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Secondary cooling in the direct-chill casting of light metals Etienne, Caron 2008

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SECONDARY COOLING IN THE DIRECT-CHILL CASTING OF LIGHT METALS  by ETIENNE CARON B.A.Sc., Université Laval, 1999 M.Sc., Rheinisch-Westflulische Technische Hochschule Aachen, 2002  A THESIS SUBMITTED iN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2008  © Etienne Caron, 2008  ABSTRACT The Direct-Chill (DC) casting process is used in the non-ferrous metals industry to produce ingots, blooms and cylindrical billets. During DC casting, primary cooling in the mould is followed by secondary cooling, in which the cast product surface is directly cooled by water jets. The formation of defects during the direct-chill casting process can be reduced by controlling the heat extraction in the secondary cooling zone during the start-up phase. The control and optimization of this process requires an accurate knowledge of the boundary conditions and their relationship with casting parameters. This research project studied the effect of different parameters on the heat transfer in the secondary cooling zone of the direct-chill casting process. This process was simulated by quenching instrumented samples of industrial DC-cast aluminum AA5 182 and magnesium AZ3 1 with water jets and recording the thermal history within the sample using sub-surface thermocouples. An inverse heat conduction algorithm specifically developed for this research project converted this thermal history into surface heat fluxes and surface temperatures. The relationship between heat flux and surface temperature was expressed by a boiling curve. Cooling experiments showed the influence of the cooling water flow rate on characteristic features of the boiling curve. The effect of thermophysical properties, initial sample temperature and water temperature on high temperature boiling regimes was also quantified. The influence of other parameters such as the water jet velocity and the surface roughness was determined in a qualitative fashion. Results from the quench tests were used as boundary conditions in a finite element model for the direct-chill casting of AZ3 1 billets. Simulations of the process start-up phase showed the critical role played by stable film boiling and water film ejection in determining the thermal history within the billet. 11  TABLE OF CONTENTS Abstract  ii  Table of contents  iii  List of tables  vi  List of figures  vii  List of symbols  x  List of acronyms and subscripts  xiii  Acknowledgements  xiv  1  Introduction: The direct-chill casting process  1  2  Literature review  6  2. .1  Secondary cooling  6  2.2  Boiling water heat transfer  7  2.3  Modelling of secondary cooling  11  2.3.1  Forced convection  12  2.3.2  Nucleate boiling  14  2.3.3  Film boiling  15  Experimental methods  17  2.4.1  Quench tests  17  2.4.2  Instrumented samples  18  2.4.3  In-situ measurements  19  2.4  2.5  Effect of parameters on secondary cooling  20  2.5.1  Waterflowrate  21  2.5.2  Water temperature  23  2.5.3  Water quality  24  2.5.4  Dissolved gases  25  2.5.5  Water jet impingement angle  26  2.5.6  Distance from water jet impingement zone  26  2.5.7  Start temperature  27  2.5.8  Surface condition  27  2.5.9  Thermophysical properties  29  2.5.10  Ingot size and geometry  30  2.5.11  Casting parameters  30 111  2.6  Scope and objectives  3  4  5  6  .33  3.1  Objectives  33  3.2  Methodology  34  Experimental  36  4.1  Set-up  36  4.2  Cooling tests  39  4.3  Thermocouples  41  Inverse Heat Conduction analysis  43  5.1  Formulation of the IHC algorithm  43  5.2  Direct heat conduction problem  44  5.3  Function specification method  47  5.4  Characterization of surface heat flux profile  50  5.5  IHC analysis with moving boundary  54  5.6  Validation of the IHC algorithm  59  Results and discussion  63  6.1  General observations  63  6.2  Impingement zone height  69  6.3  Forced convection regime  74  6.4  Nucleate boiling regime  76  6.5  Critical heat flux  78  6.5.1  Impingement zone  78  6.5.2  Free falling zone  81  6.6  Transition boiling  85  6.7  Film boiling  91  6.7.1  Minimumheatflux  91  6.7.2  Leidenfrost point  96  6.8 7  Summary. 31  Water film ejection  Effect of secondary cooling on direct-chill casting of light metals  100 110  7.1  Applicability of cooling experiment results  110  7.2  Finite Element Model of cooling experiments  111  7.2.1  Idealized boiling curves  112  7.2.2  Comparison between idealized and experimental boiling curves  116 iv  7.2.3  Progression of rewetting front  7.2.4  Effect of thermophysical properties  7.3  Finite Element Model of DC casting  117 118 119  7.3.1  Model development  119  7.3.2  Calculation domain and finite element mesh  120  7.3.3  Thermal boundary conditions  121  7.3.4  Initial conditions  124  7.4  Thermophysical properties  124  7.5  Importance of secondary cooling  126  7.5.1  Relative importance criterion  126  7.5.2  Importance of secondary cooling during process start-up  127  7.6 8  .  Effect of casting parameters  Summary and conclusions  130 135  8.1  Summary  135  8.2  Conclusions  137  8.3  Recommendations  139  Bibliography  141  Appendix A: Cooling test conditions  147  Appendix B: Finite Element Modelling  150  V  LIST OF TABLES Table I: Thermophysical properties of AZ3 1 and AA5 182 alloys  46  Table II: Thermophysical properties used in the FEM simulation of a cooling test  53  Table III: Thermophysical properties used in the FEMLAB simulation  59  Table IV: Pool boiling contribution to nucleate boiling regime  78  Table V: Heat transfer coefficient hDB for dummy block cooling  122  Table VI: Heat transfer coefficient  123  for primary cooling in the mould  Table VII: Thermal conductivity k of AZ3 1 used in the FEM model  125  Table VIII: Specific heat C ofAZ3l used in the FEM model  125  vi  LIST OF FIGURES Figure 1: Schematic of the vertical DC casting process  2  Figure 2: Typical fracture modes of casting ingot  4  Figure 3: Geometry of cooling water jet  6  Figure 4: Typical pool boiling curve for water  8  Figure 5: Evolution of the rewetting front during start-up  10  Figure 6: Effect of cooling water temperature Tf on forced convection regime  13  Figure 7: Effect of cooling water flow rate  14  Q’ on forced convection regime  Figure 8: Heat transfer coefficient with pulsed water  28  Figure 9: Schematic of the water jet experimental rig  36  Figure 10: Typical as-cast surface of AZ3 1 ingot  38  Figure 11: Typical as-cast surface of AA5 182 ingot  39  Figure 12: Effect of filtering algorithm on temperature data  42  Figure 13: x-z two-dimensional plane at the center of the instrumented sample  45  Figure 14: Section of FEM mesh used to model instrumented samples  46  Figure 15: Constant heat flux P for the evaluation of the “base” temperature  47  Figure 16: Linearly increasing heat flux P for the evaluation of sensitivity coefficients  48  Figure 17: Algorithm of the inverse heat conduction analysis  49  Figure 18: Heat flux profiles  51  Figure 19: Boiling curves for the FEM simulation of a cooling test  51  Figure 20: Section of FEM mesh and boundary conditions for cooling test simulation  52  Figure 21: Typical heat flux profile during cooling test with stationary sample  53  Figure 22: Linear heat flux profile with moving boundary  55  Figure 23: Heat flux profile with moving boundary and FEM mesh with thermocouples  55  Figure 24: Second derivative of modelled temperature signal 2 T/dt d  57  Figure 25: Wetting front progression during FEM simulation of a cooling test  57  Figure 26: Calculated heat flux with different values  59  Figure 27: Applied and calculated heat fluxes  Of t,eak  for validation of IHC algorithm  60  Figure 28: Boiling curves calculated with one- and two-dimensional FEM models  62  Figure 29: Typical measured cooling curves for experiment with stationary sample  64  Figure 30: Typical calculated boiling curves for experiment with stationary sample  65  Figure 31: Measured cooling curve for thermocouple 7 in the impingement zone  65 vii  Figure 32: Measured cooling curve for thermocouple 4 in free falling zone  66  Figure 33: Equivalent boiling curves for the impingement zone  67  Figure 34: Equivalent boiling curves for the free falling zone  68  Figure 35: Equivalent boiling curves for the impingement zone  68  Figure 36: Equivalent boiling curves for the free falling zone  69  Figure 37: 40 x 30 mm close-up of impingement zone  70  Figure 38: Typical measured cooling curves for rewetting test  71  Figure 39: Second derivative of temperature d T/d? measured during rewetting test 2  72  Figure 40: Impingement zone height H 1 as function of water flow rate  73  Q’  Figure 41: Calculated boiling curves for machined and as-cast samples  74  Figure 42: Relationship between h / (Q )“ and surface temperature  76  Figure 43: Calculated boiling curves for different start temperatures T 0  79  Figure 44: Effect of cooling water flow rate  80  Q’ on critical heat flux PCHcJZ Figure 45: Calculated boiling curves for different water flow rates Q’  82  Figure 46: Measured heat fluxes cP as function of time  83  Figure 47: Critical heat flux as function of the distance from the impingement zone d 1  84  Figure 48: Calculated boiling curves for different sample moving speeds v  86  Figure 49: Heat flux P as function of time t for different sample moving speeds v  86  Figure 50: Distribution of the transition boiling slope for the free falling zone  87  Figure 51: Distribution of the transition boiling slope for the impingement zone  88  Figure 52: Effect of thermal conductivity k 3 on transition boiling slope d/dT  89  Figure 53: Calculated boiling curves for different sample moving speeds v  90  Figure 54: Modelled minimum heat flux  93  Figure 55: Effect of water flow rate  MHF  for different water temperatures T 1  Q’ on Leidenfrost point TLpt  Figure 56: Modelled Leidenfrost point TLpt as function of water flow rate  98  Q’  100  Figure 57: Effect of initial temperature T 0 on rewetting temperature Twet  103  Figure 58: Effect of dry surface temperature Td, on rewetting temperature Twet  103  Figure 59: Effect of dry surface temperature Td, on relative rewetting temperature  104  Figure 60: Relative rewetting temperature for two cooling water flow rates  105  Figure 61: Effect of water flow rate  Q  ‘  Q’ on minimal relative rewetting temperature  Figure 62: Modelled relative rewetting temperature for different water flow rates Figure 63: Modelled relative rewetting temperature for different alloys  106  Q  ‘  107 109 viii  Figure 64: Modelled boiling curve for the impingement zone  113  Figure 65: Modelled boiling curve for the impingement zone  114  Figure 66: Modelled boiling curve fOr the free falling zone  115  Figure 67: Modelled and experimental boiling curves for the impingement zone  116  Figure 68: Modelled and experimental boiling curves for the free falling zone  117  Figure 69: Modelled and experimental progression of wetting front  118  Figure 70: Modelled boiling curves for two alloys  119  Figure 71: Model boiling curves used by Hao  120  Figure 72: Section of FEM mesh used to model the AZ3 1 455 mm billet  121  Figure 73: External boundary conditions used in the FEM model  122  Figure 74: Thermal resistance for surface cooling and heat conduction as function of time.... 127 Figure 75: Relative importance of surface cooling  as function of time  128  Figure 76: Relative importance of surface cooling R 5 as function of time  129  Figure 77: Relative importance of surface cooling  130  as function of time  Figure 78: Critical casting speed v for water film ejection as function of flow rate Figure 79: Critical casting speed v and cracking frequencyfc  Q  ‘  132 133  ix  LIST OF SYMBOLS Latin symbols Bi  Biot number  ... 3 2 , 1 C  Various coefficients  1 C  Coefficient in Rohsenow’s nucleate pooi boiling model  C  Specific heat  D  Billet diameter  1 d  Distance from water jet impingement zone  fc  Cracking frequency  Fo  Fourier number  g  Gravitational acceleration  2 rn/s  1 H  Impingement zone height  mm  h  Heat transfer coefficient  K 2 W/m  heq  Equivalent heat transfer coefficient  K 2 W/m  i  Latent heat of evaporation  -  J/kgK m mm -  -  J/kg  Least-squares criterion  (various)  k  Thermal conductivity  W/mK  L  Characteristic length in various non-dimensional numbers  PIP  Dynamic pressure at water jet impingement point  2 N/m  Q  Volumetric cooling water flow rate  /s 3 m  m  x  Latin symbols (continued) Volumetric cooling water flow rate per unit of perimeter 2 R  Square of the Pearson correlation coefficient Relative importance of secondary cooling  Re  Reynolds number  r  Exponent in Rohsenow’s nucleate pool boiling model  r  Surface roughness factor  S  Sensitivity coefficient  0 T  Initial temperature  Td,  Dry surface temperature  Tf  Water bulk temperature  TLpt  Leidenfrost point  T  Surface temperature  Tsat  Water saturation temperature  L/miwm -  %  -  /W 2 °Cm  OC  Temperature at thermocouple location Twet  Rewetting temperature  °C  Casting time  s  tCHF  Time at which the critical heat flux is experienced  s  tpeak  Time at which the heat flux peak arrives at a thermocouple location  s  0 v  Water film initial velocity  v  Casting or sample moving speed  Viet  Water jet velocity  rn/s mm/s rn/s  xi  Greek symbols af  Water thermal diffusivity  /s 2 m  zIH  Distance between thermocouples  mm  At  Time step length  s  Ax  Distance between thermocouple junction and surface  m  Thermal effusivity Heat flux Wavelength of water / steam interface Water viscosity jet 8  Water jet impingement angle  p  Density Variance of a population  Jfg  Surface tension at the water / steam interface  Ks° 2 J/m 5 2 W/m m kg/ms  3 kg/m (various) N/m  xii  LIST OF ACRONYMS AND SUBSCRIPTS ACF  Advanced cooling front  AVG  Average  CHF  Critical heat flux  DB  Dummy block  DC  Direct-chill  f  Fluid (water)  FB  Film boiling  FC  Force convection  FFZ  Free falling zone  g  Gas (steam)  IC  Initial cooling  IHC  Inverse heat conduction  IZ  Impingement zone  LPt  Leidenfrost point  MHF  Minimal heat flux  NB  Nucleate boiling  NPB  Nucleate pool boiling  PC  Primary cooling  s  Solid (metal)  sat  Saturation  SC  Secondary cooling  TB  Transition boiling  TC  Thermocouple  xlii  ACKNOWLEDGEMENTS The author wishes to thank Gary Lockhart, Serge Milaire, Ross McLeod, Carl Ng and David Torok, all of the Department of Materials Engineering at the University of British Columbia, for their help with the setup and instrumentation of the water jet experimental rig. The collaboration of Dimitry Sediako and Steve Hibbins of Timminco Metals, and of André Larouche at the Arvida Research and Development Center (Alcan), must also be acknowledged. Their experience with the direct-chill casting process and its modelling was greatly appreciated. The author would also like to express his gratefulness to John Grandfield at the Comalco Research Centre for his valuable input. Financial support for this research work was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Cy and Emerald Keyes Fellowship in Materials Engineering, the J. Keith Brimacombe Memorial Scholarship as well as the David W. Strangway Fellowship. Finally, the author would like to find the words to thank Dr. Mary A. Wells, who supervised this research project. The completion of this work would never have been possible without her constant support and enthusiasm.  xiv  1  INTRODUCTION: THE DIRECT-CHILL CASTING PROCESS Since its commercial viability was established in the 1930’s, the Direct-Chill (DC)  casting process has been widely used in the production of non-ferrous metals and their alloys [1, 2, 3, 4]. DC casting is a semi-continuous process which can produce ingots and blooms for rolling as well as cylindrical billets for extrusions and forgings [5, 6]. Typical dimensions for the cast products are 1500 x 500 mm for ingots and 200 mm in diameter for billets. Compared to conventional casting processes, direct-chill casting allows a higher productivity, a lower scrap rate, a greater product uniformity and significant energy savings. Figure 1 shows a schematic representation of the direct-chill casting process. At the start of this process, a bottom “dummy” block is partially inserted into a water-cooled copper or aluminum mould which is then filled with molten metal through a feeder. Once the metal level in the mould reaches a given height, the bottom block is gradually lowered into a casting pit to withdraw the ingot [7]. The corresponding casting speed generally lies between 1.0 and 2.0 mm/s. The process stops once the desired ingot length (ca. 4  -  10 m) has been obtained.  The removal of heat through the mould wall is referred to as primary cooling. In the primary cooling zone, liquid metal initially contacts the mould wall but then pulls away from the mould as it solidifies to form a solid metal shell, leaving an air gap with reduced heat transfer to the mould. This solid shell tends to bend inwards because of the solidification shrinkage and ensuing thermal contraction during cooling [5, 8, 9]. The cooling water is usually circulated through the mould and exits the mould bottom through a series of holes or slots to produce a series of water jets. Typical steady-state water flow rates are 100 to 200 L/min for every meter of wetted perimeter. Start-up water flow rates are generally lower, e.g. 50 L/minm.  1  Floating Mould  Primary water inlet  Secondary water Solidus Dummy  Descending table Figure 1: Schematic of the vertical DC casting process [10] The direct contact between cooling water and the ingot surface constitutes secondary cooling. Secondary cooling is responsible for the largest amount (Ca. 80%) of heat extraction during steady-state operation [11, 12] and is associated with the formation of significant thermal gradients. Finally, heat is also removed through the bottom “dummy” block during the start-up of the direct-chill casting process. This phenomenon is referred to as tertiary or bottom block cooling. During steady-state, heat conduction within the metal is the limiting factor for the secondary cooling process, so that variations in the surface heat flux have only a limited influence on the internal temperature profile [13, 14]. This is especially true for ingots or billets of large dimensions [15]. On the other hand, heat removal in the secondary cooling zone is of primary importance during the transient start-up phase, where the relatively high surface temperatures can lead to the formation of an insulating vapour film [16]. Due to its 2  semicontinuous nature, approximately 10 to 15% of the direct-chill casting process takes place in the transient start-up phase [17, 18]. This phase has been found to be critical for the formation of defects, yet models for the temperature evolution and defect formation during start-up are scarce. Despite being the subject of scientific investigation since its inception in the 1930’s, the DC casting process still presents many technical challenges. The high rates of heat extraction during start-up are associated with very high thermal stresses and can lead to the deformation of the thin metal shell [19]. One example of such deformation is butt curl, which consists in a pulling away of the ingot from the bottom block due to the thermal contraction of the sides [7]. From a heat transfer perspective, the effect of the butt curl is complicated. Intuitively, butt curl reduces the contact area between the ingot and the bottom block and decreases the heat transfer. It can also cause the narrow sides of the ingot to pull in and create an even larger air gap in the mould. Butt curl also allows cooling water to flow along the bottom of the ingot and accumulate in the bottom block. To alleviate this, bottom blocks usually have drain holes to allow the water to drain away. The lowered heat flux at the bottom of the ingot can lead to the formation of a metal crater and eventually to a breakout [19]. It can also contribute to the formation of external cracks [15]. In rolling ingots, transverse cracks can occur when high temperature gradients develop between the surface and the center [20, 21]. In the extrusion billets, internal cracks known as “hot tears” have been known to form at the bottom of the metal sump during start-up [22, 23]. Figure 2 illustrates various modes of cracking due to thermal stresses for rectangular ingots. The presence of hot tears increases the length of the billet butt cut-off and can even lead to a rejection of the billet. The reject scrap rate is one of the main cost drivers of the DC casting process [24, 251. Correspondingly, a good control of the secondary cooling during the critical 3  start-up phase could help in reducing the production costs. Moreover, the thermal history within the metal ingot is not only responsible for the formation of defects such as butt curl and hot tears, but also for the microstructure development, the occurrence of segregation and the surface quality [13, 15, 16, 17, 26, 27]. Knowledge of this thermal history can thus be a very important tool to design, control or optimize the direct-chill casting process.  Center crack  Quarter-point crack  J-crack  Cracking in ingots  Hot tears in billets  Figure 2: Typical fracture modes of casting ingot [1] A significant portion of research work conducted on direct-chill casting nowadays is of a fundamental nature and focuses on the mathematical modelling of the process [28]. A thermal model, for instance, predicts the temperature within the ingot by solving a transient heat conduction problem. The accuracy of numerical models has been found to be related not to their ability to solve the governing partial differential equations, but rather to the simp1ifiing assumptions regarding the ingot geometry, the material properties and the external boundary conditions [4]. When developing thermal models for the DC casting process, a precise knowledge of the boundary conditions, and of their variation both spatially and temporally, is therefore critical. 4  The first mathematical model of the DC casting process was developed in 1943 by Roth [29]. The analytical model assumed a constant surface temperature in the secondary cooling zone because of the very high heat transfer coefficients in that region. Subsequent attempts at modeling [30, 31] also relied on external boundary conditions of the Dirichiet type. Later models used external boundary conditions of the Cauchy type, for which the heat transfer coefficient was specified [6, 14, 32, 33, 34], or a combination of the Dirichlet and Cauchy types [4, 35]. Recent models [1, 7, 36, 37] evaluate the external boundary conditions with the help of boiling curves which describe the relationship between the surface temperature and the heat flux or heat transfer coefficient. Because of the highly non-linear nature of boiling water heat transfer, a good understanding of the factors which affect the surface heat flux are necessary from a process control point of view. This research project studied the effects of various factors (cooling water flow rate, initial sample temperature, casting speed, sample thermophysical properties, surface roughness, cooling water temperature) on the heat transfer which takes place in the secondary cooling zone of the direct-chill casting process. A particular emphasis was put on high temperature boiling water heat transfer phenomena, which take place during the transient start-up phase of the process and have been known to influence the formation of defects.  5  2  LITERATURE REVIEW  2.1  Secondary cooling This section describes the secondary cooling zone of the DC casting process from a  macroscopic perspective; the microscopic phenomena associated with boiling, such as bubble nucleation, growth, detachment and collapse, are discussed in Section 2.2. In the secondary cooling zone, the metal surface is cooled by direct contact with a vertical water film flowing down the ingot or billet. This water film is formed by jets which emerge from an array of holes in the bottom part of the mould. The impingement angle of the water jets Oet is typically in the range 15 30° [38]. For the casting of rectangular ingots, it is -  possible to obtain different water flow rates along the narrow and wide sides by adjusting the hole spacing in the mould or the hole size [1]. The water film can also be obtained with a single water curtain flowing out of a slot along the mould bottom [161. The different geometrical features of the water jet are presented in Figure 3. Impingement angle  ‘ater jet  surf:ce Climbing height  t 4,  ,::/‘  “ Jet diameter  /  Impingement point  Falling film 4  Film thickness  Figure 3: Geometry of cooling water jet [16]  6  As shown in Figure 3, the secondary cooling zone can be divided into an impingement zone (IZ), where the water jets hit the ingot surface, and a free falling zone (FFZ), in which the water film flows down the surface [7]. The impingement zone is characterized by a drop in water pressure (corresponding to the variation in momentum) as the water falling velocity increases. On the other hand, the water pressure in the free falling zone is constant and the water velocity in this region only depends on the gravitational acceleration. The impingement zone can be further subdivided into the two regions above and below the impingement point [38]: the upper impingement zone corresponds to a region of back flow with a relatively low heat transfer coefficient, whereas the heat flux is maximal at the impingement point [26, 39, 401. The height of the impingement zone H 1 depends on a number of factors such as the climbing height, the water jet impingement angle and the cooling water flow rate. Depending on the way it is defined, the impingement zone height can be attributed a wide range of values (anywhere between 5 and 300 mm) [7, 16, 36, 37, 41, 42, 43]. The water film free falling zone (sometimes referred to as “streaming water zone” [5, 36]) corresponds to the area between the impingement zone and the ingot bottom and thus takes up the greater part of the secondary cooling zone. Therefore, even though the heat removal rate is greater in the impingement zone because of the high momentum of the water jet in this region, most of the heat is extracted in the free falling zone.  2.2  Boiling water heat transfer The cooling involved in the direct-chill casting process can be classified as sub-cooled  boiling with falling film convection [16], because it involves water with a bulk temperature below its boiling point. Boiling heat transfer occurs according to one of the following regimes: film boiling, transition boiling, nucleate boiling and forced convection. These four boiling 7  regimes are shown in Figure 4 on a boiling curve, which relates the surface heat flux P. 2 or MW/rn ) to the surface temperature T (in °C). Figure 4 2 (generally expressed in W/m illustrates pool boiling, in which a heated surface is submerged in a large volume of stagnant liquid, so that the forced convection regime is replaced here with natural convection. Boiling modes AB B’C DE EF  %% s*’ec.. \\  forced convection nucleate boiling transition boiling film boiling  -  -  AB \  \  \\\‘%\\\%\\  S\\\\  Critical temperatures B onset of nucleate boiling (ONB) E Leidenfrost point  B’C  Extreme heat fluxes D critical heat flux (CHF) E minimum heat flux (MHF)  E U  Q43 CD  Ca Ca Ca U Ca  t  (0  \\\\\‘%\\\\\ \  \\  \\\  0 70  100  200 400 700 1000 Heater surface temperature, °C  2000  Figure 4: Typical pooi boiling curve for water  \‘%‘.