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Boiling water heat transfer study during dc casting of aluminum alloys Li, Dianfeng 2000

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BOILING WATER HEAT TRANSFER STUDY DURING DC CASTING OF ALUMINUM ALLOYS by Dianfeng Li B.Eng., Northeastern University, China, 1988 M.Eng., Northeastern University, China, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF METALS AND MATERIALS ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1999 ©Dianfeng Li, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MMAJ The University of British Columbia Vancouver, Canada Date yW , 2^crCrO DE-6 (2/88) Abstract During Direct Chill (DC) casting of aluminium alloys, the majority of the heat (-80%) is extracted in the secondary cooling zone where water contacts the periphery of the solidifying ingot as it is drawn from the mould. The high heat extraction rates during secondary cooling induce thermal gradients and mechanical contraction within the thin newly solidified shell and result in deformation known as "butt curl" and a contraction (pull-in) of the rolling face. Since aluminium has a relatively low melting point, heat extraction to the cooling water is strongly dependent on the temperature of the aluminium sheet ingot surface, and will vary dramatically during cooling as different heat transfer regimes are encountered (i.e. film boiling, transition /nucleate boiling and convection cooling). For the aluminium DC casting process, the heat transfer to the cooling water is particularly complicated as ingot surface temperatures at the initial point of water contact cause transition/film boiling to occur followed by nucleate boiling and convection cooling as the surface temperature of the ingot is cooled. To model the DC casting process accurately, it is necessary to develop accurate boiling curve data (i.e. heat flux Vs surface temperature) for as-cast alurninium as it is being cooled. This study investigated the influence of the ingot surface topography, sample starting temperature and water flow rate on the boiling curve for three commercially significant aluminum alloys namely: AA1050, AA3004 and AA5182. The project involved both experimental measurements (using industrial as-cast aluminium samples and an experimental set-up designed and built at UBC), a 2-D LHCP (inverse heat conduction problem) model to calculate the heat flux on the sample surface as it is was being cooled and measurements at ii NRC (National Research Council) to quantify the surface roughness for each sample using a laser profilemeter. The experimental test facility was designed and built at UBC and included: a vertical furnace to heat the samples to the desired temperature, a pneumatically operated lowering platen to move the sample out of the furnace and position it in front of the water box and a water box, which was built out of Plexiglas and duplicated a typical section of an aluminium mould used for industrial DC casting. For each test, a sample, instrumented with a number of thermocouples, was put into the furnace and heated to the desired temperature. The sample was then lowered in front of the water box (2~3 mm away from the water box), the water was turned on to the desired flow rate and the sample was cooled. During cooling of the sample the data acquisition system recorded the sample temperature as a function of time at a frequency of 20 Hz. The inverse heat transfer model was developed to calculate the boiling curves for direct water chill cooling using the thermal response of the ingot and the application of a 2D finite element based heat conduction model to iteratively calculate the heat flux at the surface of the sample. This technique was verified using both analytical solutions and hypothetical data obtained using a known heat flux profile in the commercial FEM code ABAQUS. The results from the study indicate that a variation in alloy surface morphology (machined versus as-cast), water flow rate and sample initial temperature all dramatically influence the calculated boiling curve. The intensity of the heat extraction was found to be enhanced as the surface of the sample became rougher because nucleation and growth of bubbles became easier thereby enhancing the heat transfer. Sample starting temperature also had a significant influence on the calculated boiling curve and it was found that a unique iii boiling curve for a given surface temperature did not exist for each of the alloys studied. A few tests were also run whereby the sample was moved slowly down into the water spray. Comparison of the experimentally calculated heat fluxes using the test rig at UBC to those calculated in industry by freezing thermocouples into solidifying ingots were similar in shape and it was found that the calculated boiling curves from the industrial data fell in between the boiling curve calculated using a stationary sample and the boiling curve calculated using a moving sample. Indicating that the boiling curves calculated using the UBC rig reflect the heat transfer phenomena occurring in industry during water spray cooling. iv Table of Contents Abstract ii Table of Contents V List of Figures viii List of Tables xii List of Symbols xiii Acknowledgment xvi 1.0 Introduction 1 1.1 DC casting process 1 2.0 Literature Review 5 2.1 Boiling heat transfer 5 2.2 The influence of casting parameters on DC cast ingot quality 7 2.2.1 Cast start sensitivity to h variation 9 2.2.2 Steady state heat flow and sensitivity to h 11 2.3 Boiling heat transfer in direct chill casting 12 2.3.1 Water jet and falling film physical description 12 2.3.2 Quantifying heat transfer during steady state operation 14 2.3.3 Effect of the cooling water velocity 17 2.3.4 Effect of water temperature 19 2.3.5 Effect of water flow rate 20 2.3.6 Effect of water quality 21 2.3.7 Effect of surface properties 23 2.4 Summary 25 3.0 Scope and Objectives 27 3.1 Objectives 27 3.2 Methodology 27 4.0 Experimental 29 4.1 Start material 29 4.2 DC casting water cooling simulator 30 4.3 The number and position of thermocouple 33 V 4.3.1 Thermocouple distribution in y direction 33 4.3.2 The distance of thermocouples from sample surface 35 4.4 Quantifying as-cast roughness 35 5.0 Inverse heat transfer analysis 37 5.1 Formulation of the inverse problem 37 5.2 Development of 2-D FEM heat conduction model 37 5.2.1 Basic heat transfer problem 37 5.2.2 Inverse heat transfer analysis 39 5.2.3 Comparison of FEM model prediction to an analytical solution 40 5.3 Mathematical technique used in Inverse Heat Conduction 43 Problem(IHCP) 5.3.1 Data noise smoothing 43 5.3.2 Single future time step 45 5.3.3 multiple future time step method 47 5.4 Verification of inverse heat conduction model 49 5.4.1 Temperature dependent material properties 49. 5.4.2 Verification of thermal model formulation 50 6.0 Experimental results and discussion 53 6.1 Typical experimental cooling curves 53 6.1.1 Water flow on the surface of the sample 53 6.1.2 The cooling curves 55 6.1.3 Impingement zone and free falling zone 58 6.2 Effect of sample parameters on the calculated boiling curves 59 6.2.1 Alloy composition 59 6.2.2 Surface morphology of the sample 62 6.3 Effect of process parameters on the calculated boiling curves 67 6.3.1 Water flow rate 68 6.3.2 Starting temperature 71 6.4 Comparison to industrial DC casting 73 7.0 Summary and conclusions 76 7.1 Recommendations for future work 78 vi References 80 Appendix A - Definition of roughness parameters 84 Appendix B - 2-D transient FEM model of heat transfer 89 Appendix C - Data acquisition system noise 103 vii List of Figures 1.1 Schematic of the DC casting process [ 1 ]. 1 1.2 Schematic diagram indicating the variation in heat flux along the length of an aluminum ingot as it is being water cooled. 3 2.1 Boiling curve describing different regions in boiling heat transfer [4]. 6 2.2 Schematic showing the Upstream conduction distance (UCD) and pool depth on a conventional open top sheet ingot mould [5]. 9 2.3 The predicted UCD as a function of h and alloy conductivity [5]. 12 2.4 Typical schematic of a water jet used during DC casting of aluminum [15]. 13 2.5 The heat flux of DC casting process from Weckman [16]. 14 2.6 The boiling curves used by J. B. Wiskel for: a) lapped surface and b) liquated surface [21]. 16 2.7 The high near surface temperature on 155 mm 6063 billet caused by film boiling [15]. 18 2.8 Effect of water temperature on the cooling curves [24]. 19 2.9 Effect of water temperature on boiling heat transfer [15]. 20 2.10 Effect of water flow rate on the cooling curves [24]. 21 2.11 Effect of water flow rate on water cooling heat transfer [15]. 21 2.12 Effect of surface roughness on nucleate and transition boiling [28]. 24 2.13 Boiling curves for acetone boiling on teflon-coated, mirror-finished and rough copper surface [29]. 24 3.1 Methodology used for this study. 28 4.1 Photos of the as-cast aluminum alloy surface. 29 4.2 Photograph of the experimental set-up at UBC. 30 4.3 Method used to attach the thermocouples to the sample: a) original method and b) the modified method. 32 4.4 Cut test sample indicating thermocouple positions. 32 4.5 The array of thermocouples used in each sample. 33 viii 4.6 The contact of cooling water with the sample surface during the experiments. 34 4.7 Typical results of the surface morphology as measured by NRC (AA5182). 36 5.1 Thermocouple positions within the sample. 38 5.2 Transient heat flow in: a) a semi-infinite solid and b) a quarter-infinite solid. 41 5.3 The domain used in the FEM program to calculate the temperature distribution. 42 5.4 Comparison of results from the FEM model with the analytical solution at point (0.01, 0.01). 43 5.5 Effect of the number of smoothing iterations on the accuracy of the cooling curves. 44 5.6 Effect of the magnitude of the time step (t) on the stability of the solution (temperature data is smoothed). 45 5.7 Effect of noise in the temperature data on the minimum time step required for convergence (temperature noise is about 1°C). 46 5.8 The multiple future time step method. 47 5.9 Effect of magnitude of the future time steps r on the solution for a) solution convergence and b) solution accuracy. 48 5.10 The comparison of calculated heat flux with assumed one for a) constant heat flux, and b) varying heat flux as a function of surface temperature. 51 5.11 Comparison of quenching curves from experiment and temperature profiles from the calculation of IHCP model. 52 6.1 Evolution of water wetting the sample surface during the quenching process. 53 6.2 Typical thermal profile in the sample during spray quenching. 54 6.3 Effect of surface morphology on the time required to wet the entire sample surface. 55 6.4 Quenching curve showing the different heat transfer regimes. 56 ix 6.5 Typical quenching curves during spray water cooling indicating various heat transfer regimes for: a) nucleate boiling, b) transition boiling and c) film boiling. 57 6.6 Measured cooling curves and the associated calculated boiling curves for both the impingement zone and the free falling zone (AA5182 machined, water flow rate = 6.0 gl/min). 58 6.7 Effect of alloy composition on the calculated boiling curves (machined surface for: a) the impingement zone and b) the free 60 falling zone. 6.8 Thermal conductivity of AA1050, AA5182 and AA3004 alloys as a function of temperature. 61 6.9 Effect of alloy composition on the peak heat flux of boiling curve. 62 6.10 Surface roughness as measured by the mean deviation of the surface for each of the samples tested. 63 6.11 Effect of sample morphology on the calculated boiling curves (AA5182) for: a) the impingement zone and b) the free falling zone. 64 6.12 Effect of surface morphology on boiling curves of sample (AA1050) for a) the impingement zone and b) the free falling zone. 65 6.13 Effect of surface morphology on the calculated peak heat flux in the boiling curves for a) AA5182 and b)AAl050. 66 6.14 Effect of water flow rate on the calculated boiling curves (as-cast AA5182, starting sample temperature=450°C) for: a) the impingement point and b) the free falling zone. 68 6.15 Effect of water flow rate on the calculated boiling curves (machined surface Aa5182, starting sample temperature=450°C) for: a) the impingement point and b) free falling zone. 69 6.16 Effect of initial sample temperature (as-cast AA5182, water flow rate=6.0gl/min) on the calculated boiling curves for a) the impingement point and b) the free falling zone. 70 6.17 Effect of initial sample temperature (as-cast AA1050as-cast sample, water flow rate=6.0gl/min) on the calculated boiling curves for a) the x impingement point and b) the free falling zone. 72 6.18 The cooling curves and heat flux of moving test for a) the cooling curve from a moving sample test and b) the heat flux of moving 74 sample test. 6.19 Comparison of the calculated boiling curves for moving and static tests with industrially measured data for AA5182 [35]. 75 B . l 2-D, 4- node elements and its iso-parametric element. 90 xi List of Tables 2.1 The peak heat flux measured by various researchers during DC 17 casting of aluminum. 5.1 Temperature dependent material properties. 50 A. 1 The measured 3D surface parameters by NRC. 87 A.2 The measured 3D surface parameters of AA5182 as-cast samples by 88 NRC. C.l The error in temperature reading at different temperature range (K type thermocouple). 104 xii List of Symbols Latin symbols Bi Biot number C p Specific heat J/Kg°C d The dimension of water jet cm H The sample height m h Heat transfer coefficient W/m2 oC [J] Jacobian matrix. |jj the determinant of the Jacobin, det.J K Thermal conductivity W/m°C k, The material thermal conductivity in x direction W/m°C ky The material thermal conductivity in y direction W/m°C L Ingot dimension (pi 1) • m L The sample thickness (p38) m M The peak number in x direction N The peak number in y direction Nj The interpolating polynomial or the shape function nx, iiy, The directional cosines to the surface Pe Peclet number Q Water flow rate m3/s q Heat flux W/m2 q C H F Critical Heat Flux W/m2 R Diffusion path length (pi 1) m R Relaxation factor (p40) R Residual error (p92) r Magnitude of multiple future time step Sa The arithmetic mean deviation urn Sq The RMS deviation urn xiii Sz The ten point height Lim Ssk The skewness of topography height distribution Sku The kurtosis of topography height distribution Sds The density of summits 1/mm2 Sal The fastest decay autocorrelation length mm Str The texture aspect ratio S^ The RMS slope of the surface SJC The arithmetic mean summit curvature 1/pm T w Water temperature °C T s Surface temperature °C T^, The saturation temperature °C T w Cooling water temperature °C Tj Measured temperature at the i thermocouple position and j time °C step T c i j Calculated temperature at the i thermocouple position and j °C time step T 0 Initial temperature °C Tj Node temperature °C AT The temperature difference °C At Time incremental s Y The approximate solution for temperature °C UCD Upstream conduction distance cm V Casting speed m/s Wj The weighted functions — Greek symbols a Thermal diffusivity k/pCp m2/s 8 The film thickness mm xiv " ( X i . V j ) The deviation of peak at point (i, j) p Density kg/m3 x Time s Mathematical symbols d Differential operator V Vector operator Del {} A column vector. [] A row vector xv Acknowledgment I would like to express my sincere thanks to my supervisor Dr. Mary. A. Wells for giving me the opportunity to work with her at UBC. Her support, trust and professionalism throughout the project have made it both a memorable and enjoyable learning experience. I would also like to thank Gary Lockhart for the work on the laboratory facilities, and help in obtaining data and materials when required. I would also like to acknowledge and thank the National Research Council (NRC) for doing the sample surface roughness tests for me. I would also like to thank all the people in the department of Metals and Materials Engineering at UBC who have helped me during this project. Specifically, thanks are given to the machine shop technicians (Ross McLeod and Carl Ng) who fabricated my test specimens, and all the graduate students with whom I have had such interesting conversations. xvi 1. Introduction Chapter 1 - Introduction 1.1 DC Casting process Since the commercial viability of the Direct Chill (DC) semi-continuous casting process was established in the 1930's, it has been used most extensively by the non-ferrous metal industry. DC casting is widely used in the aluminium industry to produce a wide range of semi-finished products, such as billets for extrusions or forgings and ingots for rolled products [1] (Figure 1.1). As the name implies the defining character of this casting method is the extraction of heat by the direct contact of water on the periphery of the solidifying ingot as it is drawn from the mould. Figure 1.1: Schematic of the DC casting process [1]. 