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Heat transfer model of the hot rolling runout table-cooling and coil cooling of steel Hernandez-Avila, Victor Hugo 1994

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HEAT TRANSFER MODEL OF THE HOT ROLLING RUNOUT TABLE-COOLING AND COIL COOLING OF STEEL by VICTOR HUGO HERNANDEZ-AVILA B. Sc., The National University of Mexico, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER IN APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Metals and Materials Engineering) We accept this t^ e^ s^conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1994 © Victor Hugo Hernandez-Avila, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of fjdnk W Ma&r/d iCnrji^r/n^ The University of British Columbia Vancouver, Canada Date t/an 3 1995 DE-6 (2788) ABSTRACT The controlled cooling of steel strips is required to attain the high quality standards of flat-rolled steels employed in important industries such as the automobile, petro-chemical, house-appliances and construction. Strict control of the temperature of the strip during cooling in the hot-strip runout table is necessary, but little success has been reached in the optimization of the heat removal since no real understanding of the physical mechanisms involved has been attained. Given that the experimental measurements of the local heat-transfer coefficients may involve very complex procedures, the modeling of the boiling mechanisms is presented as the best way to obtain the local thermal response of steel strips during their processing, and mathematical models for the runout table and subsequent coil cooling are presented as powerful tools to predict the thermal and the microstructural response of the steel. The runout table model is unique in the sense that it is mechanistic in nature and predicts the local heat-transfer coefficients during cooling. The model adopts the extrapolation of the "macrolayer evaporation mechanism" into the forced-flow transition boiling regime. The analysis in terms of the nucleation process, fluid flow, liquid-solid contact area, and the liquid-vapor interface instability allow succesful prediction of pilot-plant and full-scale operations and of the most fundamental microscopic parameters measured elsewhere. The liquid-solid contact found in the transition boiling regime is responsible for most of the heat released, and explains why previous assumptions with regard to film boiling failed to account for the effect of variables such as water temperature or strip velocity on the cooling process. This study shows that bottom jet cooling is much lower than top cooling not only because of the smaller contact but also because of the inherently lower stability of the liquid-vapor interface of the latter. ii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS ix DEDICATION x CHAPTER 1. INTRODUCTION 1 CHAPTER 2. LITERATURE REVIEW 5 3.1 Runout Table Model ' 5 3.2 Coil Cooling Model 9 3.3 Water Jet Cooling 14 3.3.1 Fluid How 16 3.3.2 Convection Heat Transfer 20 3.3.3 Nucleate Boiling 21 3.3.4 Critical Heat Flux 23 3.3.5 Transition Boiling 24 3.3.6 Film Boiling 28 3.4 Convective Transition Boiling Modeling 31 3.4.1 Mechanism. 32 3.4.2 Modeling Fundamentals 35 3.4.3 Liquid-Solid Contact Heat Transfer 38 3.4.4 Film Boiling 43 CHAPTER 3. SCOPE AND OBJECTIVES 46 CHAPTER 4. EXPERIMENTAL PROCEDURE 47 4.1 Pilot-Plant Trials 47 4.2 Full-Scale Measurements 51 CHAPTER 5. MATHEMATICAL MODEL 53 5.1 Runout Table Model 53 5.1.1 Air cooling 56 iii 5.1.2 Parallel Flow Transition Boiling Model 57 5.1.3 Pressure Gradient Flow Transition Boiling Model 62 5.1.4 Phase Transformation Model 63 5.1.5 Grain Size Model 65 5.2 Coil Cooling Model 65 CHAPTER 6. MODEL VALIDATION 69 6.1 Runout Table 69 6.1.1 Comparison of Model predictions against Pilot-Plant 69 measurements 6.1.2 Comparison of Model predictions with Full-Scale 73 measurements 6.2 Coil Cooling Model 90 CHAPTER 7. SENSITIVITY ANALYSIS 93 7.1 Runout Table Operating Parameters 93 7.1.1 Effect of Water Flow Rate 93 7.1.2 Effect of Water Temperature 95 7.1.3 Effect of Strip Velocity 97 7.1.4 Effect of Initial Strip Temperature 100 7.2 Coil cooling parameters 102 CHAPTER 8. SUMMARY AND CONCLUSIONS 106 BIBLIOGRAPHY 108 LIST OF SYMBOLS 115 APPENDIX 120 iv LIST OF TABLES Page Table I. Comparison between pilot-plant and typical full-scale similarity 48 parameters. Table JX Operating data for A36 Steel 52 Table III. Operating parameters for A36 steel (coil 934848) 52 v LIST OF FIGURES Page Fig. 1. Typical Hot Strip Mill. Layout of the 84-inch continuous hot-strip 1 mill at the Gary Works of U. S. Corporation (2) Fig. 2. Typical Laminar Jet Bar flow system (3) 2 Fig. 3 Typical Top and Bottom Planar Jets (3) 3 Fig. 4 Typical Temperature-Controlled and Heat-Flux Controlled 15 Boiling Curves. Fig. 5 Fluid Flow in jet cooling 17 Fig. 6. Experimental Boiling Curves for a planar water jet (27) 25 Fig. 7. Heat Transfer Coefficient as a function of 26 position from the jet (36) Fig. 8. Parameter F as a function of solid supeheat (49) 37 Fig. 9. Heat Flux during solid-liquid contact (50) 39 Fig. 10. Macrolayer Evaporation Mechanism 40 Fig. 11. Schematic Diagram of Pilot-Plant Runout Table, and Thermocouple Placement for Surface Temperature Measurements. 48 Fig. 12. Time response for Type-J thermocouple in different 50 cooling conditions (65) Fig. 13. Effect of the thermocouple material on the time constant (66) 50 Fig. 14. Runout table model reference system 54 Fig. 15. Runout table model flowchart. 57 Fig. 16. The macrolayer evaporation mechanism in jet boiling 58 Fig. 17. Schematic Coil for the Coil Cooling Model 68 vi Fig. 18. Comparison between Pilot-Plant Measurements and Model Predictions 70 Fig. 19. Typical Boiling Curve for a Single Jet Cooling in the Pilot-Plant 72 Table Fig. 20. Temperature Measurements for the A36 steel 74 Fig. 21. The Macrolayer Evaporation Parameter for the A36 steel 77 Fig. 22. Effect of metp (Thickness) on the Heat-Transfer Coefficients 79 in the Stagnation Line of a Series of Circular Jets in the Runout Table. Fig. 23. Effect of metp (Thickness) on the Heat-Transfer Coefficients in the Parallel Flow Region Series of Circular Jets in the Runout Table. Fig. 24. Comparison of Runout Table Model Predictions with Measured 81 Exit Temperature for the A36 Steel. Fig. 25. Comparison between Model Predictions and A36 Strip (Head) 83 Fig. 26. Comparison between Model Predictions and A36 Strip (Middle) 84 Fig. 27. Comparison between Model Predictions and A36 Strip (Tail) 85 Fig. 28. Comparison of the Liquid-Solid Contact Heat Flux in a 87 Falling Drop (50) and in Jet Cooling for the A36 Steel (Coil 934848). Fig. 29. Comparison of the Liquid-Solid Fractional Contact Area in 88 Laboratory Measurements and Model Predictions for the A36 steel (Coil 934848) Fig. 30. Heat-Transfer Coefficients during the Cooling of the A36 89 Steel (Coil 934848) Fig. 31. Comparison of the Coil Cooling Model Predictions with the 91 Analytical Solution of a 1-D problem in the r-direction. Fig. 32. Comparison of the Coil Cooling Model Predictions with the 92 Analytical solution of a 2-D problem in the r and z directions. vii Fig. 33. Predictions of the Effect of Jet Velocity Variations on the Thermal and Microstructural Response of A36 steel. 94 Fig. 34. Predictions of the Effect of Water Temperature on the Thermal 96 and Microstructural Response of A36 Steel. Fig. 35. Effect of Water Temperature on the Heat-Transfer Coefficients 97 forA36 Steel. Fig. 36. Effect of the Strip Speed on the Thermal and Microstructural 99 Response of A3 6 Steel. Fig. 37. Effect of the Strip Speed on the Parallel Flow Zone and Pressure 100 Gradient Zone Heat-Transfer Coefficients for A36 Steel Fig. 38. Effect of the Initial Surface Temperature on the Thermal and 101 Microstructural Response of A36 Steel. Fig. 39. Thermal Response of a DQSK Steel Coil Cooled in Still Air 103 at 20 °C Fig. 40. Thermal Response of a DQSK Steel Coil Cooled in Still Air 104 at 10 °C Fig. 41. Thermal Response of a DQSK Steel Coil Cooled in Still Air 105 at 20 °C with a lower Radial Thermal Conductivity viii ACKNOWLEDGEMENTS I would like to express my sincere appreciation to Prof. J. K. Brimacombe and Prof. I. V. Samarasekera for their support, supervision and encouragement to complete succesfully my studies. Also I am grateful to Prof. Jose Antonio Barrera Godinez and Prof. Fidel Reyes Carmona for encouraging me to pursue graduate studies, and for all the advice and observations during the time of my studies. I also thank Craig Hlady, Chris Davis and Neil Walker and for their valuable assistance. Finally, the financial support from The National University of Mexico and the International Council for Canadian Studies was greatly appreciated. ix DEDICATION This work is specially dedicated to the most precious crystal, the most refreshing morning dew, the girl more perfect than any law, my wife, Patricia, who really has been the greatest contribution to this work (This thesis is also yours!). Osita, I thank you for everything, you...are so incredible!. The present work is also dedicated to the greatest "osito" ever born, my son Andre, whose energy goes beyond anything, and to Jehovah God, who created the entire universe (Rom 1:20). x Chapter 1. INTRODUCTION Chapter 1. INTRODUCTION The controlled cooling of steels on the runout table has become a common practice in the production of hot rolled products. The introduction of water jet cooling by BISRA in 1957 opened a new technology to produce HSLA steels (1). Consequently, fine ferrite grain size, combined with precipitation strengthening, could be manipulated to achieve high strength steels with a reduced carbon content. As a consequence of reduced carbon contents, a smaller amount of pearlite is present in the microstructure, improving weldability and toughness. Accelerated cooling refines ferrite grain size because of the lower transformation temperature and the resulting undercooling of austenite. A typical layout of a Hot Strip Mill is presented in Fig. 1. As shown, the Runout Table and Coiler are the final steps in the strip production. SlAS-RCCCtVIMG AND CWAWOIWG TAQLC S L A B - C O N D I T I O N I N G A R E A CsWITciD F I L T R A T I O N P L A N T - V A R O | L S L A 6 - S T O R A G E A R E A r— H C C t i n C M M O T O R Ua R O O M C " ° * ® HOKPMC T A B L C - M J W O U T T A B L C -<a<3 C O N T I N U O U S i&ittrer H O T - S T R I P M I L L ^ " H Q T - C C T L C O W V O O A C01LC«S M A I N T E N A N C E A N D R O L L S H O P E=l I 1" • a LEGEND 0 - ® SL*e-*«£*Tmc fufm*ccs <S> VERTICAL COCCfl (?) HORIZONTAL SCACC © - ® r i V C KOUCMWC STANOS Q f—SIMMS SCALC SRCAKCR © - O M V C K rmiSMIMC STANOS Fig. 1. Typical Hot Strip Mill. Layout of the 84-inch continuous hot-strip mill at the Gary Works of U. S. Corporation (2) The purpose of using water jet cooling is to obtain a very high heat extraction rate to produce a fine ferrite grain size (~10um). A uniform temperature field in the thickness Chapter 1. INTRODUCTION 2 direction is desirable; however, the thermal resistance of the steel is an impediment Inhomogeneity in mechanical properties and distortion of the strip results from localized cooling. Therefore, optimization of the cooling pattern is required to avoid such problems. The earliest water jet systems consisted of arrays of bars impinging on the top surface of the strip. Water jet cooling is sometimes called "laminar cooling" because of the streamline flow and transparent glassy appearance of the jet, though the jet is not necessarily in laminar flow. Spray cooling, or water bar cooling at the bottom surface has been used, in an attempt to assure a more uniform cooling. Intrinsic inhomogeneity of the water flow in the width direction was expected to cause non homogeneous properties, but this effect was greatly reduced by alternating the array pattern. Figure 2 shows a schematic diagram of a typical jet flow system. Fig. 2 Typical Laminar Jet Bar flow system (3) In order to improve heat extraction with water, planar water jets (sometimes called water curtains) issuing from slot-type nozzles mounted in both upper and lower low-pressure headers were developed. According to Kohring (3), significant advantages of Chapter 1. INTRODUCTION 3 planar jets include: improved temperature control, elimination of clogged nozzles and nozzle erosion, and simple varying which is compatible with computer control. More uniform temperature profiles are expected from these systems. Figure 3 shows top and bottom jets. Fig. 3 Typical Top and Bottom Planar Jets (3) The success of water jet cooling results from the direct liquid water contact with the high temperature surface of the strip. The jet momentum assures solid-liquid contact thereby, enhancing heat transfer as compared to film boiling which is produced by spray cooling. Chapter 1. INTRODUCTION 4 For a given steel grade, the most important operating parameters in the runout table operation are: finishing mill exit temperature, coiling temperature, layout of the active cooling jets, water flow rate, water temperature, strip velocity, and gage. Extensive research work has been conducted to improve this critical stage of strip production. However, contradictory conclusions have been obtained with respect to the effect of different operating parameters on the cooling of the strip. Due to the lack of fundamental knowledge of the basic heat transfer mechanisms present in water jet cooling, only empirical analysis of the cooling performance of jets has been accomplished. This work focuses on the study of planar jet water cooling, due to the increasing importance of these systems in modern hot strip mills. The results, (i.e. the fundamental knowledge obtained) are also applicable to water bar jet cooling. Experimental and theoretical work carried out to obtain a mathematical model to describe the temperature field of a strip during its processing are presented. An additional purpose of this work was to give a better understanding of the different heat transfer mechanisms involved in the boiling phenomena. The final properties of the strip are also dependent on the possible grain growth after the runout table cooling. A mathematical model for the coil cooling is also presented. Chapter 2. LITERATURE REVIEW 5 Chapter 2. LITERATURE REVIEW 3.1 Runout Table Models. Many mathematical models have been developed to predict the temperature field in the strip during laminar cooling. Most of the approaches to thermal field predictions for the strip are based on the solution of the 1-D heat conduction problem by applying a specific heat-transfer coefficient (HTC) distribution to the strip surfaces, according to the cooling zone along the runout table. The main distmguishing aspects are the values and distribution of the heat-transfer coefficients selected. Basically, the problem of creating a runout table model is the definition of an expression for the local heat flux (HF) during water jet cooling as a function of the most important operating parameters such as: water temperature and flow rate; strip velocity; and the local surface temperatures. Other parameters of importance are: jet arrangement along the table; nozzle shape, dimensions, height and angle. In jet cooling, there are two main fluid flow zones: 1. - Parallel flow, with and against the direction of the strip. 