ç%%%\\  L441  In the film boiling regime, a stable vapour film covers the entire surface and heat transfer occurs principally by conduction and convection through the vapour. The Leidenfrost point (identified as E in Figure 4) marks the boundary between the transition boiling (DE) and film boiling (EF) regimes. This point corresponds to the breakdown of the vapour film and to a minimum heat flux (MHF), The transition boiling regime is characterized by the presence of an unstable vapour blanket and intermittent wetting of the surface. The critical heat flux (CHF) at point D corresponds to the upper limit of the boiling curve and the boundary between the nucleate boiling (B’D) and transition boiling regimes. In the nucleate boiling regime, vapour bubbles are formed at the surface. Vapour structures vary from a few individual nucleation sites 8  at low temperature to patches of coalesced bubbles and vapour columns as the heat flux is increased. Below the onset of nucleate boiling (ONB, point B’), no vapour nucleation occurs and the heat is removed by forced or natural convection (AB) [44]. In the case of DC casting of light metals, nucleate boiling is the regime which prevails both in the impingement zone and in the free falling zone during steady-state [7, 38]. Film boiling was also reported to take place in the free falling zone, where the water velocity is lower, and could be detected by its characteristic hissing sound and opaque water film effect [16]. Such observations have not been reported for the impingement zone, but this zone has nevertheless been known to experience the typical low surface heat fluxes associated with the film boiling regime [13, 16]. Film boiling at the ingot surface is sometimes promoted during start-up in order to minimize the formation of thermal gradients and the occurrence of butt curl. This can be done by reducing the cooling water flow rate, using pulsed water jets [16, 26] or dissolving gases in the cooling water [191. During this period, the casting rates tend to be very low. As the casting speed is ramped up towards its steady-state value, the water flow rate is increased so that the main boiling heat transfer mechanism is nucleate boiling. Film boiling can take place if the casting speed is increased rapidly, but it usually occurs at start-up, when the surface temperatures are relatively high (e.g. above 500°C) [9, 16]. A vapour blanket is then formed at some point of the free falling zone. The ingot surface thus experiences nucleate boiling in the impingement zone, film boiling in the portion of the free falling zone covered by the vapour film, and transition boiling (or “unstable film boiling”) between these two regions [45]. Eventually, the axial conduction within the ingot or billet reduces the surface temperature below the Leidenfrost point, and the unstable vapour film breaks down [13]. In steady-state, however, the relatively low surface temperatures and the  9  higher water flow rates generally ensure nucleate boiling or forced convection for the whole free falling zone [7, 46]. During start-up, film boiling can lead to the ejection of the water film away from the metal surface [7]. This phenomenon must be distinguished from the bounce-off of the water jet which takes place in the impingement zone when the jet velocity vjet is too high or when the impingement angle  e( 8  is too large [5, 16]. As the ingot cooling progresses, the temperature at  the point of water ejection drops below the Leidenfrost point and the vapour film moves downward to regions where the surface temperature is higher [13, 26, 47]. The evolution of the rewetting front is illustrated in Figure 5. In steady-state, film boiling and water ejection do not occur, and the water film flows down the whole surface of the ingot.  Water jet coming out of the mould  Nucleate boiling and convection  Convection only  START OF COOLING: Film boiling prevails  Film of vapour moving downward  STEADY STATE: Nucleate boiling and convection  TIME Figure 5: Evolution of the rewetting front during start-up  [71  Local film boiling is also thought to be responsible for the formation of hot spots and subsurface cracks [7]. When the casting speed is increased, the rise in surface temperature can also cause a sudden jump from nucleate boiling to film boiling, which can then lead to remelting 10  of the surface and liquid metal breakouts [4]. The transition boiling regime, in which a decrease in temperature leads to an increase in the surface heat flux, is also thought to be detrimental to the casting process, as it exacerbates the temperature differences on the ingot surface. Precise control of the boiling water heat transfer phenomena taking place in the secondary cooling zone can help prevent the formation of defects during the transient start-up phase of the DC casting process. For instance, butt curl can be reduced by limiting the cooling rate during process start-up, i.e. by promoting film boiling [15]. In industrial processes, this has been achieved by lowering the water flow rate [26, 36], using pulsed water [9, 16] or injecting noncondensable gases such as air or carbon dioxide in the cooling water [16, 19]. Hot tears can be prevented by limiting the cooling rate at the ingot surface, so that the difference between the cooling rates at the surface and in the center of the ingot is minimized [201. This is achieved, for instance, with the so-called delayed quench method, which uses two series of water jets for the secondary cooling: a first series with a relatively low water flow rate just at the mould exit, and a second series ofjets with a much higher water flow rate positioned approximately at the same level as the bottom of the liquid metal sump [21].  2.3  Modelling of secondary cooling Different semi-empirical models have been developed to quantify the surface heat flux  during DC casting. These models do not distinguish between the impingement zone and the free falling zone, i.e. they do not take into account the effect of the increased momentum at the impingement point. Equations are available for three boiling regimes (forced convection, nucleate boiling and film boiling), but no model has been developed for the transition boiling regime.  11  2.3.1  Forced convection  The heat transfer in the free falling zone was quantified by Weckman and Niessen [4] for the case of forced film convection. The following equation is based on the equation for the gravity flow of a water layer in a vertical tube of internal diameter D (in m), and is applicable to the flow of a water film at the surface of a billet of diameter D [48]: /(pg(4pjQcpjij  -  FC  l  i flI 2 pr L Pf  ()  I f  where k 2 and , P1’ C and p are the properties of water, g is the gravitational acceleration in mis 1 /s. 3 Q is the volumetric cooling water flow rate in m  The heat transfer coefficient hFc given by  Equation (1) is given in W/m K. The properties of water (thermal conductivity Icj in W/mK, 2 density Pf in kg/rn , specific heat C 3 1 in J/kgK and viscosity fif in PaSs or kg/ms) must be evaluated at the temperature of the water film boundary layer. This temperature is generally estimated as the average between the cooling water bulk temperature Tf and the billet surface temperature T . The coefficient C 3 1 given by Weckrnan and Niessen is equal to 0.0100. Equation (1) is valid for turbulent flow, i.e. Reynolds numbers Re above 2000. This Reynolds number is calculated with a characteristic length which corresponds to the billet diameter D [48]. For a billet with a diameter of 154 mm cooled with a water flow rate per unit of perimeter  Q’  of 200 L/minm and an average water temperature of 10°C, the Reynolds  number is approximately 12000 and the free-falling water film is thus fully turbulent for typical DC casting conditions. The heat transfer coefficient is strongly dependent on the water temperature, and Equation (1) was simplified to the following equation:  (  hFc  (T+T  704 = L  2  Q fJ+2.53.104J[__J  1/3  (2)  12  where the ratio Q/irD corresponds to a water flow rate per unit perimeter but, unlike  Q’,  is  /s. 2 expressed in m The heat flux for forced convection by a falling film of water in turbulent flow is thus equal to: FC  -  (3)  hc VS—TI)  1 and flow rate Figure 6 and Figure 7 illustrate the effects of water temperature T the heat flux for the forced convection regime  ‘TFC  Q’ on  as calculated with Equations (2) and (3).  They also indicate the onset of nucleate boiling (ONB), which is given by the following equation: ONB  =  (4)  56 3910.0 (T •’ 2 Tsat)  in which the water saturation temperature Tsat can be taken here as 100°C. The onset of nucleate boiling corresponds to the boundary between forced convection and nucleate boiling.  1 .2E+06 —.-  10°C  —E-2O°C  1 .OE+06  --  40’C  —ONB  2 8.OE+05  /  ‘4-  (‘3  z  6.OE+05  U) U) C-) (‘3  t  v  4.OE+05  D  Cl)  2.OE+05  0.OE+00 0  20  40  60  80  100  120  Wall temperature (°C)  Figure 6: Effect of cooling water temperature Tjon forced convection regime (Q’ = 100 L/minm)  13  1.2E+06 —.--200 L/minm -*-  I .OE+06  100 L/minm  —A—50 L/minni  (‘I  E  —ONB 8.OE+05  6.OE+05  72/  ‘f:  4.OE+05  C’)  2.OE+05  0.OE+0O 0  20  40  60  80  100  120  Wall temperature (°C)  Figure 7: Effect of cooling water flow rate Q’ on forced convection regime (Tf= 20°C)  2.3.2  Nucleate boiling The total heat flux removed from the ingot surface by nucleate boiling is obtained by  adding the contributions from forced film convection and nucleate pool boiling. The heat flux for nucleate pool boiling can be estimated by Rohsenow’s semi-empirical equation [4]: Cp,f  (  —  tat) =  fg 1  in which  lg  I !  Cii Cf INPB pflfgJg(pf—pg) kf }  1.7  .  is the latent heat of evaporation in J/kg and  crjg  (5) the surface tension between the  vapour and liquid phase in N/rn. The coefficient Cf for the cooling of an A6063 aluminum alloy with a free falling film of water is given as 0.0 16 [48]. This coefficient has been found to be strongly dependent on the cooling medium and the surface roughness of the cooled sample, and there is some uncertainty surrounding the exponents used in Equation (5) [49]. Because the  14  , uj crfg, C) of water at its saturation temperature are 1 different thermophysical properties (k constant, the heat flux for nucleate pool boiling on an A6063 sample can be simplified to: NPB  3 =20.8(Ts_Tsaj  (6)  in which the temperature difference is (7  -  Tsat), whereas is was (1.  -  Tf) in Equation (3),  because the heat flux is independent of the subcooling in the nucleate boiling regime [4]. The total heat flux for nucleate boiling in the free-falling zone of the direct-chill casting process is then equal to:  NB  =[704[  T 2  1/3  +2.53.1O4J1..J  (i _Tf)+2o.8 (i iag) 3  (7)  The effects of the water flow rate and temperature on the heat flux for the nucleate boiling regime are generally considered to be of secondary importance.  2.3.3  Film boiling The heat transfer through the vapour blanket in the film boiling regime was investigated  by KOhler, Specht and Jeschar [50]. The two main mechanisms of heat transfer for the film boiling regime are heat conduction through the vapour blanket and heat transfer to the water film by convection. The heat losses by radiation, which can be significant in the continuous casting of steel, can be neglected here because of the relatively low casting temperatures of aluminum and magnesium alloys [50]. The use of cooling water whose temperature lies well below saturation leads to the collapse of the steam bubbles within the water film, so that only an insignificant amount of water actually escapes from the system in the form of steam and heat losses by vaporization can also be neglected [50].  15  The heat flux to the water film by convection is proportional to the heat transfer coefficient and the temperature difference in the water film: FB  =  hFB (Iat  —  ) 1 T  (8)  in which Tsat represents the temperature at the steam / water interface and T 1 the temperature in the bulk of the water film. The heat transfer coefficient hFB in Equation (8) depends on the water velocity, which varies locally along the height of the ingot surface because of gravitational acceleration. In many cases, it is not so much the local heat flux which is of interest, but rather the average heat flux over the length of the vapour blanket L. This average heat flux is given by the following equation:  —  FB,AVG  2  kf /(Pe2+2Gnr2_pe3(T \ sat —Tf Gn  (9)  where Fe is the dimensionless Peclet number and Gn the dimensionless gravity number  af Gn=—  (10)  (11)  af  0 is expressed in mis, the characteristic length L is in which the initial water film velocity v /s. 2 measured in m and the water thermal diffusivity af is given in m According to Equation (9), the heat flux for the film boiling regime is a function of the cooling water flow rate via the initial water film velocity v , the water temperature via the 0 thermal diffusivity af and the length of the film boiling zone L, but is independent of the ingot  . This means that for a fixed set of casting parameters, the heat flux is 5 surface temperature T constant in the film boiling region of the boiling curve, and only changes once the Leidenfrost point is reached and transition boiling begins [50]. Experiments conducted by KOhler et al. with 16  a water film flowing down a nickel sample showed a very good agreement with the heat flux calculated with Equation (9).  2.4  Experimental methods Various experiments have been conducted to study the secondary cooling in the DC  casting process. These different experiments can be classified into three types: quench tests, cooling of instrumented samples and industrial measurements of ingot temperature with east-in thermocouples.  2.4.1  Quench tests Quench tests, also referred to as “missile tests”, involve the rapid cooling of a metallic  probe by dropping it into water. The whole body of the sample acts as a transient calorimeter and its temperature is measured by a single thermocouple at the probe center [45, 47, 51, 52, 53]. Missile tests are generally used to quantify the ability of the cooling water to extract heat. It is impossible to define a single number which perfectly describes the complex cooling processes in the secondary cooling zone, but cast shop personnel are usually interested in knowing whether a given cooling water is suitable for a specific casting operation [9, 47]. Heat fluxes measured in quench tests differ significantly from the results obtained with the cooling of larger samples with water jets [42]. This discrepancy stems from the fact that missile tests lack the impingement and convective cooling features which are characteristic of the DC casting process. The immersion of a probe in water thus corresponds more to pooi boiling than to forced convection boiling, so that the measured heat fluxes and transition temperatures between the boiling regimes are different [16].  17  2.4.2  Instrumented samples The secondary cooling of aluminum ingots has also been studied by submitting  instrumented samples to cooling conditions similar to those of the DC casting process (e.g. water jets, water curtain). These samples are generally made of an aluminum alloy, but instrumented samples of silver [26] and nickel [50] have also been used to allow for higher start temperatures. The common technique used for these experiments is to instrument samples with a number of subsurface thermocouples. The temperatures measured by the thermocouples are used as input in an inverse heat conduction (IHC) algorithm to evaluate the heat flux at the boundaries [54]. Samples with a single thermocouple have also been used in other experiments [9]. The measured cooling rate is then compared to a reference cooling rate to evaluate the effect of various cooling arrangements (water flow rate, impingement angle, cooling water quality, etc.). The cooling of the instrumented samples is conducted with water jets which simulate the cooling conditions of the DC casting process. The water jets are generally stationary with respect to the sample [13, 26, 40, 47, 53, 55]: the water then hits the surface at the top to simulate the impingement zone, and the greater part of the sample is covered by a free falling water film. This arrangement, however, does not lead to the vertical temperature distribution in the ingot surface during DC casting, where the water film meets colder regions during its fall. In the instrumented sample cooled by stationary water jets, the temperature is lowest at the top because of the higher heat fluxes in the impingement zone [45]. Experiments have also been conducted with a sample moving downwards or with ascending water jets to simulate the actual cooling conditions of the DC casting process [37, 41, 42]. Boiling curves obtained with moving  18  water jets have been found to differ significantly from the boiling curves for stationary samples [42]. Most experiments conducted with instrumented samples study the cooling in a transient state, i.e. the samples are preheated to a certain start temperature and cooled down to relatively low temperatures (usually below the onset of nucleate boiling) with the water jets. The study of secondary cooling in steady-state, which occurs during the greatest part of the casting process, necessitates the use of a heat source at the back of the instrumented sample [16]. This heat source simulates the latent heat of solidification provided by the metal sump during the actual casting process.  2.4.3  In-situ measurements The measurement of temperatures during the casting of billets or blooms requires the  development of reliable, easy-to-operate instrumentation [47] The usual method consists in allowing an array of thermocouples to freeze into the solidifying metal. Experiments had thermocouples supported by a wire frame [4, 56], inserted in a steel tube [5, 20] and stretched out between the legs of a U-shaped frame [38]. Because of the very high thermal gradients below the surface (especially in the impingement zone), the location of the thermocouples within the ingot must be precisely known [6]. Even when using rigid steel frames to position the thermocouples in the molten sump, the final thermocouple location must be determined in the solid ingot after the experiment. Methods to evaluate the thermocouple location include X-ray photography [5, 56], ultrasound and the careful machining of the ingot in thin slices [20, 38]. Most experiments [4, 6, 8, 13, 38, 54] with cast-in thermocouples were conducted in steady-state, i.e. the thermocouples were immersed in the melt once steady-state had been  19  established. Reports of thermocouples being immersed during the initial filling of the mould in order to measure the temperature during the transient start-up phase are scarce [7, 17]. When measuring the ingot temperature during casting, the arrays of cast-in thermocouples are generally placed on a single level: these thermocouples thus measure the temperature at different depths, but at the same height. Some authors [8, 17, 38] have been using such measurements to evaluate the heat flux at the ingot surface by assuming a one-dimensional heat flux: (12) in which Ax represents the distance between the thermocouple junction and the surface. This assumption is not valid, as axial heat conduction plays a significant role in the direct-chill casting process of light metals. If one-dimensional heat flux is assumed, relatively high surface heat fluxes are measured in the bottom part of the mould but above the impingement zone, where natural air convection should provide only a very small surface heat flux [6]. This temperature drop which takes place before water contact is referred to as the advanced cooling front (ACF) and has been observed by some authors [4, 5, 8, 46]. The ACF is caused by the very high thermal conductivity of aluminum and magnesium alloys, which allows significant heat conduction in the axial direction.  2.5  Effect of parameters on secondary cooling This section presents the results of experiments previously conducted to study the  secondary cooling in the direct-chill casting process. It must be pointed out that the effects of most parameters on the secondary cooling are only reported in a qualitative fashion in the literature. A given parameter will be found to increase or decrease the surface heat flux, or to  20  influence the transition temperature between boiling regimes, but this effect is rarely quantified and thus cannot easily be used in modelling the DC casting process. It must also be noted here that the influence of a given parameter on the heat flux is not always straightforward. For instance, a factor which promotes bubble nucleation will increase the heat flux in the nucleate boiling regime, but may also have an effect on the transition temperature for the occurrence of film boiling and the low heat fluxes which are associated with this boiling regime. Thus this factor may increase or decrease the heat flux, depending on the surface temperature [161. Quantitative experimental results are generally presented as a plot of the heat flux (or the heat transfer coefficient) along the surface, or as a relationship between the heat flux and the surface temperature via a boiling curve. A plot of the heat flux as a function of position can only be used to model steady-state conditions, because the surface heat flux at a given distance below the mould was found to vary significantly during start-up [171. The use of boiling curves for the modelling of the transient start-up phase would appear to be preferable to relationships between the heat flux and the position.  2.5.1  Water flow rate The effect of the water flow rate  Q  on the heat flux varies according to the different  boiling regimes. In the forced convection mode, the water velocity has an important influence on the heat flux. The heat transfer coefficient for falling film convection in Equation (2) (see p. 12) is proportional to the cubic root of the water flow rate  [41.  On the other hand, the heat flux  corresponding to nucleate pooi boiling is independent of the water velocity, so the effect of the flow rate becomes negligible in the nucleate boiling regime [16]. This is valid for the free falling zone of secondary cooling, but also for the impingement zone [57]. An increase in the water  21  1 in the free falling zone, which is known to flow rate also leads to a lower water temperature T improve the heat transfer by convection [161. The effect of the water velocity on the heat flux in the film boiling regime can be evaluated from Equation (9) on p. 16. If the effect of the gravitational acceleration is neglected, Gn tends towards zero and the equation becomes  FB,AVG  _T)  (13)  i.e. the heat flux is proportional to the square root of the water flow rate [50]. Such a non-linear relationship was observed for the heat flux in the film boiling regime [26], but also in the transition boiling regime and for the critical heat flux (CHF) at the end of the nucleate boiling regime [581. Other authors found a linear [43, 59] or quadratic [1] relationship between the heat transfer coefficient and the water flow rate. Generally, the influence of the water flow rate on the heat flux has been found to be more important for the impingement zone than for the free falling zone [36, 41]. However, there seems to exist a limit for the heat flux increase due to the water flow rate, because the heat removal is eventually determined by other limiting factors (e.g. heat conduction within the ingot). Since the heat flux for a given water velocity is higher in the impingement zone than in the free falling zone, this maximal heat flux is reached at lower water flow rates [53]. This phenomenon might explain why some authors found a strong correlation between the water velocity and the heat flux for the free falling zone, but not for the impingement zone [42]. The water flow rate not only influences the heat flux, but also has an effect on the transition temperatures between the various boiling regimes. An increase in water velocity slightly raises the onset of nucleate boiling (ONB) temperature, thus promoting forced convection over nucleate boiling [4]. The effect on the Leidenfrost point is more important, as  22  the increased water velocity promotes the breakdown of the vapour blanket [9, 36, 41]. The relationship between the Leidenfrost point for the impingement zone and the water flow rate has been found to be approximately linear by some authors [26, 59]. Yu [43] identified third-order equations to quantify the Leidenfrost point temperature in the water jet impingement zone and in the water film free falling zone: TLPIJZ  3 —888.56(g) =276.4(g) 2 +1046.2 (Q’)—135.56  (14)  g)+ 94.98  (15)  TLP(FFz  =  2 +612.2 3 —560.11(g) 168.93(g)  where the Leidenfrost temperature TLp( is expressed in °C and the water flow rate per unit of perimeter  2.5.2  Q’ is expressed in gal/minin.  Water temperature The temperature of the cooling water T 1 was found to exert an influence on the heat flux  for all boiling regimes [42]. Water sub-cooling (i.e. lower temperature) increases the heat flux in the forced convection mode [57]. A water temperature above 40°C was found to promote bubble formation (lower ONB), but led to lower heat fluxes in the nucleate boiling regime. In particular, the critical heat flux  CHF  decreases when the water temperature is increased [16, 58,  60]. A higher water temperature decreases the heat flux in the transition boiling regime [53] and promotes the start of film boiling by lowering the Leidenfrost point [61]. The effect of water temperature on the heat transfer coefficient in the film boiling regime hFB was found to be relatively small [26] or even negligible [50], but the heat flux  PFB  is nevertheless decreased with  a higher water temperature, because it is proportional to the heat transfer coefficient hFB and the temperature gradient [16]. Even though the influence of water temperature on the heat flux is generally small compared to the effect of other factors [42, 56], seasonal variations in the water temperature can 23  play a significant role in the secondary cooling of the DC casting process [161. Finally, it must be pointed out here that the water temperature in the secondary cooling zone is not constant but actually increases as the water film flows down and removes heat from the ingot or billet surface [36]; the effects discussed here concern the initial temperature of the cooling water as it comes out of the direct-chill mould.  