1 1. Introduction At the start of the cast, a hollow stool cap is partially inserted into the mould and filled with molten aluminium. When the aluminium meniscus touches the mould, the stool cap is gradually lowered into the casting pit while water from jets in the mould rapidly cool the solidified shell of the .extracted ingot. Initially the liquid is cooled by the mould wall which extracts -10-20% of the heat from the ingot. The majority of the heat (-80-90%) is removed in the secondary cooling zone, where the ingot is exposed to the direct contact of water. The high heat extraction rates during secondary cooling induce thermal gradients and mechanical contraction within the thin newly solidified shell and result in deformation known as "butt curl" and a contraction (pull-in) of the rolling face. The most critical phase of the DC casting process is start-up during which time the casting parameters are gradually adjusted to reach a uniform steady state condition where they are kept constant until the end of the cast. As a result, the start-up practice and heat extraction from the solidified shell will influence the level of stress in the ingot and the resulting butt curl as well as hot tearing tendencies in some of the crack-sensitive alloys. Final quality in DC cast aluminium products depends primarily on the thermal stress distribution during casting, which influences the occurrence of cracks, ingot shape and dimensions, macrosegregation and ingot surface condition. The thermal profile during casting depends on many process parameters including: casting speed, ingot size, alloy composition and the intensity of water cooling as measured by a heat transfer coefficient (h) at the surface of the ingot. Depending on the aluminium alloy being cast a large variety of surface morphologies can form. In turn, this can influence" the intensity of the water cooling, as it will influence the behaviour of the water on the surface of the ingot. 2 1. Introduction Since aluminium has a relatively low melting point, heat extraction to the cooling water is strongly dependent on the temperature of the aluminium sheet ingot surface, and will vary dramatically during cooling as different heat transfer regimes are encountered (i.e. film boiling, transition /nucleate boiling and convection cooling). For the aluminium DC casting process, the heat transfer to the cooling water is particularly complicated as ingot surface temperatures at the initial point of water contact cause transition/film boiling to occur followed by nucleate boiling and convection cooling as the surface temperature of the ingot is cooled (Figure 1.2) Mould Liquid Figure 1.2: Schematic diagram indicating the variation in heat flux along the length of an aluminium ingot as it is being water cooled. As shown in Figure 1.2, the heat transfer during DC casting is extremely complicated and can be influenced by such factors as: alloy composition, water quality, water temperature, water nozzle size and position. 3 1 . Introduction This research project examined the influence of various factors (aluminium composition, water flow rate, sample temperature, and surface topography) on the heat transfer which occurs during the secondary water cooling regime of the DC casting process. 4 2. Literature review Chapter 2 - Literature Review In order to develop accurate thermal-strain models of an aluminium ingot as it solidifies, it is critical to have accurate boundary conditions. Among these boundary conditions, a quantitative assessment of the heat exchanged between the water film and the surface of the slab is essential, as ~80-90% of the total heat is removed by water spray cooling [2]. In turn, the water spray heat transfer is controlled by the boiling behaviour of the water on the hot ingot surface and involves complex physical phenomena such as nucleate and transition boiling. 2.1 Boiling heat transfer Boiling heat transfer occurs during DC casting of aluminium as the newly solidified shell which enters the spray cooling region is typically in the order of ~400-500°C during the start-up stage [3] and 200-300°C during steady state [2]. Boiling heat transfer is said to occur when evaporation occurs at a solid-liquid interface; (i.e., when the temperature of the surface (Ts) exceeds the saturation temperature (Tsat) at a given liquid pressure) [4]. The different heat transfer regions that take place during boiling heat transfer are typically represented using a boiling curve (Figure 2.1). As shown in Figure 2.1, the resulting heat transfer is strongly dependent on the surface temperature of the object which is being cooled and the curve can be broken down into distinctive regimes. • Free convection (AT < ATA) • Nucleate boiling (ATA < ATS < ATc) • Transition boiling (ATc <ATS <ATD) 5 2. Literature review • Film boiling (ATs>ATD). Forced Convection Nucleate Boiling Transition Boiling Film Boiling A x a Critical Heat Flux I— Leidenfrost Point Onset of Nucleate Boiling Wall Superheat log AT, Figure 2.1: Boiling curve describing different regions in boiling heat transfer [4]. In the forced convection regime there is no vapour in contact with the liquid phase to cause boiling at the heated surface. As the temperature is increased, bubble inception will eventually occur. Below point A (referred to as the onset of nucleate boiling, ONB), fluid motion is determined principally by convection effects. In this region, heat transfer coefficients can be estimated using standard convection heat transfer correlations. In the nucleate boiling region, two different flow regimes may be distinguished. In region A-B, isolated bubbles form at nucleation sites and separate from the surface. This separation induces considerable fluid mixing near the surface, substantially increasing the heat transfer coefficient and the heat flux. In this regime most of the heat exchange is through direct transfer from the surface to liquid in motion at the surface, and not through the vapour bubbles leaving 6 2. Literature review the surface. As AT is increased beyond ATB, more nucleation sites become active and increased bubble formation causes bubble interference and coalescence. In the region B-C bubbles can grow very quickly, as they begin to coalesce they start to escape as jets or columns, thus decreasing the rate of increase of the heat flux; Interference between the densely populated bubbles inhibits the motion of liquid near the surface. Point P (Figure 2.1) corresponds to an inflection point in the boiling curve at which the heat transfer coefficient (h) is a maximum. Point C corresponds to a maximum heat flux point, because in most industrial application it is important to know the maximum heat flux, which is called the Critical Heat Flux (CHF). In the transition boiling region, (ATc <ATS <ATD), unstable film boiling, or partial film boiling occurs. Bubble formation, in this regime, is so rapid that a vapour film or blanket begins to form on the surface. At any point on the surface, conditions may oscillate between film and nucleate boiling, but the fraction of the total surface covered by the film increases with increasing ATS. Because the thermal conductivity of the vapour is much less than that of the liquid, the heat transfer coefficient (and heat flux) decrease with increasing ATS. At point D of the boiling curve, referred to as the Leidenfrost point, the heat flux is a minimum, qo = qmin, and film boiling occurs as the surface is completely covered by a vapour blanket. Heat transfer from the surface to the liquid occurs by conduction through the vapour. As the surface temperature is increased, radiation through the vapour film becomes significant and the heat flux increases with increasing AT. 2.2 The influence of casting parameters on DC cast ingot quality The influence of casting parameters such as withdrawal rate, water flow rate and bottom block design on defect formation is complex and difficult to rationalize experimentally. Part of 7 2. Literature review the problem lies in the fact that boiling water heat transfer, occurring at the ingot surface, is highly dependent on the surface temperature which is varying with time at the point of water contact during the start-up phase of the casting process. In addition, the base of the ingot undergoes a permanent macroscopic deformation know as butt curl due to the large thermal stresses present. The deformation upwards of the base corners of the ingot tends to draw the sides inwards altering heat transfer in the mould leading to a further variation in ingot surface temperature and secondary cooling heat transfer. Moreover, the displacement of the rim of the ingot base from the bottom block can lead to water incursion along the base of the ingot resulting in further variability in heat transport. To understand the influence of water cooling, during DC casting, on final ingot quality, a link is needed between the process performance (i.e., ingot surface quality and recovery rate) and the temperature distribution in the solidifying ingot. The following parameters have been used as a measure of ingot quality and are all influenced by the temperature distribution in the ingot which in turn is controlled primarily by the water cooling of the ingot during solidification [5]: a) the molten metal liquid pool depth, b) upstream conduction distance (UCD) (the distance from the water spray impact point to the solidification front position above that point — Figure 2.2), c) surface temperature at the water impact point, d) mushy zone thickness, e) cooling rate at the centre and the surface of the ingot. These parameters can be linked directly to the quality of the final ingot; for example, pool depth has been used as a criterion for cracking [6] and UCD can be related to surface 8 2. Literature review microstructure formation [7,8]. Figure 2.2: Schematic showing the Upstream Conduction Distance (UCD) and pool depth on a conventional open top sheet ingot mould [5]. The UCD can also be used to determine the likelihood of a bleedout [5], If the UCD is very small (i.e., high casting speeds), the liquid front is getting very close to the bottom of the mould and a bleedout is likely. 2.2.1 Cast start sensitivity to h variation The heat flow in the casting process can be divided into two distinct phases: the start-up (transient phase) and steady state. During the start-up the surface temperature of the ingot is changing with time and the casting speed is extremely low. When the steady state phase is maintained the ingot surface temperature remains relatively constant at a given position and the casting speeds are much higher. The requirements of the process for water cooling intensity are quite different during start-up versus during steady state casting. During the transient start-up 2. Literature review phase, low heat extraction rates, achieved by promoting film boiling, are desired to reduce butt curl and cracking in sheet ingot casting [5]. In sheet ingot, cracks can form easily and in large quantity, due to butt deformation. This deformation has been found to be related to the severe direct quench when the butt of the ingot first emerges into the water spray [9] thereby inducing large thermal gradients in the base of the ingot. Butt curl can be decreased by reducing the heat transfer coefficient (h) between the newly solidified ingot shell and the water spray. In industry many methods have been developed and are currently used, during the start-up phase, to achieve this. For example, the Alcoa CO2 process injects CO2 into the cooling water [10] and the Alcan Pulsed water process uses a special rotary valve to turn the water on and off [11 ]. At the water impingement point, the transition from film to nucleate boiling should be controlled carefully as sudden changes in this region can cause transverse cracks in ingots to form [12]. For example, Grandfield et al.[5] found that during the transition from the start-up phase to the steady state phase if the Turbo air was suddenly turned down and the water turned up the film layer collapsed rapidly causing transverse cracks to form. If, however, the air and water mixture were gradually changed, then the transition front slowly moved across the ingot and these cracks were avoided. Another example of the importance of water cooling on ingot quality was shown in Jensen and Schneider's research [13]. They found that the maximum pool depth in the ingot depends on the heat transfer between the ingot and the starting block. In turn, this is influenced by the water cooling as it will influence the amount of butt curl which occurs. Loic Maenner, et al [14] compared the thermal field obtained after a cast length of 200mm by using different water cooling heat transfer laws and found the temperature distribution 10 2. Literature review in the ingot during the start-up phase was very sensitive to the water cooling conditions. 2.2.2 Steady state heat flow and sensitivity to h During steady state, the ingot temperature remains relatively constant at a given position as there is a balance between the convective heat input (a function of the energy content of the material and the casting rate, pVRCpL) and heat extraction by diffusion (determined by diffusion path length, R and thermal conductivity, k) and convection cooling (described by the heat transfer coefficient, h). Two non-dimensional numbers characterize this balance, the Peclet, (Pe) and the Biot (Bi) numbers: Pe = CpVR/k (2_j) Where the Peclet number equals the ratio of advection to diffusive heat flow and, B i = h R / k (2-2) Where the Biot number equals the ratio of resistance to heat flow from conduction to convective cooling. Typical values for aluminium DC casting are: 1.8<Pe<4.5 and 2<Bi<60 [5]. In DC casting, the heat flow on the surface of the casting just above the water spray is predominantly in one direction, i.e., toward the water spray impact point. Although some heat flows across the air gap between the casting and the mould, this is negligible compared to the water spray heat flow. Therefore, Grandfield [5] used the heat flow at the water cooled surface to examine the sensitivity of the UCD to casting speed, alloy thermal conductivity and water cooling coefficient h (Figure 2.3). He found that the UCD and the ingot surface temperature were not very sensitive to h values above about 20kW/m2K. The calculation also shows that bleedouts are likely for heat transfer coefficients of less than about 5 kW/m2K. 11 2. Literature review 70 S SO 60 c et f 9 I i o 10000 20000 30000 40000 Heat Transfer Coefficient (W/m2*K) 50000 Figure 2.3: The predicted UCD as a function of h and alloy conductivity [5]. h values for normal nucleate boiling conditions are around 20-50 kW/mK and around 10kW/m2K for film boiling [15]. So under normal nucleate boiling conditions the UCD is not affected by water cooling, but if film boiling occurs, the UCD is predicted to change by about 10 mm. This is highly likely to cause casting surface defects and possibly even bleedouts. The study by Flood et al [8] also shows that when h is very small (less than about 15 kW/m2K for a 200 mm billet), it will have some effect on pool depth. 