2. - Pressure gradient flow, in both strip and countercurrent directions. Additionally, depending on the jet arrangement, there can be a stagnant zone where two opposing parallel flows meet As might be expected, a function for the HTC for the runout table has to be general to be reliable. However, given the nature of the two-phase heat transfer mechanism, and the different flow regimes, it is unlikely that a simple expression could include all important variables, and remain general. Chapter 2. LITERATURE REVIEW 6 In order to simplify the definition of a HTC expression, typical models such as those by McCulloch (4), and Kumar et al. (5), suggest a constant heat-transfer coefficient (HTC) for each bank of jets along the runout table. The HTC is selected to fit some experimental results. Clearly, this approach gives only a general idea of the influence of jet cooling on the thermal field, and cannot account for any specific operating parameter. However, due to the "smoothing" nature of these models, they are useful to predict the microstructural evolution. In an attempt to account for the intrinsic characteristics of jet cooling, Colas et al. (6) applied a constant HTC for the parallel flow zone, whereas for the impingement zone another HTC was used. Both heat transfer coefficients and the length of the impingement zone were fed to the model in an attempt to fit full scale measurements. An additional cooling efficiency term was included. However, it was concluded that in no case could a combination of conditions be found which resulted in a finishing temperature in reasonable agreement with observations, while at the same time reducing the surface temperature under the curtain to a level which could appear black, as observed in practice. To overcome this problem, an isolating oxide layer of an arbitrary thickness was assumed to be responsible for the differences. Even though this approach can fit some measured exit temperatures, the model cannot give an idea about the effect of an operating parameter, or of the strip surface temperature, which, as will be discussed later, has a strong effect on the HTC. Yashiro et. al. (7) suggested the use of an empirical HTC for jet cooling in a control system model which includes the effect of parameters such as: surface temperature, nozzle configuration and, water flow rate and temperature. Four empirical coefficients have to be obtained from experiments for the specific operating conditions to be employed. Chapter 2. LITERATURE REVIEW 7 Evans et al. (8) developed a model in which both top and bottom cooling were considered independently. An average HTC for the impingement zone was calculated for turbulent convective heat transfer based on the fluid friction analogy (9). The fluid thermal properties were evaluated at the average temperature between the water and the strip surface. Given that an equation from an analysis of a single phase turbulent boundary layer gives a HTC proportional to Reynolds number to the power 0.8, the jet water flow rate can be taken into account through the velocity term. The temperature dependence of the HTC is included in the evaluation of the fluid properties. The parallel flow region HTC includes radiation heat transfer through the vapor layer (emisivity of 1.0), and a constant film boiling HTC. The results of the model agree well with production data. However, the single-phase heat convection mechanism assumed in the impingement zone of an arbitrary length, contradicts all experimental evidence obtained for water jet impingement at high temperatures, and boiling is completely neglected. Although the shape of the HTC function with strip temperature is similar to those due to boiling, this is only a coincidence. As a consequence of the lack of generality of this formulation, predictions for non-conventional cooling patterns are not likely to be accurate. Filipovic et al (10) developed a more fundamental model, adopting a local HTC in the parallel flow region which includes the effect of the strip motion on the heat transfer and radiation heat transfer through the vapor layer. The authors developed an analytical turbulent two-phase double-boundary layer model based on the model by Zumbrunnen et al. (11) for laminar flow. No arbitrary impingement zone is assumed, but to define its length, it was considered that it coincides with the pressure gradient zone. The heat flux in the impingement zone is taken from experimental results for the stagnation line of a planar water jet, impinging on a static surface under heat flux control. This model does not use any "fitting" parameter, and includes the effect of most of the important operating Chapter 2. LITERATURE REVIEW 8 parameters. Comparison against only two measured results is presented. The model underpredicts the cooling of the strip, giving temperatures 60°C higher than measured. To compensate for this, the effect of the heat transfer to the rolls is included, and very good agreement was found. Perfect contact between the strip and rolls was assumed, and the oxide layer present on the rolls was neglected. However, both assumptions seem very unlikely to be realistic. Fried (12) shows clearly that the actual contact area is orders of magnitude smaller than the nominal contact area, and the thermal properties of the oxide layer might reduce significantly the heat transport to the rolls. Therefore, the effect of rolls cooling should be much smaller. Another important shortcoming of this work is the omission of the heat of the austenite-to-ferrite transformation. The work previously quoted by Kumar (5), carefully accounts for the effect of such transformation, and proves that this could produce an increase of about 50°C in the final temperature. To overcome the difficulties and limitations of the modeling of all such details, Guo (13) suggest a statistical approach to obtain the HTC distribution in multiple-jet cooling. The HTC's of air and water cooling were determined in this study based on data for a total of 75 coils from an 86" (2184 mm) hot strip mill. Power-law equations as a function of coolant flow rate, strip speed, surface temperature and thickness of the type: [3.1.1] where K — curve fit constant v = strip velocity Chapter 2. LITERATURE REVIEW 9 t = strip thickness Ts = strip surface temperature q= header flow rate a,b,cyd= experimental exponents 0 = subindex for reference values were employed. Data with top header cooling were used to estimate the power exponents of the presumed heat transfer equation. Instead of trying to define the impingement zone, a triangular shape distribution of the heat transfer coefficient was assumed. It was concluded that the impingement length was too small to affect the overall runout table cooling. The thicker gauge leads to a larger HTC, which may result from faster surface temperature recovery. Higher mill speeds provide greater HTC's, deeper cooling penetration, and better cooling efficiency but smaller total heat transfer. The warmer finishing temperature give rise to higher heat transfer due to larger heat transfer coefficient and temperature difference. The statistical nature of this analysis impedes the study of the heat-transfer and phase-transformation mechanisms involved during cooling, and the application may be confined to the specific operating conditions of the study. From his results, Guo (13) suggested that forced convection heat transfer accounts for more than the 90% of the entire heat transfer, but this seems unlikely to be realistic at hot strip mi l l normal temperatures. Therefore, from this work, it can be concluded that the mechanism is still not understood. 3.2 Coil Cooling Models. The final step of processing, the cooling of the coil has received very little attention. The strip is coiled at the end of the mill, and the coil is left to cool in a separate area of the mill in either horizontal or vertical position. Chapter 2. LITERATURE REVIEW 10 Colas et al. (14) developed a 2-D transient explicit finite differences scheme. Constant cooling media were considered: still air and immersion in water. The heat transfer coefficients used in this model were not presented. The heat transfer problem involved in coil cooling is directly related to a much more studied process, the batch annealing of steel coils. Perrin et al. (15) concluded that in the batch annealing process, the main barrier to transferring heat to the charge is the low thermal conductivity in the radial direction of the coils (kr). A regular coil contains hundreds of wraps, and each one represents an additional resistance to heat transfer. This decreases production rates and causes high thermal gradients in the steel. To assess the appkcability of Perrin's conclusion in coil cooling a brief analysis of the heat-transfer characteristics of both processes is presented. The heat-transfer coefficients at the surface of the coils are approximately one order of magnitude greater in the batch annealing process (highly forced convection) than in coil cooling (natural/forced convection). Using the ratio of the internal-to-external thermal resistances (Biot number, Bi) as an indicator of the rate-controlling mechanism, it is clear that for each process the thermal resistance due to surface heat-transfer is inversely proportional to the correspondant heat-transfer coefficient Consequently, the ratio of the Biot numbers between both processes is given by: Assuming that the internal resistance of the coils are approximately the same (similar thermal conductivities and coil diameters), the external resistance ratio between the coil properties evaluated at the mean temperature in the coil were employed, and two different h 'HA [3.2.1] Chapter 2. LITERATURE REVIEW 11 cooling and the batch annealing processes is proportional to the ratio of the Biot numbers in Eq. [3.2.1]. Therefore, Perrin's conclusion, not necessarily, can be applied to coil cooling, and hence, an additional analysis is needed to obtain a conclusive result regarding this matter. The heat flux at the surface of the coil is a function of parameters such as: coil placement (horizontal or vertical), and environmental conditions (temperature and air velocity). The heat transfer mechanism may be either natural or forced convection in air. Obviously, it is not possible to account for any forced convection mechanism with confidence, since a bulk flow velocity of the air cannot be defined for such a system. On the other hand, natural convection can be taken into account easily using published correlations. Therefore, it is expected that the expression for the heat-transfer coefficients in air cooling should be of the form of those found in natural convection. The thermal conductivity in the radial direction (kr), is a controlling parameter in the overall cooling process. Its estimation is difficult, since no theory to calculate the contact conductance at the interface is satisfactory. To overcome this problem, experimental work has been done to obtain kr as a function of several parameters. Lisogor et al. (16) pointed out that early experimental studies are highly contradictory due to different experimental conditions, and values of kr from 0.582 to 5.82 W/(m°C) were reported. Fried (12) considered that heat transfer through the contacts between the wraps is the main rate controlling mechanism of heat transfer. Therefore, the main parameter to estimate should be the true contact area. However, the experimental measurement of this parameter as a function of the important operating, or physical, parameters might require a very extensive experimental work. Nevertheless, some attempts have been made to obtain simpler relationships. Chapter 2. LITERATURE REVIEW 12 The true area of contact between surfaces of sheets in coils was measured in relation to the applied load for the case of annealed and nonannealed cold-rolled sheets at room temperature (16). The contact increases almost proportionally to the applied load. It is important to note that the true contact area is of the order of 1% of the nominal area. Differences between annealed and nonannealed conditions were about 100%. The higher contact of the annealed steel may be caused by the lower flow stress which enhances the plastic deformation of the asperities with the applied load. Therefore, the metallurgical condition of the steel is also important in the heat transfer. The thermal conductivity is a very strong function of the degree of compacting of the coil, for compacting values (ratio of the theoretical to actual coil volume) of 0.9-1.0 (16). For values smaller than 0.9, the radial thermal conductivity is almost constant and approximately equal to 0.1 W/(m°C), and for compacting of 0.99, kT is 2.77 W/(m°C). This implies that at typical conditions of coiling (compacting values higher than 0.9) the contact between wraps is controlling, and the heat transport by the gas trapped should be negligible. There is experimental evidence that kr is also a function of temperature and sheet thickness (17). The thicker the sheet, the higher the thermal conductivity, because of the reduced number of wraps, whereas the higher the temperature, the higher is the increase of the thermal conductivity with temperature. This should be related to the steel thermal conductivity dependence with temperature, but also to the conduction or convection in air and the possible contribution of radiation through air, which are more important at higher temperatures. From this review, it can be concluded that kr is a function of: 1) Metallurgical characteristics of the steel (grade, mechanical properties and surface condition). Chapter 2. LITERATURE REVIEW 13 2) Stress state of the coil, which is determined by the coiling force; environment; strip thickness; geometry of the coil; and the thermal gradients (18). 3) Coil temperature. 4) Environmental conditions. In order to include such parameters, some theoretical models have been developed. Rao et al. (19) suggested that the coil radial thermal conductivity is the result of three modes of heat flow: radiation between wraps, heat conduction across the gas layer, and conduction at the contacts, and it is given by: K = knd+kair+kamd [3.2.2] where k= thermal conductivity, and they were adopted as constants. Few of the variables described previously can be included in this expression. Potke et al. (20) neglected radiation and obtained: ^ = - r - T T . [3.2.3] K kg„ s where k = k A c o n t a a +k gas s A g A J contact ^nominal ^nominal V. b = space between wraps s = strip thickness ks,kg= Thermal conductivity of steel and gas However, the problem of the contact area estimation is not solved. Liesch and co-workers (21) analyzed more closely this problem. For the specific case of a hydrogen environment, the expression: Chapter 2. LITERATURE REVIEW 14 kr(H2,r,T) = AQ(ks -A) + BQ(A-ks) [3.2.4] where A = + 2.15 x 10 - 3 T(r)]f(r) B = K f(r) = a function for local, changing "effective" heat transfer coefficient was given, and the stress condition can be included. No explicit form for /(r) is given, and, therefore, cannot be used for modeling purposes. 3.3 Water Jet Cooling. The heat transfer behavior of a system in which boiling occurs is usually best described by a plot of heat flux as a function of the difference between the surface temperature of the solid being cooled and the saturation temperature of the coolant (superheat). This kind of curve is commonly called a "boiling curve". It is worthwhile to differentiate. between the boiling curves obtained from temperature-controlled and heat-flux controlled conditions. Typically, temperature-controlled boiling curves are obtained by transient cooling, but it is possible, in principle, to generate steady state curves. On the other hand, heat-flux controlled boiling curves are usually obtained from steady-state cooling experiments. Likewise, the heat transfer versus superheat curve is not the same for boiling and condensation. In general, it is very important the path followed during cooling, and careful consideration of boiling data is of prime importance. The typical shape of temperature-controlled and heat-flux controlled boiling curves are shown in Fig. 4. The basic difference between this, and the heat-flux controlled boiling curve is that in the latter the transition boiling regime does not exist Chapter 2. LITERATURE REVIEW 15 Under heat-flux control, to maintain a higher heat flux than the critical heat flux, a large increase in the surface temperature is required because of the very high thermal potential needed to sustain the heat flow by pure film boiling. In contrast, with the smaller temperature gradients, the liquid contacts the surface, and the cooling is very effective during that contact, generating a very high heat flow. The transition boiling mechanism is a combination of both film boiling and nucleate boiling, which is also intermediate in effectiveness of cooling. I. Pure Convection II. Fully Developed Nucleate Boiling in . Departure of Nucleate Boiling IV. Critical Heat Flux V. Transition Boiling VI. Minimum Heat Flux VII. Film Boiling q=Fqi+(l-F)qv Heat-Flux Controlled Temperature Controlled VII AT Fig. 4 Typical Temperature-Controlled and Heat-Flux Controlled Boiling Curves Chapter 2. LITERATURE REVIEW 16 The nomenclature used in the literature to describe the different mechanisms of heat transfer during boiling is not the same. In this work, the nomenclature shown in Fig. 4 will be used to describe the different boiling regimes. Jet cooling in the runout table is employed in a very wide range of surface strip temperatures, typically from 900°C down to 300°C; and therefore, the cooling is carried out by different boiling mechanisms. More specifically, nucleate boiling, the critical heat flux, transition boiling and film boiling must be fully characterized in order to develop a general model. A comprehensive review of jet impingement boiling for different coolants and jet characteristics was presented by Wolf et al. (22). Nevertheless, a more specific overview for free surface water jet cooling will be presented, as well as the basic characteristics of the fluid flow involved. 3.3.1 Fluid Flow. Basic understanding of the fluid flow phenomena involved in jet cooling is very useful to study not only the convection heat transfer, but also the boiling mechanism. Basically, there are two different kind of flows during single-phase jet cooling, as shown in Fig. 5. In the zone adjacent to the jet centerline, the free stream flow changes direction, and must develop in a finite length (impingement zone) in a direction parallel to the strip motion (cocurrent and countercurrent). This involves a change in the pressure energy (pressure gradient) which is also parallel to the strip motion. From the Falkner-Skan power-law (23), for top jets the free stream velocities are given by: [3.3.1] Chapter 2. LITERATURE REVIEW 17 WATER J E T VELOCITY PROFILES AFTER IMPINGEMENT Wj <• l<1 -J IMPINGING JET VELOCITY PROFILE D I R E C T I O N O F S T E E L STRIP M O V E M E N T -K-P A R A L L E L F L O W Z O N E S T J E E J L S 1 W x * = x / w j = 1.75-2.5 S T A G N A T I O N Z O N E ->r P A R A L L E L F L O W Z O N E and for inclined jets Fig. 5 Fluid Flow in jet cooling [3.3.2] where m = Chapter 2. LITERATURE REVIEW 18 This zone will be called "impingement zone" or "pressure gradient zone", and it is typically about two jet widths in both directions, but varies with the impingement angle, and the velocity profile of the impingement jet. The zone where the free-stream velocity is fully developed, will be called "parallel flow zone", and the free-stream velocity is: «„ = Uj [3.3.3] The free-stream velocity gradient C, is only a function of the intrinsic characteristic of the nozzle and not of position. From the measurement of the pressure distribution along the distance for the impingement of a planar water jet, Zumbrunnen (24) obtained the following pressure distribution: 2 — - 3 x. + 1 [3.3.5] J which is related with the free-stream velocity by the well known equation d{uj)= dP dx dx [3.3.6] and from both the free-stream velocity is \0£x<x. - 2 yX.j [3.3.7] and the dimensionless velocity gradient is: Chapter 2. LITERATURE REVIEW 19 dx = — [3.3.8] x=0 X* For single-phase jet coohng on a stagnant plate, a similar flow solution of the momentum and energy equations for the flow near the solid at the stagnation line (jet centerline) can be easily obtained for laminar flow (23), by either the direct solution of the Navier-Stokes equation (Fliemenz solution), or by integral methods (Falkner-Skan wedge flow solution). Even though the flow might be turbulent, the impinging of the jet tends to lam in arize the flow, and usually laminar flow solutions are accurate enough. Zumbrunnen (35) solved the Navier-Stokes equation for a moving impingement surface, to obtain the horizontal flow velocity near the surface (in the boundary layer). A modified velocity which includes the surface motion of the form: u = Cxh'(y) + L(y) [3.3.9] where the second term includes the effect of the strip motion. The fluid flow dimensionless equations to solve are finally: _ -=-dH(ry) , u=Cx——+usI(r\) dry lV2 v = -H(r|)| d3H Re, + H dlH dry2 dH _-l2 dry* dry2 "dry dry dry + 1 = 0 [3.3.10] [3.3.11] [3.3.12] Chapter 2. LITERATURE REVIEW 20 dty D H 1 j n Tj -> oo; >1 ; 7 - ^0 [3.3.13] which are solved by the Runge-Kutta method. 