2.5.3  Water quality The interfacial tension forces in the three-phase system (solid surface, water, vapour)  play a significant role in the boiling phenomena associated with secondary cooling. In particular, the progression of the triple contact lines (where the three phases meet) determines the size and duration of wet contact zones on the ingot surface, which strongly influence the heat flux in the transition boiling regime [45]. Since the surface tension and wetting properties of the water can vary with the addition of very small amounts of oil or dissolved solids, the effect of water quality on the secondary cooling must be considered [47]. This is especially important when cooling water recirculation systems are used: contaminants such as lubricants, water treatment chemicals, dissolved salts and solid particulates can then accumulate in the cooling water during each cycle [9, 52]. An organic fluid (generally castor oil) is used in DC casting as a mould lubricant to prevent the tearing of the ingot surface. The lubricant then enters the cooling water recirculation system and forms a water-oil emulsion [52]. The presence of oil in the cooling water was found to lower the Leidenfrost point, thus promoting film boiling [58]. The effect of lubricant strongly depends on its concentration [9]. At very high oil concentrations, the heat flux decreases and bubble formation becomes the determining factor in the heat transfer, i.e. the thermophysical properties of the metal surface have no effect on the heat flux [53]. A surface tension lowering 24  surfactant is often used as an oil dispersant in the cooling water recirculation system. This surfactant promotes bubble formation: it lowers the Leidenfrost point and dramatically lowers the heat flux in the transition boiling regime [52], but increases the heat flux in the nucleate boiling regime [61]. The term “hard water” refers to the total dissolved solids concentration in the water. The severity of a quench is known to be greater with hard water than with deionized water. 3 were found to raise the Leidenfrost point and thus Dissolved solids such as NaC1 and CaCO promote nucleate boiling over film boiling [16, 61]. Ions can also be purposefully introduced in the cooling water system to serve as coagulants. Whereas positively charged ions (cations) such as Fe , Al 3 3 and organic cationic polyectrolytes reduce bubble coalescence and slightly increase the critical heat flux, the negatively charged ions from anionic polyelectrolytes promote the coalescence of bubbles. The formation of a vapour blanket is then rendered easier and the Leidenfrost point is significantly lower. The critical heat flux is also decreased [52]. The influence of coagulants is however negligible if the concentrations are kept low (e.g. below 100 ppm) [9]. Suspended solids, which are not dissolved in the cooling water and remain in the form of particulates, can act as nucleation sites for the formation of bubbles. They therefore promote film boiling [52] and increase the heat flux in the nucleate boiling regime [61].  2.5.4  Dissolved gases Dissolved, non-condensable gases reduce the vapour pressure required for bubble  2 in the nucleation and thus promote bubble formation [16]. For instance, the injection of CO cooling water has been found to promote film boiling and thus to reduce the cooling rate [9, 61]. Dissolved air also exerts an influence on the secondary cooling: quench tests conducted with 25  boiled water (i.e. free of dissolved air) presented a higher onset of nucleate boiling temperature, as the formation of bubbles was rendered more difficult. On the other hand, the critical heat flux and the heat fluxes in the transition boiling regime were higher [52].  2.5.5  Water jet impingement angle The surface heat flux in the impingement zone depends on the water jet impingement  angle  Ojet.  At low water jet velocity vjeg (ca. 2.0 mIs), the effect of an impingement angle  variation from  900  (perpendicular) to  00  (parallel flow) was found to be significant only in the  forced convection regime; high impingement angles are associated with higher heat fluxes. The heat flux in the impingement zone was thus independent of the water jet angle when the surface temperature was above the onset of nucleate boiling. At higher jet velocities (ca. 10 mIs), high impingement angles lead to high heat fluxes in the forced convection mode, but lower heat fluxes in the nucleate boiling regime [57]. It can be assumed that the greater dynamic pressure of high water jet angles hinders the bubble formation. Moreover, the heat flux for nucleate boiling in the impingement zone was found to be maximal for a water jet angle of 30° [58].  2.5.6  Distance from water jet impingement zone The heat flux generally decreases with increasing distance from the impingement zone  . In the impingement zone, the water jet momentum removes the bubbles and breaks down 1 d the vapour blanket, so that the heat flux is higher than in the free falling zone [36, 41]. The transition temperatures between the different boiling regimes have also been found to be slightly lower in the impingement zone [13]. In the free falling zone, the heat flux generally keeps decreasing with increasing distance from the impingement zone [43, 53, 60]. This is probably due to an increase in the cooling water temperature. Maenner et al. report the observation of a 26  heat flux peak in the free falling zone [13]. This local maximum is attributed to a sharp rise in the turbulence level which accompanies the transition from the stagnation region at the impingement point to the accelerating flow in the free falling zone [62].  2.5.7  Start temperature In tests with instrumented samples, the temperature at the start of an experiment T 0 was  found to exert an influence on the surface heat flux. The heat flux measured during the test is generally smaller if the start temperature is lower [46]. This phenomenon can be observed in the transition and nucleate boiling regimes, where the bubble formation at the surface is known to depend on the thermal gradient within the sample, but not in the forced convection regime [41]. On the other hand, increasing the temperature at the start of an experiment can lead to film boiling and decrease the heat flux [42]. The effect of the start temperature is especially striking in tests with pulsed water: as shown in Figure 8, each pulse corresponds to an independent cooling cycle with a different start temperature and a distinct boiling curve [26].  2.5.8  Surface condition The angles of contact in the three-phase system (solid surface, water, vapour) are not  only a function of the water quality (as described in section 2.5.3) but also of the surface condition. For instance, the very thin aluminum oxide layer at the surface of aluminum ingots is generally associated with a high surface energy and thus a small contact angle. However, this contact angle increases dramatically if the surface is covered by contaminants such as oil [63]. Moreover, the actual area of the solid / liquid interface is greater in the case of rough surfaces. Correspondingly, the equilibrium between the surface tension forces is established at a different contact angle. 27  1.2.  1.0a) C.,  a) 0 C)  0.8-  -  O.6  0.4a) 0.2-  u.u100  I 200  I  I  I  I  300  400  500  Surface temperature (°C)  Figure 8: Heat transfer coefficient with pulsed water [261 (Water cooling flow rate not reported) The relationship between the contact angles for smooth (j and rough (Or) surfaces is given by: cosOr =rcosO  (16)  in which r, the roughness factor, is greater than unity. According to Equation (16), roughness will increase the wettability of a wettable surface (Or < 0<  900)  but increase the contact angle of  a non-wettable surface (Or> 0> 90°) [64]. The effect of the ingot surface roughness on secondary cooling has often been considered to be related to bubble nucleation [16]. However, the surface roughness of cast aluminum or magnesium ingots is generally in the order of millimeters, whereas bubble nucleation sites are a few micrometers in size. The influence of surface roughness has much more to do with the attachment of macrobubbles and stabilization of vapour pockets (because of the reduced drag along curved surfaces) [45]. This is why temperature fluctuations were observed at the surface of smooth, machined samples: several cycles of build-up and collapse 28  were required (because of the instability of the vapour film and the ease of bubble detachment) before a stable cooling rate was established [47]. For the water film free falling zone of secondary cooling, surface roughness decreases the critical heat flux because the formation of large bubble patches is promoted by the surface irregularities [47]. On the other hand, surface roughness increases the heat flux for the nucleate boiling and transition boiling regimes in the impingement zone [41], whereas is has no significant effect in the forced convection mode. It must be noted here that the influence of surface roughness is reported to be small in comparison to the effect of other factors such as water flow rate, casting speed and thermal conductivity of the cast metal [42].  2.5.9  Thermophysical properties The thermophysical properties of the cast metal such as thermal conductivity k, specific and density Ps determine the amount of heat which can be transferred to the ingot  heat  surface. The heat flux in the impingement zone thus increases with increasing thermal conductivity of the aluminum or magnesium alloy. This result is valid for both the nucleate boiling and transition boiling regimes [41, 42]. Quench tests have shown that the dependence of the heat transfer coefficient on the thermophysical properties can be expressed by the following equation [53]: h  =  K  (17)  where K is a constant. The square root term on the right-hand side of this equation is referred to as the intrusion coefficient, heat diffusivity or thermal effusivity  h,  and is expressed in units of  Ks° [65]. An increase in this intrusion coefficient was found to decrease the Leidenfrost 2 J/m 5 temperature, i.e. higher thermal conductivity and specific heat increase the probability of the vapour film breakdown [59]. 29  2.5.10 Ingot size and geometry The shape and size of the cast product (rectangular ingot or billet) seems to have an effect on the heat flux during secondary cooling. Critical heat fluxes  ‘PCHF  of 5 MW/rn 2 and 3.5  2 were respectively found for a 1600 x 600 mm rolling ingot and a 216 mm diameter MW/rn billet [8, 16, 38]. The critical heat flux also appears to take place at slightly lower temperatures for smaller billet diameters [8]. The dimensionless Biot number, which is calculated with the following equation: Bi=--  (18)  indicates that the effect of changes in the heat transfer coefficient h is more important in ingots with a small diameter D, whereas the heat conduction plays a determining role in larger ingots with a low thermal conductivity k and large Biot number Bi. Particular care must be observed when determining the influence of ingot size on the heat flux, as smaller ingots are usually cast with a greater speed. The apparent effect of the ingot size could thus be actually related to the casting speed [15].  2.5.11 Casting parameters A higher casting speed v requires a corresponding increase in the heat flux, as more heat must be removed from the ingot [4]. One consequence of an increase in the casting speed is higher surface temperatures in the secondary cooling zone, especially in the impingement zone [42]. This increases the time spent in the film boiling regime at start-up, but increases the heat flux once steady-state has been established and nucleate boiling takes place [26, 38]. The mould filling time also influences the surface temperature at the mould exit and determines the importance of film boiling during the process start-up [13].  30  2.6  Summary The boiling water heat transfer in the secondary cooling zone is the result of phenomena  which take place on a relatively small scale: forced convection, formation and collapse of steam bubbles, vapour blanket breakdown, etc. It would thus make sense, when analyzing the effects of these phenomena, to only consider parameters which are meaningful on this scale. Such parameters are the cooling water flow rate, the different water properties, the alloy thermophysical properties at the ingot surface, the surface roughness and geometric considerations such as the distance from the impingement zone and the water jet impingement angle. On the other hand, parameters such as the ingot size, the casting speed and the casting temperature relate to the casting process as a whole and ought to be meaningless when studying the different boiling water phenomena which occur at the ingot surface. The relationships developed to determine the surface heat flux in the secondary cooling zone should therefore be independent of these large-scale parameters. Finally, the effects of the initial temperature, which were observed in cooling tests with instrumented samples, cannot be directly used to model the full-scale DC casting process, for which there is no clearly defined initial temperature. A different approach would thus have to be developed to take this parameter into account. Boiling curves for the direct-chill casting process were evaluated by various authors and compiled first by Maenner et al. [13], then by Opstelten and Rabenberg [42]. Different boiling curves show very good agreement with each other in the forced convection and nucleate boiling regimes, at low surface temperatures. However, those curves present a significant scatter in the high temperature boiling regimes (transition boiling, film boiling). During the steady-state phase of the DC casting process, the surface temperatures are relatively low and the corresponding heat fluxes are well known. Because of the very high heat flux in the nucleate boiling regime, heat transfer is not limited by the heat extraction by the 31  cooling water, but rather by the heat conduction within the cast product (ingot, billet) [14, 15]. During the start-up phase, on the other hand, high surface temperatures can lead to film boiling, transition boiling and to the ejection of the water film. Accurate knowledge of these phenomena and the conditions in which they take place is required in order to precisely control the rate of heat removal in the secondary cooling zone and avoid the initiation of defects. Finally, it must be pointed out here that the greater part of the reviewed literature concerns the heat flux at the surface of an aluminum alloy ingot. Data for the direct-chill casting of magnesium is limited to a single article [20] which estimates the heat transfer coefficient as a function of position and for different casting parameters (casting speed, water flow rate). The effect of material properties such as thermal conductivity and specific heat on the surface heat flux is not known with sufficient precision to allow a translation of the results obtained for the DC casting of aluminum into boiling curves for the case of magnesium casting. The different surface roughness of aluminum and magnesium ingots may also play a role in determining the heat flux in the secondary cooling zone.  32  3  SCOPE AND OBJECTIVES  3.1  Objectives The overall objective of this research project was to develop a greater and more  quantitative understanding of the boiling water heat transfer which takes place in the secondary cooling zone of the DC casting process. A particular emphasis was put on high temperature boiling regimes (transition and film boiling), for which data from the literature showed significant discrepancies. High temperature boiling mechanisms are also of critical importance during the start-up phase of the direct-chill casting process, when defects are known to be initiated. The effects of parameters on different characteristics of the boiling curve were identified in a quantitative fashion so as to be implemented in a mathematical model of the DC casting process. Specifically, the sub-objectives of this research work included: • Quantifr the influence of cooling water flow rate, thermophysical properties and surface roughness on the boiling curve; • Obtain boiling curves for the secondary cooling of magnesium alloys; • Investigate the role of parameters such as the casting speed and the initial sample temperature on the boiling water heat transfer; • Develop an algorithm to produce boiling curves for a wide range of operating conditions; • Evaluate the significance of secondary cooling in the transient start-up phase.  33  3.2  Methodology The methodology adopted for this research project was multi-faceted and involved  experimental measurements, inverse heat conduction analysis and finite element modelling of the direct-chill casting process. The experimental measurements were conducted with two light metal alloys: aluminum AA5 182 (4.5% Mg, 0.35% Mn) and magnesium AZ31 (3.0% Al, 0.8% Zn, 0.3% Mn). These measurements thus provided two sets of thermophysical properties so as to study the effect of thermal conductivity, specific heat and density on the different boiling water heat transfer mechanisms. Secondary cooling was simulated by cooling stationary and moving samples with jets of water. Instrumented samples provide a more realistic representation of the thermal history experienced during DC casting compared with quench missiles while remaining easy to model in the inverse heat conduction analysis. Instrumented samples also generally allow a wider range of experimental conditions than in-situ measurements. The inverse heat conduction analysis was conducted using an algorithm specifically designed for this research work. This algorithm was able to take into account the particularities of secondary cooling such as the advanced cooling front and the ejection of the free falling water film. The two-dimensional inverse heat conduction algorithm was validated using a commercial finite element package prior to its use. A two-dimensional axi-symmetric Finite Element Model (FEM) of the full-scale DC casting process was developed to study the effect of secondary cooling during the transient start up phase. This model was based on an existing FEM model [661 for the direct-chill casting of an AZ3 1 billet, but implemented the findings of this research work as boundary conditions in the secondary cooling zone. The FEM model only took into account the thermal field within the  34  billet. The potential for defect formation was not investigated using a stress-strain model but rather with simple correlations observed in industrial settings. Finally, the results obtained from cooling experiments on AA5 182 and AZ3 1 alloys were complemented by data from previous research work conducted by Li [41] on the same experimental water jet rig. This provided data for two more aluminum alloys (AA1O5O and AA3004) and thus for two extra sets of thermophysical properties.  35  4  EXPERIMENTAL  4.1  Set-up The experimental water jet rig used to investigate the heat transfer in the secondary  cooling zone consisted of an electrical furnace, a pneumatic displacement system, a water box and an instrumented sample. This setup is illustrated in Figure 9.  Water box  Water jets Test sample Impingement point -Water ejection Thermocouples  As-cast surface  Figure 9: Schematic of the water jet experimental rig The electrical furnace was used to heat up the instrumented sample to the desired start temperature. Initial test temperatures T 0 between 250 and 600°C were investigated. The pneumatic displacement system could move the instrumented sample in order to simulate “casting speeds” v between 10 and 40 mm/s. These values had to be approximately one order of magnitude greater than the speeds used in the actual direct-chill casting process because of the much smaller amount of heat contained in a solid sample compared to a cast DC ingot with a liquid metal core. Lower sample moving speeds would have led to a rapid cooling of the surface as if the sample was stationary. The water box acted as a section of the direct-chill mould and was designed to represent actual industrial conditions for DC casting. Baffles within the water box ensured a uniform 36  distribution of the water flow across its width, and holes at the bottom of the water box produced a series of distinct water jets (rather than a water “curtain”). The water flow rate was adjusted with a flow meter and kept constant during a cooling experiment. When the water supply was turned on at the start of an experiment, a shield prevented water from entering into contact with the sample until steady-state flow conditions were attained. The shield was then removed and the sample was cooled by the water jets for a given amount of time. The original water jet rig could provide cooling water with a linear flow rate density (per unit width)  Q’ between 85  and 150 L/minm. In the original design of the experimental cooling  rig, lower flow rates did not produce distinct water jets: because of the lower hydrodynamic pressure in the water box, the cooling water simply dripped out of the holes. In order to expand the capabilities of the water jet rig, the holes were made smaller by the addition of tubular inserts. This led to a higher water jet velocity vjet for the same water flow rate. The greater jet velocity allowed linear flow rate densities as low as 50 L/minm, but the increased pressure drop within the water box limited this flow rate density to 150 L/minm. Typical cooling water flow rates in the DC casting process are 50 L/minm during the start-up and 150  -  200 L!minm in  steady-state. The water temperature during a cooling test could not be measured. The very high temperatures maintained in the electrical furnace generally led to a heating up of the water box and, correspondingly, a variation in the cooling water temperature. However, this water temperature stabilized at the beginning of each quench experiment before the shield was removed. The cooling water quality (e.g. dissolved solids, gas content, contamination by oil) was not evaluated, but assumed to be constant for all tests as it was taken from the main water supply at the University of British Columbia.  37  The experimental samples were taken from an industrially DC cast AZ3 1 bloom produced at Timminco Metals in Haley, Ontario, and an industrially DC cast AA5 182 ingot produced at Alcan in Jonquière, Québec. Both samples were taken from positions in the industrial as-cast material that would be produced under steady-state conditions, i.e. the surface roughness of the experimental samples was typical of that produced during steady state industrial DC casting for these two alloys. The characteristic “wavy” surface roughness of direct-chill cast AZ3 1 ingots is shown in Figure 10. The distance between two consecutive “waves” is approximately 20 mm. AA5 182 ingots, on the other hand, present a “pitted” surface which is illustrated in Figure 11. The industrially cast AZ3 1 bloom had a slight taper along the rolling side from the centre of the rolling face to the edge. AZ3 1 samples were taken from the mid-width location, so that the curvature along the experimental sample was minimized. This was not an issue with the AA5 182 ingot material.  a  i  6  2  7  8  3  9  O  4  11  12  13  5  14  15  17  16  6 H  Figure 10: Typical as-cast surface of AZ31 ingot 38  The samples had dimensions of 100 x 150 x 250 mm along the x, y and z axis respectively. The sample blocks were instrumented with sub-surface thermocouples and fixed to the pneumatic displacement system so that the as-cast surface would be exposed to the water cooling. The samples were also insulated at the bottom to minimize heat losses through the pneumatic displacement system.  Figure 11: Typical as-cast surface of AA5182 ingot  4.2  L411  Cooling tests Three different types of tests were performed in this research project. These experiments  included cooling tests with a sample which was stationary with respect to the water box and two types of tests with a moving sample: wetting and rewetting tests. In tests done with a stationary sample, the water jet impingement point was located in the upper part of the sample and a free falling film of cooling water went down from this impingement point. Because of the high initial 39  sample temperature, the water film was ejected from the surface and an ejection front progressed downwards as the surface cooled down. Stationary sample experiments provided much information about the free falling zone, but only a limited amount of data for the impingement zone. This is because the heat removal in the very small impingement zone could only be estimated based on a single thermocouple. Wetting tests were conducted with a relatively low initial temperature T 0 and a low moving speed v. These tests simulated the secondary cooling in the steady-state phase of the direct-chill casting process. In this type of test, the water jet impingement point started at the very bottom of the sample and went up along the surface as the sample was brought down by the pneumatic displacement system. The heat removal in the impingement zone could thus be measured by each sub-surface thermocouple. However, information about the water film free falling zone was limited to low temperature boiling regimes, i.e. forced convection and nucleate boiling, because the surface temperature was always relatively low once this zone was reached. Rewetting tests were done with a fast moving sample v and a high initial temperature . This led to film boiling and limited the amount of cooling taking place in the impingement 0 T zone. At the top of the free falling zone, the surface temperature was still relatively high and the water film was ejected from the surface. When the displacement system reached the end of its course, the sample became stationary and a rewetting front progressed downwards along the surface. Rewetting tests constituted a very realistic simulation of the start-up phase of the DC casting process. They provided information for the film boiling regime in the impingement zone as well as conditions for the rewetting of the surface by the free falling water film. All three types of cooling tests led to a gradual wetting of the sample surface and the formation of an Advanced Cooling Front (ACF). Stationary sample experiments featured a downward ACF when the water film ejection front progressed along the surface. Wetting 40  experiments involved an upward advanced cooling front because of the relative motion between the water jets and the instrumented sample. Finally, rewetting experiments led to two ACF phenomena: an upward advanced cooling front as the water jet impingement point went up along the surface and a downward advanced cooling front when the water film rewetted the sample surface. In total, 20 tests were conducted with AZ3 1 samples (16 stationary tests and 4 wetting tests), whereas 31 tests were done with an AA5 182 sample (17 stationary tests, 8 wetting tests and 6 rewetting tests). A table summarizing the different cooling test conditions can be found in Appendix A.  4.3  Thermocouples The internal temperature of the sample was measured by type “E” thermocouples which  were inserted from the back of the sample. As shown in Figure 9, the thermocouples were positioned at mid-width along the height of the sample. 1.6 mm diameter holes were drilled in the sample and the thermocouple tips were positioned approximately 1 mm from the quenched surface. Because of the surface roughness of AA5 182 and AZ3 1 alloys, the thermocouple depth could not be set with sufficient precision. The distance between the tip of a thermocouple and the quenched surface was therefore measured after the experiment by cutting the sample in slices. An accurate knowledge of the thermocouple depth is critical when conducting an inverse heat conduction analysis [41, 67]. The data acquisition rate during a cooling test was set to 50 Hz. The temperature signal presented a random noise amplitude of approximately 0.2°C. In order to further smooth the temperature data, the signal was filtered before being input in the inverse heat conduction program. This was rendered necessary by the very high sensitivity of the inverse heat 41  conduction analysis to noise. The filtering method used a combination of statistical tools: a 5point median, in which each temperature was replaced by the median of five consecutive measurements, removed individual peaks while conserving the general trend of the original data, whereas a 5-point average, in which each temperature was replaced by the average of five consecutive measurements, led to a general smoothing of the temperature data. The effect of the filtering algorithm is shown in Figure 12. 