2.3 Boiling heat transfer in direct chill casting 2.3.1 Water jet and falling film physical description During secondary cooling, the water jet emerges from holes or slots in the mould, water will hit the ingot and then form a falling film down its surface. The impact point exhibits different cooling behaviour to the falling film region and although the impact region is small (3-4mm) it has a large impact on heat flow because this is where the initial boiling regime is 12 2. Literature review determined [•« _Anglftf>fTftt Figure 2.4: Typical schematic of a water jet used during DC casting of aluminium [15]. Where the water jet hits the ingot, water flows both up and down the surface. At the impact point, velocities are high and steam bubbles are rapidly swept away making it hard for a stable film to form. As the water impacts the ingot surface, it climbs above the impact point such that there is a pool of low velocity water sitting above the impact point. The important features of the water cooling are: a) The angle of the jet relative to the ingot surface - if the angle is too perpendicular to the wall the water will bounce off the ingot surface without forming the falling film. b) The velocity of the water - the velocity increases with increasing water flow rate and decreasing outlet area. c) The diameter of the jet (d). 13 2. Literature review d) The height of the stagnant region above the impact point (climbing height). e) The film thickness (5). f) The average water velocity in the film. 2.3.2 Quantifying heat transfer during steady state operation Many industrial or lab measurements of steady state DC casting have been published of near surface temperatures, fluxes and heat transfer coefficients [16, 17, 18, 19, 20]. These enable us to understand the heat transfer conditions that occur during DC casting. For example, Weckman and Niessen's [16] modelling of heat flow during DC casting of AA6063 ingots was the first to include an analysis of water spray heat transfer coefficients. Taking industrial thermocouple data from two casts they did an inverse calculation and got a heat flux versus temperature curve from the model (Figure 2.5). Weckman and Niessen used the following equation to fit their data: AT » (TW A L L- T W A T E R ), K Figure 2.5: The heat flux of DC casting process from Weckman [16]. 14 2. Literature review h = 0L/3(352(7; +rj-167000) +20.8(7; -273)3 (2-3) The typical magnitude which has been measured for h is 4xl06 W/m2K under steady state conditions. In equation 2.3, the exponent of one third for the sensitivity to flow rate shows that the h values are not very sensitive to flow rate in the nucleate boiling regime. These equations used by Weckman and Niessen produced a good fit between predicted and experimentally measured ingot temperatures (Figure 2.5). However, these equations are appropriate only for the surface temperature below the critical temperature at which the maximum heat flux occurs. Another limitation of this correlation is that it does not take into account many other parameters which can influence the heat transfer (i.e., impingement point or free falling zone, water impinging velocity, water temperature, ingot surface condition et. al). Baken et al. [17,18], measured heat transfer coefficients during industrial casting of AA6063, by freezing in thermocouples and calculating the surface temperature and fluxes. The peak h values were 50 kW7m2K with a maximum flux of 5 MW/m2 for a sheet ingot and 20 kW/m2K and 3.5 MW/m2 for a 216 mm diameter extrusion billet. These numbers are of the same magnitude as predicted by Weckman and Niessen [16]. But the reasons behind the difference in heat flux between the sheet ingot and extrusion billet are still unknown. They found the critical temperature to be about 150°C and that the maximum heat flux they measured increased with increasing casting speed and increasing water flow rate. Tarapore [19] modelled casting of an AA2024 aluminium alloy 396 mm diameter billets, and found that to get the best fit with measured frozen in thermocouples, the peak h was 20 kW/m2K with values of 1.25-1.68 kW/m2K in the convection cooled region. Watanable and Hayashi [20] found a good fit between predicted and actual pool profiles using a critical temperature of 150°C and a peak value of h using 30 kW/m2K. 15 2. Literature review J. B. Wiskel and S. L. Cockcroft [21] used a finite element based inverse heat-transfer technique to calculate the heat fluxes in the DC aluminium casting process for an AA5182 alloy. They found that the heat flow conditions vary significantly during the start-up phase, with the bottom zone exhibiting a higher maximum flux than the top zone. They believe this is caused by the effect of increased surface roughness (lapped to liquated) dominating the effect of increasing the water flow. Another finding is that the top zone nucleate boiling line is shifted to lower temperatures relative to the curves based on data from the bottom and middle of the ingot They also attributed this to the effect of changing surface morphology. As a result, they developed two cooling curves to described the heat flux during start-up: one for conditions of low water flow and a lapped surface morphology and the other for high water flow and an liquated surface (Figure 2.6). 3.0661 • • i • i i . • • | , • • , • | . , . Sarin IfempeoomfC) Surface Temperature (*C) a: Lapped surface. b: Liquated surface Figure 2.6: The boiling curves used by J. B. Wiskel for a) lapped surfasceand b) liquated surface [21]. The compilation of some results found in the literature shows that the authors generally agree on the heat flux in the low surface temperature domain, but that the dispersion is great 16 2. Literature review above 200°C (Table 2.1). The domain of high surface temperature is of great interest for the start-up phase of DC casting., since at start-up the aluminum ingot is quite hot. Table 2.1: The peak heat flux measured by various researchers during DC casting of aluminium. Researcher Aluminium Alloy Peak heat flux (W/m2) Experimental Method D.C. Weckman [16] AA6063 (sheet ingot) l.OxlO6 Industrial trial J.A. Bakken [17] Ai-Mg-Si-Fe (sheet ingot) 3.5-5.0xl06 Industrial trial Y. Watanabe [20] AA1100 (Billet) 2.5-3.0x10b Industrial trial E.D. Tarapore [19] AA2024 (396 mm billet) l.OxlO6 Industrial trial J.B. Wiskel [21] AA5182 (sheet ingot) 2.5-3.0xl06 Industrial trial H. Kraushaar [22] A1AA1100 4.0-8.0xl06 Laboratory experiment 2.3.3 Effect of the cooling water velocity By freezing in thermocouples on a gas pressurised mould during casting, Grandfield and Baker found [15] that under normal water flow for this mould (0.002 m3/s) the surface temperature at the impact point was around 250~300°C and nucleate boiling occurred with a peak h of 40 kW/m2K and a critical temperature of around 150°C. Under typical casting conditions the cooling water impact velocity is ~2 m/s and reduces to ~1 m/s in the falling film. By doubling the slot width to 2mm wide and using the same flow rate (60 1/min), Grandfield [15] was able to reduce the impact velocity to lm/s. Under the low velocity condition, film boiling occurred giving higher surface temperatures (Figure 2.7). This 17 2. Literature review result is interesting because it shows the independent effect of the impact velocity. The flow rate was still the same and the falling film characteristic were the same, but because the impact velocity on the sample was lower, the heat transfer coefficient at the impact point was reduced and the surface temperature remained above the critical temperature. TEMPERATURE CC) 800 700 600 500 400 300 200 1C8oO) 0 200 400 600 800 DISTANCE BELOW THE MOULD (mm) Figure 2.7: The high near surface temperature on 155 mm 6063 billet caused by film boiling [15]. A 1996 paper by Matsuda et al [23] also reported the effect of slot width (1-5 mm) on the heat flux. The experiment involved heating an aluminium billet, instrumented with thermocouples, up to the desired temperature and lowering it into a water spray. The boiling curves clearly showed that the heat flux decreased with decreasing impact velocity. In addition, the maximum flux decreased with decreasing impact velocity. Since the velocity of the cooling water at the impingement point is higher than in the falling film, the transition to film boiling is more likely to occur in the falling film. Grandfield et al [15] observed this during casting, either going from nucleate to film or film to nucleate boiling at a point below the impacting jet. A clearly visible transition front moves up or down the ingot 18 2. Literature review in the falling film. 2.3.4 The effect of water temperature In 199S Langlais et al [24] put a hot block of aluminium in front of the water spray in an open top block mould to test the plant water at the point of use. They found that the quenchability of DC casting water is strongly affected by temperature, especially when the water temperature is high (over 30°C). However, when the water temperature is very low (below 20°C), the change in water temperature will have very little effect on the water quenchability (Figure 2.8). 600 r 1 Time (sec) 3 ° C 10 °C 20 °C 30 °C 40 °C 60 °C Figure 2.8: Effect of water temperature on the cooling curves [24]. Hamilton and Chen [25] have recently conducted quench studies using a 200 mm long 75 mm square centrally preheated 460°C aluminium block immersed into water. Results show that increasing water temperature promoted film boiling at lower surface temperatures. Similar to Langlais, Grandfield et al [IS] observed no discernible differences in the boiling heat transfer for water temperatures in the range 15-35°C . However, when they tested 19 2. Literature review 43°C water they found that the transition from nucleate boiling to film boiling occurred at a lower surface temperature and heat transfer was increased in the nucleate boiling regime (Figure 2.9). I 6.000.000 . 5.000.000 . . . . . . . 43dOQ c e 32degC 4.000.000 c 3.000.000 2.000.000 i / » \ 1.000.000 0 • \ 100 200 300 CoM f «c« Temp *C 400 Figure 2.9: Effect of water temperature on boiling heat transfer [IS]. 2.3.5 Effect of water flow rate Langlais et al [24], studied the effect of water flow rate by choosing three different flow rates (14, 18 and 22 1/min with corresponding velocities of 0.55, 0.70, -and 0.86 m/s). When the flow rate is low, increasing flow rate will enhance the heat transfer. When the flow rate is high, this effect will decrease (Figure 2.10). Using their test rig, Grandfield et al [15] found that the water flow rate had very little influence on the measured heat transfer coefficient in the nucleate boiling regime (Figure 2.11). However, the flow rate was seen to have a significant influence on where the transition to film boiling occurred; as the flow rate increased, the transition occurred at a higher surface temperature. 20 2. Literature review 6 0 0 5 0 0 U 4 0 0 f 3 0 0 8. E £ 2 0 0 1 0 0 -V — — " \ \ \ V \ —" v \ V \ 14 Umin \ » -I \ V \ - 18 Umin - - -22 Umin \ x . ^ ^ y ^ ^ — — — _ i 1 — • — « - — - — 1 0 1 5 TifiM(s«e) 2 0 2 5 Figure 2.10: Effect of water flow rate on the cooling curves [24]. 100 200 300 Surface T«mp«rtturt (oC) 400 Figure 2.11: Effect of flow rate on water cooling heat transfer [15]. 2.3.6 The effect of water quality Re-circulating cooling water systems are widely used in DC casting. Water treatment chemicals are added to the water system to maintain the cleanliness and reduce the corrosive 21 2. Literature review nature of the cooling water. As a result, water quality is an important factor affecting the boiling heat transfer during DC casting. In 1985 Ho Yu [26], used a hot aluminium block dropped into water to show that various additives affect the boiling heat transfer. He discovered that, dissolved air, surfactant, cationic poly-electrolyte and dissolved castor oil reduced the boiling heat transfer. The results indicate that boiling heat transfer rate can be increased by: a) de-aeration, and b) addition of inorganic cations or cationic poly-electrolytes. Whereas boiling heat transfer rate can be retarded by: a) presence of suspended solids, b) dissolving a non-condensable gas into cooling water, c) addition of surfactants, and d) addition of anionic polye-lectrolytes. In 1995 Langlais et al [24] using a hot aluminium block (slug) examined the effect of plant water quality by placing the slug in front of a water spray in an open top block mould. They found the following results: a) oil concentration reduces cooling rate markedly above 10 ppm, b) flocculent has little effect up to 100 ppm. c) increasing concentration of solids in the water reduced cooling rate. Hamilton and Chen [25] have recently conducted quench studies using a 200mm long 75mm square centrally preheated 460°C aluminium block immersed into water. Results 22 2. Literature review consistent with previous work were obtained. Grandfield [15] conducted tests on the effect of various combinations of additives on the boiling curves. The results showed the difference in critical heat flux (CJCHF) is large, but in the convection region and nucleate region, the difference in heat flux is very small. 2.3.7 The effect of surface properties Although some researchers have pointed out the importance of aluminium ingot surface morphology on the boiling curve, no literature could be found correlating the ingot surface properties to the heat transfer which occurs during DC casting of aluminium. However, some studies on the effect of surface roughness for some other materials during horizontal spray cooling have been done. Berenson [27] did a study examining the influence of two copper materials with different roughness on the measured heat flux and found the effect of increased surface roughness was to shift the nucleate boiling curve to the left. This means the rougher surface has a higher heat flux at a given wall superheat. It was noted that rougher surfaces also yielded slightly higher maximum heat fluxes. Bui and Dhir [28] obtained nucleate and transition boiling data for vertical copper surfaces of different finishes (Figure 2.12). The behaviour observed in nucleate boiling is similar to that reported by Berenson. However, during transient cooling the rough surface produced transition boiling heat fluxes lower than those on smooth surface. 23 2. Literature review O S T I A O Y S T A T S I T f t A N A I K N T C O O L I M a l 1 I I A O T • T A T C T * A M « I C H T C O O L I H 9 I SURFACES ARE CLEAN ANO NO OXIOE RUN AS RUN SO _ l I I I I too 200 A T ( K ) Figure 2.12: Effect of surface roughness on nucleate and transition boiling [28]. 