3.3.2 Convection Heat Transfer. Single-phase jet impingement is of special interest in applications such as cooling of electronic devices, cooling of glass, drying of textiles, and drying or cooling of paper. For these reasons, research work has focused in this area, and a good understanding of the effect of different cooling parameters has been achieved. Convective heat transfer has been claimed to be the main mechanism of heat transfer during jet cooling (8),(13), and a brief analysis of this possibility in view of the literature information available is presented. Zumbrunnen et al. (24) obtained an integral solution of the momentum and energy equations for laminar flow due to a free-surface nonuniform planar jet impinging on a moving plate, assuming that the effect of plate velocity on the flow is restricted to the velocity boundary layer and symmetry of the flow in the bulk flow is maintained. They concluded that the heat transfer coefficient in the stagnation line increases with a nonuniform velocity profile (fully developed flow prior to impingement), but this effect is less significant at greater distances from the stagnation line. The surface motion affects only when the surface temperature varies spatially. Far from the stagnation line, as the plate speed increases so do the heat transfer coefficients. In general, a nonuniform surface temperature affects the thermal boundary layer growth, and therefore, alters the heat flow. For a non-moving plate, the heat transfer results published (25) for the stagnation line gives a relationship such as: Chapter 2. LITERATURE REVIEW 21 Nux = CRe" Pr' [3.3.14] where n= 0.5 - 0.8, and m= 0.33 - 0.4, and the coefficient C is a constant for a fixed system and position. The coefficient C decreases as the distance to the stagnation line increases. This result is not surprising, since for laminar flow the solution of the transport equations should render n= 0.5 and m= 0.33, and for turbulent flow n- 0.8. According to the experiments by Miyasaka and Inada (26), at the impingement line the heat flux is given by for A7^<100°C, where this mechanism is predominant It is very important to note that regardless of the fluid flow conditions, pure convective heat transfer cannot exist at the temperatures typical of the runout table. Therefore, runout table models based on the assumption of a predominant convective heat transfer cannot be considered realistic. Probably, the reason why such models might work, is that the heat fluxes in this region are close to those obtained by transition boiling in the temperature range typical of the runout table. 3.3.3 Nucleate Boiling. For runout table applications, the characterization of the onset of the nucleate boiling regime is unimportant. Even fully developed nucleate boiling characterization is of very little application. However, departure of nucleate boiling might define the lower limit of achievable temperatures, and its study is important No work has been done specifically focused on departure of nucleate boiling. Given that this is intermediate between nucleate boiling and the critical heat flux, a brief review of the fully developed nucleate boiling (FNB) mechanism is presented. conv ~ATS [3.3.15] Chapter 2. LITERATURE REVIEW 22 For the range of jet velocities of interest, nucleate boiling is not affected by the jet velocity, and it depends only on the wall superheat. This regime can be considered a linear combination of convective heat transfer and pure boiling. For FNB, the convective term takes the boiling curve to higher superheats, and heat fluxes, but only as an extension of pool nucleate boiling curve. Water subcooling, A7^,, has no effect in the heat flux (22)(26). The effect of the strip speed has been analyzed by some researchers, and it seems that the heat flux increases slightly, but up to know, there are no conclusive results (22). The heat flux in FNB is given by a relationship such as: qFNB = CATs" [3.3.16] has been obtained. Miyasaka et al. (26) reported C= 79 and n= 3.0 for a wall superheat of 26-90°. Wolf et al. (22) from data by Ishigai et al. (27) obtained C= 42 and n = 3.2. The parameters of this equation are reported for other jet configurations. Values of n from 1.42-7.4 are found, and if jet FNB is an extension of pool FNB, then an important parameter not taken into account yet, should be considered. This disagreement is of very important consequences for model predictions, as will be explained in another section, and very careful analysis of the exponent n has to be carried out. In order to have a better understanding of the origin of the n parameter, let the extension of pool boiling results be a good approximation of jet boiling, so pure boiling is the main mechanism of heat transfer, therefore nucleation and growth of bubbles involved in boiling are the important processes. Given that boiling occurs at nucleate sites, and the number of nucleate sites is very dependent upon the physical condition of the surface, the wetting characteristics of the fluid and the efficiency of air trapped displacement, it is expected that a theoretical approach to calculate the heat flux might be a very difficult Chapter 2. LITERATURE REVIEW 23 task. Nevertheless, a simple analysis is presented here. Whalley (28) expressed the heat flux as: 1.2 r \0-33 N [3.3.17] where the nucleation site density, N/A,^, is dependent on the heat flux (or wall superheat). From Eq. [3.3.17] it is clear that n is directly related to the mechanism of activation and deactivation of nucleation sites, which is a function of the substrate temperature. Equation [3.3.17] is of special importance in this work, and a closer analysis of this will be presented in section 3.4.2. The parameter C has been studied with some detail. The well-known correlation by Rohsenow (28), which was developed in terms of single-phase convective heat-transfer, contains a modified velocity and length scale in the Reynolds and Nusselt numbers. The velocity is taken as the liquid velocity towards the surface supplying vapor, and the length scale is given by the most unstable wave on a liquid-vapor interface. However, the parameter C is related to specific conditions of evaporation, and usually an additional constant has to be evaluated experimentally. 3.3.4 Critical Heat Flux. According to Wolf et al. (22) several investigators have proposed the existence of four different critical flux CHF regimes, referred as V, I, L and HP. In each regime, dependence of the critical heat flux on parameters such as jet velocity, density ratio, and heater geometry has been shown to differ markedly. However, at atmospheric pressure only the L and V regimes have been observed, and the later encompasses the majority of flow conditions, where the fraction of water evaporated during cooling is small compared to the supply. Chapter 2. LITERATURE REVIEW 24 In the case of water bar cooling heat-transfer expressions such as: [3.3.18] have been proposed (22), but there is no correlation for planar water jet boiling under temperature controlled conditions. Ishigai et al. (27) examined the effects of subcooling on the complete boiling curve. An increment in the CHF by over a factor of 4 was reported increasing subcooling from 5 to 55°C at a jet velocity of 2.1 m/s. On the other hand, the effect of strip speed on the CHF has not been reported in the literature. 3.3.5 Transition Boiling. The transition boiling regime may be defined as the boiling zone where discarding the CHF point Figure 6 shows that most of the runout table jet cooling (on the stagnation line of each jet) lies in the transition boiling regime, assuming that the strip speed does not shift the CHF to superheats higher than 200 °C approximately. High water subcoolings tend to stabilize the transition boiling regime, and this regime is extended to higher superheats and heat fluxes. It is clear that nucleate boiling or forced convection could appear in the jet impingement zone only when the strip temperature is below 300 °C, which may be unlikely to occur in lower-thickness strip cooling, and that probably only occurs in higher-thickness strip operations at the last bank of the jets. Consequently, transition boiling is the main heat transfer mechanism of cooling in the impingement zone. <0 [3.3.19] dAT Chapter 2. LITERATURE REVIEW 25 Relevant knowledge on transition boiling is very limited. The experimental studies of Hatta et al. (29-30) and Kokado et al. (31) on planar jet boiling on a non-moving plate (based on macroscopic observation) showed that two different boiling regimes are present during jet cooling: The one closer to the stagnation line gives the highest heat transfer because of the total wetting of the surface (according to visual observation); and the second, in the parallel flow zone, lower heat transfer rates are consequence of the non-wetting conditions on the surface, which seems to indicate that film boiling was present. Nevertheless, the liquid-solid contact area requires a closer approach to the liquid-solid interface, as will be seen in section 3.4.2, and macroscopic observations cannot be considered conclusive. Consequently with their observations, in the analysis of the cooling data of the impinging region a relationship such as Eq. [3.3.14] was adopted whereas boiling was discarded. Thus, the numerical results cannot be used confidently. A T s a t (°C) ; Fig. 6. Experimental Boiling Curves for a planar water jet (27) Chapter 2. LITERATURE REVIEW 26 On the other hand, Otomo et al. (32) indicated that in the parallel flow zone of industrial-scale jet cooling, transition boiling exists at a distance up to 0.45 m. away from the stagnation line. Takeda et al. (36) showed that in the impingement and parallel flow zones, transition boiling is the mechanism of heat removal in industrial applications, except at very large superheats and far away from the stagnation line, as shown in Fig 7. The CHF point shifts to higher superheats decreasing the distance, from the jet centerline, whereas the HTC increases, and the condition is satisfied in jet cooling. 200 400 600 O Plate surface temperature (°C) Fig. 7. Heat Transfer Coefficient as a function of position from the jet (36) Chapter 2. LITERATURE REVIEW 27 Ishigai et al. (27) reveled that on the stagnation line of a planar jet, the heat flux increases with water subcooling significandy, and film boiling is not present at superheats lower than 1000°C and water subcoolings above 55°C. For a given jet velocity, increasing subcooling from 5 to 55°C the heat flux increases more than ten times. Similarly, at large subcoolings the heat flux increases with jet velocity; however, the effect is less important, and it is almost negligible close to the CHF (Fig. 6). Heat-transfer correlations on this regime were not presented. The effect of the strip motion on the heat flux was studied by Hatta et al. (33), but their results are unclear with respect to the real effect of motion on the boiling curve. From their data, it is likely that the strip-jet contact time determines the cooling rate, and the effect of the strip motion in the boiling curve is small. Concerning to the orientation of the jet during impingement, Raudensky et al. (34) reported experimental results of the cooling of a vertical plate by an arrangement of planar and circular jets. Their results showed that the superheat at the CHF is 200°C approximately; whereas water flow rate and distance to the stagnation line are more important variables than jet pressure. Oblique incidence of a single jet increases the heat transfer significantly. A similar effect, but not so important, was observed in experiments with a set of nozzles. It was noted that if the starting temperature for such experiments is increased from 600 to 900°C, the results obtained for the temperature range 100-500°C are different The initial state of the heat-transfer mechanism influences the whole heat removal process. According to the definition of transition boiling (Eq. [3.3.19]) the "minimum heat flux" (MHF) is part of the transition boiling regime. On the stagnation line the MHF for a planar jet is given by (27): Chapter 2. LITERATURE REVIEW 28 qMHF = 0.054 * 106M;°-607(1 + 0.527A7VJ [3.3.20] for 0.65 <U; <3.5m/s, and 5 < ATmb <55°C. A similar expression for the stagnation point of a free-surface circular jet is (22): 0.828 qMHF = 0.318*106(^-J ( H - O ^ A T ^ ) [3.3.21] for 2.0 <«,. <7.0m/s, and 5< AT^ <45°C. Equation [3.3.21] shows that the MHF decreases linearly with water temperature, and the heat transfer at the stagnation point is dependent on the nozzle dimensions. The effects of these parameters have not been studied for planar jets yet 3.3.6 Film Boiling. The film boiling regime has been considered to be the most predominant in runout table cooling (6,8,10). In section 3.3.5, it was shown that only at high superheats and in the parallel flow region film boiling could be present. Despite of this fact, film boiling in the pressure gradient zone is of prime importance in the modeling of the pressure-gradient transition boiling regime, and it will be considered. Ishigai et al. (27) and Nakanishi et al. (37) studied experimentally and mathematically respectively, the film boiling mechanism on the stagnation line of a planar water jet impinging on a non-moving plate. Figure 6 shows that the heat flux increases with superheat in film boiling. Nakanishi et al. (37) solved numerically the momentum, energy and mass equations for both liquid and vapor phases, using a similarity transformation (Hiemenz Flow) assuming a smooth vapor-liquid interface. Their model predicted accurately the experimental results of reference (27) when: Chapter 2. LITERATURE REVIEW 29 qFB = VJ5qfflmeom+0.15qjamrad [3.3.22] where qfilmcom = Heat flux due to pure forced convection heat transfer Qfdmrad ~ Radiation heat flux across the vapor layer was assumed, which resembles to a parallel-resistance heat transfer. A vapor film thickness of 10-100 |im was calculated. A more general solution of forced-convection film boiling on a non-moving surface in the presence of a pressure gradient was presented by Nakayama (38). An integral procedure based on the two-phase boundary-layer theory was proposed, and analytical solutions of the momentum and energy equations in wedge flow were presented. Asymptotic analysis of the general solution was performed, adopting an arbitrary impingement angle and large subcoolings, and expressions such as: Nux Ref , xl/2 m ( l + 3 m ) r r f ^ i k t A k t J 3 m ^ 10 ^ l ^ / P i y ; f o r r > /P r7« 2 1 + m \if [3.3.23] where m 7t „ , (5 1+m V 2 = (- = V3 for righ, angle); 4 ^J^j [3.3.24] r = V C , A T s [3.3.25] Nu Re V2 2(1+8m)^2 15 1/2/ ^ - r ^ / P r y ) ; f o r r A / P r 7 » M-v y 3 m [iv ^2\ + m\x.f [3.3.26] Chapter 2. LITKW ATIJWF, REVIEW 30 rJSUjnVp ; f o r P r / » i ^ [3-3.27] were obtained. Regardless of the method of solution of the laminar flow momentum and energy equations, in the Eq. [3.3.22] the constant 1.75 means that in film boiling there is an additional contribution to the heat flow which was not considered yet. For the parallel-flow region, the analysis of the transport equations is easier since no pressure gradient is present. Zumbrunnen et al. (11) solved the momentum and heat equations for laminar flow over a moving surface, using an integral method, assuming a smooth vapor-liquid interface. More recently, Filipovic et al. (39) obtained a similarity solution for the same conditions. The transition point between laminar and turbulent fluid flow has not been defined yet, but probably because of the counteracting effects of the water fluid flow on the top surface of the strip and the different flow disturbances found in real processing, the fluid flow might be turbulent thoroughly. Therefore, Filivopic et al. (10) applied the same method as Zumbrunnen et al. (11) but for turbulent flow, and equations such as: k.AT. Nu = ^ = 0.0195 ^ P(2u i + 7 ) ° - 2 «rRerPr f [3.3.28] for up>uj, and N u = l f ^ = 0 . 0 1 9 5 ^ W + 7)°2 Re 0 / P i f [3.3.29] * kvATs Chapter 2. LITERATURE REVIEW 31 for up <u„, where Pr„ r l+p Pr, -p— (parallel flow) [3.3.30] 1-P Pr, u, =-l + p Pr, ^p— (countercurrent flow) [3.3.31] ' « „ ' • p r / c p v a t ; [3.3.32] were obtained. Radiation heat transfer was neglected. 