467  467  465  465  463  463  2  2 461  461 9  5 459  459  457  457  455  455 5  6  7  8 Tinle (s)  9  10  5  6  7  8 Time (s)  9  10  5  6  8  7  9  10  Time(s)  Figure 12: Effect of filtering algorithm on temperature data a) Raw signal; b) After 5-point median; c) After 5-point average The presence of sub-surface thermocouples inserted perpendicular to the cooled surface disturbs the temperature field and introduces errors in the temperature measurements. This is due to the significantly lower thermal conductivity of the thermocouple hole compared to the high thermal conductivity of aluminum and magnesium alloys. When the surface in front of a thermocouple is cooled, the transfer of heat from the back of the sample is rendered more difficult by the presence of the thermocouple hole. This leads to a significantly greater cooling at the thermocouple tip and a temperature discrepancy compared to the undisturbed case of a sample without any thermocouple holes. This phenomenon was corrected using the “equivalent thermocouple depth” method recently developed at the University of British Columbia [68, 69].  42  5  INVERSE HEAT CONDUCTION ANALYSIS Direct heat conduction problems are relatively straightforward: the boundary conditions  and material properties are known and the temperature profile (or, in the case of transient problems, the temperature history) is sought. The inverse heat conduction (IHC) problem consists in evaluating either specific material properties or the external boundary conditions using internal temperature measurements [70]. Examples of boundary conditions sought in an IHC problem are surface temperatures, heat transfer coefficients and heat fluxes. Transient inverse heat conduction problems are ill-posed and a regularization method must be used to avoid instabilities [54].  5.1  Formulation of the IHC algorithm The IHC algorithm used to evaluate the heat flux at the surface of the water-cooled  sample was based on the linear relation between the measured thermocouple temperature TTC and the unknown surface heat flux q• Such a relation is commonly observed in analytical solutions for simple initial and boundary conditions [711. However, this relationship is not linear if thermophysical properties such as thermal conductivity and specific heat are temperature dependent. The non-linear heat conduction problem was simplified by assuming that temperature dependent thermophysical properties (ks, C,,,, Ps) were constant for a given time step. This assumption is valid provided the time step length ut is short enough and the temperature variation within this time step does not significantly modify the different thermophysical properties. Whereas most IHC algorithms use a least squares approach to minimize the difference between the measured and calculated temperatures, the algorithm developed for this research work relied on exact matching, i.e. the temperatures calculated in the IHC analysis were set to 43  be exactly equal to the temperatures measured by the thermocouples. With n sub-surface thermocouples in the instrumented sample and thus n temperature measurements at each time step, n heat flux values are required to obtain a linear system with only one valid solution. These discrete n heat flux values were used to model a continuous heat flux profile along the sample surface. The linear system of n equations and n unknowns was solved using sensitivity coefficients, which represent the temperature change at a given thermocouple location for an arbitrary heat flux change. The relationship between the actual heat flux change and the measured temperature change was evaluated by inverting the n x n sensitivity coefficient matrix using the Gauss-Jordan matrix inversion technique.  5.2  Direct heat conduction problem The n x n sensitivity coefficient matrix was built by solving a series of direct heat  conduction problems using the finite element method (FEM). A two-dimensional thermal model was used to represent the cooling of the instrumented sample. This two-dimensional model considered the central section of the sample, where the thermocouples were located, and in which heat was conducted in the x-direction towards the quenched surface and in the z-direction along the height of the sample. Heat flow in the y-direction was considered negligible because the water flow was evenly distributed across the width of the sample during cooling experiments. Figure 13 illustrates the two-dimensional plane considered in the IHC analysis. Heat losses by natural convection and radiation on the sample sides were considered insignificant compared to the boiling water heat transfer at the quenched surface. Typical heat fluxes for natural air convection are in the order of 10  -  40 kW/m . The maximal heat loss by 2  . The heat fluxes 2 radiation (assuming an emissivity of 1.0) for a 600°C surface is ca. 30 kWIm 44  associated with boiling water heat transfer, on the other hand, are generally in the order of , i.e. two orders of magnitude greater than the ones for natural air convection or 2 MW/rn radiation. Moreover, the different boiling water heat transfer phenomena take place at the quenched surface, i.e. 1 mm away from the thermocouple junctions. In comparison, heat losses by natural convection and radiation occur 75 mm away from the central section and thus do not significantly affect the temperature measured by the thermocouples, even though they may exert influence on the thermal history in other regions of the sample.  y  Figure 13: x-z two-dimensional plane at the center of the instrumented sample The direct heat conduction problems were solved using the finite element method (FEM) algorithm presented in Appendix B. The central section of the instrumented sample was modelled with a mesh made of rectangular linear elements. Preliminary trials showed that a 1651 element mesh with 1 mm thick elements below the quenched surface provided an accurate solution for the direct heat conduction problem, i.e. the solution did not significantly change if a finer mesh was used. Figure 14 shows how the mesh elements are larger (up to 20 mm) at the back of the sample.  45  0.050 0.045 0.040 0.035 0.030 0.025  —  —  0.020  —  0.015 0.010  —  0.005 0.000 0.00  —  0.02  0.04  0.06  0.08  0.10  x-axis (m) Figure 14: Section of FEM mesh used to model instrumented samples A heat flux profile formed with the n discrete heat fluxes was applied on one face of the mesh, whereas the other boundaries were adiabatic. The sample was at a relatively uniform temperature before the start of a cooling test, and the initial temperature at any given node was set as the first temperature measurement at the closest thermocouple location. The time step length /Jt chosen for the transient heat conduction problem was equal to the data acquisition period, i.e. 0.02 s. Temperature dependent thermophysical properties of AA5 182 and AZ3 1 are presented in Table I:  Table I: Thermophysical properties of AZ31 and AA5182 alloys [20, 41, 66] Property  AA5182  Thermal conductivity k (W/ntK) Specific heat  (J/kgK)  Density 3 (kg/m p )  AZ31  118.3 + 0.1094 T 88.0 + 0.0800 T 897 + 0.452 T  1014 + 0.500 T  2650- 0.1940 T  1772- 0.360 T  46  5.3  Function specification method The inverse heat conduction algorithm developed for this research work was based on  the Beck’s function specification method: it assumed a linear form for the variation of the surface heat flux in the next m future time steps [72]. Sensitivity coefficients were determined by first evaluating the “base” temperature at the different thermocouple locations. Direct heat conduction problems were solved for m future steps using the FEM model described earlier. This first series of FEM simulations assumed constant heat flux values, as shown on Figure 15. The “base” temperature corresponded to the temperature obtained at a thermocouple location if none of the heat fluxes changed during the next m time steps.  D (U (U  z  t-3  t-2  t-1  t  t+1  t+2  t+3  •..  t+rn  Time steps Figure 15: Constant heat flux ‘P for the evaluation of the “base” temperature For each of the n heat fluxes, a number of m direct heat conduction problems were also solved while assuming an arbitrary linear increase of the heat flux z1, as shown in Figure 16. The sensitivity coefficient for a given thermocouple location and a given heat flux was obtained  47  by substracting the “base” temperature from the temperature calculated after “m” time steps of heat flux increase, then dividing by the applied heat flux variation z1 s.  = “i,lin  “i,base  —  (19)  f 4 Z  where T , is the temperature in °C at thermocouple location i when heat flux j is increased by 1 7 (in W/m z12 ) at every time step, whereas Tj,base is the temperature at location i when all heat 2 fluxes are kept constant.  >c:  D  ci)  z  F  t-3  t-2  t-1  I  t+1  1+2  t+3  •..  t+rn  Time steps  Figure 16: Linearly increasing heat flux for the evaluation of sensitivity coefficients Each series of m direct problems thus allowed the calculation of n sensitivity coefficients, one for each thermocouple location. The actual heat flux increases were evaluated by inverting the n x n sensitivity coefficient matrix: (20) where the (uT) temperature variation vector was obtained by subtracting the “base” temperature at thermocouple location i from the temperature actually measured by the corresponding  48  thermocouple TTC after m time steps. The heat flux values were calculated by adding the heat flux variation vector {zFZ} to the previous heat flux values, after which a final direct heat conduction problem was solved using those new heat flux values in order to evaluate the temperatures after one single time step. The complete temperature profile within the sample and the calculated surface heat fluxes were kept in memory for the next cycle. Figure 17 shows a schematic representation of the IHC algorithm. Input of mesh information and material properties  Heat flux and surface temperature data  FEM problem (constant heat fluxes) FEM problem (linear increase of &D)  FEM problem (calculated heat fluxes) ‘  Calculation of sensitivity coefficients S 1 Sfl  Calculation of heat fluxes  FEM problem (linear increase of I)  Inversion of sensitivity coefficient matrix I  Calculation of sensitivity coefficients S 12 S 22 S n2 ...  Calculation of sensitivity coefficients S in S 2n S nfl ...  FEM problem (linear increase of LW,)  Figure 17: Algorithm of the inverse heat conduction analysis The number of future time steps m to consider in the function specification method depends on the time step length, the thermocouple depth, the noise level in the temperature measurements and the thermophysical properties. A short time step length, a large thermocouple depth or a low thermal diffusivity all reduce the temperature change measured by a thermocouple. If the number of time steps is insufficient, this temperature change cannot be distinguished from the signal noise and the calculated heat flux values are unstable. An  49  exaggerated number of time steps, on the other hand, considers an average of the future temperature history and can miss significant but short temperature variations. This phenomenon tends to “flatten” the boiling curve and underestimates peak heat flux values. The optimal number of future time steps was found by trial and error. An IHC analysis was first conducted with two future time steps, and this number was gradually increased until the calculated heat flux values were stable, i.e. did not diverge. A number of three to seven future time steps was generally found to produce good results.  5.4  Characterization of surface heat flux profile The inverse heat conduction algorithm developed for this research relied on an exact  matching scheme: it considered temperature measurements from a discrete number n of thermocouples and used n heat flux values to describe the continuous heat flux profile along the cooled surface. Figure 18 illustrates two ways in which the surface heat flux can be modelled using a limited number of discrete values. In order to determine the best way to model the surface heat flux profile at the surface of a stationary sample quenched with water jets, a typical cooling test was modelled using the finite element method. The FEM model used two boiling curves to evaluate the surface heat flux profile: a boiling curve for the impingement zone (IZ) at the top of the sample and a boiling curve for the free falling zone (FFZ) for the rest of the sample surface. As can be seen on Figure 19, the boiling curve for the water film free-falling zone featured a very small heat flux for surface temperatures above 350°C. This small heat flux corresponded to natural air convection below the water film ejection point.  50  0.30  0.30  0.25  0.25  0.20  0.20 2  x 4  0.15  0.15  4 0.10  0.10  0.05  0.05  0.00 0.OE+00  0.00 0.OE+00  1.5E+06  3.OE+06  4.5E+06  Heat flux (W/m ) 2  1.5E+06  3.OE+06  4.5E+06  Heat flux (W1rn ) 2  Figure 18: Heat flux profiles a) Stepwise function; b) Linear function 7.OE÷06 6.OE+06 5.OE+06 2  4.OE+06 3.OE+06 2.OE+06 I .OE+06  0.OE+00 0  100  200  300  400  500  Surface temperature (C)  Figure 19: Boiling curves for the FEM simulation of a cooling test  51  The FEM simulation of the cooling test used a 100 x 250 mm rectangular body formed of 1792 nodes and 1651 rectangular elements. The boundary conditions were adiabatic on three of the four sides. A heat flux calculated using the impingement zone boiling curve was applied on a 4 mm long segment of the fourth face. The boundary condition on the rest of that face was evaluated using the free falling zone boiling curve. These two types of boundary conditions are indicated on Figure 20, which shows a section of the FEM mesh used in the simulation of a 0 of each node was 400°C, i.e. above the water film ejection cooling test. The start temperature T temperature. Simplified constant thermophysical properties were used in the FEM model and are given in Table II. 0.26  IP  ——  0.25  0.24  N  FFZ  ——-  0.23  0.22  0.21 0.00  0.02  0.04  0.06  0.08  0.10  x-axis (rn) Figure 20: Section of FEM mesh and boundary conditions for cooling test simulation Figure 21 shows the heat flux along the cooled surface after 18.0 seconds and constitutes a typical example of the surface heat flux profile in a cooling test with a stationary sample. The 2 in the water film heat flux is relatively high in the impingement zone, drops to ca. 1.0 MW/rn free falling zone, then presents a very sharp peak which corresponds to the nucleate and 52  transition boiling regimes. Below that peak, the surface temperature is above 3 50°C, i.e. the water film is ejected and the dry surface is cooled by natural air convection.  Table II: Thermophysical properties used in the FEM simulation of a cooling test Property  Value  Thermal conductivity k (W/mK)  150  Specific heat  1100  (J/kgK)  1750  Density Ps (kg/rn ) 3 0.25  0.20  I  0.15  0.10  0.05  0.00 0.OE+00  1 .0E+06  2.OE+06  3.OE+06  4.OE+06  5.OE+06  Heat flux (W1rn ) 2  Figure 21: Typical heat flux profile during cooling test with stationary sample Such an irregular heat flux profile cannot be accurately represented by the simple functions illustrated in Figure 18 on p. 51. If a stepwise or linear function is used to model the surface heat flux, the temperature drop measured as the heat flux peak approaches the level of a given thermocouple is translated into an increased heat flux in front of this thermocouple. In other words, the effect of the advanced cooling front is interpreted as boiling water heat transfer taking place at the surface. This in turn leads to significant inaccuracies in the resulting boiling 53  curve, especially for the high temperature boiling regimes. The IHC algorithm used to analyze a cooling test thus needs to take into account this rapid increase in the heat flux which takes place at the rewetting front and track the position of the rewetting front during each cooling test.  5.5  IHC analysis with moving boundary The FEM model used in the IHC analysis was modified to keep track of the peak in the  heat flux profile at the rewetting front and its progression during a cooling test. This model formed the continuous surface heat flux profile using n+1 heat flux nodes. The first n heat flux nodes were stationary and positioned at the surface, right in front of the n thermocouples, i.e. they corresponded to n given nodes in the FEM mesh. The last node was not fixed in space but followed the position of the heat flux peak, i.e. it corresponded to the moving boundary between the wet and dry regions of the surface. The heat flux value at each stationary node was calculated by inverting the sensitivity coefficient matrix, as explained earlier, whereas the value of the heat flux at the extra node was set as the maximum heat flux encountered previously in the heat flux node above it. The introduction of this extra heat flux node therefore did not change the number of unknowns in the linear system of n equations. The heat flux between adjacent heat flux nodes was calculated with a linear interpolation, except for the region below the moving heat flux node where it was constant. Figure 22 shows an example for the heat flux profile used in the inverse heat conduction analysis. Figure 23 shows the position of heat flux nodes with respect to the thermocouples and the FEM mesh. When the heat flux peak approached a stationary heat flux node, the IHC analysis could differentiate between the two cooling mechanisms: axial heat conduction towards the heat flux peak and transverse heat conduction towards the quenched surface.  54  0.30  0.25  0.20  E 0  0.15 boundary 0.10  0.05  0.00 0.OE+00  1.5E+06  3.OE+06  4.5E+06  ) 2 Heat flux (W/m  Figure 22: Linear heat flux profile with moving boundary 0.050  . .  0.045  .  .  .  .  :  0.040 0.035  0.030 0.025  —  0.020  —  -  —  -  —  -  0.015 —  • Thermocouple  0.010  o Heat flux node 0.005  0.000  1/  0.06  —  0.08 x-axis (m)  —  o Moving heat flux node  —  0.10  Figure 23: Heat flux profile with moving boundary and FEM mesh with thermocouples  55  The position of the wetting front and corresponding heat flux node was determined prior to the actual IHC analysis by evaluating the second derivative of the measured temperature with respect to time during the cooling test. The second derivative was calculated using discrete temperature measurements according to Equation (21): —  —  ÷ —27; 1 T  +  (21)  2 (At)  This second derivative (i.e. the rate of change in the cooling rate) reaches a minimum when the heat transfer mechanism is changed from axial conduction to boiling water heat transfer [73). The time at which this minimum was observed for a set of temperatures measured by a sub-surface thermocouple thus corresponded to the arrival of the wetting front at the level of this thermocouple. At this specific time, the stationary heat flux node in front of this thermocouple and the extra heat flux node representing the heat flux peak were at the same position. The value of the heat flux peak, which was defined as the maximum value experienced at the previous heat flux node, was then reset to the current heat flux value at the stationary node. Figure 24 shows an example for the temperature data analysis based on the calculation of the second derivative. Each set of temperature measurements from a sub-surface thermocouple provided a time at which wetting was observed, and the progression of the wetting front between consecutive thermocouples was assumed to proceed at a constant speed. As shown in Figure 25, which compares the times at which the minimum second derivative was observed and the progression of the wetting front (i.e. the point at which the surface temperature is 3 50°C in accordance with Figure 19), this assumption is valid for the case of a stationary sample.  56  1000  450 400  500  350 0  C”  (I)  300  ?  -  250  -  E I-  -1000 200 -1500  -2000 10.0  150  11.0  12.0  13.0  14.0  15.0  100 16.0  Time (s)  Figure 24: Second derivative of modelled temperature signal d T/d? 2  0.25  0.20  0.15  0.10  0.05  0.00 0.0  2.5  5.0  7.5  10.0 Time (s)  12.5  15.0  17.5  20.0  Figure 25: Wetting front progression during FEM simulation of a cooling test Equation (21) on p.56 uses three discrete temperature measurements to evaluate the second derivative of the temperature with respect to time. Correspondingly, the noise in the 57  original temperature signal was multiplied by three for the discrete second derivative calculations. It then became difficult to accurately pinpoint the minimum in the second derivative and thus the time of arrival of the wetting front. Filtering of the temperature signal or of the second derivative calculations could reduce the amount of noise, but only to a certain degree: an error of several seconds could still exist for the time at which the wetting front reaches a given thermocouple. The inverse heat conduction analysis was used to assess whether the calculated time of arrival was accurate or not. Figure 26 illustrates the effects of an error in the heat flux peak arrival time  tpeak.  If the heat flux peak corresponding to the wetting front reaches a given heat  flux node too early during the IHC analysis  (tpeak  =  11.66 s  <  11.90 s), i.e. before the actual  effects of the advanced cooling front are measured by the corresponding thermocouple, the IHC analysis must compensate for this exaggerated cooling by decreasing the heat flux value at the stationary heat flux node. When the advanced cooling front causes the temperature at the thermocouple location to drop, the IHC analysis interprets this cooling as the result of boiling water heat transfer and produces a “flat” boiling curve with a relatively low heat flux value. On the other hand, if the heat flux peak reaches the heat flux node too late  (tpeak  12.20  S>  11.90  s), the effects of the ACF have already been attributed to boiling water heat transfer, and the actual boiling water heat transfer is partially attributed to the advanced cooling front. The IHC analysis then produces a “flat” boiling curve with a sharp heat flux drop. The accurate times for the progression of the wetting front had to be found by trial and error: after each run of the IHC algorithm, the wetting front progression data were adjusted in accordance with the shape of each boiling curve obtained.  58  6.OE+06  6.OE+06  D tpak  11.66  S  *tpeak  11.90  S  +tpeak  12.20  S  4.OE+06  2 3.OE+06  Z 2.OE+06  1.OE+06  0.OE+00 11.0  11.5  12.0  12.5  13.0  Time (s) Figure 26: Calculated heat flux øwith different values of Ipeak  5.6  Validation of the IHC algorithm The IHC algorithm was validated using the FEMLAB commercial finite element  package. A 2-D 50 mm x 200 mm rectangular body was meshed with 2483 triangular quadratic elements and subjected to a variable heat flux on one of its 200 mm faces. The thermophysical properties were assumed to be constant for the FEMLAB simulations and are given in Table III.  Table III: Thermophysical properties used in the FEMLAB simulation Property  Value  Thermal conductivity k (W/mK)  96.0  Specific heat  1200  (J/kgK)  Density Ps (kg/rn ) 3  1800  59  0 was uniformly set at 500°C. The surface heat flux The initial temperature of the body T was applied over 5 seconds using time steps of 0.01 seconds. The heat flux varied slightly along the surface according to an arbitrary function f(z), but presented a significant variation with time. Equation (22) was used to determine the surface heat flux: 6 [f(z)][sin2.J] ø(y,t)=—l.10  (22)  2 andf(z) a function of the height z where t is the time in seconds, the surface heat flux in W/m (in m) and given by Equation (23): (23)  2 f(z) = 0.700 + 5.00 z —20.0 z  Temperature data was interpolated for ten thermocouple locations 1.0 mm below the cooled surface and used as input in the IHC analysis. Since the FEMLAB simulation did not feature the progression of a rewetting front, the IHC algorithm was used without the additional moving boundary. As can be seen in Figure 27, the calculated heat flux followed the applied heat flux closely, except in regions where the slope changes rapidly.  1 .2E+06 1 .OE+06  8.OE+05 6.OE+05 4.OE+05 2.OE+05 O.OE+OO 0.0  1.0  3.0  2.0  4.0  5.0  Time (s)  Figure 27: Applied and calculated heat fluxes 0 for validation of IHC algorithm 60  The IHC algorithm was also validated using the FEM model described in Section 5.4. A 100 x 250 nmi rectangular body initially at a uniform temperature of 415°C was cooled on one side according to the idealized boiling curves shown in Figure 19 (see p. 51). The thermophysical properties used in this analysis are presented in Table II on p. 53. The FEM simulation provided temperature data at discrete locations 1 mm below the cooled surface. This temperature history was used as input in the IHC analysis. Figure 28 compares the original boiling curve for the free falling zone, and three calculated boiling curves. The first calculated boiling curve was obtained with a onedimensional IHC analysis. It is shifted to higher temperatures compared to the original boiling curve, because it attributes the effect of the advanced cooling front to boiling water heat transfer. The second boiling curve was calculated using a two-dimensional model of the sample and a linear heat flux profile as illustrated in Figure 1 8.b) (see p. 51). This curve is also shifted to higher temperatures, because it cannot accurately model the peak in the heat flux profile. The third calculated boiling curve, on the other hand, presents a very accurate temperature for the ejection of the wetting front. As shown in Figure 27 and Figure 28, the IHC algorithm using a moving boundary could generally provide a very good approximation of the heat flux profile along the cooled surface. However, because the value of the heat flux at the moving boundary was defined as the maximum heat flux encountered in the previous node, the heat flux profile was “flattened” when the wetting front passed in front of a sub-surface thermocouple. Such a flat profile has the same potential to remove heat as a triangular profile with a higher maximum heat flux. Correspondingly, the IHC analysis ended up underestimating the critical heat flux in the free falling zone. This phenomenon can be observed in Figure 28.  61  4.5E÷06 4.OE+06 3.5E+06 3.OE+06  @4  E 2.5E+06 2.OE+06 1.5E+06 tOE+06 5.OE+05 O.OE+OO 0  100  200 300 Surface temperature (C)  400  500  Figure 28: Boiling curves calculated with one- and two-dimensional FEM models  62  6  RESULTS AND DISCUSSION This chapter presents the results of the various cooling tests conducted in this research  work. First of all, the distinction between the water jet impingement zone and the water film free falling zone is precisely defined. Using this clarified definition, the height of the impingement zone is measured. The relationship between the cooling water flow rate and the impingement zone height is presented. The different boiling regimes (forced convection, nucleate boiling, transition boiling, film boiling) are then studied separately and the effects of different parameters on the boiling water heat transfer are discussed.  