50 00 WALL SUPERHEAT 6T„t.K Figure 2.13: Boiling curves for acetone boiling on teflon-coated, rnirror-finished and rough copper surface [29]. Ramilison & Lienhard' [29] performed experiments using acetone on a horizontal copper surface and clearly showed the influence of surface roughness on the heat transfer coefficient (Figure 2.13). The heat transfer coefficient in nucleate boiling is significantly increased with 24 2. Literature review increasing roughness. Furthermore, the critical heat flux and Leidenfrost point are higher and shifted to lower temperatures. Although there are some studies on the effect of surface roughness on the boiling heat transfer curve, most have been done on copper and to date, no one has studied a case similar to DC casting. Furthermore, the rough surfaces used in the experiments were usually only a few microns in difference, whereas the surface roughness of as-cast aluiminum alloys can vary by many millimetres. 2.4 Summary As shown, there has been a large amount of work done to measure the heat transfer coefficient during the DC casting of various aluminium alloys. Many of the studies have been done by freezing thermocouples into the molten aluminium during the industrial casting process and calculating the heat flux using an inverse heat transfer technique. In general, the measured peak heat flux ranged from l-8xl06 W/m2. In addition, there have been a number of laboratory studies done using a hot aluminium block and a water spray to assess the influence of various parameters (water flow rate, water quality, water temperature, angle of water jet, and impact velocity) on the boiling curve. Specifically, the following points can be made about secondary water spray cooling during DC casting of aluminium based on the literature review: • The water spray cooling during DC casting of aluminium belongs to the free falling subcooled boiling heat transfer. It is very complicated process, and although much research has been done to investigate the heat transfer, the influence of ingot surface parameters on the ability of the water to extract heat from the ingot remains unclear. • Water spray cooling plays an important role in DC casting. It can affect the process 25 2. Literature review operation and final ingot quality, especially during the start-up phase. There are a lot of factors which influence the water cooling heat transfer. Among these factors, surface roughness plays an important role, however to date, no one has done a comprehensive study quantifying the influence of ingot surface roughness on the heat extraction from the ingot to the water sprays during industrial DC casting. 26 3. Scope and objectives Chapter 3 - Scope and Objectives 3.1 Objectives The objectives for this research are: • Quantify the influence of sample temperature, water flow rate and alloy composition on the calculated boiling curves for aluminium during water spray quenching, • Quantify the influence of surface topography on the calculated boiling curves at both the impact point and in the streaming zone. Although much research has been done to quantify the effect of some DC casting process parameters on the heat transfer between the surface of the ingot and the cooling water, none of the studies to date has quantified the effect of the surface roughness on the boiling curve behaviour for different aluminium alloys. In addition, all of the laboratory experiments which have been done to date have typically used machined samples. 3.2 Methodology The methodology used to do this research involved both experimental measurements (using industrial as-cast aluminium samples and an experimental set-up designed and built at UBC), a 2-D EHCP (inverse heat conduction problem) model to calculate the heat flux on the sample surface as it is being cooled and measurements at NRC (National Research Council) to quantify the surface roughness for each sample using a laser profilometer (Figure 3.1). 27 3. Scope and objectives As-cast & machined samples (AA5182. AA3004. AA1050', Water Experimental set-up Measure Surface roughness (NRC) / Temperature I history data As-cast surface \ roughness data J Inverse heat conduction calculation c Boiling curves Results analysis CEffect of surface roughness on boiling curve Figure 3.1: Methodology used for this study. 28 4. Experimental Chapter 4 - Experimental 4.1 Start material The experimental work was carried out on three commercially significant aluminum alloys, namely: AA1050 (Al>99.5%), AA3004 (Al-l%Mn-l%Mg) and AA5182 (Al-4.5%Mg) which were supplied in the as-cast form by Alcan International Arvida Research & Development Laboratory and Reynold Metals. These alloys were chosen, as they represent a great difference in terms of their as-cast surface roughness, with the AA1050 exhibiting the smoothest surface and the AA5182 exhibiting the roughest (Figure 4.1). AA5182 AA3004 AA1050 Figure 4.1: Photos of the as cast aluminium alloy surface. 29 4. Experimental In addition, for some samples, the as-cast surface was removed by machining in order to compare the response, to the cooling water, of a machined surface and the as-cast surface for the same alloy. The sample size used for each of the experiments was: 5.08cmxl2.7cmx30.48cm (2'xx5'\12w). 4.2 DC casting water cooling simulator The experimental program was performed using a custom built system which included: a vertical furnace to heat the samples to the desired temperature, a pneumatically operated lowering platen to move the sample out of the furnace and position it in front of the water box and a water box, which was built out of Plexiglas and duplicated a typical section of an aluminium mould used for industrial DC casting. Also included in the set-up was a data acquisition system that was used to record the temperature change in the sample (Figure 4.2). 4. Experimental For each test, a sample, instrumented with a number of thermocouples, was put into the furnace and heated to the desired temperature. The sample was usually held for an hour at temperature to ensure uniform heating. The sample was then lowered in front of the water box (2-3 mm away from the water box) and the water was turned on to the desired flow rate and the sample was cooled. During cooling of the sample the data acquisition system recorded the sample temperature as a function of time at a frequency of 20 Hz. In order to get accurate temperature data, holes were drilled through each sample and a number of thermocouples were inserted into the sample within five millimetres of the surface contacted by the water. To ensure precise, thermal histories for each sample, it is important that the thermocouples make good contact with the sample and have fast response time. Initially, we attempted to use chromal and alumel wires (0.28mm) and have the aluminium sample act as a hot junction. To insure that the thermocouple wire made good contact with the sample an aluminium screw was used to fix the thermocouple inside the sample (Figure 4.3a). This method did not work for two reasons: (1) during heating, as the sample expanded, poor contact occurred between the thermocouple wires and the sample and (2) the aluminium screw could not be tightened at high temperature as it softened. After several tests, we developed a second method to fix the thermocouple (Figure 4.3b). This method used the chromel and alumel wires which were spot welded together with a ~lmm bead. A chromel wire was used to fix the thermocouple (1.8mm) to the back of the sample over the insulation cover and this wire exerted a constant force on the thermocouple to confirm that it had good contact with the sample. Even though, at high temperature, thermal expansion can reduce this force, we reinforced it by tightening the chromal wire again before cooling the sample. 31 4. Experimental a) Original method b) Modified method Figure 4.3: Method used to attach the thermocouples to the sample: a) original method and b) the modified method. After the tests were complete, the samples were cut through the centre so that the exact location of the thermocouples could be measured (Figure 4.4). Figure 4.4: Cut test sample indicating thermocouple positions. 32 4. Experimental 4.3 The number and position of thermocouple The temperature profile of the sample was obtained from the thermocouples in the sample. The array of thermocouples in the sample is shown in Figure 4.5. Thermocouples y 4 Sample surface Figure 4.5: The array of thermocouples used in each sample. 4.3.1 Thermocouple distribution in y direction When calculating the surface heat flux, the estimated heat fluxes are used as a boundary condition at the surface of the sample, and the temperature distribution of the next time step was obtained from an FEM model. After comparing the calculated temperatures at thermocouple positions with the measured temperatures, the corrected heat fluxes replaced the estimated heat fluxes, and a new temperature distribution of the sample was obtained by redoing the FEM calculation. This iterative procedure was repeated until the difference between the calculated temperatures and the measured temperatures met the accuracy requirement for the solution (generally 0.2°C). In order to calculate the surface heat flux by using FEM, the every surface element should correspond one thermocouple (Figure 4.5). If the change of surface heat fluxes in the y-direction 33 4. Experimental varies smoothly, then the element size in the y-direction can be large, and there will be a average heat flux ( q i , q 2 , q 3 , q 4 , q$ etc) corresponding to each thermocouple. Time Figure 4.6: The contact of cooling water with the sample surface during the experiments. In our tests, the water did not wet the whole sample surface immediately and it usually ran down the surface gradually (Figure 4.6). This means the heat flux on one surface element (qi) is very large, the heat flux on the neighbour element (qi+i) maybe very small. When using FEM to calculate the surface heat flux, larger element size can make the calculated heat flux deviate from the real one. So, in order to get the correct result, the elements in y direction should be refined. From this point, we should have as many as possible thermocouples in the sample. But there are some limits. One is the data acquisition system. Another limit is the sample. If two thermocouples are too close, the space between them will be very small, and the material used for fixing thermocouple or thermocouple itself will have relatively large effect on the temperature filed of the sample. 34 4. Experimental As a result, there is a balance between the maximum number of thermocouples along the length of the sample (so that the dramatic change in the heat flux (q) can be measured) and ensuring there is a enough space between the thermocouples so that it will not influence the thermal profile in the sample. A total of nine thermocouples were used for a sample, such that the vertical distance between the thermocouples was 2 cm. 4.3.2 The distance of thermocouples from sample surface The thermocouples were placed ~5mm from the surface of the sample. This distance was chosen so that the thermocouples were as close as possible to the sample surface where the water was being contacted but still far enough away from the surface so that the large surface variation of as-cast samples did not impact on the temperature measurements. 4.4 Quantifying as-cast surface roughness The surface roughness of the samples was quantified using a laser profilometer at NRC (Figure 4.7) Using the results from the profilometer a number of parameters can be used to describe the surface roughness. Appendix A shows the parameter measured and gives a definition of each of these parameters as well as a schematic of each of the surfaces, which were measured. 35 4. Experimental Figure 4.7. Typical results of the surface morphology as measured by NRC (AA5182). 36 5. Inverse Heat Transfer Analysis Chapter 5 - Inverse Heat Transfer Analysis In order to calculate the associated boiling curves for each test condition performed on the test rig, it was necessary to develop a two-dimensional FEM mathematical mode capable of solving the governing heat conduction equation associated with the inverse heat analysis. This method assumes an initial heat flux at the surface of the sample and then compares the predicted temperature distribution within the sample to that which was measured by the thermocouples. The heat fluxes are then adjusted and the analysis run again to determine the new temperature profile. This process is performed iteratively until the difference between the predicted and measured temperatures meet the accuracy requirement (~0.2°C). 5. / Formulation of the inverse problem Solution of the boiling curves for direct chill water cooling of a hot aluminum sample requires solving an inverse heat transfer problem as the surface heat flux and temperature histories of the solid are determined from transient temperature measurements at an interior location. The inverse heat conduction problem (IHCP) is much more difficult to solve than a direct heat conduction problem [30]. One of the reasons for this is because this type of problem is extremely sensitive to measurement errors and the use of small time steps frequently introduces instabilities in the solution of the DHCP unless restrictions are employed. 5.2 Development of 2-D FEM heat conduction model 5.2.1 Basic heat transfer problem 37 5. Inverse Heat Transfer Analysis To analyse the heat transfer which occurs in the heated as-cast aluminum sample which is surface-quenched using a water spray, a two-dimensional thermal heat transfer analysis was viewed sufficient as the majority of the heat was conducted through the material in the x-direction towards the water and along the length of the sample in the y-direction. Since the thermocouples were located in the middle of the sample, we did not consider heat flow across the width of the sample in the z-direction (Figure 5.1). Insulation Thermocouples As-cast surface Figure 5.1: Thermocouple positions witJiin the sample. The governing partial differential equation (PDE) for heat conduction in two dimensions is: dx{ x dx) by{ y dy) ^ p dt with the following boundary conditions: 38 5. Inverse Heat Transfer Analysis 1) At the back of the sample, since it is insulated: -k^\ = 0 (5-2) JT=0 dy dx x=L dx\ 2) At the top of the sample, where heat losses are small (h = ~20W/m2K) relative to heat loss due to the spray cooling (h = l-5xl04 W/m2K). = 0 (5-3) 3) At the surface of the sample which is exposed to the water cooling:: = h(Timface-Twttter) = q (5-4) Assuming that the sample starts at a uniform temperature To, the initial condition is: T(x,y)\^ = T0 ( 5 5 ) 5.2.2 Inverse Heat Transfer Analysis The heat flux at the surface of the sample was calculated using an inverse analysis technique. Using this technique, an initial heat flux is assumed at the surface of the sample and the finite element method is used to calculate the resulting temperature distribution in the sample. After the calculated temperature distribution is obtained, it is compared with the temperature distribution measured in the sample and the temperatures and hence heat fluxes are adjusted iteratively so that the difference between the measured and calculated temperatures at each thermocouple position is minimized. In the 2-D inverse analysis, this is achieved by simultaneously adjusting all of the surface heat fluxes. The difference between the measured and calculated temperatures at the current time step 39 5. Inverse Heat Transfer Analysis for each thermocouple position is minimised by modifying the surface heat flux by an amount, Aq. If the difference between the calculated and measured temperatures meet the accuracy requirements, then the calculated temperatures will be used as the initial temperature for the next time step and the corresponding surface heat fluxes will be used as the starting heat fluxes for the next time step. The Aq can be determined using the following relationship: *qtJ =-RC,pVj-Tj) ( 5 . 