3.4 Convective Transition Boiling Modeling. The generality of the boiling curves required in the modeling of the runout table cooling cannot be achieved based on the present experimental or theoretical work reported. Experimental measurement of the heat flux from a moving surface under different operating conditions requires an amount of work that goes beyond the scope of this study. On the other hand, the modeling of this phenomena arises as a cheaper and easier source of useful information for modeling purposes. This section presents an overview of the modeling work done in transition boiling. Chapter 2. LITERATURE REVIEW 32 3.4.1 Mechanism. The fundamental knowledge in transition boiling for different systems have been summarized by Kalinin et al. (40), and more recently by Auracher (41). Only in recent years the interest in this boiling regime has increased, mainly in connection with the safety of nuclear reactors. Other fields of interest include: the material quenching processes, and the design of high performance evaporators heated by a liquid or a condensing fluid, which may also be operated in the transition region without the danger of instabilities because the heat transfer is temperature controlled (41). The experimental results on the transition boiling mechanism and the estimates of heat transfer rates show that at each instant some part of the hot surface is wetted by the liquid and the remainder is covered by a vapor film. Consequently, each point of the heating surface is alternately in contact with the liquid and vapor phases of the boiling medium. The mean duration of the heating surface contact with the liquid depends on the superheat, the properties of the boiling fluid, wall material and surface conditions. Since the heat transfer to liquid is much higher than that to vapor, the processes at the wall-liquid contacts are dominant in the case of transition boiling (40). Kalinin et al. (40) distinguished three zones in the transition boiling region on the wetted part of the heating surface: 1. - A low superheat zone near the CHF where the duration of the liquid-wall contact is rather large and nucleate boiling occurs at the contact place. 2. - A high superheat zone near the MHF where nucleate boiling cannot develop because of the small contact time, the heat transfer from the wall to the liquid dominates, and unsteady heat conduction occurs. Chapter 2. LITERATURE REVIEW 33 3.- A mean-superheat zone where the contributions of nucleate boiling and unsteady heat conduction are comparable. Increasing the superheat, the duration of the periodic liquid-solid contacts is decreased. The disturbance of the hydrodynamic stability of the vapor film and the conservation of the thermodynamic stability of the liquid at the contact place are necessary conditions for the liquid-solid contact Stable equilibrium of the vapor-liquid interface is possible only when the less dense phase is above the more dense one (for a lower-side horizontal surface, and small free-flow velocity of both phases). For all the other cases, (film boiling on vertical, inclined, cylindrical, spherical surfaces and above a horizontal surface), the interface boundary is unstable as the more dense phase is above or to the side of the less dense one. The instability initiates a transverse motion of the interface boundary. However, at high superheats, the vapor film is thick, and the liquid does not touch the surface. Decreasing the superheat, the thickness of the vapor film decreases, and the vibration amplitude of the interface boundary may coincide with it, and the liquid-solid contact becomes possible from a hydrodynamic viewpoint. Whether this contact occurs depends on the thermodynamic condition or their combination with the hydrodynamic ones. When the wave peak is close to the surface, and when the liquid temperature is much higher than the saturation temperature, the intense vaporization produce a reactive force that can throw the liquid from the surface, and contact will not occur. If contact takes place, and if the liquid reaches a higher temperature than a limiting metastable liquid heating temperature, then the explosive boiling of the thinnest layer occurs and the liquid is thrown from the surface. When the liquid temperature is lower, then wetting occurs, and unsteady heat conduction (small contact time) or nucleate boiling (large contact time) will appear. Chapter 2. LITERATURE REVIEW 34 Regardless the properties of the solid or the surface orientation, increasing the liquid subcooling, the transition boiling region becomes wider and shifts to higher superheats, enhancing the heat transfer. In contrast, heat removal in the NB region does not depend on subcooling, which may be attributed to two compensating factors such as increasing the temperature drop between the surface and liquid and the decreasing rate of bubble growth due to subcooled liquid condensation at their caps. In the transition region these factors act in the same direction since increasing the temperature difference between the surface and water increases the heat release during the sokd-liquid contacts while the falling bubble growth increases the duration of this contact The influence of the thermal properties and surface conditions (roughness and wettability) of the solid is quite important in transition boiling. Such influence has been examined mostly in pool boiling experiments, but there is no reason to doubt that the effects are similar in flow boiling (41). According to Kalinin et al. (40), the experimental data supports the conclusion that decreasing the thermal effusivity (pcpk) the nucleate and transition boiling curves shift to regions of higher superheats, while film boiling is not affected. However, Auracher (41) states that the MHF point shifts to the right also. Since nucleate boiling is very dependent on the nucleation site density, with decreasing the height of the microroughnesses on the surface, NB and CHF shift to higher superheats, while the MHF remains the same, if the micro roughness height is smaller than the vapor layer thickness. Therefore, under the same superheat, in transition boiling the heat flux increases with smoother surfaces (40)(41). Wettability has a strong influence on transition boiling. Enhancing wettability, increases the heat transfer rate by increasing the liquid-solid contact time. The resulting Chapter 2. LITERATURE REVIEW 35 increase in heat flux with decreasing the wetting contact angle includes both the CHF and the MHF, but this effect is greater in the MHF. The wettability also changes with oxidation or deposition. Contamination may, in a very complicated way, simultaneously affect roughness, wettability and the thermal properties of the surface, thus causing non-reproducibility in much of the experimental data available. However, at least in flow boiling, an oxidized surface shifts the CHF point to higher heat fluxes and superheats, thus higher transition boiling heat transfer rates (40)(41). A very important factor is the steadiness in the experimental boiling curves. Very different boiling curves are obtained from steady state experiments compared to transient ones. Kalinin et al. (40) pointed out that the general tendency is the slower the unsteady process is the less the boiling curves differ from the steady-state curve obtained under the same conditions. An analysis by Auracher (41) showed that the instantaneous interface temperature and the cooling rate at the surface are the primary parameters in the description of this problem. The heat flux increases as the cooling rate at surface increases in heat-up processes, but the opposite is true for cool-down processes. Finally, the transient boiling curves can generally be characterized by the cooling rate in addition to the steady-state expressions. 3.4.2 Modeling Fundamentals. Most of the modeling approaches to transition boiling are based on the assumption that the heat flux can be expressed as a combination of the heat fluxes during the liquid-solid and solid-vapor contacts, as follows: Qn=^F+qv_t(l-F)' [3.4.1] Most of the experimental and modeling efforts have been focused on the evaluation of the three parameters of this equation. Given the complexity of estimation of all the Chapter 2. LITERATURE REVIEW 36 parameters at a time, usually assumptions regarding to the heat flux values are adopted whereas measurements or modeling efforts are focused on the evaluation of F. Ragheb and Cheng (42) assumed that qx_s=qcm, q^^q^. This is a good approximation, but usually the CHF and the MHF are also unknown, thus it is of little use for modeling purposes. Kalinin et al. (43) proposed that q,_s =qNB, qv_s =qFB, evaluating them as the extrapolation to real superheats of known correlations for both, nucleate boiling and film boiling. However, direct extrapolation of the present NB heat flux correlations to transition boiling superheats in convective boiling might overestimate in orders of magnitude the liquid-solid contact heat flux (41). Kostiuk et al. (44), Pan et al. (45), and Farmer et al. (46) suggested a variation of Eq. [3.4.1], but includes the effect of the transient conduction before bubble formation, which is expressed as: *ITB ~ xt(lt'^ xNB <1NB'^ XFBQFB [3.4.2] This approach has the advantage of being able to include the thermal characteristics of the surface. Nevertheless, Pan et al. (45) using a similar approach showed that the nucleate boiling mechanism contact time is of the order of 10"3s, whereas the transient conduction lasts about 10~*s, and since the heat flux is much higher in nucleate boiling than in transient conduction, the later mechanism is negligible. However, Pan et al. (45) includes this effect to evaluate the local superheat in nucleate boiling. On the other hand, the estimation of F has been the subject of most of the research done. Auracher (41) has reviewed the work done to evaluate this parameter and only a review of the relevant information for this research is presented. Chapter 2. LITERATURE REVIEW 37 Direct measurement of F has been accomplished in pool boiling by Lee et al. (47), Dhuga et al. (48) and Shoji et al. (49). Results are presented in Fig. 8. = 1 V ' A ' ' ' 1 1 1 1 1 1 1 : =: a •*• = ; o * a * = - o A a -limn i i \ • i O 1 o _ °o = - \ " = 5 O : F (present) ' • \ = z A :F(Ohuga) • : Fa (Shoji) o -• : Fa (Ohuga) o : Ft (present) z - a : Ft (Lee) i 1 t t t t 1 . . . . 1 . : 0.0 AT 3 0 t = AT«r-ATofr 1.0 Fig. 8. Parameter F as a function of solid supeheat (49) Figure 8 shows that there is liquid-solid contact not only in transition boiling, but in the film boiling region also. Likewise, the surface at the critical heat flux is not completely wetted, but about 0.7 is in contact with liquid. A remarkable observation is that the wetted area fraction Fa and the wetted time fraction Ft are not equal. In fact, Ft is generally much larger than the area fraction Fa„ hence the ergodic assumption, F = — [3.4.3] according to Shoji et al. (49) is not appropiate. Consequently, it is concluded that the modeling work (43)-(46) based on that assumption may be erroneous. Chapter 2. LITERATURE REVIEW 38 Still, as Auracher pointed out, the scattering in the data shows that better estimates F are needed. 3.4.2 Liquid-Solid Contact Heat Transfer. References (47) and (48) presented measurements of the heat flux during liquid-solid contact that show that the heat flux does not increase with superheat as expected, but rather decreases. However, under the circumstances of their experiments, no conclusive behavior can be obtained from their measurements. In a more recent work (50), q,_s was direcdy measured for a falling water drop on an Inconel 600 surface, and the results appear in Fig. 9. Figure 9 shows that water subcooling increases the solid-liquid contact heat flux, and for a large subcooling q,_s increases monotonically. For saturated liquids, the observations in references (47)-(48) agree with those in Fig. 9, but it is clear that different results are obtained for subcooled liquids. The sokd-liquid contact heat flux increases with the drop velocity, but alike jet cooling (see Fig. 6) the water fluid flow is less important than the water subcooling on the results. It is remarkable that the contact heat flux does not follow any nucleate boiling correlation available (where n~3 in the typical power law correlations), but n is smaller. Therefore, if the nucleate boiling behaviour is to be extended to the transition boiling solid-liquid contact, a different approach to q,_s calculation is necessary. The mathematical modeling of the nucleate boiling extension to transition boiling may be the simplest way to solve this problem. Nevertheless, the more detailed nucleate boiling models (51)(52) require very complex information to calculate the heat flux, and may be of little usefulness in the formulation of the transition boiling model. Chapter 2. LITERATURE REVIEW 39 SUM m v-0.5 m/« • v - l m/« • v-2 m/i OTCSubeooUng * . O v-OJ m/t « 10' ' o v-2 m/» « c * / a 0 / « i Y icr X • / ^ * m u. //% "N." « a / \ 41 CD 10* / ° <3 / * * U < 1 ° * \ • • 10» 3 100 200 300 400 500 Initial Surface Superbeal. AT. *C Fig. 9. Heat Flux during solid-liquid contact (50) On the other hand, the need of calculating the CHF in nuclear reactor operations has led to a great effort, and development, in the modeling of the nucleate boiling mechanism near the CHF. A review of such models is presented by Lee and Mudawar (53). Three basic mechanism were suggested: 1. - Boundary-layer separation. 2. - Near-wall bubble crowding. 3. - Sublayer dryout. There is a very strong experimental evidence, in pool boiling, of the formation of a very thin liquid sublayer trapped beneath a blanket formed by the coalescence of several bubbles at the surface, which seems to support the sublayer dryout mechanism, that has been also called "macrolayer evaporation" mechanism. Basically, this consists on a mushroom-shaped bubble which after being formed by coalescence, grows by vapor Chapter 2. LITERATURE REVIEW 40 supply through little vapor stems present in a liquid layer (macrolayer) adjacent to the solid surface, as shown in Fig. 10. MACROLAYER U l ' VAPOR BUBBLE Uv 1 Fig. 10. Macrolayer Evaporation Mechanism Haramura and Katto (54) suggested a hydrodynamic model based on the mechanism shown in Fig. 10. The thickness of the macrolayer was obtained from the Helmholtz instabihty criterion imposed on the vapor-liquid interface of the columnar vapor stems distributed in the macrolayer. This vapor-liquid system is collapsed wholly by the instabihty, but due to the suppression of the solid surface, a thin liquid film including vapor stems is left stable on the surface with a certain definite thickness relating to the Helmholtz critical wavelength. Accordingly, adopting the general Kelvin-Helmoltz instabihty equation, the folowing wave velocity is obtained: 1 2KG P/Pv Pz + Pv ^ ( P , + P v ) ' -("v+"/)2 V2 [3.4.4] Chapter 2. LITERATURE REVIEW 41 In order to obtain the critical wave length, the condition c = 0 is applied on Eq. [3.4.4], and through a heat balance in the macrolayer, the Helmholtz wavelength is: Katto and Yokoya (55) observed that the heat transferred from the surface is completely absorbed by evaporation, and by the application of this observation, Haramura and Katto (54) were able to make predictions of the CHF in pool boiling and forced convection boiling. The forced convection boiling model was based on the assumption that the heat transferred from the heated surface is equal to the latent heat of the total evaporation of the liquid flowing into the liquid film. The work by Haramura et al. (54) was extended following the analysis of Lee and Mudawar (53) on subcooled flow boiling in round tubes at normal pressures (56), and in a wider pressure range (57). According to Celata et. al. (58) a very good prediction of experimental CHF data is provided by this model (72.3% of predictions are within ± 25%), but it is limited to a void fraction in the boiling layer less than 0.7. The model, although mechanistic in nature, still requires empirical parameters introduced in order to describe the dynamics of bubble behavior, as well as the velocity of the flow close to the surface. [3.4.5] and, the macrolayer thickness is assumed to be given by: [3.4.6] Chapter 2. LITERATURE REVIEW 42 Based on the model by Haramura and Katto (54), Monde (59) developed a general correlation for saturated CHF in jet impingement that successfully predicts the experimental data. Galloway and Mudawar (60) in a more recent work, examined the conditions of FC-87 fluid flowing in a channel and boiling on a 0.16 x 1.27 cm. pure copper surface. Photographic evidence is presented of the solid-liquid contacts, and the characteristics of the liquid-vapor interface in flow boiling. From their observations, the authors concluded that there was not minute vapor jets stemming from a liquid sub-film, but rather violent boiling and evaporation from the liquid sub-film. For the specific application of nucleate boiling predictions, Pasamehmetoglu et al. (61) carefully analyzed the macrolayer evaporation model by Haramura and Katto (54) and solved the coupled transient two-dimensional conduction equation the heater and the liquid microlayer, while allowing for the time-wise thinning of the macrolayer. They concluded: 1) The dominant evaporation occurs at the liquid-vapor-solid contact point (triple-point), or near this area. Quantitatively, this is in agreement with the existence of a microlayer under a stem found by other researchers. 2) Transient conduction within the macrolayer cannot account for the high fluxes. 