6.1  General observations Figure 29 gives an example of the temperature history measured during a cooling test  with a stationary sample. Four thermocouples (2, 4, 5 and 6) were positioned in the water film free falling zone, whereas thermocouple 7 was placed in the water jet impingement zone. Thermocouples 1 and 3 were also in the free falling zone but failed to provide a signal and are thus not considered in this discussion. During the cooling experiment, the impingement zone where thermocouple 7 is located initially experiences film boiling, because the very high surface temperature prevents the water from wetting the surface. This causes the water jet to be ejected away from the surface. As the sample cools down during the test, the water film begins to wet the surface and moves down the sample surface until the entire sample is covered by water. As shown in Figure 29, thermocouples 2, 4, 5 and 6 experience very slow cooling at first, then a sharp temperature drop. Even though water cannot wet the corresponding locations at the sample surface initially, heat is lost by natural air convection and, as the advanced cooling front (ACF) approaches the  63  thermocouple location, by conduction towards the ACF. The rapid cooling shown in Figure 29, on the other hand, is characteristic of boiling water heat transfer at the surface.  jet  50 0.0  5.0  I 10.0  15.0  20.0  I 25.0  30.0  I 35.0  40.0  Time (s)  Figure 29: Typical measured cooling curves for experiment with stationary sample (#25: AA5182, T 0 = 475°C, Q’=87.5 L/minm, v=O mm/s) The cooling curve for thermocouple 7 shows a moderate cooling rate at the beginning of the experiment, then a relatively constant temperature plateau around 420°C, and finally extremely rapid cooling. The relatively constant temperature phase can be attributed to film boiling, in which a vapour blanket insulates the surface from the cooling water. As can be seen in Figure 30, the boiling curve for the impingement zone is very different from those for the free falling zone. The different boiling regimes are easy to recognize in the cooling curves of Figure 29. The next figures identifr those boiling regimes for thermocouple 7 (see Figure 31) and 4 (see Figure 32). Forced convection (FC) and nucleate boiling (NB) are lumped together in these figures because the boundary between these two regimes could not be pinpointed. The position of the boundary between natural air convection (NC) and advanced cooling (ACF) in Figure 32 was arbitrarily chosen as the point where the slope of the temperature curve changes. 64  6.OE+06  5.OE+06  <ç- 4.OE+06 2 3.OE+06  I 2.OE+06  1 0E+06  0.OE+O0 0  100  200  300  400  500  Surface temperature (°C)  Figure 30: Typical calculated boiling curves for experiment with stationary sample (#25: AA5182, T 0 = 475°C, Q’=87.5 L/minm, v=0 mm/s) 500 450 400 350 300 250  2  ci) F- 200  150 100 50 0.0  5.0  10.0  15.0  20.0  25.0  30.0  35.0  40.0  Time (s)  Figure 31: Measured cooling curve for thermocouple 7 in the impingement zone (#25: AA5182, T 0 = 475°C, Q’=87.5 L/minm, v=0 mm/s)  65  500 450 400 350 300 250  2  ci)  I  150 100 50 0.0  5O  10.0  15.0  20.0  25.0  30.0  35.0  40.0  Time (s)  Figure 32: Measured cooling curve for thermocouple 4 in free falling zone (#25: AA5182, T 0 = 475°C, Q’=87.5 L/minm, v=O mm/s) When film boiling takes place on a horizontal heated surface, gravity keeps the cooling water in contact with the vapour blanket. But because secondary cooling takes place along a vertical surface in direct-chill casting, the conventional film boiling regime was only observed in the impingement zone, where the horizontal water jet momentum provides a force in the x direction to compensate for the pressure associated with the formation of steam. The cooling water is thus “trapped” between the vapour blanket on one side and the water jets on the other side, so that relatively stable film boiling can take place. In the water film free-falling zone, however, the formation of steam is not opposed by any other horizontal force, and the cooling water is ejected from the surface. Stable film boiling is therefore impossible in this zone. Correspondingly, the constant temperature plateau which is characteristic of film boiling is never observed in the water film free falling zone. For the purpose of this research work, the distinction between the impingement zone and the free falling zone was defined as such:  66  The water jet impingement zone is the area of the cooled surface where stable film boiling can be observed in favourable conditions (e.g. low water flow rate, high surface temperature, etc.). The water film free falling zone, on the other hand, is the area of the cooled surface where stable film boiling never takes place and from which the water film can be ejected As the next four figures show, cooling experiments conducted with the same cooling water flow rate  and initial sample temperature T 0 produce similar boiling curves. The  Q’  difference between two equivalent boiling curves is maximal at the critical heat flux and corresponds to a 5.8 1% error in Figure 33, 11.04% in Figure 34, 4.63% in Figure 35 and 4.53% in Figure 36. A significant error is also observed for the wetting temperature in the water film free falling zone: 1.25% in Figure 34 and 3.27% in Figure 36. The data scatter for the water film ejection temperature is further discussed in Section 6.8. Overall, the reproducibility of cooling tests is found to be very good. 6.OE+06  5.OE÷06  c-I  E  4.OE+06  x 3.OE+06 44-,  a,  2.OE+06  1 .OE+06  0.OE+00 0  100  200  300  400  500  600  Surface temperature (CC)  Figure 33: Equivalent boiling curves for the impingement zone 0 = 325°C, Q’=lOO L/minm, v=0 mm/s) (#30, 31: AA5182, T  67  6.OE+06  5.OE+06  4.OE+06 2 x  3.OE+06  -I-,  2.OE+06  1 .OE+06  0.OE÷00 0  100  200  300  Surface temperature  400  500  600  (00)  Figure 34: Equivalent boiling curves for the free falling zone (#30,31: AA5182, T 0 = 325°C, Q’=lOO L/minm, v=O mm/s) 6.OE+06  5.OEi-06  4.OE+06 2 x  D  3.OE+06  4-,  Cu  ci)  z  2.OE+06  1 .OE+06  0.OE+00 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 35: Equivalent boiling curves for the impingement zone (#33, 34: AA5182, T 0 = 475°C, Q’lOO L/minm, v=O mm/s)  68  6.OE+06  5.0E+06  4.OE+06 E 3.OE+06  Z  2.OE+06  tOE+06  0.OE+00 0  100  200  300  400  500  600  Surface temperature (CC)  Figure 36: Equivalent boiling curves for the free falling zone (#33, 34: AA5182, T 0 = 475°C, Q’=lOO L/minm, v=O mm/s)  6.2  Impingement zone height Boiling curves for the water jet impingement zone have been known to significantly  differ from those for the water film free falling zone. It is therefore critical, when modelling the direct-chill casting process, to know the height of the impingement zone H . This is particularly 1 important during the transient start-up phase of the process, because high temperature phenomena such as stable film boiling can only take place in the impingement zone, whereas water film ejection can only occur in the free falling zone. Figure 37 illustrates the water flow in the secondary cooling zone. This picture was obtained by replacing the instrumented sample by a transparent sheet of plexiglas. The holes at the bottom of the water box and the cooling water jets can be seen at the top of the picture. A close-up shows how the distinct water jets merge into a common water film. The impingement zone is recognizable as an oval shape of ca. 10 mm height which is highlighted in Figure 37. 69  Such visual observations are not precise enough for the purpose of modelling the DC casting process, but provide a rough estimate of the impingement zone height.  Figure 37: 40 x 30 mm close-up of impingement zone (Q’lOO L/min.m) Rewetting tests constitute another method of measuring the height of the impingement zone. These tests were conducted with a high sample moving speed and a high initial temperature. In the first phase of such a test, the free falling water film was always ejected very rapidly and cooling only took place in the impingement zone. Figure 38 shows an example of the temperature history measured in a rewetting test. The first thermocouples show a temperature drop as the cooling water jets hit the bottom of the instrumented sample. When the sample is lowered, the impingement point correspondingly moves up along the surface and the  70  water film is ejected in the free falling zone. Because the surface is then only cooled by natural air convection, heat flows to the surface and causes it to reheat. At a given thermocouple location, the cooling rate is thus very low at first (natural air convection above the impingement zone), then relatively low (stable film boiling) then very low again (natural air convection below the water film ejection front). Correspondingly, the second derivative of the measured temperature with respect to time presents a minimum at the top of the impingement zone, when the rate of heat removal increases, then a maximum at the bottom of the impingement zone, when the rate of heat removal decreases to its original value. 600 550  I  400 350 300  (6  H50  150 100 0.0  I 2.0  I 4.0  6.0  8.0  10.0  12.0  Time (s)  Figure 38: Typical measured cooling curves for rewetting test (#56: AA5182, T 0 = 500°C, Q’=lOO L/minm, v=30 mmls) Figure 39 shows the second derivative of temperature with respect to time for a sub surface thermocouple. Despite the significant amount of signal noise, the minimum and maximum are easy to identify.  71  1000  4  800 600 (N  0  400  0 0  200  >  0  >  .  I  -200 0 U  W*4&*  -400  w  C!)  -600 -800 -1000 0.0  2.0  4.0  6.0  8.0  10.0  12.0  Time (s)  Figure 39: Second derivative of temperature d T/d? measured during rewetting test 2 (#56: AA5182, T 0 = 500°C, Q’lOO L/minm, v=30 mm/s) The height of the impingement zone could be calculated by dividing the time interval between the minimum and the maximum by the sample moving speed: 1 H  =  tmc  —  tmin  (24)  vc  in which the sample moving speed v was determined by dividing the distance between two consecutive thermocouples by the time interval between the corresponding two minima in the second derivative of the measured temperature: v  —  1 H  (25)  =  tmin 2  —  tmin I  The height of the impingement zone Hj was measured for different water flow rates  Q’.  Because of the signal noise, a certain amount of scatter was observed for the impingement zone height. Average values for a given rewetting test, however, present a good correlation with the water flow rate (R 2  =  0.939). This is due to the fact that the error for a given time  tmjfl,j  leads to 72  two errors of opposite sign for the sample moving speeds  v  and  +i. 1 v,  These two errors thus  compensate each other and their average value is not affected by measurement errors. Figure 40 shows the influence of the water flow rate on the height of the impingement zone. The values obtained by analyzing the second derivative of the temperature with respect to time are slightly larger than the oval shape which was observed visually and presented earlier in this section. The height of the impingement zone (in mm) is given by: (26)  =6.5+0.1lQ’ 1 H  where  Q’ is expressed in L/minm. 45 40 E ci)  3  0 N  4-.  ci)  E  U)  25  E  20 ci) N  D  15  S  ± 100  10 20  40  60  80  120  140  160  180  Water flow rate (L!rninm)  Figure 40: Impingement zone height H 1 as function of water flow rate 0 = 475 500°C, v =30 -40 mmls) (#53 59: AA5182, T -  Q’  -  For the purpose of modelling the DC casting process, Equation (26) is more accurate than visual observations of the impingement zone, because it is based on the very definition of the impingement zone: the portion of the surface where film boiling can take place and the water film is not ejected.  73  6.3  Forced convection regime The semi-empirical equation developed by Weckman and Niessen [4] and presented in  Section 2.3.1 has been shown to evaluate the heat flux in the forced convection regime accurately, as it takes into account both the cooling water flow rate  PFC  very  Q and the water temperature  Tf. However, it was developed from the equation for a film of water flowing downward in a vertical tube, and is therefore a valid representation of the water film free falling zone, but not the impingement zone. It also does not take into account the surface roughness. Previous research work by Li [41] on stationary samples with a machined surface showed the influence of surface roughness on the heat flux for the forced convection regime. Figure 41 presents experimental boiling curves for AA5 182 samples with the typical as-cast surface and with a machined surface. 3.5E+06 3.OE+06 2.5E+06 E 2.OE+06 >4: ‘4-  1 .5E+06  a)  I  1 .OE+06 5.OE÷05 0.OE+00 0  50  100  150  200  250  Surface temperature (°C)  Figure 41: Calculated boiling curves for machined and as-cast samples 0 = 450°C, Q’=lOO L/minm, v=O mm/s) (AA5182, T  74  The heat flux is significantly higher for the sample with a rough surface. It is also higher in the impingement zone than in the free falling zone. Both phenomena can probably be attributed to the greater turbulence in the cooling water flow caused by surface roughness and the impinging water jets. This increased heat flux can be taken into account by modifying the coefficients in Equation (2) (see p. 12). This was done by re-arranging this equation as: hFc  3 ’ 1 (Q’)  FC  —  -CT 2 1 s CT —  27  C3  )” 3 (TS—Tf)(Q’  3 are coefficients. These coefficients can easily be determined by 2 and C , C 1 in which C conducting a linear regression, as illustrated in Figure 42. Oscillations in the temperature signal for the forced convection regime lead to small loops in the boiling curve, but the general trend of the curve is nevertheless easy to quantify. Each quenching experiment allows the calculation 7 for a given alloy and a given region of the . The different values of C 7 of a distinct coefficient C secondary cooling zone were found to be very similar and an average value was determined for the purpose of modelling the forced convection regime. The residual of this first regression was then compared to the water temperature Tf to evaluate the other two coefficients. The forced convection regime was also studied on stationary AZ3 1 samples with an ascast surface. Equations (28) to (31) summarize the influence of water temperature, wall temperature, cooling water flow rate and distance from the impingement point on the surface heat flux. They also take into account the combined effects of surface roughness and thermophysical properties of the cast alloy, which could not be distinguished from each other. FC,IZ,AA5182  FC,IZ,AZ31  =  (14.6 T +68.5 Tf +1230)  (i’s  —Tf)  3 _Tf) l. —541)J(T 7 + T =(16.6 7’ 6  FC,FFZ,M5182 FC,FFZ,AZ31  3 _7’) =(20.2T +38.5T +799)j(T  3 _i’) =(13.2 I’ +39.5 Tf +88)J(T  , 2 where is in W/m  (28) (29) (30) (31)  are in °C. 1 3 and T Q’ is in L/minm and T 75  5.OE+03  4.5E+03  40E+03  3.5E+03  3.0E+03  2.5E+03 70.0  80.0  90.0  100.0  110.0  120.0  Surface temperature (C)  3 and surface temperature Figure 42: Relationship between h /(Q9” 0 = 475°C, v=0 mmls) (#33, 35, 37, 40, 43: AA5182, T In the impingement zone, the turbulence caused by the water jets appears to be the determining factor and the coefficients are the same. The coefficients for the free falling zone, however, are lower for the magnesium AZ3 1 alloy. This lower heat flux in the free falling zone is probably due to the smoother as-cast surface of AZ3 1 compared to the “pitted” as-cast surface ofAA5l82.  6.4  Nucleate boiling regime Nucleate boiling can be subdivided into two regimes: low temperature nucleate boiling,  in which individual steam bubbles are formed at the surface, and high temperature nucleate boiling, in which steam bubbles merge together to form columns or patches. On the boiling curve in Figure 4 (see p. 8), the boundary between these two subregimes is represented by the point C, which corresponds to the maximum heat transfer coefficient. Low temperature nucleate  76  boiling takes place along the B’C segment, from the onset of nucleate boiling to the maximum heat transfer coefficient. High temperature nucleate boiling is represented by the segment CD, in which the heat transfer coefficient  hNB  decreases, but the heat flux value  increases because of the increase in the temperature difference  ’s 7 (  -  PNB  nevertheless  Tsa,). This section discusses  the effect of parameters on the low-temperature nucleate boiling. The influence of parameters on the critical heat flux is covered in the next section. Equation (7), which was developed by Weckman and Niessen [4] for nucleate boiling (see p. 15), is obtained by adding the contributions of forced convection (already discussed in the previous section) and pool boiling. The heat flux for pooi boiling is given by Rohsenow’s semi-empirical equation: CPf(TS Tsat)  _Cf  (  1  Ufg  (PNPBI  iPflfg)Ig(pfpg)  fg 1  r(CfP  I  1.7  (32)  i  in which the water properties are evaluated at the saturation temperature and are therefore constant at atmospheric pressure. The parameters Cf and r given by Weckman and Niessen are 0.0 16 and 0.33 respectively, so that the relationship between the heat flux for pool boiling and the wall temperature becomes: NPB 0  =  20.8 (i’  1at)  (33)  However, both Cf and r have been known to depend on the thermophysical properties of the ingot material as well as the surface roughness [4]. In particular, macroscopic surface roughness features such as scratches and scores generally increase the exponent r and thus decrease the slope of the boiling curve in the low temperature nucleate boiling regime [49]. This can be observed in Figure 41: the heat fluxes for the sample with a machined surface are relatively low in the forced convection regime, but increase very quickly past the onset of  77  nucleate boiling point. The heat fluxes for the sample with the as-cast surface, on the other hand, increase almost linearly and the onset of nucleate boiling is difficult to pinpoint. Table IV presents the corresponding equations for the pooi boiling term in Equation (32) for the two alloys studied. These equations were determined by subtracting the contribution of forced convection, which was given by Equations (28) to (31), from the global heat flux. A linear regression was then conducted using the logarithms of the pool boiling heat flux  NPB  and  2 obtained were all between the temperature difference (7 Tsat). The regression coefficients R -  0.85 and 0.98. As Table IV shows, the nucleate boiling regime for AZ3 1 can be quantified using a relatively low exponent compared to the one suggested by Weckman and Niessen. This is probably due to the relative smoothness of AZ3 1 compared to the AA5 182 aluminum alloy.  Table IV: Pool boiling contribution to nucleate boiling regime PNPB  AA5182  AZ31  6.5  Critical heat flux  6.5.1  Impingement zone  IZ  9.47 (T  FFZ  5 —TSat 2.33 33.0 (T (35) I  TZ  4120 (T 5  FFZ  —TSat, 3 1.96(T  ‘2.59  Tsat;  (34)  ‘1.40  (36)  ‘1.35  (37)  Tsat)  Previous research work with instrumented samples has shown that the initial temperature plays a significant role on the measured surface heat flux [26, 41, 46]. Figure 43 gives an example of this phenomenon: the heat flux in the transition boiling and upper nucleate boiling regimes increases with an increase in the start temperature. Beyond a certain point, however, the 78  initial sample temperature does not influence the boiling curve. In Figure 43, the boiling curves for T 0  =  400°C and T 0  temperature  475°C present a very similar heat flux at the critical heat flux  =  CHF.  6.OE+06  50E+06  <r 4.OE+06 E 3.OE+06  I  2.OE+Q5  I .OE+06  0.OE+00 100  150  200  250  300  350  400  450  500  Surface temperature (°C)  0 Figure 43: Calculated boiling curves for different start temperatures T (#28 33: AA5182, Q’lOO L/minm, v=O mm/s) -  The effect of the start temperature is partially responsible for the large amount of scatter in Figure 44, which plots the critical heat flux in the impingement zone as function of the cooling water flow rate. However, a clear trend can be identified if only the maximum values for each water flow rate are considered; these critical heat fluxes correspond to the cooling tests conducted with a high initial temperature (e.g. above 480°C) and are circled in Figure 44. This trend can be quantified with the following second-order equation: CHF,IZ  Q) Q’—3.3.10 5 =1.0.10 ( 2  (38)  2 can be calculated from the least squares for which an equivalent regression coefficient R criterion J : 2  79  ‘Z  )]2  (Q  (39)  R2=1_—L  (40)  in which n is the number of data points considered and op the variance of the heat flux data. 2 for Equation (38) is equal to 2.11.1012, which corresponds to a The least squares criterion J regression coefficient of 0.783. Previous research work by Du et al. [1] and Jeschar et a!. [59] also found that the relationship between the critical heat flux and the cooling water flow rate could be expressed by a quadratic equation. 8.OE+06 7.OE+06 6.OE+06 E 5.OE+06 4.OE+06 U)  .c 3.OE+06 C-) 4-.  C.)  2.0E+06 I .OE+06 0.OE+00 40  60  80  100  120  140  160  Water flow rate (L/minm)  Figure 44: Effect of cooling water flow rate Q’ on critical heat flux (AA5182, vO mm/s)  IPCHF,Jz  Equation (38) presents a maximum at 150 L/minm, the highest cooling water flow rate investigated in this research work. This equation should therefore not be used to quantify the critical heat flux in the impingement zone with a water flow rate higher than 150 L/niiwm.  80  The critical heat flux at the surface of the magnesium AZ3 1 alloy was also found to be a function of the cooling water flow rate. The quadratic equation which provides the best fit 2 = 0.915) with the results of cooling tests conducted with a high start temperature is given (R below: CHF,IZ  Q’—2.5.10 4 =8.8.10 ( 2 Q’)  (41)  and reaches a maximum at 175 L/minm. It must be pointed out here that Equation (41) is very similar to Equation (38), i.e. the critical heat flux at the surface of AZ3 1 and AA5 182 for a given cooling water flow rate may not be significantly different. Thus the effect of thermophysical properties on the critical heat flux could not be accurately determined. Recent research work conducted by Yu [43] with instrumented samples identified an effect of the water jet velocity. Such an effect was not observed in this research work, and the results presented in Figure 44 include the low-velocity, as well as the high-velocity, cooling experiments. It must however be pointed out here that this could be due to a particularity in the range of experimental conditions: high-velocity cooling tests were all conducted with a higher initial temperature than low-velocity cooling tests. The effect of the water jet velocity could thus be impossible to distinguish from the influence of the start temperature.  6.5.2  Free falling zone As previously discussed in Section 5.6, the IHC analysis used in this research work is  very precise in evaluating the water film ejection temperature, but has a tendency to underestimate the critical heat flux. The study of the CHF in the water film free falling zone was therefore conducted by using a simple one-dimensional IHC analysis. As Figure 28 on p. 62  81  shows, an IHC analysis conducted without a moving boundary generally produces a more accurate value for the critical heat flux. The CHF in the water film free falling zone was found to be a function of the cooling water flow rate, as Figure 45 illustrates. The critical heat flux also depends on the distance from the impingement zone d , because the cooling water temperature increases as the water film 1 flows down along the surface. Figure 46 shows the heat flux measured at six locations as function of time. Location 7 corresponds to the impingement zone and presents the highest critical heat flux. The other locations are positioned in the free falling zone and experience a maximum heat flux shortly after the arrival of the water film ejection front at this position. Whereas the critical heat flux  CHF  at location 6, which is 10 to 20 mm away from the  , the CHF at location 2, ca. 100 mm from the impingement 2 impingement zone, is 4.45.106 W/m . 2 zone, is only 3.51.106 W/m 6.OE÷06  5.OE+06  ‘  4.OE+06  2 3.OE+06  2.QE+06  I .OE+06  O.QE+OO 150  200  250  300  350  400  450  500  550  Surface temperature (°C)  Figure 45: Calculated boiling curves for different water flow rates (# 24, 25, 35, 45: AA5182, v=O mmls)  Q’  82  6.OE+06  5.OE+06  4.OE+06 E 3.OE+06  2.OE+06  1.OE+06  O.OE+OO 0.00  5.00  10.00  15.00  20.00  Time (s) Figure 46: Measured heat fluxes 0 as function of time (#33: AA5182, T 0 = 475°C, Q’=lOO L/miwm, v=O mm/s) Experiments conducted by Matsueda et al. [60] on copper and carbon steel showed that the relationship between the critical heat flux in the free falling zone and the position was characterized by a constant slope of -1/3 when plotted on a log-log scale. The following equation was designed in such a way that the critical heat flux in the free falling zone would be equal to 0 1 equal to zero, and decrease according to a -1/3 power CHF,Jz at a distance d relationship: ‘‘CHF,FFZ  —  0 CHF,IZ  k  iZ  794 U  +  ,  42  where . 79 is the distance from the impingement zone (in mm) where the critical heat flux is d 4 equal to 79.4% or (2.0)hh13 of 0 cHpJz. Equation (42) is equal to  CH)JZ 0  when d 1 is equal to zero and tends towards 0 when d 1  tends towards infinity. Larouche et al. [26] also observed a steep decrease of the heat transfer coefficient followed by a relatively stable value in the free falling zone. Yu [43] quantified the  83  heat flux in the secondary cooling zone with a linear decrease in the first 12 inches, followed by a constant value. Compared to Yu’s results, Equation (42) presents a sharper drop in the heat flux at the top of the water film free falling zone, but a higher critical heat flux a long distance away from the impingement zone. Figure 47 illustrates the relationship between the distance from the impingement zone and the ratio  Since the exact position of the impingement zone is generally  PCHFFFZ  not known, the distance between the impingement zone and the first thermocouple in the free falling zone was estimated as d IZ,1  TC 1 ‘  H 1 2  where iiHrc is the distance between two consecutive thermocouples (expressed in mm) and H 1 is the height of the impingement zone as given by Equation (26) on p. 73. The value calculated with Equation (43) corresponds to the average between the minimum (H Hjz) and maximum -  (HTC) possible distances from the impingement zone. 1.0 0.9 0.8 0.7 !t 0.6 L1  I C-)  e  0.4 0.3 0.2 0.1 0.0 0  20  40  60  80  100  120  140  Distance from impingement zone (mm)  Figure 47: Critical heat flux as function of the distance from the impingement zone dj (AA5182, vO mm/s) 84  2 Figure 47 presents a considerable amount of scatter (J  =  1.428, R 2  =  0.299). This is  partly because of the uncertainty in evaluating the distance from the impingement zone, but mostly because of the scatter in  lCHFFZ  and cj-jpjz. Despite this relatively poor correlation,  the parameter . 79 in Equation (42) was evaluated by determining the best fit for this equation d 4 and is equal to 27.5 mm for the aluminum AA5 182 alloy. The same analysis conducted on 79 parameter d 4 results for the AZ3 1 alloy gave a very similar value of 29.6 mm for the . 