6 ) where Cp is the specific heat of sample, p is the density of the sample, is the measured temperature at the i thermocouple position and j time step, TV is the calculated temperature at the i thermocouple position and j time step, and R is a relaxation factor, which ranged from 0.001 to 0.1 in our analysis. If the difference between the measured and the calculated temperatures is positive, the surface heat flux needs to be decreased. Whereas if the difference between the measured and the calculated temperatures is negative, the surface heat flux needs to be increased. The relaxation factor, R, is needed to ensure a convergent solution and stems from the ill-conditioned and highly non-linear nature of the problem. If, for example, no relaxation is used (R=l), then the solution oscillates and eventually diverges, whereas if R is set too small, the solution will converge very slowly. 5.2.3 Comparison of FEM model predictions to an analytical solution Details of the finite element solution to the PDE are given in Appendix B. The transient two-dimensional inverse heat transfer model was verified using a number of techniques. Initially verification of the two-dimensional heat transfer FEM model was done by comparing the model 40 5. Inverse Heat Transfer Analysis solution to an analytical solution for a semi-infinite solid with a uniform initial temperature, Tj which is suddenly exposed to a constant surface heat flux qo. For constant thermal properties, the differential equation for the temperature distribution T(x,t) is: #LmL?L (5.7) dx2 a dt qo qo a) semi-infinite solid b) quarter-infinite solid Figure 5.2: Transient heat flow in: a) a semi-infinite solid and b) a quarter-infinite solid. The boundary and initial conditions are: T(x, 0) = T 0 (5.8) <7o = dT dx For t > 0 x=0 (5.9) The analytical solution for this case is [31]: T-T0 = 2q0 -latin; exp f-x^ l-erf 2 Var (5.10) 41 5. Inverse Heat Transfer Analysis Let S= T/Ti and rearranging the above equation, we get: T0 T0\ k exp K4a* J <7o£ k f \-erf (5.11) The approximate analytical solution for a quarter-infinite solid, can be derived by combining two semi-infinite solids into a two-dimensional problem [32]: S(x, y, t) = S(x, t) x S(y, t) (5.12) The temperature prediction using the analytical solution can now be compared to the temperature prediction using our two-dimensional FEM model run under the following conditions (Figure 5.3): 0.1m 0.1m Figure 5. 3: The domain used in the FEM program to calculate the temperature distribution. • small time variable - At (5s), • heat flux in Face 1 and Face 4, and Face 2 and Face 3 are insulated. 42 5. Inverse Heat Transfer Analysis Agreement between the FEM model and the analytical solution was excellent as shown in Figure 5.4 providing us with confidence that the model can be used to accurately calculate two-dimensional heat conduction. 550 -500 -u o 450 -4> 3 400 -h 2-i 350 -300 -250 -FEM solution at (0.0,0.0) FEM solution at (0.01,0.01) •••"FEM solution at (0.05,0.05) X analytical solution at (0.0,0.0) A analytical solution at (0.01,0.01) • analytical solution at (0.05,0.05) —r— 3.5 —r— 4.5 0.5 1.5 2.5 Time (s) Figure 5.4: Comparison of results from the FEM model with the analytical solution. 5.3 Mathematical Technique Used in Inverse Heat Conduction Problem(IHCP) 5.3.1 Data noise and smoothing The inverse heat conduction problem is very sensitive to measurement errors, but the temperature data from the experimental tests inevitably has some degree of noise. In order to minimize data noise, we must smooth the data. The temperature data from the experiment had some noise as indicated in Appendix C. Appendix C shows how an estimate of the noise was made based on the data acquisition system. The magnitude of this noise was ~±5°C. As a result a smoothing operation had to be performed on the data. 43 5. Inverse Heat Transfer Analysis The technique we used to smooth the data was the two-point averaging method. This method was used over a number of iterations to smooth the data to an acceptable level. Figure 5.5 indicates the effect of the smoothing iterations on the measured temperature/time curve. 296 original curve iteration number=2 iteration number=3 iteration number=5 iteration number=10 iteration number=50 9.2 9.4 9.6 9.8 Cooling time (s) 10 10.2 10.4 Figure 5.5: Effect of the number of smoothing iterations on the accuracy of the cooling curves. From Figure 5.5 it is obvious that smoothing the data through 5 iterations was reasonable as the temperature data noise was reduced to 1°C and the smoothed curve is representative of the original data. Hence all the experimental temperature data was smoothed through 5 iterations using the two-points average method. 44 5. Inverse Heat Transfer Analysis 5.3.2 Single future time step The earliest methods proposed for the solution of the LHCP used a single future time step. A single heat flux component was estimated at each time step, producing a sequential algorithm. In this method, the calculated temperatures are made equal to the measured values. This can be called "exact matching" to distinguish it from approximate matching obtained by using least squares. 2.0E+06 15 20 25 Cooling time (s) Figure 5.6: The effect of the magnitude of the time step (At) on the stability of the solution (temperature data is smoothed). An advantage of this method is its simplicity which permits easy understanding of the 45 5. Inverse Heat Transfer Analysis method and the development of relatively simple and computationally efficient algorithms. The major weakness is its extreme sensitivity to measurement errors particularly as time steps are reduced, which is the usual difficulty (if any) in finite methods (Finite Control Volume (FCV), and FEM) [30] (Figure 5.6). Figure 5.6 shows the effect of the time step magnitude on the stability of the solution. In order to eliminate the effect of temperature noise, we have smoothed the temperature profiles thereby reducing the amount of noise. From Figure 5.6 we can see that a time step of 1.2 seconds is required to ensure convergence of the solution. When there is a lot of noise in the temperature data the smallest time step required for convergence of the solution will be larger (Figure 5.7). 3.0E+06 -i ; 1 - - - - 1.0s w -l.OE+06 — ^ ; • -1.5E+06 J -i 1 1 1 1 1 1 1 0 5 10 15 20 25 30 35 40 45 Cooling time (s) Figure 5.7: The effect of noise in the temperature data on the minimum time step required for convergence (temperature noise is about 1°C). The single future time step method is extremely sensitive to time step and measurement 46 5. Inverse Heat Transfer Analysis errors because of the nature of inverse heat conduction problem. In our experiments, the sample surface temperature and heat fluxes change very quickly in a short time, and it is necessary to use an extremely small time step to get a detailed and accurate profile of the boiling curve. Therefore we must use the multiple future time step method to solve the IHCP. 5.3.3 Multiple future time step method In the multiple future step method, it is assumed initially, that the future heat fluxes are constant with time as shown in Figure 5.8. It is assumed, that the estimated heat flux components qi, q.2, — . <lm-i are known and the objective becomes to estimate qm. In order to add stability to the IHCP algorithms, the heat flux components qm, qm+i, qm+M are assumed equal, i.e., Qm = Qm+l ~ Qm+2 = = 4m+r-l Since r "future" heat flux components are made equal, the time step of multiple future step method becomes rxAt. After the calculation, the heat flux qm is obtained and added to known heat flux group and the next calculation will begin at qm+i-Heat Flux q Time t Figure 5.8: The multiple future time step method. qm qm+i qm+2 qm+T Q4 qm-i q i q2 47 5. Inverse Heat Transfer Analysis 4.0E+06 i -1.0E+06 * J Cooling time (s) a) Solution convergence. 2.5E+06 350 Surface temperature (°C) b) Solution accuracy. Figure 5.9: The effect of magnitude of the future time steps r on the solution for a) solution convergence and b) solution accuracy. 48 5. Inverse Heat Transfer Analysis In order to obtain a stable solution, we need to choose a suitable value for r. The magnitude of r depends on both the thermocouple position and the magnitude of the time step At. When the thermocouple is far from the surface and time step is small, the value of r will have to be large in order to get a stable solution. Otherwise, when the thermocouple is near surface and the time step is large, the value of r can be smaller. As shown in Figure 5.9 the value of r will also influence the magnitude of the predicted boiling curve. Therefore it is necessary to use as low a value of r as possible while still maintaining a stable solution. In our case, because the thermocouple position is about 5-7 mm from the surface of the sample, and we used a time step of 0.1s, we used a value or r = 5 for our calculations. The other advantage of the multiple future time steps method is that it reduces the sensitivity of the solution to the experimental noise of the data. When the data acquisition system is determined, the magnitude of noise of experimental data will be certain (Appendix C). ff the noise of temperature is 5°C, the temperature change rate is 100°C/s and the time step is 0.05s, the noise will be 100% of temperature change when r=l. This magnitude of noise will make the solution divergence. The noise will be reduced to 5% of temperature change if we choose r=10, in this case, the noise will have relative little effect on the solution. 5.4 Verification of Inverse Heat Conduction model 5.4.1 Temperature dependent material properties The thermal physical properties used in the heat conduction equation, specifically thermal conductivity (k), specific heat (Cp) and density (p) were obtained from the literature [33,34]. The values used in the model are given in Table 5.1. 49 5. Inverse Heat Transfer Analysis Table 5.1. Temperature dependent material properties. Alloy Conductivity (W/mK) Density (kg/m3) Specific Heat (J/kgK) AA1050 [33] 245-0.043xT 2647-0.194xT 892.35+0.46 lxT AA3004 [34] 115.86+0.1425xT 2647-0.194xT 897+0.452xT AA5182 [34] 118.26+0.1094xT 2647-0.194xT 897+0.452xT 5.4.2 Verification of thermal model formulation The transient two-dimensional inverse heat transfer model was verified by using a commercial FEM code, ABAQUS, in two different ways. Firstly, ABAQUS was run to simulate the thermal response in the aluminium block at a particular location in the sample, subject to a known surface heat flux. The ABAQUS predicted thermal history of the position was then input into the two-dimensional inverse heat transfer model so that a heat flux could be calculated (Figure 5.10). Secondly, the model was validated by inputting the estimated surface heat flux from a given test into ABAQUS and then comparing the ABAQUS temperature predictions at the thermocouple location in the block to those measured in the experiment. Both verification tests indicated that the model worked extremely well (Figure 5.11). 50 5. Inverse Heat Transfer Analysis i 1.5E+06 1.4E+06 1.3E+06 1.2E+06 1.1E+06 * 1.0E+06 5= •JS 9.0E+05 X 8.0E+05 7.0E+05 6.0E+05 5.0E+05 • Model predicted Input to ABAQUS 10 15 20 25 30 Time (s) 35 40 45 50 a) Constant heat flux. 5.0E+05 O.OE+00 100 200 300 400 Surface temperature (°C) 500 600 b) Varying heat flux as a function of surface temperature. Figure 5.10: The comparison of calculated heat flux with assumed one for a) constant heat flux, and b) varying heat flux as a function of surface temperature. 51 5. Inverse Heat Transfer Analysis Figure 5.11: Comparison of quenching curves measured by an experiment and predicted by ABAQUS. 52 6. Experimental results and discussion Chapter 6 - Experimental Results and Discussion 6.1 Typical experimental cooling curves 6.1.1 Water flow on the surface of the sample As described early, there are four modes of heat transfer between water and a hot surface: stable film boiling at high temperature, transition boiling, nucleate boiling and convective heat transfer. The situation becomes even more complicated, as these modes can exist simultaneously during spray cooling of a hot sample surface at different locations on the ingot surface. In our experimental set-up as the water touches on the hot sample surface, it will contact the sample surface at the impingement point first, and then will run down the sample surface. As a result, the sample can be divided into two different regimes: the impingement point and the free falling zone. As cooling progresses, film boiling will gradually be replaced by nucleate boiling at the front of the water stream as it moves down the surface of the sample and eventually wets the entire surface (Figure 6.1). Time Figure 6.1: Evolution of water wetting the sample surface during the quenching process. 53 6. Experimental results and discussion A typical measured temperature/time response in a sample is shown in Figure 6.2. As shown, the thermocouple response at the impingement point is different from that in the free falling zone. The spray water does not cool the surface of the sample at the location above the impingement zone, however, heat will conduct form this part of the sample towards the impingement zone, and as a result the temperature decrease in the impingement zone will be lower. This is shown in Figure 6.2, where the temperature of the impingement point becomes higher than the temperature of the other locations as cooling progresses. 500 i o-i , , , , , 1 0 20 40 60 80 100 120 Time (s) Figure 6.2: Typical thermal profile in the sample during spray quenching. The amount of time it took for the water to wet the entire sample surface varied dramatically from test to test and depended on such factors as: water flow rate, surface roughness, initial temperature and alloy type. Figure 6.3 shows the effect of sample surface roughness on the time it takes for the water to completely wet the sample surface. When the surface is rough, water 54 6. Experimental results and discussion will wet the surface relatively quickly. In contrast, when the surface is smooth, it will take a much longer time for the water to wet the entire sample surface 60 250 300 350 400 450 500 550 Surface initial temperature Figure 6.3: The effect of surface morphology on the time required to wet the entire sample surface (AA5182, water flow rate 6.0gl/min). 6.1.2 The cooling curves While the boiling curve has found wide use in most boiling studies, the temperature-time or cooling curve shown in Figure 6.4 is more frequently used in quenching studies. The curve displays the cooling history for a surface and exhibits the four heat transfer regimes identified on the boiling curves. The heat transfer mechanisms and corresponding cooling performance in each regime are characteristically different, manifested by slope changes in the temperature-time 55 6. Experimental results and discussion cooling curve. The patterns of liquid-surface contact and ensuing heat transfer are characteristically different in each of the four regimes. In our experiments, the quenching curves we measured were similar in shape to Figure 6.4, but there were some differences depending on the condition each test was run under (Figure 6.5). Vapor Blanket 2 I & E .2 Single-Phase Regime Isolated '— Bubbles Minimum Heat' Critical Haat Onset of Sngle-Phase Cooing Time Figure 6.4: Clenching curve showing the different heat transfer regimes. 