. A key finding of the work was that an anisotropic thermal conductivity, applied for temperatures above the liquidus, was needed in the ABAQUS-based forward conduction model in order to correctly approximate both the thermocouple data and the pool profile. The anisotropy needed reflected a significant increase in the radial heat transport and a reduction in the vertical transport. This manipulation of the conductivity was reasoned to be a result of the significant fluid flow occurring in the melt pool and highlights the limitations of a conduction only model in this application. Potential drivers for the fluid 83  flow were put forward, but not verified as this was deemed beyond the scope of this work.   7.2 Recommendations for Future Work In this study, the limitations of a diffusion only heat transfer analysis were identified. Future work could include the coupling of the inverse code with a CFD model. The boundary conditions implemented in the heat transfer model to account for heat loss from the block were assumed. Parameters like the temperature of the gas and the hearth wall will influence the heat loss from the block and will affect the calculated heat flux. Additional measurements on the operating process could be used to reduce the inaccuracy associated with these assumed parameters.   84  Bibliography [1]  Titanium Alloying and Heat Treatment. www.totalmateria.com. 2005. Retrieved on July 5th, 2016. [2] Zhang Chunjiang. Titanium cutting processing technology. Xi'an Northwestern University Press. 1986. [3] W.L. Masterton, C. N. Hurley. Chemistry: Principles and Reactions 6th Edition. 2008. [4] C. Leyens, M. Peters. Titanium and Titanium Alloys. 2005. [5] F. Froes. Will the titanium golf club be tomorrow's mashy niblick. Journal of metal. 51:18-20. 1999. [6] W. Bramer. Biocompatibility of dental alloys. Advanced engineering materials. 3(10): 753-761. 2001. 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Grandemange, Y. Combres, B. Champin, A. Jardy, S. Hans, D. Ablitzer. The Modelling of Heat, Momentum and Solute Transfers to Optimize the Melting Parameters of VAR Titanium Ingots: an Application to the "Wafer" Practice. International Symposium on Liquid Metal Processing and Casting. 204-213. 1999. [15] Duan Junwei. Titanium Alloy Technology and its Application with Cold Hearth Melting. Nonferrous Metals Processing. Vol: 40 P42. 2011. [16] Progress in Titanium-Alloy Hearth Melting. Retrieved from http://www.industrialheating.com/articles/86369-progress-in-titanium-alloy-hearth-melting. 2002. Retrieved on July 5th, 2016. [17] R.M. Ward, T.P. Johnson, M.H. Jacobs. Liquid pool shapes during the plasma remelting of a nickel-based superalloy. International Symposium on Liquid Metal Processing and Casting. 97-109. 1999. [18] X. Huang, J.S. Chou, and Y. Pang. Modeling of Plasma Arc Cold Hearth Melting and Refining of Ti Alloys. 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Effect of mold coating materials and thickness on aluminum alloys heat transfer in permanent mold casting of aluminum alloys. Metallurgical and Materials Transactions. Vol 38A.1303-1316. 2007. [24] Liqiang Zhang, Carl Reilly, Luoxing Li, Steve Cockcroft, Lu Yao. Development of an inverse heat conduction model and its application to determination of heat transfer coefficient during casting solidification. Heat Mass Transfer. 50: 945–955. 2014. [25] K.C. Mills. Recommended Values of Thermophysical Properties for Selected Commercial Alloys. ASM International. 2002. [26] Riley Evan Shuster. Modeling of Aluminum Evaporation during Electron Beam Cold Hearth Melting of Titanium Alloy Ingots. Master’s thesis, The University of British 87  Columbia, 2013. [27] Boran Xue. Heat Transfer Characterization of Secondary Cooling in the Horizontal Direct Chill Casting Process for Aluminum Alloy Remelt Ingot. Master’s thesis, The University of British Columbia, 2010. [28] Helen Hols. Study of the Influence of the Plasma Torch in a Plasma Arc Melting Furnace. Master’s thesis, University of Birminghama, 2014. [29] R.T.C. Choo, J. Szekely and R.C. Westhoff. Modeling of High-Current Arcs with Emphasis on Free Surface Phenomena in the Weld Pool. Welding Research Supplement. 346-361. 1990.   88  Appendix A: Table Parameter input form of density applied in ABAQUS. T (°C) Density  (kg/   ) 25  4198  1300  4198  1481  4194  1600  4198  1618  4120  1622  4034  1623  3941  1696  3889  1789  3825  1899  3748  1992  3748    Table Parameter input form of specific heat capacity applied in ABAQUS.  T (°C) Cp (J/kg·K) 24  546  133  571  300  607  461  645  599  667  789  690  912  675  998  642  1145  672  1349  703  1587  745  1615  773  1620  831  1753  831  1895  829  89     Table Parameter input form of isotropic thermal conductivity.  T (°C) kc (W/m·k) 35  7  91  7  233  9  521  13  795  18  907  19  1013  19  1215  23  1377  24  1595  27  1626  33  1646  34  1783  34  1894  34      

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