3) Evaporation at the macrolayer upper surface and stem interfaces are not significant, except near CHF. Their calculations show that close to the CHF more than 80% of the heat transferred comes from evaporation at the triple-point. Given, the complexity to define a mechanism of evaporation which enable us to account for so many different variables encountered in Chapter 2. LITERATURE REVIEW 43 nucleate boiling, the authors suggested that the heat flux may be calculated by the relationship: Qmpu-po^m^H^lTO^^-Tj)^- [3.4.7] where the evaporation rate parameter is by definition: hAr m„=- f— [3.4.8] Pasamehmetoglu et al. (61) results show some discrepancy in the behavior of the boiling curve, but the values for the heat fluxes were very close. The authors suggested that it was possible to improve the shape of the boiling curve by adjusting the evaporation-rate parameter, but they believed that this is only a function of the liquid-solid contact angle, and therefore should be constant for a fixed system. 3.4.3 Film Boiling. Film boiling heat transfer is of interest for many different technologies. A comprehensive review of the research advances was presented by Kalinin et al. (62). In film boiling (FB), the fluid is separated from the surface by a vapor blanket, on which surface bubbles form and break away. The shape of the interface can be extremely variable: continuous or discrete, stable or unstable. The shape of the interface is determined by a great number of various parameters. Heat and mass transfer, and also Chapter 2. LITERATURE REVIEW 44 phase transformation are always unsteady, and the formation of metastable phases is Kalinin et al. (62) suggest some conclusions from various experimental results: 1) The saturation temperature at the liquid-vapor interface is assured at any wall temperature. 2) The MHF and the rninimum vapor film thickness possible correspond to steady laminar vapor film with a smooth steady interface. 3) Enhancing the vapor removal, the higher the heat flux (decrease thermal resistance due to a smaller thickness). 4) Heat transfer is increased by a reduction of the hydrodynamic stability of the liquid-vapor interface (increase the interface wave amplitude). Subcooling tends to stabilize the interface, but increases the heat flux through an increase in the sensible heat required for evaporation. 5) The oscillation behavior of the interface leads to fluctuations in the solid surface temperature, and unsteady conduction in the solid occurs. It is clear that the shape of the liquid-vapor interface is a determining factor in the heat transfer process. The basic theory of the hydrodynamic instability in a two-phase flow can be found elsewhere (63). The Kelvin-Helmholtz instabihty, or instabihty of two superposed invisid fluids flowing irrotationally over a horizontal flat surface, can be expressed for the case of a two-dimensional disturbance as: possible. P/Pv [3.4.9] (P*+Pv) Chapter 2. LITERATURE REVIEW in which c0 is the wave velocity in absence of currents given by: 45 2K{ p,+p v + a 2K Pz + Pv ^ [3.4.10] In film boiling over a horizontal surface the interface is unstable when c 2 <0 [3.4.11] therefore the wavelength is given by: X> A + V a [3.4.12] For finite vapor and liquid thicknesses (64) the density should be corrected according to: It is important to realize that for any interface and fluids that are accelerated normal to the interface, the acceleration has to be added to the gravitational term. Most of the film boiling and nucleate boiling correlations are based on a length scale equal to the critical wavelength (when cl is a minimum), showing the importance of wave phenomena in boiling. [3.4.13] Chapter 3. SCOPE AND OBJECTIVES 46 Chapter 3. SCOPE AND OBJECTIVES The objective of this work is to develop a mathematical model to predict the temperature field of a steel strip during its processing in the hot rolling runout table, and subsequent coil cooling under diverse operating conditions. Fundamental knowledge of the boiling phenomena in water jet cooling is of prime importance to develop a versatile model able to reproduce full-scale results consistently. In this study, the "macrolayer" evaporation mechanism is assumed to be the physical mechanism involved in water jet cooling, given the successful application of this mechanism in other similar systems. Consequently, through the application of the fundamental fluid-flow and heat-transfer principles to model the boiling mechanism, the effect of each of the most important parameters during cooling are to be estimated. Although the coil thermal field prediction is not critical in steel processing, certainly has an influence on the final mechanical properties of the coiled steel, and a model of the temperature field during cooling is an important goal of this work. Chapter 4. EXPERIMENTAL PROCEDURE 47 Chapter 4. EXPERIMENTAL PROCEDURE 4.1 Pilot Plant Trials. In order to measure the local heat transfer coefficients during cooling under each jet of an array similar to a full-scale bank, pilot-plant trials were carried out at the USS Research Center pilot-plant runout table. A schematic layout of the experimental setup is presented in Fig. 11. Three Type-K thermocouples (0.51 mm diameter) were installed in a stainless steel 304 plate (see Fig. 11) with the objective of measuring the plate response at different locations through the thickness of the sample. No cleaning or special treatment was applied on the surfaces of the testing plates. Temperatures were recorded using the Lab Tech-Notebook data acquisition system with a sampling frequency of 500 Hz during the cooling of each sample plate under an array of six planar water jets. Each plate was placed on a.sled and accelerated to constant speed before being drawn through the jet array. A total of twelve temperature responses were obtained for top and bottom jet cooling. The selection of the operating parameters was chosen to satisfy similarity criterion with respect to: • The jet velocity profile, that is, similar jet Reynolds numbers (Re,) [35]. • The water velocity profiles during and after impingement, represented by the ratio Up/Uj [35]. • Thermal profiles, selecting a similar heat transfer driving force. Chapter 4. EXPERIMENTAL PROCEDURE 48 Water Boxes Fig. 11. Schematic Diagram of Pilot-Plant Runout Table, and Thermocouple Placement for Surface Temperature Measurements. Comparison between selected similarity parameters between the pilot-plant experiment and typical full-scale operation (13) is shown in Table I. Limitations in the maximum plate velocity attainable precluded a closer scaling of the fluid flow. The scaling of the fluid flow was limited to the maximum attainable plate speed (1.45 m/s). Table I. Comparison Between Pilot Plant and Typical Full Scale Similarity Parameters. Rej Up/Uj (Ts-Tsat) (Tsat-Tj) Pilot Plant 6000 1.0 700-400 75 Full Scale 12800-38400 2.5-15.3 800-400 75 Chapter 4. EXPERIMENTAL PROCEDURE 49 It is worthwhile to mention the characteristics of the thermocouple installation. In order to measure the surface temperature directly (avoiding the solution of an inverse heat conduction problem and the effect of the "thermal capacitance" of the plate on the thermal response of the thermocouples) two thermocouple wires were spot-welded on the surface to be cooled down, whereas an additional thermocouple was placed at the center (see Fig. 11). The relatively heavy wire gage and large cooling rates in these experiments have a large effect on the time constant of the thermocouples, which may be critical in view of the high frequency of the termperature changes during the experiment The effect of the characteristics of the cooling medium on the time response of a Type-J thermocouple as a function of the thermocouple diameter is shown in Fig. 12. Similarly, the effect of different wire materials is shown in Fig. 13. Valvano (66) solving the 1-D heat conduction problem for a spheric contact under forced convection conditions obtained the following expression for the time constant: x = x — [4.1.1] Equation [4.1.1] was adopted to fit the data of reference (65), and the results are shown in Fig. 12 (lines). It is clear that the Eq. [4.1.1] describes well the behaviour of the thermocouple response. However, the constant % is a function of the cooling medium, and for the specific conditions of this work, % is unknown and a , is not available. A time constant of 0.02 sec is estimated from Fig. 13 assuming: (1) a , =0.001 cm 2/sec (thermistor), (2) small thermal driving force, and (3) constant thermocouple surface temperature. Nevertheless, this value of time constant sets the upper limit of the expected time constant during the present experiments because it decreases dramatically with increasing the heat flux (see Fig. 12). Chapter 4. EXPERIMENTAL PROCEDURE 50 1 0 - 3 10- 2 10- 1 Thermocouple diameter (inches) Fig. 12. Time response for Type-J thermocouple in different cooling conditions (65) to1 10 ••* 101 101 radius squared a (cm2 ) Fig. 13. Effect of the thermocouple material on the time constant (66) Chapter 4. EXPERIMENTAL PROCEDURE 51 In view of the previous analysis, probably the time constant is smaller than required for these tests. On the other hand, the sampling frequency was selected to register at least ten temperature readings in the impingement zone of each jet 4.2 FuIl-Scale Measurements Data from the USS Gary Works normal operation were gathered for the processing of A36 steel strips of different gages, cooled under to different jet patterns. The operating conditions are presented in Tables II, and these data will be employed to calibrate some model parameters for this specific steel grade. Additional information for an A36 strip (coil 934848) was obtained in order to compare the model predictions. The total length of the strip was divided into 35 sections to access the effect of variations in the operating conditions along the length of the strip on the cooling performance. The top-surface temperature was measured at the entrance and exit positions of the runout table and the correspondent cooling pattern to each section was recorded. Samples from the head, middle and tail were cut for microstructural analysis. General operating data for this coil are shown in Table Ul. Chapter 4. EXPERIMENTAL PROCEDURE 52 Table II. Operating Data for A36 steel A36 Thickness I. Speed I. Temp Top Jets T. Vernier Bottom J. B. Vernier Exit Temp Exit Speed mm m/s °C °C m/s Coil ID smp hx vfx tlx main-t vernier-t main-b vernier-b tvp vce 907272 2 2.55778 10.09396 875 25 2 27 2 662.222 10.68324 37 2.58318 11.41984 892.78 33 2 36 2 656.111 12.6746 71 2.59334 13.08608 902.78 42 3 46 3 653.889 13.15212 907454 2 4.26466 7.25932 895.56 27 2 30 2 657.222 7.92988 21 4.17068 7.87908 890 31 2 33 2 657.778 8.636 41 4.16052 8.636 895 36 2 39 2 658.333 8.82396 908390 2 9.77392 4.12496 938.89 34 2 37 2 676.667 4.64312 17 9.95426 4.51104 931.67 39 2 42 4 667222 5.00888 32 9.9568 4.8768 951.11 45 3 49 4 668.889 5.00888 908391 2 7.94004 4.62788 930 29 2 31 2 677.222 5.12064 21 7.94512 5.08508 909.44 32 2 35 2 666.667 5.63372 41 7.95528 5.6134 919.44 41 2 45 2 662.222 5.8166 908440 2 5.30352 6.21792 912.22 28 2 32 2 656.667 6.76148 30 5.31876 6.95452 887.22 33 2 36 2 655.556 7.67588 57 5.32384 7.94512 894.44 43 2 46 2 656.111 8.0772 908495 2 4.52882 7.25424 92722 37 2 40 2 651.111 7.9502 49 4.6101 8.93064 927.22 48 2 55 2 655.556 9.6266 96 4.61518 9.84504 930 54 6 60 5 645 9.8552 913581 2 7.874 4.58216 899.44 24 2 26 2 669.444 5.09016 15 7.95528 4.93776 890.56 28 2 31 2 660.556 5.21716 28 7.95274 5.21716 895.56 30 1 33 1 673.889 5.21716 913582 2 7.94512 4.6228 901.11 24 2 26 2 671.667 5.09524 15 7.93242 4.91744 906.67 28 2 30 2 666.667 5.22732 28 7.96544 5.22732 894.44 29 2 31 2 665 5.22732 913583 2 7.9756 4.69392 896.67 23 2 25 2 678.889 5.09016 15 7.94258 4.91236 895 26 2 29 2 671.667 5.23748 28 7.93496 5.23748 891.67 28 2 31 2 676.667 5.23748 914699 2 5.5245 6.18236 911.11 30 2 32 2 660 6.53288 16 5.54482 6.4262 918.89 33 2 36 2 657222 6.731 30 5.5372 6.64464 915.56 35 2 38 3 663.333 6.731 Table HX Operating Parameters for A36 steel (Coil 934848) Sample number 35 (tail) Strip thickness (mm) 9.54 Initial strip velocity (m/s) 4.9 Final strip velocity (m/s) 4.9 Number of top jets used 40 Number of bottom jets used 45 Top nozzle dimension(mm) 18.6 (diameter) Bottom nozzle dimension(mm) 10.4 (diameter) Top jet speed (m/s) 1.84 Bottom jet speed (m/s) 1.89 Water temperature (°C) 24.5 Bottom nozzle angle 15° (to vertical) Chapter 5. MATHEMATICAL MODEL 53 Chapter 5. MATHEMATICAL MODEL 5.1 Runout Table Model The Runout Table Model solves the heat-conduction equation for a moving solid (67): p , c ^ = V ( ^ v r ) + g [5.1.1] where the variables are ps = ps(T); CPi=CPs(T); T=T(r,t);g = g(r,t) and the differential operators for a rectangular coordinate system (Fig. 14) are defined as: d d a a a _ * a a f a — =—+ur—+ w„ \-u,— ; V = l hi hk— Dt dt xdx ydy zdz Bx Jdy dz i, j, k, are the unit direction vectors along the x,y and z directions r is the position with respect to a fixed coordinate system under the following assumptions: (1) The temperature field is in steady state, « 0 [5.1.2] dt Chapter 5. MATHEMATICAL MODEL 54 (2) The heat flux in direction of the width of the strip is negligible, and the thermal profile in the same direction is not required. dT dz = 0 [5.1.3] (3) Strip speed condition ux =up; uy=uz=0 [5.1.4] Fig. 14. Runout table model reference system (4) Heat transfer due to bulk motion is much larger than the heat conduction in the same direction. From the dimensional analysis (neglecting the heat generation term) of Eq. [5.1.1] the following dimensionless equation is obtained: Chapter 5. MATHEMATICAL MODEL 55 ar 1 ia 2 r r o 2 a 2 r where dx+ Pe[d2x+ ViJ a y j y + = y/l; ^ = T + = 7 - T^./AT^; = Ux/{d I lc) Pe=ConVeCti°n=U2xW; (IJ if =3900 Diffusion for the typical pilot-plant conditions. Therefore, f ( * ~ l = 0 [5.1.5] dx\ dx (5) Constant strip velocity within the time interval of each calculation, then the coordinate tranformation x = upt can be applied. Consequently, the runout table differential equation: n r ar a p C>aTa? v 9yy [5.1.6] subject to the initial condition: t = 0, 0<y<Ls, T=T0(y) [5.1.7] and boundary conditions: dT y = 0; -k—+h0T = h0T0 dy [5.1.8] Chapter 5. MATHEMATICAL MODEL y = L*; k^r+KT=KT~.L, 56 [5.1.9] was solved by the Crank-Nicholson finite difference scheme (67). The local heat-transfer coefficients were calculated in the jet cooling zones using the Parallel How and Pressure Gradient Flow Transition Boiling Models to be presented in this chapter, whereas the air cooling zones, convection and radiation were considered. The heat generation term in Eq. [5.1.6] was computed from the phase transformation model that will be presented later. A flow chart of the runout table model is shown in Fig. 15. The model was run with 100 through-thickness nodes and a variable time step depending on the cooling zone (2000 for the pilot-plant trials, 4000 for the full-scale predictions). 5.1.1 Air cooling The heat-transfer coefficients during cooling under the convective conditions induced by the strip motion are computed using the expression for laminar flow (68): Re fP r " 2 [5.1.10] 10 | 20r 3 27(PrAj >V2 V or up > uair; Rex > 5x10s; Nux = 0.019(9-7r)02 Re [5.1.11] r = \-u„lup\ A = 1/(0.3-0.0074r) for turbulent flow conditions (10). The radiation heat transfer was calculated assuming large surroundings at room temperature and the strip emissivity is taken from Seredinski (69): Chapter 5. MATHEMATICAL MODEL 57 e = 1.1 + (T 2 7 3 )[1.25*10-*(r-273)-0.381; Tin K 1000 1 ' J [5.1.12] The heat conduction to the runout table rolls is neglected according to the analysis presented in section 3.1. Start J [rural Conditions j Plant Layout Node Discretization Cooling zones Model parameters Finite Differences Solver » ( 1=1 Jit ) ^ C E i D y Prim [Results Define cooling zone Boiling Curve model J=0,Nx > Thermal Properties Phase Transformation Model y Finite Differences Coefficients Tndiagooal Solver Ferrite Grain Size Calculation Fig. 15. Runout table model flowchart. 5.1.2 Parallel Flow Transition Boiling Model The model assumes that the macrolayer evaporation mechanism is the boiling mechanism in flow nucleate and transition boiling. This mechanism has been confirmed Chapter 5. MATHEMATICAL MODEL 58 experimentally recently during pool transition boiling of water on copper by Shoji (70), and it is extended to flow boiling conditions. Figure 16 shows the application of this mechanism in jet boiling. Strip Motion > U=U(x) U=Uj 5co =5T L b L K-- - - - * K-K - ->K-Pressure Gradient Flow Zone Parallel Flow Zone Fig. 16. The macrolayer evaporation mechanism in jet boiling The heat-transfer coefficients in parallel flow boiling on both surfaces in the Runout Table Model are calculated using the following transition boiling heat-transfer coefficient: hTB — AT+AT [5.1.13] sub where the heat flux is taken from Eq. [3.4.1]: <liB=<li-.F+qv-.<l-F) [3.4.1] and the parameter of this equation are: Chapter 5. MATHEMATICAL MODEL 59 Ql-s ~ QNB ~ Qtriple-point [5.1.14] Qv-s QFB <lFB=<lFB+-^(lrad [5.1.15] [5.1.16] The variable qtriple_pomi is given by Eq. [3.4.7] assuming that the surface under the bubble is isothermal (7^  = 7 ,^); whereas qFB is computed from the Eqs.[3.3.28] and [3.3.29]. Radiation in the vapor-film (28) is computed by: =eai((r, + 273)4-(rMr+273)4) [5.1.17] with the emissivity as defined in Eq. [5.1.12]. Moreover, the parameter Rs in Eq. [3.4.7] is calculated using the expression: [5.1.18] where the ratio of vapor-solid to liquid-solid contact areas is (70): ^ = 0.165 [5.1.19] A critical variable is the nucleation site density (NI A,a,) in Eqs. [3.4.7] and [5.1.18]. It is well known that the nucleation site density is a function of the heat flux, and thus is related to the superheat. From Eqs. [3.4.7] and [5.1.18], and expression such as: f N V* v A» j [5.1.20] can be obtained, and it compares well with empirical expressions such as N . 