56 6. Experimental results and discussion 350 " 250 p h 200 s £ 150 I 100 50 0 40 45 50 55 60 Time (s) 65 t ^ ^ ^ ^ Cooling curve — — Nucleate boiling ^^^^^^^^^•^••k^ ») i i ~ i i i 70 75 3.5E+06 3.0E+06 2.5E+06 2.0E+06 £ 1.5E+06 I 1.0E+06 5 5.0E+05 0.0E+00 80 a) Nucleate boiling (AA5182 machined sample, water flow rate 6.0gl/min, impingement zone). 600 500 t 5.0E+06 - 4.0E+06 • 3.0E+06 1 3 - 2.0E+06 t i a X - 1.0E+06 + 0.0E+O0 100 b) Transition boiling (AA5182 as-cast sample, water flow rate 6.0 gl/min, free falling zone). 500 „ 400 u o £ 300 2 & 200 E 100 Cooling curve Film boiling SrkmJ ^ ^ ^ ^ _ ~ • — i 1 1 1 1 1 1 4.0E+06 I 2.0E+06 g s + 1.0E+06 1 O.OE+00 50 55 60 65 70 75 80 Time (s) 85 90 95 100 c) Film boiling (AA5182 machined sample, water flow rate 4.0gl/min, impingement point). Figure 6.5: Typical quenching curves during spray water cooling indicating various heat transfer regimes for: a) nucleate boiling, b) transition boiling and c) film boiling. 57 6. Experimental results and discussion 6.1.3 Impingement zone and free falling zone The sample/water interaction at the impingement point is different from that in the free falling zone. Figure 6.6 shows the temperature profile and associated boiling curve for the sample 600 60 80 100 Time (s) 120 140 160 a) Measured cooling curves. 6.0E+06 5.0E+06 4.0E+06 I g 3.0E+06 £ 2.0E+06 1.0E+06 O.OE+00 100 -impingement zone -free falling zone 200 300 400 Surface temperature (°C) 500 600 b) Calculated boiling curves. Figure 6.6: Measured cooling curves and the associated calculated boiling curves for both the impingement zone and the free falling zone falling zone (AA5182 machined, water flow rate =6.0 gl/min). 58 6. Experimental results and discussion at both the impingement point and in the free falling zone indicating the differences between these two areas. The heat flux at the impingement zone was calculated from the temperature change at the impingement point, and the heat flux in the free falling zone usually refers to the heat flux in the middle position of the free falling zone. From Figure 6.6 we can see that in the impingement zone, the heat transfer is much higher starting in the nucleate boiling regime as compared to the free falling zone. In the free falling region, the heat transfer between the sample and the water starts in the transition boiling regime and then moves into the nucleate boiling regime. The peak heat flux in the two regions is also very different with the impingement zone exhibiting a much higher heat flux. Part of the reason for this is that in the impingement zone the water impacts the surface at such a high velocity that it penetrates and removes any steam or bubbles which may have formed. The impingement zone refers to the narrow band around the water impingement point (~ 2.0 cm), its size depends on the water flow rate as well as the impinging angle. As the water flow rate increases the impingement zone also increases. Whereas when the impingement angle is small, i.e. the water nozzle tends to parallel the sample surface, the impingement zone decreases. When the impinging angle is large, i.e., the water nozzle tends to perpendicular to the sample surface, the impingement zone also decreases because the water impinged on the surface can rebound back. 6.2 Effect of sample parameters on the calculated boiling curves 6.2.1 Alloy composition During the experiment we detected that the alloy type had a significant influence on the boiling heat transfer, so we specially studied the effect of alloy type on the boiling curves. Figure 59 6. Experimental results and discussion 6.7 shows the effect of the different aluminum alloys on the calculated boiling curves in both the impingement and free falling zones. From this figure we can see that the experiments done with AA1050 alloy had the highest calculated heat flux, followed by the AA3004 alloy and the AA5182 alloy. We can also see that in the impingement zone, the Leidenfrost temperature for the AA5182 alloy is lower than the other two alloys, but for free falling zone there is no large difference. 200 300 400 Surface temperature (°C) a) Impingement zone. 7.0E+06 6.0E+06 5.0E+06 200 300 Surface temperature (°C) b) Free falling zone. 600 600 Figure 6.7: The effect of alloy composition on the calculated boiling curves (machined surfaces) for: a) the impingement zone and b) the free falling zone. 60 6. Experimental results and discussion Another important point from Figure 6.7 is that the influence of the alloy composition on the boiling curve appears to disappear in the convective cooling regime. The reason the phenomenon occurs is because when the sample surface contacts the water, the surface temperature of the sample will decrease as the water extracts heat from the sample. As this process proceeds, a temperature gradient builds up in the sample. For the same temperature gradient, higher heat conductivity means more heat is supplied from inner part of sample to the surface. Since more heat can be supplied to the sample surface at a faster rate in the alloys with a higher conductivity, bubbles can nucleate and grow quickly. As described above, the heat transfer in the transition and nucleate regimes is dominated by bubble behaviour, so heat transfer capacity will be enhanced in the samples with a higher conductivity (Figure 6.8, 6.9). 300 *~ 50 0 -I , , , , 1 0 100 200 300 400 . 500 600 Temperature (°C) Figure 6.8: Thermal conductivity of the AA1050, AA5182 and AA3004 alloys as a function of temperature. 61 6. Experimental results and discussion 7.0E+06 AA1050 AA3004 Figure 6.9: Effect of alloy composition on the peak heat flux of the boiling curve. 6.2.2 Surface morphology of the sample As described earlier, depending on the alloy composition very different surface morphologies are produced during solidification. Using the NRC data, we can see that the average roughness of the AA5182 is significantly higher than the AA3004 and the AA1050 (Figure 6.10). To assess the influence of sample surface morphology on the boiling curve, we took one sample from each alloy group and machined the face so that it was very smooth, we then repeated the experiments to see the influence of surface morphology. We found that surface roughness has a significant impact on the calculated boiling curve behaviour, with rougher surfaces exhibiting overall higher heat fluxes. This effect is seen at both the impingement point and in the free falling zone but is more pronounced at the impingement point (Figure 6.11). 62 6. Experimental results and discussion 500 AA5182 as-cast AA3004 as-cast AA1005 as-cast machined sample Figure 6.10: Surface roughness as measured by the mean deviation of the surface for each of the samples tested. As shown in Figure 6.11 a, the two as-cast samples have almost the same heat flux, but the machined sample has a lower heat flux at temperatures in the nucleate/transition boiling regime. More importantly, film boiling occurs on the machined sample to a lower surface temperature. The effect of surface roughness on the boiling curve is most pronounced during the transition/nucleate boiling regime as this is where the ability to nucleate and grow bubbles has a significant impact on the heat transfer which occurs at the interface. During convective cooling both the machined and as-cast surfaces exhibit similar heat fluxes. Similar to the impingement point, in the free falling zone, we measured a lower heat flux at temperatures in the nucleate/transition boiling regimes in the sample with the machined surface as compared to the as-cast sample. The AA1050 has a much smoother as-cast surface than the AA5182 hence the difference between the machined and as-cast surfaces is not as great in both the impingement and free 63 6. Experimental results and discussion falling zones (Figure 6.12). 6.0E+06 5.0E+06 4.0E+06 3 3.0E+06 1 2.0E+06 1.0E+06 O.OE+00 100 a) Impingement zone 6.0E+06 5.0E+06 O.OE+00 -machined sample -as-cast sample 1 -as-cast sample 2 200 300 400 Surface temperature (°C) 500 600 -machined sample - as-cast sample 1 -as-cast sample 2 200 300 400 Surface temperature (°C) b) Free falling zone. Figure 6.11: Effect of sample surface morphology on the calculated boiling curves (AA5182) for: a) the impingement zone and b) the free falling zone. 64 6. Experimental results and discussion 7.0E+06 6.0E+06 5.0E+06 §. 4.0E+06 x 3 2 3.0E+06 S 1 2.0E+06 1.0E+06 O.OE+00 - machined sample -as-cast sample 100 200 300 400 Surface temperature (°C) 500 600 a) Impingement zone. 7.0E+06 6.0E+06 5.0E+06 I, 4.0E+06 — m a c h i n e d sample H as-cast sample 200 300 400 Surface temperature (°C) 500 600 b) Free falling zone. Figure 6.12: Effect of surface morphology on boiling curves of sample (AA1050) for a) the impingement zone and b) the free falling zone. 65 6. Experimental results and discussion O.OE+00 a) AA5182 alloy. 7.0E+06 6.0E+06 ,«-5.0E+06 E * 4.0E+06 | 3.0E+06 a- 2.0E+06 1.0E+06 O.OE+00 as-cast sample machined sample llmpingement zone I Free falling zone As-cast sample machined sample b) AA1050 alloy. Figure 6.13: The effect of surface morphology on the calculated peak heat flux in the boiling curves for a) AA5182 and b) AA1050. 66 6. Experimental results and discussion 6.3 Effect of process parameters on the calculated boiling curves 6.3.1 Water flow rate Many researcher have demonstrated that water flow rate can affect the heat transfer of the boiling process. During our study we examined the effect of water flow rate on the calculated boiling curve. Water flow rate (4-7.4gl/min) only slightly influenced the boiling behaviour of the as-cast samples in both the impingement and free falling zones (Figure 6.14), with higher water flow rates leading to higher measured heat fluxes. For the machined sample, the effect of water flow rate on the boiling curves is much more obvious (Figure 6.15). In the impingement zone, the heat flux increases as the water flow rate increases, especially in the transition boiling regime. When the water flow rate is low (4.0, 5.0 gallon/min), film boiling can occur as the water flow rate at the impingement point is low, and the water cannot penetrate the steam layer as effectively as in the high flow rate situation. In the free falling zone, the calculated boiling curve is less sensitive to the water flow rate however, film boiling can occur at water flow rates below 6 gallon/min. (Figure 6.15). 67 6. Experimental results and discussion 200 250 300 Surface temperature (°C) 350 400 450 a) Impingement zone. 500 7.4 g/min 6.8 g/min 6.0 g/min 5.0 g/min 4.0 g/min 200 250 300 Surface temperature (°C) 350 400 450 500 b) Free falling zone. Figure 6.14: Effect of water flow rate on the calculated boiling curves (as-cast AA5182, starting sample temperature = 450°C) for: a) the impingement point and b) the free falling zone. 68 6. Experimental results and discussion 4.0E+06 500 Surface temperature (°C) a) Impingement zone. 4.QE+06 Surface temperature (°C) b) Free falling zone. Figure 6.15: Effect of water flow rate on the calculated boiling curves (machined surface AA5182, starting sample temperature = 450°C) for: a) the impingement point and b) the free falling zone. 69 6. Experimental results and discussion 6.0E+06 600 Surface temperature (°C) a) Impingement zone. 6.0E+06 -i 5.0E+06 -Surface temperature (°C) b) Free falling zone. Figure 6.16: The effect of initial sample temperature (as-cast AA5182, water flow rate = 6gVmin) on the calculated boiling curves for a) the impingement point and b) the free falling zone. 70 6. Experimental results and discussion 6.3.2 Starting temperature The starting temperature of the sample had a significant impact of the calculated boiling curve in both the impingement and free falling zones (Figure 6.16). As the starting temperature is increased, the heat flux increases in the transition and nucleate boiling regime, but remains virtually the same in the convective cooling regime. This phenomenon can be explained from boiling heat transfer mechanisms. The heat transfer in both nucleate and transition regions are dominated by bubble behaviour. The bubble behaviour (nucleate, growth and detachment of the bubbles) is dominated by: 1) the surface temperature of the sample and 2) the heat that the surface can supply to the cooling liquid. The heat the surface supplies to the water is determined by the temperature gradient in the sample near the surface. Under transient conditions as in the DC casting process and our experiments, where the water heats up and produces steam as the sample cools down, bubble formation at the interface is extremely dependent on the starting temperature of the sample. As the initial sample temperature increases, the ability to form bubbles increases and hence as the sample cools down the heat flux to the water is higher than if the'sample had started at a lower initial temperature. For steady state processes, such as pool boiling, the situation is different because the surface temperature is fixed, and the bubble behaviour is only related to surface temperature; one surface temperature has one unique heat flux. When the initial sample temperature is high enough, so that heat transfer occurs initially in the film boiling regime, the boiling curves in the nucleate and transition regions are identical. This can be see from Figure 6.17 in the free falling regime when the starting sample temperature was greater than 400°C. In the impingement zone, because film boiling does not occur, the calculated boiling curves are different for different initial sample temperature. 71 6. Experimental results and discussion 7.0E+06 Surface temperature (°C) a) Impingement zone. 7.0E+06 0 100 200 300 400 500 600 Surface temperature (°C) b) Free falling zone. Figure 6.17: The effect of initial sample temperature (AA1050 as-cast sample, water flow rate = 6gl/min) on the calculated boiling curves for a) impingement zone and b) free falling zone. 72 6. Experimental results and discussion 6.4 Comparison to industrial DC casting Using the test-rig built at UBC, all of the tests to calculate the boiling curve were done with a stationary sample with the water spray contacting the top of the sample at an impingement point and then falling down the face of the sample. In reality during a DC casting process the newly solidified ingot moves into the water spray and then down from it. In order to assess the difference of having a stationary sample versus one which moved into the water spray, a few tests were done using a moving sample. During these tests, the sample was moved down very slowly past the water spray box (-8.5 cm/min). As shown in Figure 6.18, as the water hits the sample the temperature at the impingement point drops very quickly and the surface temperature of the sample above the impingement point starts to decrease below the starting temperature. In addition, as there is a climbing height associated with the water spray, the sample will initially get cooled in this area before it moves into the spray water impingement zone. Comparing the boiling curves for both the static and moving tests as well as data collected in industry by Wiskel and Cockcroft [35] for an AA5182 alloy we can see that the industrial measured data lies between the static and moving tests we performed using our test-rig, indicating that our calculated boiling curves are reasonable (Figure 6.19). 73 6. Experimental results and discussion 200 a: The cooling curve from a moving sample test. 3.0E+06 2.5E+06 ~ 2.0E+06 E i x 1.5E+06 1.0E+06 5.0E+O5 O.OE+00 • < — Impingement point Temperature profile \ V .Heat flux Initial water contact >-50 T ime(s ) 100 b: The heat flux of moving sample test. 600 500 400 ? 