2 = 0.324). It must be pointed out here that Bamberger and Prinz [53] found similar levels of (R scatter in their study of water blade jet cooling: normalized heat transfer coefficients for a given water flow rate and a given distance from the impinging line could take any value between 0.5 and 1.75.  6.6  Transition boiling The transition boiling regime was investigated with stationary experiments as well as  wetting experiments with a moving sample. Figure 48 shows boiling curves obtained with different sample moving speeds, but with the same cooling water flow rates and initial temperature. One boiling curve corresponds to the heat flux in the impingement zone during an experiment with a stationary sample (v  =  0 mm/s, T 0  =  400°C). The other curves represent  experiments conducted with a moving sample. Despite the relatively high sample moving speeds (v  10 mm/s and 20 mm/s), the transition boiling and high temperature nucleate boiling  regimes both took place in the impingement zone. The free falling zone was thus the site of only low temperature nucleate boiling and forced convection. Because of the advanced cooling front phenomenon, the boiling curves for moving samples start at a lower temperature. However, the slope of the boiling curve in the  85  transition boiling regime is the same for all three experiments. As Figure 49 shows, the heat flux s when transition boiling begins. 2 increases as a constant rate of 1.51 O W/m 6.OE+06  5.OE+06  4.OE+06 E 3.OE+06 a) I 2.OE+06  1.OE+06  0.OE+O0 100  150  200  250  300  350  400  450  Surlacetemperature (°C)  Figure 48: Calculated boiling curves for different sample moving speeds v (#32, 48, 51: AA5182, T 0 = 400°C, Q’lOO L/minm) 6.OE+06  5.OE+06  <ç 4.OE+06 E  3.OE÷06  2.OE+06  1.OE+06  0.OE+00 0.0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  Time (s) Figure 49: Heat flux as function of time I for different sample moving speeds v 0 = 400°C, Q’lOO L/minm) (#32, 48, 51: AA5182, T 86  In transient conditions, the transition boiling regime represents an intermediate state between the dry (film boiling) and wet (nucleate boiling) states. As transition boiling takes place and the surface temperature goes down, the portion of the surface which is wetted by the cooling water increases, whereas the portion covered by the vapour blanket decreases. This progression from a completely dry to a completely wet surface appears to take place at a certain speed which is independent of the surface temperature or the water flow rate. This would indicate that the nucleation and subsequent propagation of wet spots on the quenched surface is probably governed by heat conduction below the surface and not by the hydrodynamics of the water / steam dual-phase layer. As Figure 50 shows, the slope for the transition boiling regime in the free falling zone (d/dT)FFz presents a normal distribution with K. This normal distribution and the absence of correlation 2 an average value of -6.5lO W/m between the transition boiling slope and other parameters indicates that the variations in the slope of the transition boiling regime can be attributed to random variations. 25  20  0 0 D  10 LL  5  0 Lfl  .  C +  w ID ci  C + iii 0  C + Lu  0  0  Lfl  w 0  w  I  I  +  ‘+ I  +  10 I  0  + Ui  C  + Lu  C  + Lu  0  IC)  0  IC)  U)  CD  I  I  I  0  + Lu IC) CD I  C  + Ui  0 NI  0  + Ui 10 NI  0  0  C  0  0  Lu  Lu  Lu  Lu  Lu  0 CO  IC) CO  0 0)  10 0)  0  +  I  +  I  +  I  +  I  +  I  K) 2 Transition boiling slope (W/m  Figure 50: Distribution of the transition boiling slope for the free falling zone (AA5182) 87  The distribution of the transition boiling slope in the impingement zone (d/dT) jz is also K (see Figure 51). Similar transition boiling slopes 2 normal; its average value is -6.0 1 O W/m K in the water film free falling zone and -4.51O 2 W/m  were found for the AZ31 alloy:  K in the water jet impingement zone. A compilation of boiling curves for the water jet 2 W/m impingement cooling on AA1O5O [41], AA3 004 [41] and carbon steel [74] shows that the slope . Figure 52 presents 3 in the transition boiling regime is a function of the thermal conductivity k the different materials studied in this compilation. Materials with a high conductivity (e.g. aluminum alloys) allow a rapid propagation of the wet spots during transition boiling and thus a rapid increase in the heat flux. Materials with a low thermal conductivity (e.g. steel), on the other hand, tend to form low temperature wet spots and high temperature dry spots, and thus hinder the transition to nucleate boiling. 8 7 6  C) cD 4 D ci) LL  2 I 0 o + UI C c  ,.  .  .  .  .  .  + Ui C  + Lii (N  + LU  + UI  + UI C Co  + Ui  o  c6 I  c’ I  o  o  -  I  , I  o I  o  o  CD  (N  CO  r-.  I  + LU I  o  + LU CO t-. I  o  o  -  0  co  o,  + Lii I  + UI I  Transition boiling slope (Wim 2 K)  Figure 51: Distribution of the transition boiling slope for the impingement zone (AA5182)  88  -8.OE+04 -7.QE+04 -6.OE+04 0 1 0  -5.OE+04  D) -4.OE+04 0  • -3.OE+04 0  -2.OE+04 -1 .OE+04 0.:OE+00 0  50  100  150  200  250  Thermal conductivity (W!niK)  Figure 52: Effect of thermal conductivity k on transition boiling slope dø/dT The transition boiling regime corresponds to the interval between the critical heat flux and the Leidenfrost point. The slope of the boiling curve in this regime is only constant for relatively high temperatures, and decreases to zero at the critical heat flux. Knowledge of the transition boiling slope is therefore insufficient to accurately quantifr the heat flux for this boiling regime. However, the relationship between the surface heat flux and the temperature can be converted into a heat transfer coefficient equation. As Figure 53 shows, the heat transfer coefficient hTB increases somewhat linearly in the transition boiling region of the boiling curve. A comparison of Figure 48 and Figure 53 shows that the maximum heat transfer coefficient is reached at a lower temperature than the maximum heat flux. Knowledge of the heat transfer coefficient can therefore be used to quantify the heat flux for the whole transition boiling region of the boiling curve.  89  3.5E+04 3.OE+04  E 2.5E+04 ci) .  2.OE+04  41)  0  1.5E+04 0) .  1.OE+04 5.0E+03 0.OE+00 100  150  200  250  300  350  400  450  Surface temperature (°C)  Figure 53: Calculated boiling curves for different sample moving speeds v 0 = 400°C, Q’lOO L/min.m) (#33, 48, 51: AA5182, T The heat flux for transition boiling can be quantified using the following equation: TB =hTB (‘  (44)  ) 1 —T  in which the heat transfer coefficient hTB is given by the linear equation (d h TB -h MHFII  (TMHF-I)  MHF )TB 1  ( )  f 1  The minimal heat transfer coefficient hMHF in Equation (45) corresponds to the heat transfer coefficient at the beginning of either the transition boiling or initial cooling regime. It thus corresponds to the minimum heat flux at the Leidenfrost point PMHF if transition boiling takes place after stable film boiling, or to natural air convection if film boiling does not occur. The variable hMHF is also equal to the heat transfer coefficient for natural air convection in the case of initial cooling, which takes place before stable film boiling. The variable TMHF in Equation (45) corresponds to the initial surface temperature for the initial cooling regime (or for  90  the transition boiling regime if stable film boiling does not occur). It is equal to the Leidenfrost temperature  TLpt  for the case of transition boiling taking place after stable film boiling. Finally,  the boiling curve slope in the transition boiling regime (d/dT)TB is a sole function of the thermophysical properties and therefore remains constant. As Figure 53 shows, identical slopes for the heat flux in the transition boiling regime do not correspond to identical slopes for the heat transfer coefficient. Whereas the heat flux slope is constant for a given material, the heat transfer coefficient slope is also a function of the temperature range for the transition boiling regime. Thus the position of the heat flux minimum will influence the shape of the boiling curve in the transition boiling regime. In particular, a very low temperature TMHF will limit the increase in the heat transfer coefficient before the nucleate boiling and forced convection regimes are reached. In other words, a low value of TMHF will not allow the full development of the nucleate boiling structure which can generally be observed at the critical heat flux, and the corresponding maximum in the heat flux will not occur. Such a phenomenon can happen during the steady-state phase of the direct-chill casting process, in which the advanced cooling front effect leads to relatively low surface temperatures in the impingement zone.  6.7  Film boiling  6.7.1  Minimum heat flux The film boiling regime in the water jet impingement zone can be modelled with two  values: the minimum heat flux  MHF  and the Leidenfrost point TLpt. As demonstrated by Köhler  et al. [50], the heat flux is independent of the surface temperature in the film boiling regime. This heat flux is therefore equal to the minimal heat flux for any surface temperature above the Leidenfrost point. The Leidenfrost point temperature TLp( represents the boundary between 91  stable film boiling and transition boiling. Accurate knowledge of the Leidenfrost point is of critical importance when modelling the transient start-up phase of the DC casting process, because the heat flux is relatively low for stable film boiling, but increases significantly when transition boiling begins. The average heat flux for film boiling is given by Equation (46), which was already presented in Section 2.3.3:  —  2  FB,AVG  kf I(Pe2+2Gnr2_Pe3(T sat —Tf Gn  46  The length L corresponds to the height of the impingement zone and is a function of the cooling water flow rate, as expressed by Equation (26) on p. 73. This length is also used to calculate the Peclet and Gravity numbers. The Gravity number is equal to the Galilei number times the square of the Prandtl number, and is given by the following equation: =  Gn  L3 [CPfuf  2  (47) =  2 a  in which af is the thermal diffusivity of water. Gn is thus a function of the water flow rate (via  L) and the water temperature (via af). The Peclet number is calculated by: (48)  aj  /sm divided by the 3 0 is equal to the water flow rate in m in which the water film velocity v thickness of the water layer in m. The thickness of a free falling water film is itself a function of the water flow rate  =  0.0004524  Q’, the density jy and the viscosity pj [16, 48]:  g(Q’Pf 3  (49)  92  The average heat flux according to Equation (46) is therefore a sole function of the water flow rate and the water temperature. As Figure 54 illustrates, the effect of the water flow rate on the average heat flux is minimal, whereas the effect of the water temperature is significant. It must, however, be pointed out here that the water flow rate also influences the length of the impingement zone. In other words, an increase in the cooling water flow rate increases the total heat flux for the impingement zone even though the average heat flux is approximately constant. Cooling experiments conducted on AA5 182 and AZ3 1 samples provided significantly different values for the minimum heat flux.  iíi-IF 13  values between 4.2l0 and 7.1l0 W/m 2  were observed, whereas the values in Figure 54 are all above 1.0 MW/m . Results from cooling 2 experiments also presented a clear relationship between the cooling water flow rate and the minimum heat flux. 1.7E+06  1 .6E+06  1.5E+06 x ZT —10°C (U 0 C,  1.4E+06  1.3E+06  1 •A—  1.2E+06 40  60  I 80  A A  A  A  A  A  I  100  120  140  160  I 180  .200  220  Water flow rate (L/rnin..rn)  Figure 54: Modelled minimum heat flux  ‘PMHF  for different water temperatures Tf  The discrepancy between KOhler’s model and experimental results could be attributed to some of the assumptions made in developing Equation (42). For instance, the heat flux  PFB,AVQ  93  is calculated by determining the thermal gradient in the water film (dT/dx). This thermal gradient is obtained by differentiating an equation for the temperature distribution in the water film. Equation (46) thus corresponds to a case of steady-state film boiling in which the temperature profile remains constant, whereas cooling experiments are of a transient nature. Moreover, Köhler’s model does not take into account the cooling water flow rate but only the . Equation (46) therefore constitutes a model for steady-state film 0 water film initial velocity v  boiling in a water film of a given thickness rather than for transient film boiling in the water jet impingement zone. Nevertheless, Köhler’s model provides important qualitative information for the film boiling regime. It shows the significant role played by the cooling water temperature, it demonstrates how heat losses by radiation can be neglected, and it illustrates how the heat flux is independent of the surface temperature in the film boiling regime. But instead of relying on Equation (46), the minimum heat flux was quantified by modifying Berenson’s equation for film boiling heat transfer from a horizontal surface [75]: I  /  \114  (PiPg)j •fg 0 J 1 J.VYPgZfg I / ‘PfPg) ‘\g’Pf—Pg  —  This equation is based on the Kelvin-Helmholtz instability between two immiscible fluids (water and steam) flowing relative to each other along a boundary and the RayleighTaylor instability of an interface between two fluids of different densities. The maximum relative velocity between two fluids is given by:  I max  where  I  27r0 fgf p  jPgrii(P +Pg)  5l ‘  is the wavelength of the steam I water vertical column expressed in m. As shown by  Equation (51), the stability of this interface increases with the surface tension o but decreases 94  with the wavelength. The maximum wavelength of the steam / water horizontal interface according to the Rayleigh-Taylor instability criterion  RT 2  I  =  RT  is given by:  ajg  1 g(p  (52) Pg)  Equation (52) corresponds to an equilibrium between the gravity forces which tend to break down the vapour blanket (because of the density difference between water and steam) and the surface tension forces which tend to maintain this vapour blanket. Equation (50) is obtained by determining the geometrical relationship between  RT 2  and  KH, 2  then by combining Equations  (51) and (52) and converting the velocity of the steam / water column Vm into a heat flux: MHF  =  (53)  C2VmaxPglfg  2 is a constant equal to 0.09 in Equation (50). in which C Film boiling in the water jet impingement zone was modelled by replacing the gravity forces in Equation (52) by the momentum force of the water jets, because gravity does not influence the stability of a vertical steam / water interface. The momentum force of a water jet is related to the dynamic pressure at the impingement point [76]: 1 Pip  Pf (V jet)  2  (54)  where the jet velocity vjet is proportional to the cooling water flow rate. Rearranging Equations (51), (43) and (54) leads to:  I  1  MHF31’jePfjJ fg 1 1 + Pg) I Pg (p  (55)  According to Equation (55), the minimum heat flux is proportional to the water jet velocity (i.e. to the water flow rate) and to the latent heat of evaporation z. It is, however, practically independent of the cooling water temperature, because the variation of the water 95  density P1 with temperature is not significant. The effect of the water temperature can be taken into account by replacing the latent heat of evaporation  lg  by the total amount of heat transferred  to the cooling water in order to vapourize it: 1  MHF  (56)  ] (lOO—T +C ) [ijg 1 1 V 3 =C / jetPf lJ Pg (Pf + Pg)  The different terms and coefficients in Equation (56) were evaluated by determining the best fit with experimental results. Results of cooling tests conducted on AA5 182 were considered as well as experimental results presented by Grealy et al. [36] and originally obtained by Opstelten and Rabenberg [42]. These additional results provided minimum heat fluxes for different cooling water flow rates and water temperatures. The resulting equation for the heat flux in the film boiling regime is: MHF  (57)  ] =Q’[2500+44(100—T ) 1  2 of 0.953. with a regression coefficient R The linear relationship between the water flow rate  Q’ and the minimum heat flux  MHF 1  was also observed by Larouche et al. [26] for a wide range of cooling water flow rates. The extrapolation of experimental results obtained with 87.5  -  112.5 L/minm to lower flow rates is  therefore justified.  6.7.2  Leidenfrost point Surface heat flux irregularities in the film boiling regime complicate the calculation of  the temperature corresponding to the Leidenfrost point. Because of the transient nature of the cooling tests, stable film boiling was never observed as it is in steady-state pooi boiling experiments. However, the surface heat flux increase which takes place when transition boiling begins is very easy to pinpoint. The Leidenfrost point in this research work was therefore 96  defined not as the temperature at which radiation and conduction through the vapour layer increase the heat flux in the film boiling regime, but as the temperature at the breakdown of that vapour layer signals the beginning of transition boiling. A first approximation of the Leidenfrost point temperature  TLp(  was done by considering the very last local minimum in the heat flux  observed before transition boiling. Other local minima taking place earlier, i.e. at higher temperatures, were attributed to irregularities in the stable film boiling regime. Figure 55 illustrates the relationship between the Leidenfrost point TLpI and the cooling water flow rate  Q’  for experiments conducted with low velocity water jets. The correlation  between the two variables is very clear if only the maximal values, i.e. the results of cooling experiments conducted with a high initial temperature, are considered. It is important to remember that cooling tests conducted with a relatively low initial temperature did not necessarily lead to stable film boiling. The local minimum measured in those tests does not correspond to the Leidenfrost point but rather to the wetting of the surface and the beginning of transition boiling. Although the relationship between the Leidenfrost point and the cooling water flow rate appears linear in Figure 55, this is not the case for all water flow rates. Previous research work by Jeschar et al. [59] as well as Grandfield et al. [16] has shown that the Leidenfrost point is proportional to either the square root or the cubic root of the water flow rate. Thus a linear equation based on the results presented in Figure 55 would overestimate the Leidenfrost point for water flow rates outside of the range studied in this research work. This could lead to significant errors when modelling the direct-chill casting process start-up, during which very low water flow rates are used.  97  500 480 460 0 0 420 400 380 U)  360 340 320 300 80  100  120  140  160  180  200  Cooling water flow rate (L/minm)  Figure 55: Effect of water flow rate Q’ on Leidenfrost point (AA5 182, vo=O mm/s)  TLpt  2 Instead, Equation (58) was obtained by conducting a linear regression (R  =  0.892) on  the square root of the water flow rate. TLPt  =100.O+33J  (58)  Stable film boiling was not observed during quenching tests with AZ3 1 samples. This is probably due to the lower thermal effusivity of magnesium alloys compared to the effusivity of aluminum alloys such as AA5 182 (see Section 2.5.9). A material with a high thermal effusivity th 8  can transfer heat very rapidly to a wet spot on the surface and prevent it from spreading. This  phenomenon stabilizes the vapour blanket and makes it harder to break down. Correspondingly, a material with a low effusivity  h  will form stable wet spots easily and will be associated with a  high Leidenfrost point. Research conducted by Jeschar et al. [59] on the vapour blanket breakdown temperature at the surface of different materials provided data to identify a linear  98  relationship between the thermal effusivity and the Leidenfrost point for a given set of cooling conditions: TLPt  (59)  —O. 4 =C h 1 6 02  Thus the Leidenfrost point for a material can be obtained by comparing its thermal effusivity to the effusivity of AA5 182: TLP,  =  TLP,AA5182 +  0.02  (sthAA5l82  —  6th)  (60)  Figure 56 illustrates the effect of the cooling water flow rate on the Leidenfrost point for three light metal alloys. The Leidenfrost point of AA5 182 was calculated using Equation (58), whereas those of AZ3 1 and AA1 050 were evaluated with Equation (60). As Figure 56 shows, an initial temperature above 500°C would have been required to observe stable film boiling at the surface of an AZ3 1 sample. However, cooling experiments conducted with this alloy always started below 500°C due to safety concerns. Stable film boiling was also not observed during cooling experiments with high-velocity water jets. It can be assumed here that the greater jet momentum was able to break down the vapour blanket instantly and therefore that transition boiling began as soon as the cooling water hit the impingement zone. Even tests conducted with a very low water flow rate of ca. 50 L/miwm did not present the characteristic low heat flux at high temperatures generally associated with stable film boiling. This phenomenon, which was also observed in cooling experiments conducted by Yu [43], would indicate that the design of the direct-chill mould bottom plays a significant role on the occurrence of film boiling during the process start-up. It would thus become possible to dissociate the primary cooling, which is a sole flmction of the cooling water flow rate in the  99  mould, from the secondary cooling, which depends both on the cooling water flow rate and the water jet velocity. —  700 600  —  500 400 300  -.—AZ3I A  200  --M5182 (model) 0 A5182 (experimental)  100  —-AA1050 0 30  50  70  90  110  130  150  170  190  210  Water flow rate (Liminm)  Figure 56: Modelled Leidenfrost point TLpg as function of water flow rate  Q’  Finally, it must be pointed out here that this research project only investigated the secondary cooling of aluminum and magnesium alloys with a very specific set of cooling water conditions. Thus the equations presented in this section are only valid for cooling with pure water within a narrow range of temperatures, although the same general trends are expected to be observed in different conditions. Additional quenching experiments would be required to quantify the effect of dissolved gases or water contaminants such as lubricants, coagulants or suspended solids.  6.8  Water film ejection Because of geometrical considerations already discussed in Section 6.1, stable film  boiling cannot take place in the water film free falling zone. The formation of a vapour blanket 100  leads instead to the ejection of the water film. Whereas the ingot surface above the ejection point experiences boiling water heat transfer, the surface below this point remains dry and is cooled by natural air convection. Axial heat conduction towards the point of ejection also takes place. Eventually, as the surface temperature decreases, the water film is able to rewet the surface. The progression of the water film ejection / rewetting front (and the associated advanced cooling front) was studied in cooling experiments with a stationary sample as well as in rewetting tests with a moving sample. The surface temperature at which the free falling water film could rewet the surface without being ejected was defined as the rewetting temperature Twet. The rewetting temperature Twet is similar to the Leidenfrost temperature in the water jet impingement zone  TLpt,  as it represents the boundary between two regimes: transition boiling  (below the rewetting temperature) and natural air convection (above the rewetting temperature). It also corresponds to the critical water film ejection temperature. If the temperature at the bottom of the water jet impingement zone (i.e. at the beginning of the water film free falling zone) is above the rewetting temperature, the water film will be ejected and a rewetting front will progress down the surface as the surface cools down. If it is below the rewetting temperature, the water film will not be ejected. Knowledge of the rewetting or ejection temperature Twet is therefore of critical importance when modelling the transient start-up phase of the direct-chill casting process. As Figure 57 shows, the initial sample temperature T 0 appears to exert a significant influence on the rewetting temperature Twet. Cooling tests conducted with a stationary sample at very low temperature led to a water film ejection temperature which was very similar to the initial temperature. Thus data points on Figure 57 tend to follow the 1:1 relationship line. In wetting tests with a relatively high start temperature, the ejection temperature Twet can be up to 101  . This relationship between the initial sample 0 150°C lower than the initial temperature T temperature and the rewetting temperature underlines the transient nature of the ejection front progression, because the criterion as to whether the water film will be ejected or not depends on the temperature history at this point of the surface. The wetting of the surface by the water film thus requires a certain “incubation” time which increases with the initial temperature. The reliance on a relationship between the initial temperature and the water film ejection temperature is problematic when modelling the actual direct-chill casting process. This is because, contrary to the instrumented sample in the water jet experimental rig, an ingot coming out of the DC mould does not present a uniform temperature. Moreover, rewetting tests have shown no relationship between the initial sample temperature and the water film ejection temperature. These tests, in which the surface is briefly cooled by impinging water jets, then rewetted by a free falling water film, constitute a very good simulation of the process start-up phase and indicate that the initial sample temperature is not the parameter which governs the ejection of the water film. A comparison of Figure 57 and Figure 58 shows that it is the temperature of the dry surface before rewetting Td,, which determines the water film ejection temperature. In cooling tests with a stationary sample, the temperature of the dry surface is equal to . In rewetting tests, however, the surface temperature after the passage 0 the initial temperature T of the impinging water jets is lower than the start temperature. Figure 38 on p. 71 shows how the surface reheating after the ejection of the water film does not bring the surface temperature back to its original value: it reaches a maximum, then starts to go down again because of the advanced cooling front effect. The maximum surface temperature experienced between the passage of the water jets and the arrival of the rewetting front corresponds to the dry surface temperature Td, and governs the water film ejection temperature Twet. 102  500 C) -  450  D  .1  400  £ 350  300  2 ‘4-  ci)  250  200 200  250  300  350  400  450  500  550  600  Initial temperature (C) Figure 57: Effect of initial temperature T 0 on rewetting temperature Twet (AA5182)  500 C-) 2  450  ci) I 1-’  (U  1  400  2  )  350  300 2 ‘4-  250  200 200  250  300  350  400  450  500  550  Dry temperature (°C) Figure 58: Effect of dry surface temperature Td,. on rewetting temperature Twet (AA5182)  103  Another way to visualize the relationship between the dry surface temperature and the rewetting temperature is to plot the relative rewetting temperature (T, /T,,) as function of the dry surface temperature Td,. As Figure 59 shows, the relative rewetting temperature is equal to unity for very low dry surface temperatures and decreases as Td increases. A clear linear 2 relationship can be observed for dry surface temperatures below 475°C (R  =  0.929), but there  appears to be no correlation between the two variables for dry surface temperatures above 435°C (R 2  =  0.000193). This would indicate that the rewetting temperature Twet is then  independent of the dry surface temperature, i.e. becomes constant above a certain value of Td,,,. 