0 . 3 3 INB ~ A 7 I .1.2 N A J [3.3.17] Chapter 5. MATHEMATICAL MODEL 60 presented by Whalley (28). Therefore, the experimental evidence highlights that the nucleation site density is given by an expression of the type: q"m [5.1.21] where n~2 in the case of pool boiling at low and medium heat fluxes. However, the typical range of heat fluxes in jet cooling is of the order 10 7W/m 2 [O(107W7 m2)], and is at least two orders of magnitude larger than those found in pool boiling [O(105W/m2)], precluding the applicability of this value in the present model. Del Valle et al. (71) measured the nucleation site density in highly subcooled flow boiling of water on a stainless steel surface, and obtained from their experiments a value of n~l, which was concluded to represent the effect of the process of deactivation of nucleation sites observed at high heat fluxes. The range of heat fluxes measured from their experiments is still one order of magnitude [O(106W/m2)] smaller than required for the present model. Consequently, if the parameter (n) can be related to the heat flux by an expression such as ^nucleate boiling 0(1O6) n 'flow boiling then it may be assumed that a similar expression O(106) flow boiling 0(1O7) n = 2 'jet boiling can be used to estimate the parameter n for jet cooling. Accordingly, a value of n=0.5 was assumed in the present model, and the heat flux in nucleate boiling is represented by Q m - * T [5.1.22] Chapter 5. MATHEMATICAL MODEL 61 The macrolayer evaporation mechanism is also applied in the estimation of F in Eq. [3.4.1]. Following the idea offered by Haramura and Katto (54), Pan et al. (72) calculated the average liquid-solid contact area fraction in transition boiling as: [5.1.23] where parameters L and LB in Eq. [5.1.23] are the actual and initial lengths of the macrolayer (see Fig. 16). Since film boiling is the natural extension from transition boiling in a heating or in a cooling process, it is reasonable to assume that the length of the vapor mushroom is equal to the wavelength of the vapor-liquid interface when it is unstable, so LB = X [5.1.24] where X is given by Eq. [3.4.12]. The actual length of the macrolayer, L, can be calculated from an energy balance on the macrolayer, so within the length L the heat transfered from the surface is equal to the latent heat of the total evaporation of the liquid flowing into the macrolayer (conduction to the liquid water is neglected), rendering L _ P , 8 . 0 ( ^ + ^ ) ^ ( 8 , 0 ) [5.1.25] where 8 c 0 is given by Eq. [3.4.6], and £ / m e ( 8 c 0 ) is calculated using the expression by Chappidi et al. (68) for the strip motion in the direction of the water flow: . ume®c0) = uA 1+ V uiJ (2T 1-2T 1 3 +TI 4 ) [5.1.26] Chapter 5. MATHEMATICAL MODEL g2 n = ^ ; i R e ? - a r 1 - f i J , - , _ i ; R e , = M 5 x x \03R-Q.\\1AR2) up x v evaluated at a the position from the jet centerline. For the reversal flow case, Lin et al. (73) obtained a solution, but the numerical procedure employed is very complex and its applicability to this model is limited. Consequently, the following assumptions is adopted C/me(6c0) = «, [5.1.27] 5.1.3 Pressure Gradient Flow Transition Boiling Model The Pressure Gradient How Transition Boiling Model is based on the same principles and equations of the Parallel How Transition Boiling Model, but correction of the qv_s and F parameters in Eq [3.4.1] have to be considered in view of the acceleration of the flow and the pressure gradient conditions of the impinging flow. Figure 16 shows the macrolayer during boiling in this zone. As the jet flows from the nozzle to the strip, acceleration of the fluid flow is caused by gravity, and the impinging velocity is given by: Uj=(u2n+2gHn)y2 [5.1.28] and accordingly the impinging jet width is w, =^-wn [5.1.29] UJ and the local free stream velocity is given by Eq. [3.3.7]. Again, qv_s is given by Eq. [5.1.15], but qFB is computed using Eq. [3.3.20]. A more general equation might be obtained from an analysis similar to that adopted to obtain Eq. [3.3.23] or Eq. [3.3.26] extended to a moving surface, and then adding the contribution of radiation heat-transfer Chapter 5. MATHEMATICAL MODEL 63 by Eq. [3.3.22]. However, given that it is difficult to define an initial vapor-layer thickness during jet impingement on a moving surface, this analysis was not adopted in this work. The parameter F is calculated using Eq. [5.1.23], where LB is given by Eq. [5.1.24] and A, is given by Eq. [3.4.12], but correction of the gravitational term (see section 3.4.3) due to the accelerating flow is assumed to be: u) 8 =8+^r- [5.1.30] 2\Vj The L parameter is obtained from the same heat balance employed to obtain Eq. [5.1.25], but an additional input/output of water to the macrolayer due to the y-component of the fluid flow has to be considered, and the following expression is obtained: pfie0{HJk+AHmt)l/MTGE0) ^ ( l - p { ( ^ + ^ ) C / y ( 5 c 0 ) / q N B ) where Ume(bc0) is computed solving Eq. [3.3.9] to [3.3.13] by the Runge-Kutta method described elsewhere (23), the dimensionless coordinate is given by: 2 v vdi J ; dj = distance to jet center and parameters u„ and f/y(8c0)are calculated using Eq. [3.3.1]and [3.3.11] respectively. 5.1.4 Phase Transformation Model The methodology adopted is similar to that described by Kumar et al. (5), in which the continuous cooling phase trasformation was approximated by a discrete number of Chapter 5. MATHEMATICAL MODEL 64 isothermal steps (each characterized by the Avrami equation) and by invoking the additivity principle (74). Consequently, the fraction transformed is: [5.1.32] and t., = -fr,(r,+A,) [5.1.33] which are applied once the local temperature was below Tstarl .The heat generated by the phase transformation is obtained from: 8 = P,H,- [5.1.34] were Ht was taken from Campbell (75). The Avrami equation parameters and Tstart are chemistry dependent; for the DQSK steel, for example, these are (5): /-)TY>.155 Tm = 875 - 27.61 — 1 ; na = 0.79; In ba (T) = 4.279 - 0.00597 [5.1.35] The A36 steel transforms to a ferrite plus pearlite microstructure, so the ferrite transformation parameters are given by (76): Tm =827-71.01 — I ;n a=1.0; Inba(T) = 0.04(827-T)-5.95 [5.1.36] and for the pearlite transformation case these are (76) (77): Chapter 5. MATHEMATICAL MODEL 65 Tstart,p =Tm -88.0; np =3.01(%C)2-1.06(%C) + 0.5(%Mn)+0.792 [5.1.37] labp =0.0419v|/ +0.35'7vi/ I /2-10.2^%C+-^^j-1.9; y = L i f -T [5.1.38] The instantaneous cooling rate was adopted to determine the transformation-start temperature at each position, unless the local temperature arose; then the last previous positive cooling rate was assumed. 5.1.5 Grain Size Model Ferrite grain size prediction is important to characterize the final microstructure and mechanical properties of the steel. The present model computes the ferrite grain size at 5% fraction transformed, according to (78): da =[5.76-10%C-1.3%Mn]rfT°-45l^-j [5.1.39] where the same criterion for the cooling rate calculation as for the Phase Transformation Model was adopted. 5.2 Coil Cooling Model The Coil Cooling Model assumes that the coil can be considered as a continuous orthotropic hollow cylinder. Accordingly, the model solves the heat conduction equation (67): r ^ I - l A f t LIS\K PsCp'dt~rdr{rrdr)+r2 3<t>J d_( dT + dz{zdzj + 8 [5-2.1] T = T(r,$,z,t); p, = PAT); CPi = C^T); kr = kr(r,T); K = k,(z,T); ^ = k^,T); 8 = 8(r,z,T) under the following assumptions: Chapter 5. MATHEMATICAL MODEL 66 (1) The angular component of the heat flux vector is negligible M 7 y * r d<|) (2) There is no phase transformation in the coil g = 0 [5.2.3] (3) The thermal conductivity in the axial direction is that of the steel kt=k, [5.2.4] (4) The thermal resistance due to the imperfect contact between wraps may be computed by assuming an equivalent thermal conductivity (see section 3.20). Furthermore, it is assumed that the thermal conductivity in the radial direction is a fraction of that for the steel (14): kr=(fiks [5.2.5] where a value of co = 0.1 is adopted (14). Under these assumptions the Coil Cooling Model differential equation becomes: „ BT 1 B (, BT\ 3 (. BT\ r_ . which is solved subject to the initial condition t = 0; ra<r<rb;0<z<Lc; T = T0 [5.2.7] and boundary conditions: r = r ; -k— + hT = hTa [5.2.8] Chapter 5. MATHEMATICAL MODEL 67 r = rb; kr-^+hbT = h b T ^ b z = 0; -kz—+h()T = hXjT„0 dT z = 4 ; K - ^ + K T = K T ~ . L C [5.2.9] [5.2.10] [5.2.11] Figure 17 shows the coil scheme employed in this model. Equation [5.2.6] under the initial and boundary conditions [5.2.7]-[5.2.11] is solved by the ADI finite differences scheme obtained by applying the conservation principle to a control volume (67). The model was run using 20-nodes in each direction and 1000-time steps during calculations. The heat transfer coefficients adopted are calculated for natural convection of air at room pressure (79): For z = 0 and z = L M O 4 < Gr Pr < lxlO 9 ; h = 1.42 v 4 y W O 9 <GrPr<Lri0 1 3 ; A = 131 v 4 , [5.2.12] [5.2.13] for r = r lx l0 4 < Gr Pr < I J C I O 9 ; h = 1.32 \ 2 r a J [5.2.14] for r = r IxlO9 < Gr Pr < I J C I O 1 2 ; h = 1.24 'AT.*" \ 2 r » J [5.2.15] where the Grashof number is Chapter 5. MATHEMATICAL MODEL •» f * 6 8 Air convection Fig. 17. Schematic Coil for the Coil Cooling Model Chapter 6. MODEL VALIDATION 69 Chapter 6. Model Validation 6.1 Runout Table The runout table model was validated by comparison with the pilot-plant and full-scale measurements obtained by the experimental procedure previously presented. 6.1.1 Comparison of Model Predictions against Pilot-Plant Measurements The pilot-plant data allow the simplification of the analysis of the full-scale problem since no phase transformation occurs during cooling, therefore, serves to compare the heat transfer model alone before including the volumetric heat generation. The response of eleven of the twelve experiments carried out correspond to internal placement of the thermocouples, and the information generated was not used in this work for two reasons: (1) The thermal capacitance of the material between the surface and the thermocouple might be so large that precludes the possibility of solving accurately the inverse heat-conduction problem for heat flux variations from 1 to 10 M W / m 2 occuring with a frequency of 50 Hz, typical of the present experiments; (2) The required generality in the scaling-up to full-scale conditions of the local pilot-plant heat fluxes cannot be accomplished from the reduced number of experimental data. Accordingly, comparison of the model predictions with the surface temperature measurements is presented in Fig. 18. The experimental results show that the cooling rate Chapter 6. MODEL VALIDATION 70 1300 1200 A g *T 1100 § 1000 CL § 900 Parallel Row zone 0) o H 800 700 600 Pressure Gradient Zone TC1 TC3 Predictions -i 1 1 r -l 1 1 1 1 1 r 0 T 1 Cooling time (sec) -i 1 1 r Fig. 18. Comparison between Pilot-Plant Measurements and Model Predictions due to the first jet is larger than for the other jets. Also, the cooling rate is very high from the beginning of the plate contact with the jet, which corresponds to the parallel flow zone in countercurrent flow. The highest cooling rate is reached in the pressure gradient zone. Once the plate is in the parallel flow zone, in the direction of the plate motion, the jet cannot maintain the high heat flux of the pressure gradient zone and the surface temperature increases as a result of heat conduction from the interior of the plate. Chapter 6. MODEL VALIDATION 71 The model predictions agree well with the experimental results. The shape of the calculated curves agree well with those measured, but the heat removal calculated in the parallel flow zone is smaller than the experimental values. Probably, this can be explained considering that the model is based on steady-state fluid flow and boiling, and the transience of these processes is difficult to be predicted by the present model. Nevertheless, if an increase in the length of the impingement zone (x* = 4) is assumed to compensate for the transience effects, good results can be obtained. According to theoretical calculations the impingement zone length is about 5 times the jet thickness, while pressure measurements give values of 3.5 to 5 (24,80), as shown in Fig.5. Probably, the reason for a larger impingement zone found in the experiment is due to splashing during transient fluid flow. It is important to mention that a better fit of the data is possible by adopting a specific value of the macrolayer evaporation rate parameter, metp, for each jet. However, Pasamehmetoglu et al. (61) suggested that metp does not depend on the sample superheat, and should be constant for a fixed cooling configuration (material, surface conditions and thickness of the sample, cooling fluid). For pool boiling of water on a 10 mm thick copper block a value of mefp = 6.0JC10"5 ( K g m V ' C " 1 ) was obtained (61), whereas in the present case, metp = 2.5;cl0~s (Kg m"1.?-1 ° C - 1 ) was assumed. The instantaneous heat flux was obtained from the measurements shown in Fig. 18 by an inverse boundary condition technique, and typical graphs similar to boiling curves (heat flux vs. superheat) were developed, which for the case of the first jet is shown in Fig. 19. Figure 19 reveals that the heat flux increases monotonically with decreasing surface temperature, and heat fluxes higher than 1.0 MW/ m2 were obtained. These results clearly resemble the typical transition boiling curve, supporting the assumption that transition boiling is the mechanism of heat transfer. Also, there is no evidence of a niinimum heat Chapter 6. MODEL VALIDATION 72 flux nor a decrease in the slope of the curve at high superheats indicating that film boiling was present. On the other hand, no maximum heat flux was found, indicating that possibly nucleate boiling was not present. The experimental conditions themselves suggest that the fluid flow is not developed during cooling, and therefore, the heat transfer problem is unsteady as well. Then, the very high cooling rates obtained may be caused by the transience of this process; and liquid-solid contact (thermal shock) before evaporation may be responsible for the very high heat fluxes in the parallel flow zone. This agrees with the results in a similar experiment by Chen et al. (81). Fig. 19. Typical Boiling Curve for a Single Jet Cooling in the Pilot-Plant Runout Table Chapter 6. MODEL VALIDATION 73 6.1.2 Comparison of Model Predictions with Full-Scale Measurements Even though the model was developed for planar water jets, in this section the application of the model is extended to water round bar cooling systems by means of a very simple consideration, as long as unidimensional heat transfer can be assumed. In the model, the shape of the jet nozzle is considered in the calculation of the stagnation zone length, and in the mass flux of water entering the macrolayer. From visual observations of the fluid flow pattern in the impingement zone of industrial-scale cooling systems, it is reasonable to assume that the stagnation zone length is approximately the minimum for the equivalent planar jet (same width and velocity), and hence x* = 1.75 is adopted. Also, it is reasonable to assume that the amount of water flowing close to the strip surface is approximately the same in view of the interaction between neighbor jets, which tends to keep most of the water flowing in the strip motion direction. At typical runout table strip speeds, the water flowing parallel to the strip motion interacts with the countercurrent flow from the next jet downstream in such a way that the countercurrent parallel-flow region for each jet but the first is negligible (10). Measurements in the runout table were taken at each 19 ft (5.79 m) along the length of the strip, starting from the head, and the initial surface temperature (Tfx), the exit temperature (Tvp) and the coiling temperature (Tee) were recorded at fixed locations, assignning a sample number to each section recorded for each coil. The results for each Coil and sample appear in Fig. 20. Three samples were selected randomly to represent head, middle and tail conditions for each coil, and Table II includes the operating conditions employed during the processing of those specific samples. It is manifest that the initial runout table temperature is not constant, but oscillates with a an non-regular oscillation amplitude, whereas the mean value do not behave regularly for all the coils. The difference between the absolute minimum and maximum temperatures may be above Chapter 6. MODEL VALIDATION 74 40 °C, but after the runout table cooling the amplitude of such variations decreases. The exit runout table temperature (Tvp) and the coiling temperature (Tee) follow similar patterns or cycles, but not identical. They are not strongly affected by the amplitude of Tfx variations at smaller gages, but the importance increases while increasing thickness. Also, the differences in Tee and Tvp temperatures are larger while increasing the thickness, but it is not a general trend, and they may be strongly affected by the pearlite fraction transformed, which is function of the local chemistry of the strip. 0 10 2 0 3 0 4 0 5 0 6 0 70 6 0 9 0 • n o sao 670 sao 650 MO 630 620 610 600 950 0 10 20 30 40 50 60 70 60 90 100 Sample 0 10 20 30 40 SO 60 70 60 90 100 940 A 930 -] 920 910 900 _ 890 i 8 —»•] o Cort 908390 n T»p Tee 650 640 630 620 610 600 7 7. 