300 % » a 200 J 100 150 Figure 6.18: The cooling curves and heat flux of moving test for a) the cooling curve from a moving sample test and b) the heat flux of moving sample test. 74 6. Experimental results and discussion 5.0E-H)6 4.5E+06 200 300 400 Surface temperature (°C) 500 600 Figure 6.19: Comparison of the calculated boiling curves for moving and static tests with industrially measured data for AA5182 [35]. 75 7. Summary and Conclusions Chapter7 - Summary and Conclusions During Direct Chill (DC) casting of aluminum alloys, the majority of the heat (-80%) is extracted in the secondary cooling zone where water contacts the periphery of the solidifying ingot as it is drawn from the mould. The high heat extraction rates during secondary cooling induce thermal gradients and mechanical contraction within the thin newly solidified shell and result in deformation known as "butt curl" and a contraction (pull-in) of the rolling face. For the aluminium DC casting process, the heat transfer to the cooling water is particularly complicated as ingot surface temperatures at the initial point of water contact cause transition/film boiling to occur followed by nucleate boiling and convection cooling as the surface temperature of the ingot is cooled. To model the DC casting process correctly, it is necessary to develop accurate boiling curve data (i.e. heat flux vs surface temperature) for as-cast aluminum as it is being cooled. This study investigated the influence of the ingot surface topography, sample starting temperature and water flow rate on the boiling curve for three commercially significant aluminum alloys namely: AA1050, AA3004 and AA5182. The project involved both experimental measurements (using industrial as-cast aluminum samples and an experimental set-up designed and built at UBC), a 2-D FEM inverse heat conduction model to calculate the heat flux on the sample surface as it is was being cooled and measurements at NRC (National Research Council) to quantify the surface roughness for each sample using a laser profilometer. The experimental test facility was designed and built at UBC and included: a vertical furnace to heat the samples to the desired temperature, a pneumatically operated lowering platen 76 7. Summary and Conclusions to move the sample out of the furnace and position it in front of the water box, and a water box, which was built out of Plexiglas and duplicated a typical section of an aluminium mould used for industrial DC casting. For each test, a sample, instrumented with a number of thermocouples, was put into the furnace and heated to the desired temperature. The sample was then lowered in front of the water box (2-3 mm away from the water box), the water was turned on to the desired flow rate and the sample was cooled. During cooling of the sample the data acquisition system recorded the sample temperature as a function of time at a frequency of 20 Hz. The results of the inverse heat conduction model indicate that heat flow from the as-cast ingot are significantly influenced by the alloy composition, ingot surface morphology, water flow rate and starting sample temperature. Specifically, the following results were found: • The boiling behaviour at the impingement zone was significantly different than in the free falling zone, with the impingement zone exhibiting a higher heat flux than the free falling zone. In addition, it was more difficult for film boiling to occur in the impingement zone as the water would penetrate through the steam layer. • The sample initial temperature has a significant effect on the shape of boiling curves. The heat flux increases as the sample temperature increased especially when heat transfer started in the transition or nucleate boiling regimes. This has to do with the ability to nucleate and grow bubbles at the interface. • The sample surface morphology has a significant effect on the boiling curves. The heat flux appeared to increase with an increase in the surface roughness of the sample. Smoother surfaces also tended to promote the occurrence of film boiling at lower surface temperatures. This has a strong implication for modelling of the DC casting process as different families of 77 7. Summary and Conclusions aluminum (i.e. AAlxxx, AA3xxx and AA5xxx) alloys exhibit extremely different surface morphologies. • Water flow rate has an impact on the calculated boiling curve, especially when the sample surface is smooth. As the water flow rate decreased below a critical flow rate, film boiling is more likely to occur and the overall heat flux will decrease for a given surface temperature. • Due to the transient nature of the cooling process, a unique boiling curve for a given sample surface temperature does not exist, when cooling starts in the transition/nucleate boiling regime. This implies that when modelling the transient start-up phase of the DC casting process we cannot use a unique boiling curve in which the calculated heat flux is only a function of the ingot surface temperature. • Comparison of the calculated boiling curve using the test rig and IHCP model developed at UBC was similar to data measured in industry for a similar alloy (AA5182). 7.1 Recommendations for future work The results from this research indicate that the boiling heat transfer behaviour of the aluminum sample with the spray cooling water is extremely complicated and that a number of factors can influence this behaviour. The study we conducted showed that a number of factors can influence the heat transfer behaviour of water spray cooling on a hot aluminum surface. Suggestions for future work include: • Examining the influence of mould lubricant (a castor oil) on the surface of the sample on the boiling behaviour. 78 7. Summary and Conclusions • Determining the influence of the water spray angle on the heat transfer behaviour as the impingement point was seen to exhibit extremely different heat transfer behaviour than the free falling region and this will be dictated to some extent by the angle of the water spray. • Investigating the effect water velocity at the impingement point on the calculated boiling curve. Other studies have shown that at a given water flow rate, the size of the outlet area in the mould will influence the velocity of the water as it hits the ingot and hence the boiling behaviour. • Modifying the test rig so that the aluminum samples can be lowered into the water spray to better reflect what occurs during DC casting. 79 References References 1. W. Schneider & W. Reif, "Present situation of continuous casting for aluminum wrought alloys", Advance in continuous casting: research and technology, Woodhead Publishing Ltd, 1992,pl73-190. 2. D. C. Weckman and P. Niessen, "A numerical simulation of the DC casting process including nucleate boiling heat transfer", Met. Trans. B, 13B, 1982, p593-602. 3 E.K. Jensen, et al., "Development of a new starting block shape for the DC casting of aluminum sheet ingots. Part II: modeling results", Light metals 1995, p969-978. 4 L. C. Burmeister, "Convector heat transfer (second edition)" John Wiley & Sons, Inc. 1993. 5 J. F. Grandfield, et al., "Water cooling in direct chill casting: Part 2, effect on billet heat flow and solidification", Light Metals 1997, pi081-1090. 6 W. Schneider, & E. K. Jensen. "Investigations about starting cracks in DC casting of 6063 type billets. Part 1: experimental results", Light metals 1990, p931-936. 7 C. Devadas, & J. Grandfield. "Experiences with modelling DC casting of aluminium", Light Metals 1991, p883. 8 S. Flood, et al, "Modelling of casting, welding and advanced solidification process VIII", M. Cross and J. Campbell, eds., TMS, Warrendale, P. A. 1995, p801. 9 W. Deoste, W. Schneider, "Laboratory investigations about the influence of stating conditions on butt curl and swell of DC cast sheet ingots", Light Metals 1991, p945-951. 10 H. Yu, "A process to reduce DC ingot butt curl and swell", J. Metals, Vol. 32, No 11, 1980, p23-27. 11 N. B. Bryson, "Reduction of ingot bottom "bowing and bumping" in large sheet ingot casting", Light Metals 1974, p587-590. 80 References 12 Y. Caron, A. Larouche, "Importance of understanding ingot butt cooling conditions at cast start-up: A case study", Light Metals 1996, p963-968. 13 E. K. Jensen, W. Schneider, "Investigations about starting cracks in DC casting of 6063 type billets. Part II: Modelling results", Light Metals 1990, p937. 14 L. Maenner, et al., "A comprehensive approach to water cooling in DC casting", Light Metals 1997, p701-707. 15 J. F. Grandfield, et al, "Water cooling in direct chill casting: Part 1, boiling theory and control", Light Metals 1997, p691 -699. 16 D. C. Weckman and P. Niessen, "A numerical simulation of the DC casting process including nucleate boiling heat transfer", Met. Trans. B, 13B, 1982, p593-602. 17 J. A. Baken, T. Bergstrom, "Heat transfer measurement during DC casting of aluminium. Part I: Measurement technique", Light Metals 1986, p883-889. 18 E. K. Jensen, et al, "Heat transfer measurements during DC casting of aluminium. Part II: Results and verification for extrusion ingots", Light metals 1986, p891-896. 19 E. D. Tarapore, "Thermal modelling of DC continuous billet casting", Light Metals 1989, p875-880. 20 Y. Watanabe, N. Hayashi, "3-D solidification analysis of the initial state of the DC casting process", Light Metals 1996, p979. 21 J.B. Wiskel and S.L. Cockcroft, "Heat-flow-based analysis of surface crack formation during the start-up of the direct chill casting process: part II. Experimental study of an AA5182 rolling ingot", Metallurgical and materials transactions B, Vol. 27B, Feb., 1996, pl29-137. 22 H. Kraushaar, et al., "Correlation of surface temperature and heat transfer by DC casting of aluminium ingots", Light metals 1995, ppl055-1059. 23 K. Matsuda, et al, "Prevention of cast cracks in Al-Zn-Mg-Cu alloy Casting", 6th international 81 References aluminium extrusion technology seminar. VI, Aluminium association & aluminium extruders council 1996, p525. 24 J. Lamglais, et al, "Measuring the heat extraction capacity of D.C. casting cooling water", Light Metals 1995, p979. 25 S. Hamilton, "Heat transfer and water quality in D.C. casting", 4th chemical engineering project, October 1995, Chemical & Materials Engineering, The University of Auckland. 26 Ho, Yu, "The effect of cooling water quality on aluminium ingot casting", Light Metals 1985, pl331-1347. 27 P. J. Berenso, "Experiments on pool boiling heat transfer", Int. J. Heat & Mass transfer, Vol. 5,1962, p985-999. 28 T. D. Bui & V. K. Dhir, "Transition boiling heat transfer on a vertical surface", ASME, J. Heat transfer, Vol. 107,1985, p756-763. 29 J. M. Ramilison, & J. H. Lienhard, "Transition boiling heat transfer and the film transition regime", J. of heat transfer, Vol. 109,1987, p746-752 30 James V. B. Et al., "Inverse heat conduction — Ill-posed problems", A Wiley-Interscience Publication, New York, 1963. 31 J. P. Holman, "Heat transfer-fifth edition", McGraw-Hill Book Company, 1981. 32 William S. Janna, "Engineering heat transfer", 1986, PWS Engineering Boston. P282. 33 J.M. Drezet and M. Rappaz, "Modelling of ingot distortions during direct chill casting of aluminium alloys", Metallurgical and Materials transactions A, Vol. 27A, Oct, 1996, p3214-3225. 34 D. C. Prasso et al., "Mathematical modelling of heat transport and solidification and comparison with measurements on a pilot caster at Reynolds Metals Company", Light metals 1994,p871-877. 82 References 35 J. Barry Wiskel, "Thermal analysis of the start up phase for D.C. casting of an AA5182 aluminium ingot", PhD thesis, UBC, 1995. 83 Appendix A Appendix A. Definition of Roughness Parameters (1) The arithmetic mean deviation (Sa) Sa is insensitive to the sampling interval, but is sensitive to the cut-off in a 2-D filter is adopted. The 2-D counterpart of Sa is a very general and commonly used parameter in practical applications. (2) The RMS deviation (Sq) I i (A-2) Due to the squaring operation, it is more sensitive to extreme data values than Sa. It has a very definite meaning in statistics, i.e. it is the sample standard deviation. Similar to Sa,- it is insensitive to the sampling interval, but it is sensitive to the size of the sampling area and the cut-off if a 2-D filter is adopted. (3) The ten point height (Sz) I v 1! I (A-3) ZKI+Zkl S Z = M — 5 The ten-point height requires both a summit list and a valley list. The summit list and valley list must be found using an Autocorrelation area. The points used are the highest five un-correlated summits, and the lowest five un-correlated valleys. (4) The skewness of topography height distribution 1 N M Ssk = — r Y Y n 3 ( x , , y , ) (A-4) 84 Appendix A V The skewness is used to describe the shape of the topography height distribution. A negative skew shows that the outliners tend towards valleys below the mean. A positive skewness shows that outliers tend towards peaks above the mean. If there is a symmetric height distribution about the mean, the skewness is zero. (5) The kurtosis of topography height distribution S k u = — L ^ y y o w , ) ( A " 5 ) MN(Sq) fa£f 1 1 The kurtosis is used to describe the peakedness or sharpness of the surface height distribution. If (Sku>3.0), then most of the data is near the mean plane, and the outliers look like sharp spikes or pits. If (Sku<3.0), then the peaks and valleys are more rounded, meaning that less of the data is near the mean plane. (6) The density of summits The density of summits is simply the number of summits divided by the area of the map, and is calculated by: „, # Summits (A-6) Sds = (M-l)(N-\)-Ax-Ay The summit list is found using the 8 point definition. Note that the Sds is significantly influenced by the sampling interval. (7) The fastest decay autocorrelation length This parameter describes the autocorrelation character of the AACF. It is defined as the horizontal distance of the AACF which has the fastest decay to 0.2. it can be calculated by: Sal = mm^r2x+T2y) R(rx,ry)<0.2 (A"7) (8) The texture aspect ratio 85 Appendix A The texture aspect ratio helps identify texture pattern, like long-crestness or uniform texture. It is calculated by: Q c j - f r _ distance- that - the - normalized - A A CF - has - the- fastest-decay-to-0.2-in-any-direction ^ ^  / A O \ distance- that - the - nirmalized - AACF - has - the - slowest - decay -to- 0.2 - in - any - direction V. / Larger values, like Str>0.5 indicates stronger uniform texture aspect in directions, whereas smaller values, <0.3, indicates stronger long-crestness. (9) The RMS slope of the surface The RMS slope of the surface can be calculated by: (A-9) 1 N M s = — — — — y y Ax J 1 Av Please note that this formula simply sums up the net slope at each point, and calculates the RMS values of that sum. The net slope at each point is magnitude of the slope in both X and Y directions. Consequently, the above formula simplifies to: I i F ~ * r i : (A-10) = , " X X [slopeX1 + slopeY2 This parameter is sensitive to the sampling interval. (10) The arithmetic mean summit curvature The arithmetic mean summit curvature cab be calculated by: 1 •y(l(xp+t,yq) + rj(.xp.i,yq)-2tj(xl,,yll) ij(xp, yqtl) + rj(xp ,>»,_,)- 2t](x„ .y^ (A-ll) This formula uses any summit located at xp and yq> and is sensitive to the sampling interval. 