1.00 _..  1) I.... D  4-,  .  0.95 0.90  -  0  .  0.85  D  )  :  x  0.80  E.___  0.75  CU  x  4.  —ai--—--x 0  0.70  (1)  0 Stationary tests  1  U  0.65 X Rewetting tests 0.60 200  250  300  350  400  450  500  550  Dry temperature (°C)  Figure 59: Effect of dry surface temperature Td, on relative rewetting temperature (AA5182) Figure 60 presents the relative rewetting temperature as function of the dry surface temperature for two cooling water flow rates. A higher water flow rate  Q’ increases the relative  rewetting temperature (Twet /Ta’,y) and thus makes rewetting easier. This can be attributed to the increased thickness of the free falling water film, which is then harder to eject.  104  1.00 0.95 a,  I—  1.501  T/TrTd  ••  -  0.0C 628 Td  0.90  a,  2  0.85 Twet! Tdry  4-.  1.305 0.00 2:: -  0.80 ci) 1) 2 ci) > a, ci)  4-.  0.75  %  0.70 D  100 LIminj  0.65 x150 L/minrn 0.60 200  300  250  Dry  350  temperature  400  450  500  (°C)  Figure 60: Relative rewetting temperature for two cooling water flow rates (AA5182) 2 Linear regressions (R  =  Q’  0.928 and 0.908 respectively) were conducted on both sets of  data and are expressed by the following equations: For 100 L/minm: Twet /Td,Y =1.3 For 150 L/miwm:  0.0012 Td,.,  (61)  Twet /Td,Y =l.S—O.OO Td,Y 16  (62)  —  The point where Equation (62) intersects the (Twet /Td,y)  =  1 line indicates that a minimal  temperature of 310°C is required to eject the water film with a flow rate of 150 L/minm. If the dry surface temperature is below 310°C, the relative rewetting temperature is equal to unity and rewetting takes place at the dry surface temperature Td,,,. For a cooling water flow rate  Q’ of 100  L/minm, the dry surface temperature can be as low as 250°C for the water film to be ejected from the surface. Data for other cooling water flow rates and low dry surface temperatures are scarce. However, the relationships observed in Figure 60 can be generalized by assuming a linear 105  relationship between the cooling water flow rate and the coefficients of Equations (61) and (62). The equation for the relative rewetting temperature then becomes: Twet lTd,.,, =(0.91 + 0.0039 Q’)— (4.6.  +  (63)  7.8• 10Q’)Td,Y  Equation (63) is only valid for relatively low dry surface temperatures. As discussed earlier in this section, the relative rewetting temperature becomes independent of the dry surface temperature above  Ca.  435°C. It is thus possible to determine the rewetting temperature by using  Equation (63) for low values of Td,,, and a Constant value for higher dry surface temperatures. A relationship between this constant relative rewetting temperature and the cooling water flow rate was found by considering the average relative rewetting temperature for all cooling tests conducted with a given water flow rate and with dry surface temperatures Td,, above 435°C. As Figure 61 shows, an increase in the cooling water flow rate clearly increases the relative rewetting temperature for high dry surface temperatures. 0.82  .i.- 0.80 4-.  a)  cL  0.78  2(1)  ) 0.76  0.74  0.72  0.70 35  55  75  95  115  135  155  175  Water flow rate (L/minm)  Figure 61: Effect of water flow rate  Q’ on minimal relative rewetting temperature (AA5182)  106  0.962) presented in Figure 61 is expressed by:  2 The linear regression (R  (64)  (Twet /Td,Y)mjn =0.68+0.0008Q’  Figure 62 illustrates the relative rewetting temperature for three different cooling water flow rates  Q’.  The curves in Figure 62 were built using Equations (63) and (64), which were  identified for the water film ejection and rewetting of AA5 182. Similar equations were found for the rewetting of AZ3 1. These equations are: TWet/Td  (65)  ’)Td, 4.2.10 7 + 4 =(1.2+0.0011Q’)—(9.5•10 Q  (Twet /Td,y)min  =  0.76 + 0.00075  (66)  Q’  1.00 0.95 0.90  :f: 0.70  0.65 200  250  300  350  400  450  500  550  600  Dry temperature (C)  Figure 62: Modelled relative rewetting temperature for different water flow rates (AA5182)  Q’  In both cases, the relative rewetting temperature for the magnesium AZ3 1 alloy is found to be significantly higher than for the aluminum AA5 182 alloy. This is consistent with the observations presented in Section 6.7.2, where the Leidenfrost point for AZ3 1 was higher than  107  the one for AA5 182 with the same cooling water flow rate. AZ3 1 is thus generally easier to wet than AA5182. The equations for the relative rewetting temperature can be generalized to other materials by assuming a linear relationship between the coefficients and the thermal effusivity Such a relationship was already observed for the Leidenfrost point, and the wetting of the  .h.  surface in the water jet impingement zone is in many ways similar to the rewetting of the surface by the free falling water film. The relationship between the dry surface temperature, the cooling water flow rate, the thermal effusivity and the relative rewetting temperature is summarized by Equations (67) to (70): Twet 5 C  /Td,  =  =  5 C  (67)  + CoTd,  (1.86—4.3. 10 eh) + (4.7.10 5  —  0.0062) Q’  (68)  6 9 C 1.23.10 th)Q’ cth—0.0022)+(1.89.10 8 =(8.0.10 — 5 &  (69)  (Twet/Td,y)min 9 8.5.10 (h)Q’ 6th)+(6.15.10 5 =(0.99—1.46.10 + 4 c  (70)  in which the efflisivity  h  is evaluated at 450°C.  Figure 63 compares the relative rewetting temperature (Twet /Td,,,) for three light metal alloys. This figure presents the same trend as Figure 56 on p. 100, in which the relatively low Leidenfrost point for AA1O5O indicated that this alloy forms a stable vapour blanket at high temperatures and is difficult to wet. Correspondingly, the aluminum AA1O5O alloy also has a tendency to eject the free falling water film and its relative rewetting temperature is significantly lower than the relative rewetting temperatures of AA5 182 and AZ3 1.  108  1.00 AZ31 _.  *M5182  0.95  I  Ml 050  0.75 O.70 0.65 200  I 250  I 300  I 350  400  450  500  I 550  600  Dry temperature (C)  Figure 63: Modelled relative rewetting temperature for different alloys (Q’=lOO L/minm)  109  7  EFFECT OF SECONDARY COOLING ON DIRECT-CHILL CASTING OF LIGHT METALS This section discusses how the different results presented in the previous chapter can be  used to model the secondary cooling in the direct-chill casting of light metals. It presents a finite element model of the DC casting process and shows how boiling curves can be put together using various equations for the heat flux. A number of FEM simulations are used to demonstrate the importance of secondary cooling during the transient start-up phase and the critical role of the cooling water flow rate.  7.1  Applicability of cooling experiment results The different equations developed from cooling experiments and presented in Chapter 6  can be used to determine the boundary conditions in a mathematical model of the direct-chill casting process. Results obtained by quenching relatively small instrumented samples are thus extrapolated to the full-scale process even though cooling tests did not constitute a perfect simulation of the DC casting process. For instance, the sample moving speeds used in wetting experiments were between 10 and 60 mmls, i.e. approximately an order of magnitude higher  than typical casting speeds. Tests conducted with a stationary sample, on the other hand, corresponded to a casting speed equal to zero. But since both types of experiment led to the same results (e.g. slope in the transition boiling regime, critical heat flux), it can be safely concluded that a sample moving speed in the range of typical casting speeds would also have produced the same results. Another difference between cooling tests with instrumented samples and the actual direct-chill casting process is the amount of heat involved. Whereas instrumented samples are initially at a uniform temperature below the melting point, DC cast ingots and billets come out  110  of the mould as a solid shell surrounding a molten metal core. The latent heat of solidification constitutes a very large amount of additional heat to extract from the surface in the full-scale process. However, the cooling experiments presented in this research work were conducted with relatively large samples compared to the ones used by other authors [40, 41, 45, 50]. The IHC analysis of temperature data showed that heat was primarily removed from a thin section of the sample just below the quenched surface. Moreover, the calculated temperature at the back of the sample was found to remain constant throughout the test. This would indicate that experiments conducted with even larger samples, or with samples with an extra heat reservoir far below the surface, would have led to the same boiling water heat transfer phenomena at the surface. In other words, the instrumented samples used in this research work were providing an accurate simulation of a DC cast ingot. The approximation of a quenching experiment to the cooling of a semi-infinite solid is generally considered valid if the non-dimensional Fourier number given by the following equation is smaller than 0.1: Fo=  kt  in which the characteristic length L is the thickness of the sample  (71) (Ca.  0.10 m) [62]. The Fourier  numbers for the quenching tests conducted in this research work are below 0.1 for the first 20 seconds of each test, i.e. the limited sample size would only affect the heat flux measurements at the end of an experiment, when the surface is cooled by forced convection.  7.2  Finite Element Model of cooling experiments The accuracy of the results presented in Chapter 6 was evaluated by modelling cooling  experiments and comparing the results of simulations to the actual results of the cooling tests.  111  Stationary experiments were modelled using the same mesh and thermophysical properties as in the IHC analysis. The external boundary conditions were adiabatical, except on the quenched face of the sample. A surface heat flux corresponding to the impingement zone was applied over an area of height Hjz as obtained from Equation (26) on p. 73, whereas the rest of the quenched face was cooled according to a boiling curve for the free falling zone. The surface heat flux for secondary cooling was previously found to be a function of the surface temperature, the cooling water flow rate, the surface roughness, the cooling water temperature, the thermophysical properties, the surface temperature history, etc. Boiling curves were used to relate this heat flux to the sample surface temperature. The shape and magnitude of these boiling curves was a function of the other parameters listed above.  7.2.1  Idealized boiling curves The boiling curves for a given set of conditions were calculated by combining the  equations presented in Chapter 6 for each boiling regime. Examples of this process are presented here for the AA5 182 alloy. In order to put together a boiling curve for the impingement zone, the following variables must be evaluated: • Heat flux in forced convection regime; • Onset of nucleate boiling temperature; • Heat flux in nucleate boiling regime; • Critical heat flux; • Heat flux in transition boiling regime; • In the water jet impingement zone:  112  Leidenfrost point; Minimum heat flux at the Leidenfrost point; • In the water film free falling zone: Rewetting temperature. Figure 64 shows the boiling curve for the impingement zone at the surface of an AA5 182 ingot cooled with 150 L/minm of water at 15°C. Because the initial surface temperature T 0 is lower than the Leidenfrost point  TLpt,  film boiling does not occur. The surface  is cooled by natural air convection above the water jet impingement zone and by transition boiling in the impingement zone itself.  E  0  100  200  300  400  500  600  Surface temperature (°C)  Figure 64: Modelled boiling curve for the impingement zone (AA5182, T 0 = 400°C, Tj 15°C, Q’150 L/minm) Moreover, the very high critical heat flux  JCHF  which corresponds to the high cooling  water flow rate is not reached. Transition boiling begins at a relatively low temperature and cannot fully develop the steam bubble structure of the critical heat flux. The cooling mechanism 113  thus goes from partially developed transition boiling to low temperature nucleate boiling. This combination of high cooling water flow rate and relatively low initial temperature is characteristic of the steady-state phase of the DC casting process. Figure 65, on the other hand, represents the calculated or idealized boiling curve for typical start-up conditions: high initial temperature (To  550°C) and low water flow rate  (Q’  =  50 L/minm). Correspondingly, the Leidenfrost point is lower than the initial temperature and film boiling takes place. Moreover, the critical heat flux  CHF  is much lower and is reached very  quickly after transition boiling begins. Boiling curves with a critical heat flux plateau as in Figure 65 have already been successfully used to model the direct-chill casting process [36]. 8.OE+06 7.OE+06 6.OE+06 .  5.QE+06  4.OE+06 3.OE+06  2.OE+06 1 .OE+06 O.OE+00 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 65: Modelled boiling curve for the impingement zone (AA5182, T 0 = 550°C, Tf= 15°C, Q’50 L/min.m) Another point of importance is the transition from natural air convection and stable film boiling, which is not instantaneous but rather proceeds in a fashion similar to the transition boiling regime. This particular boiling water heat transfer mechanism has been referred to as  114  “initial cooling” (IC) by Li [77]. Initial cooling (IC) can also be observed in Figure 43 on p. 79. Secondary cooling for the transition boiling and initial cooling regimes can be modelled with a heat transfer coefficient which increases linearly with temperature, as presented in Section 6.6. Figure 66 illustrates an idealized boiling curve for the water film free falling zone. It corresponds to a moderate cooling water flow rate  Q’  of 100 L/minm and a high initial  temperature T 0 of 525°C. The rewetting temperature Twet for this set of conditions is approximately equal to 400°C. This rewetting temperature will determine whether the water film will be ejected at the top of the free falling zone. If water film ejection does not take place, the boiling curve shown in Figure 66 can be used to model the secondary cooling. 8.OE+06  1 CHF 6.OE+06  CI-fFd  E 5.OE+06  10mm  CHFd = 50mm  x 4.OE+06 D CHFd = 250 mm  3.OE+06 2.OE+06 I .OE+06 0.OE+00 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 66: Modelled boiling curve for the free falling zone (AA5182, T 0 = 525°C, Tj 15°C, Q’lOO L/min.m) On the other hand, if the surface temperature at the top of the free falling zone is above 400°C and the water film is ejected, another idealized boiling curve will have to be built for the new dry surface temperature Td,,,. Figure 66 also presents four lines for the critical heat flux:  115  each line corresponds to the critical heat flux at a certain distance below the impingement zone z. The boiling curve in bold represents the different boiling water heat transfer mechanisms 1 d which will take place 50 mm below the water jet impingement zone.  7.2.2  Comparison between idealized and experimental boiling curves Figure 67 compares the boiling curve for the impingement zone from Figure 30 (see p.  65) with the corresponding idealized or calculated boiling curve. The modelled curve follows the experimental one very closely, except in the high temperature nucleate boiling regime where a deviation can be observed. Figure 68, which compares boiling curves for the water film free falling zone, shows that the model is very accurate in predicting the experimental boiling curve. The largest source of error appears to be the rewetting temperature: it is 345°C on the idealized boiling curve and 335°C on the experimental one. 8.OE+06 7.OE+06 6.OE+06 n  2 4.OE+06  z  3.OE+06 2.OE+06 1.OE+06 O.OE+QO 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 67: Modelled and experimental boiling curves for the impingement zone (AA5182, T 0 = 475°C, Tf 30°C, Q’=87.5 L/minm)  116  8.OE+06 7.OE+06 6.OE+06 (N  E 5.OE+06 4.OE+06 3.OE÷06 2.OE+06 1.OE+06 0.OE+00 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 68: Modelled and experimental boiling curves for the free falling zone (AA5182, T 0 = 475°C, Tf= 30°C, Q’87.5 L/min.m)  7.2.3  Progression of rewetting front The accuracy of equations presented in Chapter 6 was also evaluated by considering the  progression of the rewetting front during a cooling experiment. Data from FEM simulations of cooling experiments was analyzed, and the progression of the rewetting front was determined by calculating the second derivative of the temperature with respect to time, as described in Section 5.5. As Figure 69 shows, the FEM model of cooling experiments based on the equations from Chapter 6 can accurately predict the speed of the rewetting front in the water film free falling zone.  117  30  \  • Experimental results  E 25  —  i20  FEM simulations  0  a)  o  0  Cl) Cl)  a) I—  2 10  I  250  I 350  300  Initial  400  450  500  temperature (°C)  Figure 69: Modelled and experimental progression of wetting front  7.2.4  Effect of thermophysical properties Figure 70 compares boiling curves for aluminum AA5 182 and magnesium AZ3 1. The  conditions used to draw these curves were the same as in Figure 65: high initial temperature (To =  550°C) and low water flow rate  (Q’  =  50 L/minm). Figure 70 clearly illustrates the effect of  thermophysical properties on the Leidenfrost point  TLpt.  The two boiling curves also present a  significant difference in the transition boiling regime. The heat fluxes in lower temperature regimes (critical heat flux, nucleate boiling, forced convection), on the other hand, are very similar for both boiling curves.  118  5.OE+06  4.OE+06  3.OE+06  2.OE+06  1 .OE+06  0.OE+00 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 70: Modelled boiling curves for two alloys 0 = 550°C, Tf= 15°C, Q’50 L/minm) (T  7.3  Finite Element Model of DC casting  7.3.1  Model development A FEM model was developed for the direct-chill casting of 455 mm diameter AZ3 1  magnesium billets. The greater part of this model was developed by Hao et al. [661 Among other simplifications, this model considered a Leidenfrost point which was independent of the cooling water flow rate and therefore remained constant at 290°C. It also did not take into account the possibility of water film ejection, but assumed instead that stable film boiling could take place in the free falling zone. Figure 71 shows the boiling curves used in Hao’s model for a water temperature Tf of 3 7.8°C and a water flow rate  Q’ of 190 L/minm. This model was updated by  using the different correlations developed in this research work as boundary conditions for the secondary cooling zone. The FEM algorithm used in the new model is described in Appendix B.  119  1.2E+07  1.OE+07  8.OE+06 E 6.OE+06 x D 44-,  ci)  I 4.OE+06  2.OE+06  0.0E+0O 0  100  200  300  400  500  600  Surface temperature (°C)  Figure 71: Model boiling curves used by Hao L66] (AZ31, Tj 37.8°C, Q’lOO L/min.m)  7.3.2  Calculation domain and finite element mesh The 1000 mm long cylindrical AZ3 1 billet with a 455 mm diameter was modelled using  an axisymmetric two-dimensional mesh with 2500 linear rectangular elements. The mesh size was 1 mm at the surface but as coarse as 35 mm at the axis. Figure 72 shows how the mesh size increases very gradually both in the r- and z-directions. The mould and dummy block were not part of the finite element mesh: heat transfer from the billet to the mould or dummy block was modelled using temperature-dependent boundary conditions. The continuous formation of the billet and its withdrawal from the mould was simulated by activating layers of elements at the top of the mesh. Unactive layers above the billet top were kept in heat conduction “limbo” by assigning them an extremely low thermal conductivity k of 0.1 W/mK.  120  0.05  0.04  V  V  V  —-  V  .gO.03  N  0.02  0.01  0.00 0.20  —  —  —  -  -  -  —  —  —  -  -  -  ———i-——  I  0.21  —  —  0.22  0.23  0.24  0.25  r-axis (m) Figure 72: Section of FEM mesh used to model the AZ31 455 mm billet  7.3.3  Thermal boundary conditions Both the top and the symmetry axis of the mesh were assumed to be adiabatical, i.e. the  temperature gradients were equal to zero as per Fourier’s law: =0  (72)  r=O  =0  (73)  z=1.O  The heat transfer between the bottom of the billet and the dummy block was governed by a temperature-dependent heat transfer coefficient: (74) z=O  121  Table V presents the heat transfer coefficient between the dummy block and billet bottom hDB as a function of temperature. The ambient temperature T-,) in Equation (74) was arbitrarily chosen as the water temperature Tf, which was equal to 37.8°C.  Table V: Heat transfer coefficient hDB for dummy block cooling 1661 Temperature range Heat transfer coefficient hDB .K) 2 (W/m (°C) 680>T>570  1200  570>T>540  26.7T-14000  540>T  400  The boundary conditions along the billet surface (r  =  0.455) were determined according  to the position of each integration point in the FEM mesh, as illustrated in Figure 73:  hp  BCFFZ  hIB Figure 73: External boundary conditions used in the FEM model  122  The primary cooling within the 100 mm high DC mould was governed by a temperature dependent heat transfer coefficient hpc as presented in Table VI. ÔT —k 3 UI  (75)  1 hpcTs_Tj r=O.455  Table VI: Heat transfer coefficient hp for primary cooling in the mould [66] Temperature range Heat transfer coefficient hpc K) 2 (WIm (°C) 680>T>590  1500  560 590>T > 5  47.5T-26500  > 530 3 560>T  75  530>T>510  2.5T-1250  510>T  25  Integration points above the DC mould, which corresponded to billet layers that had yet to be cast, were insulated: =0  (76)  r=O.455  The boundary conditions in the secondary cooling zone are governed by a boiling curve whose shape and magnitude is a function of numerous parameters. The process by which boiling curves are calculated is described in Section 7.2.1. The position of a given integration point in one of the four surface cooling zones (above the mould, in the mould, impingement zone, free falling zone) was determined by considering the casting speed v and the time elapsed since the beginning of the cast  The boundary between the impingement zone and the free falling zone  was also a function of the height of the impingement zone Hiz, which increases with the cooling  123  water flow rate. The withdrawal of the AZ3 1 billet from the DC mould was thus simulated by moving the different boundary conditions upwards along the stationary mesh.  7.3.4  Initial conditions The initial filling of the DC mould was assumed to take place in 60 s with a pour  temperature of 680°C. This phase of the direct-chill casting process was simulated by applying primary cooling and dummy block cooling for 60 s on the bottom portion of a mesh initially at a uniform temperature. The temperature profile at the end of the mould filling time was used as the set of initial conditions for the subsequent FEM simulations, which modelled the withdrawal of the AZ3 1 billet with constant casting speed v and cooling water flow rate  7.4  Q’.  Thermophysical properties The temperature dependent material properties of AZ3 1 are presented in Table VII and  Table VIII. The density s was assumed to be independent of the temperature and thus constant . As Table VII indicates, the thermal conductivity k was evaluated using linear 3 at 1780 kg!m relationships, except in the temperature interval between the solidus (424°C) and the liquidus 3 (630°C), where it was calculated by adding together the contribution of the solid (k W/mK) and liquid (k  =  =  118.5  60 W/mK) phases. The solid fraction was a function of the temperature  and given by the following equation:  f  =1.024—  5.12 (635—T)  (77)  which is equal to unity at 424°C, 0.90 at Ca. 594°C and zero at 630°C. Finally, the thermal conductivity above the liquidus was artificially set extremely high to take into account the convective heat transfer in the liquid pooi [20, 66]. 124  5 of AZ31 used in the FEM model Table VII: Thermal conductivity k Temperature range Thermal conductivity k (W/mK) (°C) 680>T>635  2.66T- 1570  635>T>630  12.OOT-7500  630 >T> 424  119.9-3001(635 -T)  424> T  0.0938 T+78.5  The specific heat C, 3 is given by a linear equation for temperature below the solidus. Between the solidus and the liquidus, it incorporates the latent heat of solidification sf of 339 kJ/kg and thus increases significantly [661. Whereas the specific heat between 424°C and 6 10°C was determined by taking into account the solid fractionfs as given by Equation (77), it was set as a constant between 6 10°C and 630°C in order to prevent instabilities in the FEM model.  Table VIII: Specific heat C,,, of AZ31 used in the FEM model Temperature range (°C)  Specific heat (JfkgK)  680>T>630  1414  630>T>610  15300  610>T>424  (635-T) 2 / 6 0.858T+891+1.73610  424> T>300  0.858T+891  300>T> 100  0.530T+989  l00>T  0.025T+ 1040  125  7.5  Importance of secondary cooling  7.5.1  Relative importance criterion As already pointed out in Section 2.6, the heat flux for nucleate boiling is of a secondary  importance when modelling the DC casting process, because heat conduction within the ingot is the limiting factor at low temperatures. The relative importance of secondary cooling can be quantified by considering the equivalent heat transfer coefficient heq , which takes into account both the heat transfer coefficient for surface cooling h and heat conduction, and is expressed by Equation (78):  heq  h  (78) eth  in which t is the time in seconds and  th 8  Ks° The equivalent 2 J/m . the thermal effusivity in 5  heat transfer coefficient thus increases with either the surface heat transfer coefficient or the thermal effusivity. As time increases and a thermal gradient is established below the quenched surface, the equivalent heat transfer coefficient decreases. Equation (78) can also be thought of as a sum of thermal resistances: the global resistance to heat transfer l/heq is then governed by the highest term on the right-hand side of Equation (78). /W during the DC casting of an 2 Figure 74 shows the different thermal resistances in Km AZ3 1 billet with a casting speed  v  of 0.8 minIs and a water flow rate  Q’  of 100 L/minm.  Whereas the thermal resistance corresponding to heat conduction within the billet is somewhat constant throughout the casting process, the thermal resistance for surface cooling experiences a significant drop around  t  =  60 s, i.e. after the mould filling time is elapsed. This drop  corresponds to the start of transition boiling, because the temperature at the top of the impingement zone T 0 is below the Leidenfrost point  TLp(.  126  0.1  (‘4  2  a) 0 0.001  .  