10 2 0 3 0 4 0 5 0 6 0 70 6 0 9 C Sample 0 10 20 30 40 SO 60 70 60 90 100 Coil 907474 T»p 0 10 20 30 40 50 60 70 60 90 100 950 Sample 0 10 20 X 40 SO 60 70 60 90 100 940 930 920 4 910 900 0 690 S 660 ^ 1 «*> £ 700 « 660 Coil 908391 iftt Tvp Tee 650 4 640 630 620 610 600 7 7 0 10 20 30 40 50 60 70 60 90 100 Sample Fig. 20. Temperature Measurements for the A36 steel Chapter 6. MODEL VALIDATION 75 9 5 0 9 4 0 9 3 0 9 2 0 9 1 0 • 9 0 0 -O 8 9 0 -<U 8 8 0 -I S 8 7 0 , S 7 0 0 , 9 " 6 9 0 -53 6 8 0 -* ~ 6 7 0 -6 6 0 -6 5 0 -6 4 0 -6 3 0 -6 2 0 -6 1 0 -6 0 0 -1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Coil 908440 • Tfx Tvp Tee > s 1 <& Coil 908495 1 0 T 1 20 T 3 0 4 0 SO 6 0 7 0 8 0 9 0 1 0 0 Sample ' | . . i i | . . , i i . . | 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Sample 9 5 0 9 4 0 4 9 3 0 9 2 0 4 9 1 0 4 9 0 0 O 8 9 0 <U 8 8 0 4 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 t • • • • i . . . . i . . . . i . . . . i . . . . i . . i 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 0 ) 0 J 8 7 0 y 7 0 0 / S- 6 9 0 6 8 0 J\ 6 7 0 \ V \ 6 6 0 -; 6 5 0 6 4 0 \ 6 3 0 6 2 0 6 1 0 6 0 0 i 1 1 1 ' i 1 i Coil 913581 — Tfx • — Tvp — • Tea 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Sample 9 5 0 -r 9 4 0 -j 9 3 0 4 9 2 0 9 1 0 Coil 913582 — Tfx • — Tvp - Tee Sample / i E-1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Fig. 20. Temperature Measurements for the A36 steel (cont.) Chapter 6. MODEL VALIDATION 76 1 0 o 950 940 930 920 910 4 900 4 890 £ 8 8 0 20 30 40 50 60 70 80 90 100 • ' ' i . . . . i i . . . . . . . . i . .1 8. E <u 870 y 700 ' 690 4 680 670 660 4 650 640 630 620 4 610 600 3 Coil 913583 Tfx Tvp Tee 1 0 r - T - r - i 2 0 3 0 4 0 111 1 ' 5 0 -rjT-, 6 0 70 80 90 100 Sample 0 10 20 30 40 50 60 70 80 90 9 5 0 | I I 1 0 0 940 4 930 920 910 900 4 O 890 2 880 15 870 > ™/ g- 690 <5 680 *~ 670 660 - , , 650 -. 640 ' 630 620 610 6 0 0 Coil 914699 Tfx Tvp Tee 1 ' I " " I " " I 11111111 • 111... i . • 111 • • 10 20 30 40 50 60 70 80 Sample " T " 9 0 1 0 0 Fig. 20. Temperature Measurements for the A36 steel (cont.) The runout table model was run using the data of Table II to back-calculate the metp parameter for each position (head, middle and tail) and for each coil, by feeding a value of this parameter until agreement within ±1°C with the measured exit temperature was reached. It is important to mention that the initial condition to solve the thermal field for each run was calculated from the regression analysis of the expected firiishing mill exit temperature profiles using an UBC code (82), fitting exacdy the measured surface temperature for each case. The metp parameter values obtained (points) are plotted versus the correspondent strip thickness in Fig. 21. According to the definition of the macrolayer evaporation parameter, metp (Eq.[3.4.8]), it is unlikely that the heat-transfer coefficient for the evaporation on Chapter 6. MODEL VALIDATION 77 9e-5 -\—1—'—1—1—1—L—J—1—1—1—1—1—1—1—1—1—'—' " i i i 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 Strip Thickness (m) Fig. 21. The Macrolayer Evaporation Parameter for the A36 steel the vapor-stem base could be a function of the thickness of the strip. However, under the assumption of equation [5.1.14] the base material underneath the bubble should be isothermal, but the results shown in Fig. 21 might indicate that this condition is not satisfied, and the back-calculated parameter is compensating for the localized cooling under each vapor stem. Consequently, it is expected that the problem of calculating T in Chapter 6. MODEL VALIDATION 78 eq. [3.4.7] should be closely related with the cooling equation of a fin of uniform cross-section (79): Ttp=c1 cosh mLs + c2 sinh mLs [6.1.1] This equation was employed for the regression analysis of the macrolayer evaporation parameter, and the results correspond to the lines in Fig. 21, where the clear agreement with the data gives support to this conclusion. Nevertheless, given the complexity of the evaporation processes on the base of the vapor stems, no further analysis of the evaporation phenomena was attemped, and the compensation for the non-uniform temperature will be included in the macrolayer evaporation parameter. The effect of variation of metp (or strip thickness) on the heat-transfer coefficients for the stagnation line and the onset of the parallel flow zone appear in Fig. 22 and 23 respectively. Within the range of the back-calculated metp values, the heat-transfer coefficients increase with metp (decrease with thickness) in the nucleate boiling regime, also the maximum heat-tranfer coefficients increase with melp, but the opposite occurs in the transition boiling regime. The superheat at the maximum heat-transfer coefficient increases decreasing metp (increasing thickness). The heat-transfer coefficients calculated may seem very large compared to other systems, however, these values can be compared with the experimental results by Chen et al. (83) for a circular jet impinging on a moving surface in the nucleate boiling regime. Minimum values of about 50000 W/m 2 / °C and maximums of 200000 W/m 2 / °C were measured, which agree very well with the present model predictions (see Fig. 22). Chapter 6. MODEL VALIDATION 79 250000 Superheat (Ts-Tsa t)(°C) Fig. 22. Effect of metp (Thickness) on the Heat-Transfer Coefficients in the Stagnation Line of a Series of Circular Jets in the Runout Table. The specific effect of the substrate thickness has been addressed only recently by Unal et. al. (84). Their numerical results show that in transient cooling experiments the nucleate boiling curve shifts to lower superheats while increasing the thickness of the specimen, and similarly the model results presented in Fig. 22 and 23 agree with this observation. Chapter 6. MODEL VALIDATION 80 1 5 0 0 0 0 Superheat (Ts-Tsat)(°C) Fig. 23. Effect of melp (Thickness) on the Heat-Transfer Coefficients in the Parallel Flow Region Series of Circular Jets in the Runout Table. Model predictions were carried out using the regression equations [6.1.1] of the data in Fig. 21 (lines), and the results are compared with the measured temperatures for the A36 steel strips in Fig. 24. Most of the predictions lie in the range ± 15°C or the measured value, however, some show large deviation, and diverse reasons for these results exist. Chapter 6. MODEL VALIDATION 81 6 0 0 6 1 0 6 2 0 6 3 0 6 4 0 6 5 0 6 6 0 6 7 0 6 8 0 6 9 0 7 0 0 6 0 0 6 1 0 6 2 0 6 3 0 6 4 0 6 5 0 6 6 0 6 7 0 6 8 0 6 9 0 7 0 0 Measured exit temperature (C) Fig. 24. Comparison of Runout Table Model Predictions with Measured Exit Temperature for the A36 Steel. The Runout Table Model is very sensitive to small changes in the heat-transfer coefficients due to the fact that it accounts for the cooling of about one hundred jets, which implies that even 1°C difference in the calculation of the temperature leaving each jet may result in several degrees of difference, and this may be the case of any model using local heat-transfer coefficients. On the other hand, the samples employed for the calculation of the macrolayer evaporation parameter were randomly selected, which may Chapter 6. MODEL VALIDATION 82 deviate the results from the general behaviour of the coil (see Fig. 20). From measurements in Fig. 20, it might be observed that when Tfx of the head, middle and tail are very close to each other, metp behaves closer to Eq. [6.1.1]. Also, it was noted that when the Tee and Tvp temperatures are very different, the me parameter deviates from the fitted curves. In order to verify the model with an independent coil, the coil 934848 was selected (see Table III). The samples representing the head, middle and tail conditions were selected randomly as well. Comparison of the model predictions with the measured values for the head, middle and tail samples are plotted in Fig. 25-27, respectively. The model predictions are in very good agreement for the middle and the tail, but they are not as accurate for the head. Figures 25-27 show that the cooling due to the bottom jets is much lower than for the top jets, for almost the same jet velocity (see Table HI), which is in agreement with the literature (34). The reason for the difference is the smaller contact area with the jet water, and lower heat fluxes. The vapor-liquid interface is much more stable for the bottom jets because there is no gravity induced instability. For this particular case, the strip moves relatively slowly (4.9 m/s) compared to typical speeds (-10 m/s) for smaller gages; consequently the larger contact time with each jet cools the top surface to a temperature below that for transition boiling in the latter stages. After the critical heat flux has been passed during cooling, the heat flux diminishes following the nucleate boiling curve. The model shows that the cooling pattern employed for this particular steel generates comparatively small thermal gradients through the thickness of the strip, and the temperature differences are smaller than 100°C in the half the thickness of the strip during the cooling process. Finally, after thermal recovery, the temperature profile is virtually uniform. Chapter 6. MODEL VALIDATION 83 0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Distance (m) Fig. 25. Comparison between Model Predictions and A36 Strip (Head) The effect of the heat of transformation is shown shown in Fig. 25-27 (dotted lines). The phase transformation is more sensitive to the heat of transformation than the thermal response of the strip. It is interesting to note that the effect on the exit temperature may be very important in some cases (Fig.27), but not in others (Fig.25). Chapter 6. MODEL VALIDATION 84 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Distance (m) Fig. 26. Comparison between Model Predictions and A36 Strip (Middle) The results show that the lower the predicted exit temperature, the more important is the heat of the transformation in the thermal problem. Therefore, the heat of transformation cannot be neglected. Chapter 6. MODEL VALIDATION 85 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 Distance (m) Fig. 27. Comparison between Model Predictions and A36 Strip (Tail) It is worthwhile to mention that the cooling performance of jet cooling improves substantially while the surface temperature is lower. In Fig. 25-27 show that the cooling performance of the last of the three top water banks is greater than the second one, and for the second bank is greater than the first one. This is also evident comparing the thermal response due to the last vernier jets (peaks near the exit). Chapter 6. MODEL VALIDATION 86 Regarding to the metallurgical parameters, the fraction transformed to ferrite and pearlite are strongly dependent on the Tvp temperature. Calculations for the head show that an exit temperature 30°C above would delay the the transformation to ferrite and no pearlite would appear in the runout table, as compared to the middle where the transformation to ferrite is complete and the the pearlite fraction transformed is roughly 25% of that expected in equilibrium. Finally, an exit temperature 15°C below accelerates considerably both transformations and they would be completed at the exit of the runout table. In this manner, the calculations show the importance of accurate temperature control in the runout table to control the homogeneity of the strip microstructure, and hence, mechanical properties. As it was mentioned in the section 4.2, samples from the independent coil 934848 were obtained and analysed. The through-thickness microstructure was very similar for the second half of the strip (from centerline to bottom surface), whereas the microstructure at the top surface was different, supporting the model calculation in regard to the expected cooling rates in each zone. The grain size model was employed for the grain size prediction at the middle of the strip. Two different initial austenite grain sizes were used (uniform through-thickness distribution) for calculations, and: (1) the austenite grain size of 27 fxm, which was calculated from a UBC finishing mill model (82) rendered a final ferrite grain size calculated of 5.3 |im in comparison to 5.2 \im measured at the centerline, whereas; (2) the austenite grain size of 49 [im, estimated from the polygonal ferrite microstructure gave a ferrite grain size of 8.0 |im. The model also predicted very similar centerline and bottom surface ferrite grain sizes, as it was corroborated experimentally. Chapter 6. MODEL VALIDATION 87 In order to verify the heat-transfer mechanism proposed, comparison of the more fundamental parameters, the liquid-solid contact heat flux q,_s (see Fig. 9); and the fractional liquid-solid contact area, F, in transition boiling (see Fig. 8), with the predicted values for the A36 steel (coil 934848) appear in Fig. 28 and 29 respectively. u. a o X _ 10' 10* 10s SarVl a v-OJ m/t • v-t m/s • v«2 m/t 80"C Subcooling O v-0.5 m/t m/t • o » - 2 m/t - ! 1 ^ « c o e i Y • / . //*^\ / o * / ° j A / ° • • • • • 0 100 200 300 400 500 Initial Surface Superheat, AT. *C E o CO c o O "o in 1 0 8 (.1 I I I | I I I I | I I I I | I I I I | I I I M X i f 1 0 7 Cl) X 1 0 6 Subcooling= 85.5°C Jet Velocity= 1.84 m/s -JQ5 I I I I I I I I I I I I I I I I I I I t I I I I I 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 Superheat ( C) Fig. 28. Comparison of the Liquid-Solid Contact Heat Flux in a Falling Drop (50) and in Jet Cooling for the A36 Steel (Coil 934848). The shape of the predicted curve for the A36 steel agrees well with measurements for a falling drop, but lies above them (Fig. 28). The subcooling in both cases is very similar, and then the difference should be related to differences in the fluid flow phenomena. It is clear that in the experimental plot, the liquid-solid contact heat flux increases following the same behaviour as the calculated curve, so increasing the falling drop velocity (compare v= 2 m/s) the experimental solid-liquid heat transfer increases, and eventually it would Chapter 6. MODEL VALIDATION 88 tend to reach the calculated values. This shows the importance of the fluid flow involved on contact heat-transfer. Probably the greater momentum associated to jet cooling contributes to the enhancement of the contact heat transfer. Also these results validate the analysis adopted to obtain Eq. [5.1.22], which confirms the nucleation site density model. io° | 1 v io-10" 10" 10" 10 10" I-3 -j—i—r—r— a a • • a O : F (present) £ A:F(0huga) i : Fo (Shoji) = • :Fa(Dhuga) 5 <» : Ft (present) I a : Ft (Lee) I it i i a o _i i—i i_ 0-0 _ AT^r-ATpf 1.0 " | 10"5 \r 03 LL 1 0 - 6 Stagnation Line Parallel Row Zone • i i i i i i ' i i < I i_ 0.0 0.5 AT* 1.0 Fig. 29. Comparison of the Liquid-Solid Fractional Contact Area in Laboratory Measurements and Model Predictions for the A36 steel (Coil 934848) Figure 29 compares the fractional liquid-solid contact area predicted for the A36 steel with the experimental Fa values already presented, and good agreement was found. The curve for the stagnation line of the jet array is very close to that of Fa (Shoji), whereas the one for the parallel flow zone is closer to the Fa (Dhuga) results. Even though direct comparison between these curves is not a conclusive proof to validate the present model, given the different conditions involved in the fluid flow, certainly it shows that the Chapter 6. MODEL VALIDATION 89 behavior of the calculated parameter is in good agreement with the present knowledge in boiling. The predictions also support the empirical observation that the jet cooling enhances the liquid contact with the strip surface compared with water running parallel to the surface. Finally, Figure 30 presents some calculated heat-transfer coefficients for coil 934848 showing that for this coil, the heat-transfer is relatively small compared to that for thinner strips (Fig. 22-23). 140000 - i 1 Superheat (°C) Fig. 30. Heat-Transfer Coefficients during the Cooling of the A36 Steel (Coil 934848) Chapter 6. MODEL VALIDATION 90 6.2 Coil Cooling Model The Coil Cooling Model is compared with the analytical solution of simpler problems due to the fact that the reported information regarding to the thermal conductivity in the radial direction is incomplete to verify the present model. The model was run to predict the interior surface temperature of a semi-infinite hollow cylinder with constant thermal properties to verify the code in the r-direction. A convective boundary condition was assumed on the interior surface (r = ra), whereas an isolated exterior surface was adopted, and comparison with the equivalent analytical solution of the same problem (85) is shown in Fig. 31. The model calculations compare very well with the analytical solution. However, the inherent nature of the ADI procedure in shown in the small deviation shown at the beginning of the heating process, which correspond to the first half-time steps. To verify the code in the r and z directions, the model was used to calculate the temperature at (r = ra+(rb-ra)/2,z = Lc/2) of a short hollow cylinder of constant thermal properties with semi-infinite interior radius subject to the same convective boundary conditions on all the surfaces. Comparison with the equivalent analytical solution is shown in Fig. 32, and again, the model predictions agree very well with the analytical solution, but in this case, the time-step deviation at early stages intrinsic in the ADI schemes does not appear. Chapter 6. MODEL VALIDATION 91 J— i—i i I • • i i ! i i i • I • • i i I • • ' • O Coil Cooling Model 0 "Cp—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—J -Oe+0 5e-8 1e-7 2e-7 2e-7 3e-7 Time Fig. 31. Comparison of the Coil Cooling Model Predictions with the Analytical Solution of a 1-D problem in the r-direction. Chapter 6. MODEL VALIDATION 92 1100 1000 -I I 1 I ' I l _ 500 H 400 0 • i • i i • i • • i i I i i < i_ O Coil Cooling Model Analytical Solution ra=500, rb=502 L=2 HTC=5 T a m b=500 -i—i—i—|—i—i—i—i—|—r 2 3 Time (sec) n—i—|—i—i—i—r 4 Fig. 32. Comparison of the Coil Cooling Model Predictions with the Analytical solution of a 2-D problem in the r and z directions. Chapter 7. SENSITIVITY ANALYSIS 93 Chapter 7. SENSITIVITY ANALYSIS 7.1 Runout Table Operating Parameters In this section a brief sensitivity analysis is performed on four operating parameters: (1) water flow rate, (2) water temperature, (3) strip speed and (4) the strip initial temperature profile, which are the most important direct operating parameters (not including the actual jet layout). In this section the effect of variations of one operating parameter at a time on the thermal response of the A36 steel coil 934848 is presented, and comparison with the original model predictions is presented to access the effect of each individual parameter. 