86 Appendix A Table A. 1: The measured 3D surface parameters by NRC. 3D surface parameters AA1005 AA5182 Machined AA3004 Machined AA3004 (area 1) (area 2) Mean Deviation of Surface, Sa (um) 13.67 39.03 4.90 4.57 120.57 RMS Deviation of Surface, Sq(um) 18.03 64.90 6.17 5.70 148.18 Ten Point Height of Surface, Sz (um) 115.11 16.08 44.27 30.77 849.30 Skewness, Ssk .884 -2.770 0.029 -1.065 -0.120 Kurtosis, Sku 4.935 11.230 3.117 3.567 3.4702 Density of Summits, Sds (l/mm2) ** 1.005 .674 2.958 6.527 0.5756 Fastest Decay AACF length, Sal (mm) 1.992 1.096 1.923 5.105 2.9888 Texture Aspect Ratio, Str 0.16 0..88 .703 .413 0.7766 RMS Slope of the Surface, Sdq** .031 .161 .0195 .00777 0.1591 Arithmetic Mean Summit Curvature, Ssc (1/um) ** .00018 .00024 .00021 .00011 0.0009 Fractal Dimension 2.281 2.205 2.360 2.368 2.1477 Surface Bearing Index, Sbi .650 1.239 .620 .886 0.6033 Surface Area Index: surface area/plane area 1.000 1.010 1.000 1.000 1.012 **: Parameter sensitive to sample intervals 87 Appendix A Table A.2: The measured 3D surface parameters of AA5182 as-cast samples by NRC. 3D surface parameters IXD38 IXD39 IXD40 IXD41 IXD42 LXD43 IXD46 Mean Deviation of Surface, Sa (um) 395.72 373.55 430.89 370.22 404.74 385.56 477.27 RMS Deviation of Surface, Sq (um) 450.56 427.15 512.81 438.90 454.37 430.45 570.67 Ten Point Height of Surface, Sz (um) 1699.92 1787.55 3123.97 2754.59 1801.23 1840.33 2538.95 Skewness, Ssk 0.5848 -0.2702 0.4671 0.3857 0.0981 0.0956 -0.149 Kurtosis, Sku 1.8796 1.8786 2.6182 2.3051 1.6663 1.6376 2.1457 Density of Summits, Sds (l/mm2) ** 1.2974 0.9804 0.6260 0.602 0.7398 0.8471 0.6666 Fastest Decay AACF length, Sal (mm) 2.1907 2.6818 2.1906 2.2922 1.9834 1.7941 2.8521 Texture Aspect Ratio, Str 0.7793 0.7274 0.7038 .7411 0.5838 0.6667 0.6022 RMS Slope of the Surface, Sdq ** 1.7647 1.5164 1.3427 1.4176 1.4087 1.6768 1.2415 Arithmetic Mean Summit Curvature, Ssc (1/um) ** 0.0132 0.0100 .0088 .0051 0.0118 0.0150 0.0173 Fractal Dimension 2.3271 2.2969 2.1851 2.2103 2.2609 2.3027 2.2112 Surface Bearing Index, Sbi 0/5794 0.7275 0.5375 0.5633 0.6494 0.6767 0.6647 Surface Area Index: Surface area/plane area 1.246 1.194 1.201 1.202 1.221 1.266 1.287 : Parameter sensitive to sample intervals 88 Appendix B Appendix B. 2-D transient FEM model of heat transfer 1. The governing equation The governing partial differential equation describing our 2-D heat transfer problem is The first term of LHS represents the heat flow of control volume in x direction and the second term represents the heat flow of control volume in y direction. The RHS term represents the heat accumulation in the control volume of samples. 2. The initial condition and boundary conditions 2.1. The initial condition d (, 8T) d , dT) „ dT — k— +— k— = pC— dx \ dx J dyydy) dt (B-l) (B-l) 2.2. The boundary conditions 0 (B-3) 0 (B-4) 0 (B-5) surface ^water ) 7^ (B-6) 3. The shape function 89 Appendix B The finite element solution to the P.D.E. is based on the method of weighted residual (Galerkin's method). This is a general method for deriving approximate solution to linear and non-linear P.D.E. The technique involves steps: A. Assume the general functional behaving of the temperature, i.e., we would define some approximate solution for the P.D.E. B. Substitute the approximate solution into the P.D.E. and requires the error or residual to be minimized. Assume a function that approximately solves the equation. T ~T = YjNi(x)Ti (B-7) 1=1 where T is the exact solution that we seek. T is the approximate solution that we have assumed. Tj is the node temperature. Nj is the interpolating polynomial or the shape function In general, the interpolating or shape functions are polynomials of degree dependent on the number of node per element. The shape functions would become very complex and cumbersome for higher order elements. u Figure B.l: 2-D, 4-node elements and its iso-parametric element. 90 Appendix B For our problem, we use 2-D, 4-node, linear temperature elements as showed in Figure 2. First the element shape function are defined in terms of the local u, v co-ordinator system, this greatly simplifies their definition, i.e. for above example: iVI.(«,v) = i ( l + « 0 )(l + v0) (B-8) 4 Where u0 = uui v0 = wi Substitute the above shape functions into equation (7), the expression for the approximate temperature becomes r(u,v) = XAr,(",v)7;e (B-9) This expression is just for an element which is in its local coordinate system. Now we need transfer it into global coordinate system. The coordinate transformation is done by simply interpolating using the same shape functions. x = fdNi(u,v)xi (B-10) 1=1 j> = ! > , ( « , v)y, (B-ll) 1=1 For cases where the interpolating polynomials applied to the coordinator transformation are the same as those adopted for the field variable, the element is referred to as iso-parametric element. 4. The F.E. expression for our 2-D transient temperature problem The deriving of our F.E. equation is based on the method of weighted residual. The method of weight residuals seeks to determine the m unknown Tj, in such a way that the error, R, 91 Appendix B over the entire domain is very small, this is completed by forming a weighted average of the errors and requirement that it vanish to 0. Substituting the approximation T for T into our P.D.E equation (1) yields , d2T , d2T _ dT k—r + k—r~pCn — 8x2 8y2 " dt dD = JJflw.rfD =0 i=l,m (B-l 2) where D refers the domain (2-D domain), m = the number of nodes, R is the residual or error introduced by our approximating function, and the W j are the weighted functions. Galerkin's method is a special case of the method of weighted residuals where W j = Nj. , d2T , d2T _ dT k —- + k — T - - pC„ dx2 dy2 H^p dt dD = o (B-13) This expression applies the entire domain D that we are undertaking to analysis. Because the proceeding equation holds for any points in the analysis domain, it must also hold for a group of points making up a substitute or element. We can shift our focus to the element level—i.e. 1 element. I K D' , d2r d2r _ a r * —— + k — — - pC. dx1 dy1 dt Uxdy = 0 (B-14) We can now integrate by parts Jw(V • v)dCi = jw(v • n)ds - ^vVudQ (B-15) Where V is the vector operator Del V T d ~ d ~ d : V = — 1 + — j + — k dx dy dz (B-16) By selecting 92 Appendix B v = N, .dT .dT u - k h k — dx dy (B-17) (B-18) Substitute into our integration by parts formula. JJ k ^ 7 ^ + k ^ r ^ r ~ IK* 7 * -Tr** D' dx dx dy dy (B-19) j { k ^ + k^i.)N:dS = o dx dy This is the F.E. expression for a general 2-D transient temperature problem. The surface integration allows us to introduce the natural boundary conditions. E.g. recall the boundary conditions have the form , dT .dT k — nr +k—«„ +o = 0 dx x dy y (B-20) on as-cast surface Also note that T'=$]§•'} (B-21) In vector notation where [ ] indicate row vector and {} indicate a column vector. Substitute the above expression into the F.E. equation, we have -II D' , dTe dN° , dTe dNf k— — + k-dx dx dy dy dxdy - llN°pCp —dxdy + jqNfdS = 0 (B-22) In the above expression, the first term is dN_ dy J (B-23) 93 Appendix B It refers to as the conductivity term (not application dependent). \\NiPCp{N]^-dxdy dt (B-24) This term refers to as the heat capacitance term (applicable in transient problems). \qNtdS (B-25) This term refers to as the specified heat flux boundary condition, representing an input or output of heat on boundary S, it is application dependent. These terms encompass the expression necessary to solve a range of general heat conduction problems. Let's take a closer look at each term individually, with a view to their evaluation numerically. 4.1. Conductivity term D' f r k J J 3v ( > Fir Ph, ( > dx dN: dx dN dy dy Vixdy i=l ,n . (B-26) After some manipulation, it turns out to be an expression of the form: [KcJ{Tf (B-27) Where [Kc] is an nxn square matrix, n is the number of nodes per element. {T}e is an nxl column vector. The conductivity or temperature influence matrix [Kc]. It is composed of the following terms: [KJ = l\[B]T[K][B]dxdy (B-28) D' Where [B] is matrix of differential operators, and [K] is matrix of conductivity coefficient. They are evaluated as follows. 94 Appendix B 2[B] = [B(x"y)] dN, 8N2 dNn dx dx dN: dN. i " " 2 dx dN. (B-29) 2[K] = Kx 0 0 K„ dy dy dy for K x =Ky =K isotopic material [Bf = dN, dN. dx dy dNn dNn (B-30) dx dy For more general element (iso-parametric element), we need to come up with expressions in terms of u and v, e.g. r(M,v) = X^("'v)r/e (B-31) dN- dN-Therefore, we need to express —'- and — - and dxdy in terms of u and v. By the chain dx dy rule dN, = dN, dx | dN, dy du dx du dy du dN, _ dN, dx [ dN, dy dv dx dv dy dv (B-32) (B-33) In matrix form 'dN,' dx dy l d N i ) 'dN,' . du dN, du du dx dy dx dN, dN, ' [ dv \ _8v dv. {dy\ [dy] (B-34) where [J] is referred to as the Jacobian matrix. 95 Appendix B We can evaluate the Jacobin matrix using our expression for the coordinate transformation. x = Y,Ni(u>v)xi n y = ^ 7V,.(M,v)y,. (B-35) (B-36) [J(u,v)] = 1=1 ±8Nt ±dN, du t? du V < W L ±dNj_ {hdvXi ^dvy' (B-37) The desired derivations are evaluated as follows 'dNt~ 'dN,' dx ' dNt du . dNt [dy \ { dv J (B-38) To complete the evaluation, we also need do something about dxdy. This is obtained from the following expression. dxdy = |J|dudv (B-39) Where [J] is defined as the determinant of the Jacobin, det.J Det J = |J| = J(1, l)x J(2,2)-J(l,2)xJ(2,1) (B-40) 7(2,2) 7(1,2)" det 7 7(2,1) det 7 7(1,1) det 7 det 7 (B-41) We can now rewrite the expression for [Kc] i i [KJ = J j[B]T[K][B]deUdudv (B-42) 96 Appendix B About this expression, we use Gauss quadratic method to do the numerical integration. General method for numerical integration is as follows. r*H/ 0/(* 0)+ *,/(*,) (B-43) Where wo and wi are referred to as the weighting functions and xn and xi are referred to as the integration points. Sum of errors tends to be cancelled by choosing suitable values of xn, X l , W 0 , W i . For 2 points Gauss quadrature, we choose w0 = w, = 1 (B-44) 1 Where n = number of integration points. Return to the conductivity matrix [IQ]. i i [KJ = J $[B]T[K][B]detJdudv - i - i i i = j \f{u,v)dudv -1 -1 For 2 points or 2x2 integration (B-45) xx = -p=xi (B-46) This yields exact solution up to a cubic function. The general form is )f(x)dx = fjwjf(xj) (B-47) = 1 £*y",/(«<».v,) (B-48) 97 Appendix B W=E i^j[B]Tk[B]detJ (B-49) 1=1 j=i i i J jdet Jdudv = area (B-50) - i - i Heat capacitance term \\NiPCp[N]^-dxdy i=l,n. (B-51) Where n is number of node per element. In matrix form [Cr^=jj{N}pCp[N]dxdy^- (B-52) In numerical notation, using Gauss quadrature, we have = H^j{N}pCp[N]dctJ (B-53) '=1 7=1 C p is possible temperature dependence that we may want to account for. During integration at given u, v. 7(M,v) = XAr/("'v)7;- (B-54) 1=1 Cp(T) = f(T(u,v)) (B-55) 4.2 The boundary term We have derived expression for the volumetric terms [K] and [Cp] (evaluated for all elements). Now, we will focus on the terms associated with the various boundary conditions. These terms are only evaluated for elements laying on the boundary of the analysis domain. We must evaluate terms of the form 98 Appendix B \qNids = \qNtdl (B-56) s' i So, we need to derive an expression that will allow us to calculate the arc length of a boundary or arbitrary length in x, y space. Consider the following example. For face (1) ds = yjdx2 +dy2 = arc length (B=57) In the local coordinate system, i \®(u)du (B-58) - i and dx = —du dy = —du (B-59) du ' du For our expression for the Jacobin, we have te= =f a y , d y _ r _ + m, du 1 1 t r a Substituting and rearranging dx r cW • . dy _ d/V, . . _ ^ N ^-=-/.. =Z^r( w ' v K ^ = ^ 2 = E^r("' v)^ ( B- 6°) du ti du du " du ds = dupu +Jf2 (B-61) Thus \qN.ds = \qN^J2n +Jf2du (B-62) In vector form {fq}=\q{N}ds (B-63) And in numerical form, we have 99 Appendix B > = Z wMNUJn + J n ( V a l i d for face (1) only) (B-64) 1=1 Similar expression must be derived for face 2, 3, 4. For face (1), i=i 0 0 ^ (v=-l,u=±0.577) (B-65) For face (2) ={/,} = 0 / , 0 f (u=l,v=±0.577) (B-66) For face (3) Z w * M * V ^ n + ^ = { / , } = 0 0 /, 1/4 J (v=l,u=±0.577) (B-67) For face (4) 0 0 I/4 J ^ (u=-l, v=±0.577) (B-68) 4.5 J/**? transient term Now, we can collect all of the terms that comprise one F.E. equation. [ C ' ] U I + ^ ] e l ( r } = { / ' r (B-69) 100 Appendix B If we were solving the steady state problem, we could k]e{Te} = {fqY (B-70) The [A]e and {B}e can be hand out to a routine to place in the globe [A] matrix and {B} vector. [A]{T} = {B} (B-71) For the unsteady state case, we need to deal with the term [Cp]dT/dt in order to arrive at an equation of the form [A]{T} = {B] (B-72) In general, the form of the equation we must linearize and solve is [C]e {T} + [k]e {T}e + {/}* = 0 (B-73) No analytical solution exists, so we must adopt some type of step by step recurrence scheme. We use finite difference in time method to deal with this problem. We can use a Taylor series expression about {T„} which is the current temperature At.dTl {Tn+l} = {Tn} + - - -1! dt + ••• + higher order terms (B-74) Where tn +i = tn + At [C]'{T}g+[kY{T)t' + {f}' =0 (B-75) With the following approximation {T)g = {r"+'}~{r"} (B-76) At {T}d=(l-9){T„} + 0{Tn+l} (B-77) td = tn + OAt (B-78) Therefore, 0 is some number between 0 and 1, i.e., O<0<1. Introduction of 9 leads a 101 Appendix B general set of recurrence schemes. Rewriting our P.D.E. Substituting into our P.D.E. [CT At + 9[k]e {R"+1 ]E = [~A7 {t- }e _ [*]'(1" 9){T" ]e + {fe} (B-79) Choosing values of 0 will increase or decrease the emphasis of {Tn}. In some situation, this can add necessary stability or improve the rate of convergence. It must be assessed on a problem by problem bases. If we choose 0=0.5, then our P.D.E. will become 2 At \ J (B-80) 102 Appendix C Appendix C. Data acquisition system noise For our test equipment, the magnitude of the data noise can be obtained by following deriving. The data acquisition system we used has a 8-bit board, so the range of measurement will Q be 2 = 256. For a gain of 1, the range of voltage is -5 V ~ 5 V, so the difference between two neighbouring data points will be \0V = 0.039F = 39/nF 256 This means there must be at least 39mv change in the voltage of thermocouple in order to make the measured temperature increase or decrease. In this case the measurement error will be ±39 mV. During our test conditions, the sample temperature rarely exceeded 550°C, therefore, the maximum voltage the thermocouple (K type) can produce is ~20 mV. If we use a gain of 1, the noise becomes larger than the thermocouple signal. As a result in our tests we used a gain of 7. For a gain of 7, the range of voltage is -25 mV ~ 25 mV. The difference between two temperature readings will be 50 mv/256 = 0.195 mV. This means that the error of the data acquisition system will be ±0.195 mV. The following table shows the temperature errors given by ±0.195 mV of voltage at different temperature ranges. 103 Appendix C Table C . l : The error in temperature readings at different temperature range (K type thermocouple). Temperature range (°C) Voltage range (mV) Voltage difference (mV) Temperature difference (°C) 1-100 0.000-4.096 +0.195 ±4.76 100-200 4.096-8.138 ±0.195 ±4.82 200-300 8.138-12.209 ±0.195 ±4.79 300-400 12.209-16.397 ±0.195 ±4.66 400-500 16.396-20.644 ±0.195 ±4.59 From the table we can see that for our data acquisition system there will be ~ ±5°C noise in the data. As a result a smoothing operation must be performed on the data.. But in order to calculate the heat flux we need the temperature data that has very little noise, because the large data noise can make the solution divergence. Suppose our cooling rate is 100°C/s, if our time step is 0.1s, the temperature change in 0.1s will be 10°C. Because we have ±5°C data noise, the noise in data will be about 50%. If first temperature is 300°C, the second temperature can be 290±5°C, the third temperature can be 280±5°C. If the second temperature is 285°C, the calculated heat flux will larger than actual one. If the third temperature is also 285°C, the calculated heat flux will not be to cooling the sample but to heat the sample. The calculated results will diverge. So, in order to get good results we must smooth the data of experiments. 104 

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