J) C’) Cu  • 0.0001 I-  0.00001 58  63  68  73  78  83  Time (s)  Figure 74: Thermal resistance for surface cooling and heat conduction as function of time (AZ31, Tf= 37.8°C, Q’=lOO L/minm, v=O.8 mm/s) The total resistance is seen to be approximately equal to the surface cooling resistance  1/h at first, then approximately equal to the heat conduction resistance. The contribution of boiling water heat transfer in the secondary cooling zone to the global thermal resistance is therefore high when h is low (e.g. in the film boiling regime), but decreases when h increases (e.g. at the critical heat flux). The relative importance of secondary cooling  expressed in %  is given by: 100  (79)  ,  hi1+ h  7.5.2  eth  Importance of secondary cooling during process start-up Figure 75 illustrates the relative importance of surface cooling R 5 corresponding to the  thermal resistances presented in Figure 74. 127  100 90  Q... 80 70 60  E  20 i0 0  58  63  68  73  7&  83  Time (s) Figure 75: Relative importance of surface cooling Rs as function of time (AZ31, Tf= 37.8°C, Q’lOO L/minm, vO.8 mm/s)  Once again, surface cooling by boiling water heat transfer is seen to be significant for approximately 5 seconds. The significance criterion R 5 then drops below 5%, which means that 95% of the global thermal resistance is provided by heat conduction within the AZ3 1 billet. The relative importance of surface cooling remains very low at the transition point between the water jet impingement zone and the water film free falling zone. Figure 76 illustrates the relative contribution of surface cooling to the overall heat transfer during a simulation with a moderate water flow rate (75 L/minm) and casting speed (0.80 mm!s). The relevance criterion R 5 remains approximately at 100% during the mould filling time, but quickly drops to ca. 60% at the beginning of the water jet impingement zone. Film boiling, characterized by a relatively low heat flux, takes place for 5 seconds, then transition boiling and nucleate boiling lead to a significant increase in the surface heat transfer  128  coefficient. The relative importance of surface cooling correspondingly drops below 5% for the rest of the process. 100 90 80 -70 60  E  20 ft 10 0  58  63  68  73  78  83  Time (s)  Figure 76: Relative importance of surface cooling Rc as function of time (AZ31, Tf= 37.8°C, Q’=75 L/minm, vO.8 mm/s) Figure 77 shows the relevance of secondary cooling for a slightly lower water flow rate (60 L/minm) and a much lower casting speed (0.40 mm/s). Film boiling is seen to take place in the impingement zone and the relative importance of surface cooling is approximately 60%. Correspondingly, the boiling water heat transfer at the billet surface and the heat conduction within the billet both play a similarly significant role in the global cooling. When the water film is ejected at the beginning of the free falling zone, the surface heat flux is greatly reduced and surface cooling by natural air convection becomes the phenomenon which governs the cooling rate.  129  100 90 80 -70  IE 0 58  63  6  73  78  83  88  93  98  Time(s)  Figure 77: Relative importance of surface cooling as function of time (AZ31, Tf= 37.8°C, Q’6O L/minm, v=O.4 mmls) A comparison of Figure 75, Figure 76 and Figure 77 indicates that water film ejection in the free falling zone and, to a lesser extent, stable film boiling in the impingement zone, are two of the most important phenomena during the transient start-up phase. It is therefore critical to determine whether the water film will be ejected or not.  7.6  Effect of casting parameters Control of the direct-chill casting process can only be achieved through a limited set of  parameters: casting temperature, mould filling time, casting speed, cooling water temperature and quality, water flow rate and water jet velocity. Each of these parameters influences the likelihood of seeing the water film being ejected in the free falling zone. For instance, a higher casting temperature will increase the surface temperature at the beginning of the water jet 0 and favour the ejection of the water film. Conversely, a longer mould impingement zone T  130  filling time will lead to a lower surface temperature and decrease the probability of water film ejection. The casting speed influences both the surface temperature in the impingement zone (because of its effect on the advanced cooling front phenomenon) and the time spent under the water jets. A greater casting speed will therefore promote the ejection of the water film. Cooling water temperature exerts an effect on the primary cooling as well as on the heat flux in the film boiling regime. It was also reported to influence the Leidenfrost point, although this was not studied in this research work. A higher water temperature will thus favour water film ejection. A greater water jet velocity, which can be obtained by designing the DC mould with smaller holes, was seen to significantly increase the Leidenfrost point  TLpg.  It would therefore be expected to  act against the ejection of the water film. Finally, the cooling water flow rate influences the likelihood of water film ejection in at least four distinct ways: • It determines the Leidenfrost point temperature TLp( for the water jet impingement zone; • It influences the heat flux in the film boiling regime I4iHF; • It affects the height of the impingement zone I-liz and thus the amount of time spent in this zone; • It exerts an effect on the water film ejection / rewetting temperature Twet. Correspondingly, the relationship between the water flow rate and the probability of water film ejection is not linear, because the different effects listed above are multiplied. Figure 78 illustrates the critical casting speed v which will lead to the ejection of the water film for different cooling water flow rates  Q’.  Whereas a 10 L/minm increase from 50 to 60 L/minm  allows a 0.14 mm/s increase in the casting speed while avoiding water film ejection in the free  131  falling zone, an additional increase to 70 L/minm will allow a casting speed of 1.24 mm!s, i.e. an increase of Ca. 1.00 mmls. 1.4 1.2  0  S..  E .  0.8  0 ci) ci)  C (I) 0  o  0.4 0.2  0.0 35  40  45  50  55  60  55  70  75  Water flow rate (L/minm)  Figure 78: Critical casting speed v for water film ejection as function of flow rate (AZ31, Tf= 37.8°C)  Q’  As Figure 78 shows, a high casting speed and a low water flow rate will lead to the ejection of the water film. This phenomenon is generally undesirable in the DC casting of magnesium AZ3 1 because of safety concerns: the ejection of the water film is associated with a reheating of the billet surface and a potential liquid metal breakout [4, 19]. A low casting speed and a high cooling water flow rate would ensure that the water film is not ejected in the free falling zone, but low casting speeds translate into low productivity. Finally, a high casting speed and a high water flow rate have been shown to favour the formation of surface cracks, because of the significant difference between the cooling rates at the surface and in the center [231. A criterion combining the casting speed and the cooling water flow rate can be used as a rule of thumb to determine the average length of surface cracks or the cracking frequency fc for magnesium AZ3 1 billets [20]: 132  f  (80)  Q’ 2 =0.0036(v)  Figure 79 presents the critical casting speed and water flow rates with respect to the ejection of the water film as well as conditions with the same surface crack criterion fc according to Equation (80). The conditions which maximize the productivity while preventing water film ejection and liquid metal breakouts correspond to the area just below the critical water film ejection curve. The casting speed and cooling water flow rate at the beginning of the DC casting process would therefore have to follow the critical water film ejection curve closely but without going over it. Figure 79 also provides the minimum cooling water flow rate to maintain in the steady-state phase of the direct-chill casting process, i.e. once the casting speed has reached its maximum. For the conditions investigated in this chapter, this minimum water flow rate  Q  is approximately equal to 75 L/minm.  1.4 1.2  C)  E 0.8 a) U)  Z)  0.6  o 0.4 0.2 0.0 35  40  45  50  55  60  65  70  75  Water flow rate (Liminm)  Figure 79: Critical casting speed v and cracking frequencyfc (AZ31, Tf= 37.8°C)  133  In the direct-chill casting of aluminum ingots, film boiling and water film ejection can constitute favourable conditions, because they prevent the formation of butt curl during start-up. Although Figure 78 was developed for the DC casting of magnesium AZ3 1 billets, it provides qualitative information for other processes as well. For instance, there exists a non-linear relationship between the cooling water flow rate  Q’  and the critical casting speed v for the  direct-chill casting of aluminum. It would therefore be possible to develop a critical water film ejection curve similar to the one in Figure 79 and design a start-up procedure which would minimize the formation of defects while maximizing the casting speed.  134  8  SUMMARY AND CONCLUSIONS  8.1  Summary The direct-chill casting process is used in the non-ferrous metals industry to produce  ingots, blooms and cylindrical billets. Molten metal is poured into a water-cooled mould where it forms a solid metal shell. Primary cooling, i.e. the removal of heat through the mould wall, is followed by secondary cooling, in which the ingot surface is directly cooled by water jets. Secondary cooling is responsible for approximately 80% of the heat extraction in the steadystate phase of the direct-chill casting process. Because of the high surface heat flux associated with the nucleate boiling regime, the factor which limits the secondary cooling during steadystate is not the boiling water heat transfer at the ingot surface but rather the heat conduction within the ingot. During the transient start-up phase, however, the surface heat flux is significantly lower and the heat extraction in the secondary cooling zone is governed by the different boiling water phenomena. This phase of the direct-chill casting process is of critical importance for the formation of defects such as butt curl, hot tears and surface cracks. Control of the heat extraction by primary, secondary and tertiary cooling can reduce or altogether prevent the formation of defects and thus decrease the amount of process scrap. The design, control and optimization of the direct-chill casting process can be conducted using a numerical model of the thermal history within the ingot or billet. Such a model is based on the solution of transient heat conduction problems and requires an accurate knowledge of the boundary conditions. Whereas earlier models used constant temperatures, heat fluxes or heat transfer coefficients for the secondary cooling zone, recent models rely on non-linear boiling curves which describe the relationship between the heat flux and the surface temperature. This research project studied the effect of different parameters on the heat flux in the secondary cooling zone of the direct-chill casting process. The secondary cooling of aluminum 135  AA5 182 and magnesium AZ3 1 was investigated by quenching instrumented samples with water jets similar to the ones used in the DC casting process. Sub-surface thermocouples recorded the thermal history at several points within the sample. An inverse heat conduction analysis converted the different thermal histories into surface heat fluxes and surface temperatures in order to draw boiling curves. The inverse heat conduction algorithm developed for this research work considered phenomena such as the advanced cooling front and the ejection of the free falling water film, which exert a significant influence on the temperature measured by the sub surface thermocouples and thus could introduce errors in the calculated heat flux and surface temperature if they were not taken into account. The inverse heat conduction algorithm was validated by comparing its results to known solutions of simple heat conduction problems. It was shown to be very precise in modelling the non-linear heat flux profile along the sample surface and identifying the water film rewetting temperature. Cooling experiments were conducted with two alloys, which provided data for two sets of thermophysical properties and two types of surface roughness. Cooling tests were also done with a wide range of initial temperatures, casting speeds and cooling water flow rates. The influences of these parameters on specific attributes of the boiling curve, such as the Leidenfrost point, the critical heat flux and the slope in the transition boiling regime, were identified. Changes made to the experimental water jet rig also allowed studying the effect of the water jet velocity. The relationship between a given parameter and a characteristic of the boiling curve was compared to and often modelled after already published equations for the secondary cooling of light metals. Equations for each boiling water heat transfer regime were combined to calculate a complete boiling curve for a set of conditions. The idealized boiling curves drawn using these equations were implemented in a finite element model for the direct-chill casting of AZ3 1 billets. Simulations of the process start-up 136  phase showed the interaction between different casting parameters such as the casting speed and the cooling water flow rate. The respective contributions of surface cooling and internal heat conduction to the global heat extraction were identified and compared. This provided information about the significance of boiling water heat transfer phenomena in the secondary cooling zone.  8.2  Conclusions The following conclusions can be drawn from this research project: • The integration of a moving boundary to the IHC analysis provided reliable and accurate data for the heat flux in the secondary cooling zone. In particular, the IHC algorithm’s ability to distinguish between the advanced cooling front effect and boiling water heat transfer was essential in correctly identifying the water film rewetting temperature. • Boiling curves for the water jet impingement zone and the water film free falling zone can be calculated using a number of equations. These equations relate different parameters to characteristic attributes of the boiling curve, such as the Leidenfrost point, the slope of the transition boiling regime or the critical heat flux. Boiling curves for the secondary cooling of magnesium AZ3 1 had never been previously calculated, and were shown to significantly differ from boiling curves for aluminum alloys such as AA5 182. • The cooling water flow rate exerts a significant effect on several aspects of the secondary cooling. Specifically, it was found to influence the height of the impingement zone, the heat flux in the forced convection, nucleate boiling and stable  137  film boiling regimes, the critical heat flux, the Leidenfrost point temperature and the water film rewetting temperature. The influence of the water flow rate on the ejection of the water film in the free falling zone is extremely important, because its effects on the Leidenfrost point, the rewetting temperature and the height of the water jet impingement zone are multiplicative. The thermophysical properties of the cast alloy play a significant role at high surface temperatures, i.e. when the heat flux is relatively low. The thermal effusivity was found to influence the Leidenfrost point as well as the water film rewetting temperature. Aluminum alloys with a high thermal effusivity are easier to wet than magnesium alloys, whose thermal effusivity is lower. Correspondingly, the Leidenfrost point and rewetting temperature of aluminum AA5 182 was significantly lower than those of magnesium AZ3 1. • The thermophysical properties also influence the slope of the boiling curve in the transition boiling regime. A relationship was observed between this slope and the thermal conductivity for the casting of various materials. Aluminum alloys with a high thermal conductivity will allow a faster propagation of the wet / dry interface and thus present a steeper slope in the transition boiling regime. • This research work highlighted the influence of various thermophysical properties (e.g. thermal effusivity, thermal conductivity) on the boiling curve and underlined the absolute necessity of taking into account these properties when modelling the DC casting process, especially during the transient start-up phase. Correlations between the thermal effusivity and various attributes of the boiling curve allow the extrapolation of experimental results obtained with aluminum AA5 182 and magnesium AZ3 1 to other light metal alloys. 138  • The influence of the surface roughness on the boiling curve could not be quantified. Its effect on the forced convection and nucleate boiling regimes was observed, but could not be distinguished from the influence of thermophysical properties. • The effect of the casting speed was linked to the initial temperature at the impingement point. This initial temperature was found to exert a significant influence on the transition boiling regime, the water film rewetting temperature and, to a lesser extent, the critical heat flux. • The role of boiling water heat transfer on the global rate of heat extraction in the secondary cooling zone is only significant if stable film boiling and water film ejection occur. Accurate knowledge of the boiling curve in high temperature boiling regimes (transition boiling and film boiling) is therefore much more important than precise equations for lower temperature regimes (nucleate boiling and forced convection). This would indicate that further research work should concentrate on the effect of different parameters on the Leidenfrost point and the rewetting temperature rather than on the critical heat flux or the nucleate boiling regime.  8.3  Recommendations  Based on the results of this research work, topics which warrant further investigation include the following: • The occurrence of stable film boiling in the impingement zone was shown to be extremely important in determining the contribution of surface cooling to the global heat extraction process. This research project quantified the effect of the cooling water flow rate and the thennophysical properties on the Leidenfrost point. It also  139  showed how the water jet velocity exerts an influence on the Leidenfrost point, but only in a qualitative fashion. Further investigation with a wider range of water jet velocities would allow the development of a relationship between this parameter and the Leidenfrost point. • Another parameter of importance which wasn’t fully studied is the cooling water temperature. The effect of the water temperature on the heat flux in the film boiling regime was obtained by comparing experimental results with data from the literature. Additional cooling tests conducted with different water temperatures could provide information regarding the Leidenfrost point, the slope of the transition boiling regime and the rewetting temperature. • Research work conducted with quench missiles showed that the water composition (e.g. oil, surfactants, ions, suspended solids, dissolved gases) can significantly affect the heat flux in the film boiling and transition boiling regimes. However, the results of these tests cannot be directly translated to boiling curves for the full-scale directchill process. 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Bathe: Finite Element Procedures in Engineering Analysis, Prentice-Hall, New Jersey, 1982.  146  APPENDIX A: COOLING TEST CONDITIONS  Test number  Type*  Alloy  Cooling water flow rate Q’ (Llminm)  Jet velocity**  Casting speed v (mmls)  Initial temperature T 0 (°C)  1  S  AZ31  50  H  0  500-540  2  S  AZ31  62.5  H  0  485-525  3  5  AZ31  75  H  0  510-550  4  S  AZ31  87.5  L  0  340-350  5  S  AZ31  87.5  L  0  440-450  6  S  AZ31  87.5  H  0  485-525  7  S  AZ31  100  L  0  350-355  8  S  AZ31  100  L  0  435-440  9  5  AZ31  100  H  0  520-555  10  S  AZ31  100  H  0  570-605  11  S  AZ31  100  H  0  520-560  12  5  AZ31  112.5  L  0  340-345  13  5  AZ31  125  L  0  330-335  14  5  AZ31  125  L  0  345-350  15  S  AZ31  125  L  0  435-445  16  5  AZ31  125  H  0  490-525  17  W  AZ31  87.5  L  20  400-405  18  W  AZ31  87.5  L  20  400-410  19  W  AZ31  87.5  L  20  450-455  20  W  AZ31  87.5  L  20  450-455  21  5  AA5182  50  H  0  485-530  22  5  AA5182  50  H  0  480-520  147  Test number  Type*  Alloy  Cooling water flow rate Q’ (L/minm)  Jet velocity**  Casting speed v (mm/s)  Initial temperature T 0 (°C)  23  S  AA5182  62.5  H  0  485-520  24  S  AA5182  75  H  0  495-530  25  S  AA5182  87.5  L  0  460-470  26  S  AA5182  87.5  L  0  425-470  27  S  AA5182  87.5  H  0  485-530  28  S  AA5182  100  L  0  240-250  29  S  AA5182  100  L  0  290-305  30  S  AA5182  100  L  0  310-330  31  5  AA5182  100  L  0  315-325  32  5  AA5182  100  L  0  395-400  33  S  AA5182  100  L  0  465-470  34  5  AA5182  100  L  0  460-475  35  S  AA5182  100  H  0  495-525  36  5  AA5182  100  H  0  485-515  37  5  AA5182  112.5  L  0  465-475  38  S  AA5182  112.5  L  0  435-465  39  5  AA5182  112.5  H  0  490-525  40  5  AA5182  125  L  0  465-475  41  S  AA5182  125  L  0  440-460  42  5  AA5182  125  H  0  490-530  43  S  AA5182  137.5  L  0  465-475  44  5  AA5182  150  L  0  380-405  45  S  AA5182  150  L  0  460-470  46  5  AA5182  150  L  0  430-460 148  *  Test number  Type*  Alloy  Cooling water flow rate Q’ (L/minm)  Jet velocity**  Casting speed v (mm/s)  Initial 0 temperature T (°C)  47  W  AA5182  100  L  10  320-325  48  W  AA5182  100  L  10  395-405  49  W  AA5182  100  L  10  470-485  50  W  AA5182  100  L  20  320-335  51  W  AA5182  100  L  20  395-405  52  W  AA5182  100  L  20  470-480  53  W  AA5182  150  L  40  455-480  54  R  AA5182  75  H  30  525-560  55  R  AA5182  87.5  H  30  475-560  56  R  AA5182  100  H  30  525-560  57  R  AA5182  100  H  30  545-580  58  R  AA5182  112.5  H  30  525-560  59  R  AA5182  125  H  30  535-565  Test types: stationary (S), wetting (W), rewetting (R)  **  Jet velocity: low (L), high (H)  149  APPENDIX B: FINITE ELEMENT MODELLING The partial differential equation governing transient heat conduction is:  dT o( 8T” o( 8T”i 81 8T” —1k —1+—Ik —1+—ik —I=pC dt 8x) 6y ôy) 8z ãz)  (81)  in which k, k and k are the thermal conductivities in the x-, y- and z-direction. For an isotropic material, the thermal conductivity k is independent of the direction. In this research work, the partial differential equation had to be solved for boundary conditions of the Neumann type:  k  ÔT ôx  —  +  k  ÔT n, 8))  —  +  k  ÔT n 8z  +  q(x, y, z, t) =0  (82)  and for initial conditions:  (x,y,z) 0 =T 0 10 T  (83)  The finite element solution to the partial differential equation is obtained by assuming a linear function for the temperature T(x,y,z) such that the residual between this approximate solution and the exact solution to the partial differential equation is equal to zero (in the weighted-integral sense). The weighted residual method used to solve the partial differential equation is Galerkin’ s method, in which the weight functions w 1 are equal to the shape functions N 1 used to model the linear temperature profile: I 1 T=N  (84)  in which T 1 corresponds to the different nodal temperatures. The four linear shape functions corresponding to a rectangular isoparametric element  (u,v)= 1 N (u,v)= 2 N  (1—uXl—v) (1—uXl+v)  (ii, v)  are:  (85) (86) 150  (u,v)z 3 N (u,v)= 4 N  (i+uXi—v)  (87)  (i+uXi+v)  (88)  The residual over the two-dimensional domain 12 is given by:  (89) and can be integrated by parts using the divergence theorem: s[kn  +  k4cny1NidS  _ss[k-+ k!L]dxdy _ss[NIPc  J1  dxdy  =  0  (90)  The first term of Equation (90) corresponds to the integration of the boundary conditions over the surface S. The two other terms are respectively referred to as the conductivity and heat capacity terms. If the unknown temperatures are expressed in vector form  T=[N1{T’}  (91)  then Equation (90) becomes dS — 11[k[]{Te }1 — JPN  +  }!‘dxdy  pc [Nlç]dxdY =0 1 — f [ii,r  (92)  in which the surface heat flux c2 is positive if heat flows into the body, and negative if it flows out of this body. The different terms of Equation (92) are evaluated numerically using a four-point Gauss quadrature integration. The four points for a rectangular isoparametric element (u, v) are (_l/J,l/J’) and coordinates (x,y)  (iiJ,iiJ). The conversion from global  to (u,v) space is done via the Jacobian matrix [J]:  151  1 oN  1 aN —x 1 Ou ON. --xi  In— 1j  ON. --y 1  so that the element area can be simplified as: 11  ffdxdy  =  Q  I IIjIUcV -1-1  (94)  where the determinant of the Jacobian matrix  Lx  =  is given by:  !iYLyJ _!yLxJ  (95)  For instance, the conductivity term in Equation (92) is integrated by multiplying the matrix of differential operators [B] by the determinant of the Jacobian matrix  hi  at each Gauss  integration point:  —  .iI[k  +  k  4  .L.dxdy  =  _[[B]T [k][B]JI]{Te  }  =  —[K]{T}  (96)  where the (2x4) [B] matrix is given by: ON 1 rBl_ Ox ON Oy -  —  ON 2 Ox 2 ON Oy  ON 3 Ox 3 ON Oy  ON 4 Ox 4 ON Oy  97  and the (2x2) diagonal [k] matrix is simply: [k] =  Ek  01  [o  kj  (98)  Correspondingly, the heat capacity term in Equation (92) is given by: PC[N]ç]dXdY = 1 _[N  [2{}c{N}TJI]dT  =  _[c1.ff  (99)  and the boundary conditions at the surface S are replaced by a load vector. For the top and the bottom faces of the mesh (v  ±1), this load vector is given by:  152  —  S fN d 1  =  +  = {f,}  (100)  ={f}  (101)  whereas it is equal to  for the side faces of the mesh (u  = ±1).  Equations (96) to (101) correspond to the case of two-dimensional heat conduction in cartesian (x,y) coordinates. In the case of axisymmetrical (r,z) coordinates, the original partial differential equation is: dT io( 6T” a( 8T” ——Irk— I+—I k— I=pC dt ôz) 8r } ôz r  (102)  The integration by parts of this equation gives: 1[kflr s  +  knz1Njds  _sf  rôr  8zz  f!-jdQ dt  =0  (103)  in which the surface element d5 is a cylinder:  = 2vrdr at the top and bottom dS = 2rrdz at the edge dS  (104) (105)  and the volume element d[2 is a ring: (106)  d2=22rrdrdz In vector form, the different terms of Equation (103) become +  —  _il[NIPCP[N]r]dQ  = _22r[ yzr[BJ T [kIB]J] {Te }= —[KJ{T} = _2Jr[ r{N}pCp{N}TIJ]J = —[c]  2 = {f} 1 = 22rr{N}iJJ?i +J dS = 2rr{N}ijJ 1 CPN 1 4 = f}  S -JN d 1  —  +  (107)  (108)  (109) (110) 153  i.e. the global radius in (r,z) space is required to calculate the matrices [K] and [C] as well as the load vector  ff}.  The transient heat conduction equation (111)  [CJ+[KJ{T}={f}  is solved using a step-by-step, recurrent calculation. The derivative of the temperature with respect to time is approximated using the two-point Crank-Nicholson method: cIT dt T=  =  {T} {T}, , 1 At  (112)  —  1 ±{T} 11 {T}  (113)  Equation (111) then becomes  [_1 + Tii]{T} , 1  =  [. —  +  {j  }  where {T}, corresponds to the known temperatures at time temperatures at time  (114) 1 t  and {T},+j to the unknown  . ÷ 1 t  Equation (114) is non-linear if the stiffhess matrix [K], the heat capacity matrix [C] or the load vector  {fp}  are temperature-dependent. However, it can be linearized by assuming that  thermophysical properties are constant within the time step zlt. This assumption is valid provided the time step chosen is sufficiently small. The algorithm used to solve the linear Equation (114) relied on compacted storage and a column reduction scheme to speed up the calculation [72].  154  

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