7.1.1 Effect of Water Flow Rate Perhaps, the most studied operating parameter in the runout table cooling is the jet velocity (water flow rate). Probably this is a consequence of: (1) The flow rate is a variable relatively easy to control, and (2) In single-phase convection heat transfer the jet velocity is the most important individual variable in determining the heat-transfer coefficients. The effect of jet velocity (flow rate) on the strip centerline temperature and phase transformation kinetics is shown in Fig. 33. Clearly, the rate at which the austenite to ferrite transformation is occurring is slower, as a consequence of lower subcoolings (with respect to the transformation-start temperature, Tstart) and the smaller residence times at temperatures lower than TstaTt. Chapter 7. SENSITIVITY ANALYSIS 94 Distance (m) Fig. 33. Predictions of the Effect of Jet Velocity Variations on the Thermal and Microstructural Response of A3 6 steel. In terms of the physical mechanism proposed in this work, the local heat flux is decreased because the liquid-solid contact beneath the vapor bubble is reduced at lower jet velocities. It is important to mention that this explanation agrees with the observations of Ishigai et al. (27) during the measurements shown in Fig. 6. Then, the convection heat transfer associated with this process is not responsible for the smaller heat fluxes observed reducing the jet velocity, because the main mechanism of heat removal is boiling and not the conduction to a moving fluid. Chapter 7. SENSITIVITY ANALYSIS 95 7.1.2 Effect of Water Temperature The experimental results by Ishigai et al. (see Fig. 6) clearly show that the effect of water subcooling is probably the most important variable in jet cooling, and relatively small changes in water temperature are responsible for large differences in the local heat fluxes observed. Model predictions of the effect of water temperature on the centerline temperature and microstructure are shown in Fig.34. Environmental conditions may affect the cooling performance in the runout table as exemplified in Fig. 34, where a decrease from 24.5 °C to 10.0 °C in water temperature accelerates the phase transformation to ferrite and pearlite, and for this case, both are completed in the runout table. The exit temperature is decreased by more than 20 °C, and this difference increases with the distance because the transformation to pearlite was not completed in the original case. In the present model the water temperature is accounted for by the parameter A / / m 6 in Equation [5.1.25], as well as in the evaluation of the water thermal and viscous properties, and has a direct effect on qt_s and F. Then, water at lower temperatures requires more energy to evaporate, increasing the solid-liquid contact length which in turn increases the local heat-transfer coefficients as shown in Fig. 35, and the effect is greater in the pressure-gradient zone than in the parallel flow region. These results agree qualitatively with the experimental data by Ishigai et al. (27), but the effect predicted is smaller, although data for the subcooling of the predictions was not obtained experimentally. Chapter 7. SENSITIVITY ANALYSTS 96 Finally, the improved heat removal is a consequence of the higher heat-transfer coefficients and not to the sligthly higher driving force associated with lower water temperatures. Distance (m) Fig. 34. Predictions of the Effect of Water Temperature on the Thermal and Microstructural Response of A36 Steel. Chapter 7. SENSITIVITY ANALYSIS 97 0 200 400 600 800 1000 Superheat (°C) Fig. 35. Effect of Water Temperature on the Heat-Transfer Coefficients for A36 Steel. 7.1.3 Effect of the Strip Speed As it was mentioned previously, the experimental difficulties associated with a moving surface precludes the determination of the effect of the strip speed on the local heat-Chapter 7. SENSITIVITY ANALYSTS 98 transfer coefficients, and this is one of the important reasons why the the modeling of the boiling phenomena is attractive for runout table predictions. The effect of strip speed on the A36 strip cooling is exemplified in Fig. 36. The cooling performance is not greatly affected by increasing the strip speed from 4.51 to 5.5 m/s for this particular case. This interesting result is a consequence of two counteracting factors: (1) It is expected that increasing the strip speed the contact time with the cooling water is reduced proportionally, and therefore, lower cooling should be obtained; but (2) the enhancement in the strip motion increases the local heat-transfer coefficients in both the parallel-flow zone and the stagnation zone (see Fig. 37), and the net effect is such that the exit temperature is just slighdy above the original. However, the difference should increase with distance, since the pearlite transformation has not even started in the second case, then the difference in the pearlite fraction transformed should be reflected in a higher temperature for the faster strip. An additional observation regarding to the ferrite fraction transformed is that even for very similar cooling patterns, the rate at which the transformation occurs is very sensitive to local small variations once the temperature is Chapter 7. SENSITIVITY ANALYSTS 99 Fig. 36. Effect of the Strip Speed on the Thermal and Microstructural Response of A36 Steel. Chapter 7. SENSITIVITY ANALYSTS 100 140000 Superheat (°C) Fig. 37. Effect of the Strip Speed on the Parallel Flow Zone and Pressure Gradient Zone Heat-Transfer Coefficients for A36 Steel. 7.1.4 Effect of Initial Strip Temperature The influence of the finishing mill exit temperatures on the centerline temperature and phase transformation is assessed by decreasing the initial top surface temperature in Chapter 7. SENSITIVITY ANALYSIS 101 12.8°C, which represents typical variations in full-scale operations (see Fig.20). Results of the runout table model for a top surface temperature of 900 °C appear in Fig. 38. 1000 o o <D 900 CO i_ CD D_ £ 800 CD \— CD C CD 700 H CD O 600 Initial Surface Temperature= 912.8°C Initial Surface Temperature= 900°C Ferrite Pearlite T—i—i—i—|—v i" i i | i—i—i—i—|—i—i—r"T—p T — i — i — r h 0.6 h-1.0 0.8 T3 CD £ CO c co 0.4 0.2 0.0 o CO III CD c. CD o 0 20 40 60 80 100 Distance (m) Fig. 38. Effect of the Initial Surface Temperature on the Thermal and Microstructural Response of A36 Steel. Chapter 7. SENSITIVITY ANALYSIS 102 7.2 Coil cooling parameters Even though the present Coil Cooling Model has not been verified with experimental data, some predictions may be carried out to investigate the possible effect of different variables, in an attempt to study the most suitable experimental procedure to measure some important heat-transfer parameters such as the radial thermal conductivity. Predictions for the DQSK steel are presented since no pearlite transformation occurs, and the heat generation associated is not present in this study. Calculations were performed for a 0.711m I.D. x 1.092m O.D x 1.803m L. coil, cooled in air at 20°C, and the results appear in Fig. 39. Figure 39 shows temperatures at three locations in the coil corresponding to the the interior surface, the middle and the exterior surface of the coil at the center of its length. Temperature gradients are of the order of 50°C in the onset of the cooling process, but they decrease with time until the temperature is virtually homogeneous after 20 hrs. approximately. The coil reaches a temperature below 100 °C after 30 hrs, but room temperature is reached after 100 hrs. Reducing the ambient temperature to 10 °C does not change the cooling curves noticeably, and the time to reach a temperature below 100 °C is approximately 28 hrs, as it is shown in Fig. 40. Therefore, variations in the environmental temperatures are not expected to be important in the cooling problem, and for experimental purposes, not very carefull control of the ambient temperature is necessary for the radial conductivity measurements. On the other hand, variations in the radial thermal conductivity lead to large thermal gradients in the coil, and for the case of a radial thermal conductivity of 5% of that for the steel, thermal gradients of more than 100 °C might be expected, as it is seen in Fig. 41. However, the time to reach a temperature of 100°C is 30 hrs again, suggesting that for the Chapter 7. SENSITIVITY ANALYSTS 103 overall cooling process is not very important, but in terms of thermal gradients and metallurgical problems associates with them may be relevant j—i—i—i—l—i i i i I i i i i i • • i i i • • • • "~j I I I I J I I I I J I "T 1 I | I I I I | I I I I J 0 20 40 60 80 100 Time (hrs) Fig. 39. Thermal Response of a DQSK Steel Coil Cooled in Still Air at 20 °C Chapter 7. SENSITIVITY ANALYSTS 104 Fig. 40. Thermal Response of a DQSK Steel Coil Cooled in Still Air at 10 °C Chapter 7. SENSITIVITY ANALYSTS 105 Fig. 41. Thermal Response of a DQSK Steel Coil Cooled in Still Air at 20 °C with a lower Radial Thermal Conductivity Chapter 8. SUMMARY AND CONCLUSIONS 106 Chapter 8. SUMMARY AND CONCLUSIONS In the present work mathematical models to describe the cooling of a steel strip in the runout table and of the corresponding coil have been presented. Given the difficulties of the experimental procedures involved in the measurement of the heat-transfer coefficients in the runout table cooling, this work presents the mathematical modeling of the heat-transfer mechanisms in jet cooling as the best option available to estimate them. The coil cooling problem has been analyzed, but due to the lack of a general equation for the radial thermal conductivity, this problem cannot be solved properly. Some trends have been obtained with regard to the possible effect of some variables, but the verification of these results was not possible owing to a lack of data. From the results of this work, the following conclusions were obtained: • The runout table model based on the analysis of the mechanisms of boiling proves to accurately predict the thermal problem, and constitutes an important step in the modeling of this process for industrial applications. • The detailed analysis of the transition boiling mechanisms allow an understanding of the effect of important operating parameters on the cooling process, and explains the reasons why previous modeling efforts failed to explain the effect of some variables such as water temperature and strip velocity on the thermal field. The macrolayer evaporation mechanism proposed allows detailed characterization of the effect of the strip surface on the nucleate boiling regime at high temperatures, and the validity of this model does not depend on any specific plant layout Chapter 8. SUMMARY AND CONCLUSIONS 107 • Top jet cooling is responsible for most of the cooling on the runout table not only because of the longer presence of water on the surface but also because of the inherent higher instabihty of the liquid-vapor interface. • The heat generated by the phase transformation is important for the thermal problem and for the kinetics of the phase transformation and cannot be neglected. • The analysis of more sophisticated variables such as the strip roughness, surface conditions and wettability on the cooling performance of water jet cooling can be studied in detail through the application of the mechanism proposed. BIBLIOGRAPHY 108 BIBLIOGRAPHY 1. - M. Korchynsky, "Development of Controlled Cooling Practice", Proceedings of the Accelerated Cooling of Steel Symposium of the Metallurgical Society, Pittsburgh, Pennsylvania, August 19-21,1985. 2. - W. L. Roberts, Hot Rolling of Steel. Marcel Dekker, Inc., 1983, p 472. 3. - F. C. 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LIST OF SYMBOLS 115 LIST OF SYMBOLS A = Area A, = Cross - sectional area of vapor stems A^ = Area of heated surface bt = Avrami parameter for the i - transformation Bi- Biot number c= Wave velocity C= Coefficient C = Free-stream velocity gradient near stagnation line -C = Dimensionless velocity gradient = WjC I Uj Cp = Heat capacity c0 = Wave velocity in absence of currents d = free-surface circular jet diameter dj = Distance to jet centerline da = Ferrite grain size dy = Austenite grain size F = A,_s I Atol - tt_s I tm, Fraction transformed g = Gravity acceleration, heat generation Gr = Grashof number h = Heat - transfer coefficient H= h{uicyy2 Konv = Convection heat transfer coefficient he = Evaporation heat transfer coefficient Hfg = Latent heat of evaporation ht = Liquid film thickness Hn - Nozzle height Ht = Volumetric heat of transformation hv = Vapor film thickness hw = Empirical heat - transfer coefficient for cooling in the runout table K (y) = Function related to flow velocity LIST OF SYMBOLS 116 I=L(y)/u, k = Steel thermal conductivity kf = Thermal conductivity of water L = Characteristic length, Macrolayer length L(y) = Function related to flow velocity on moving impingement surface l,lc = Characteristic length in the y and x directions respectively LB = Vapor bubble length Lc = Length of the coil Ls — Strip thickness m = exponent in the Falkner - Skan power - law rtifp = Triple - point evaporation - rate parameter N = Number of nucleation sites tt; = Avrami equation exponent for the i - transformation Nux = Nusselt number, hconvXj I kf 0()= Order of magnitude P = dimensionless pressure = (P - P„) I (1 / 2)p /«^ P= Static pressure Pe= Peclet number Pr = Prandtl number q = Heat flux r,§,z= Cylindrical coordinates ra,rb= Internal and external coil radius Re^ =ux/v Rex = Reynolds number u-xi /v , uxxlv Rs = Stem radius rt = Thermocouple radius T — Temperature t= Time Tslart = Temperature at 5% transformed tv = Virtual time u= velocity U = M / Uj = Water velocity at the entrance of the microlayer LIST OF SYMBOLS 117 Uj = Velocity of the impinging jet Ut = Bulk flow liquid horizontal velocity us = Strip velocity U s = U s / U j Uv = Bulk flow vapor horizontal velocity u„ = x component of free - stream velocity = Velocity ratio um I Uj v = u/ llj Wj = Width of the impinging jet x = Distance from the jet centerline along the length of the strip x = Dimensionless position = Xj I Wj Xj = streamwise position measured from the stagnation plane xi = Contact time ratio, ti I tlot x, = Value for x where P = 0 y= normal coordinate y=y/wj z = Coordinate in the direction of the width of the strip a = Thermal diffusivity a , = Thermocouple thermal diffusivity P = 2 / K times the impingement angle with respect to x, volumetric thermal expansion coefficient X ~ Constant 8 = Momentum bounday - layer thickness 5 c = Macrolayer thickness AH^ = Sensible enthalpy at actual temperature Ar= radial distance At = Time step AT; = Wall superheat, Ts - Tsat ATsub = W a t e r subcooling, Tsat - Tt e = Emissivity r l =KCRe 7 . ) 1 / 2 <() = Impingement semiangle X = Vapor - liquid interface wavelength v = water kinematic viscosity LIST OF SYMBOLS 118 v„ = y component of free - stream velocity p = Density p, = Water density a = Surface tension a s = Stefan - Boltzmann constant x = Thermocouple time constant Subscripts and superscripts: a,b = Values at r = ra and r = rb, respectively air = Enviromental medium BA = Batch annealing CC= Coil Cooling CHF- critical heat flux cond= Conduction conv = convective eq - Equilibrium / = fluid, water FB= Film boiling H = Helmholtz instability j= Impinging jet /= liquid l-s= Liquid - solid contact Lc = Value at z = Lc Ls = Value at y = Ls MHF= Minimum heat flux NB = FNB - nucleate boiling n- Value at the nozzle p = plate, strip r,<|),z= Directions rad = Radial direction ref = Reference value sat= saturation LIST OF SYMBOLS 119 s= surface, steel TB= Transition boiling tp= Triple-point tot = statistical suficient quantity at a point on a surface v = vapor v - s = Vapor - solid contact x,y,z = Coordinates 0 0 = free stream + = Dimensionless variable 0 = Initial value, value at y = 0, value at z = 0 APPENDIX 120 APPENDIX Chemical Composition of the A36 Steel %c %Mn %P %S %Si %Cu %Ni %Cr %A1 %N 0.17 0.74 0.009 0.008 0.012 0.016 0.010 0.019 0.040 .0047 Thermophysical Properties of the A36 Steel (SI Units)(*) Thermal Conductivity: ky =15.829 + 1.1566x10~2T ka =65.422-5.2176jcl0_2r+9.7673x10^ T 2 kper = 50.742 - 3.0567JCIO-2 T+1.1539xl0-7 T2 Kteel = F*K + + (1 - Fa - Fper)ky where T(°Q Density: psUel = 7882.97-3.5x10^7 APPENDIX 121 Specific Heat: Cpy = 758.69-0.28727+1.772jd0_472 Cpa = -10034.5 + 5.96687, + 5.2002JC109 7f 2; (7 > 787) Cpa =34754.5-31.91967;; (769<7<787) Cpa = -11462.6 + 12.43467,; (727 < 7 < 769) Cpa = -4704.5 + 4.5687/, +1.10577;cl09 TT2; (527 < 7 < 727) Cpa = 503.13-0.130687. -5.1702xl067,-2 + 4.4712x10^ 7,2; (7 < 527) where 7, (AT) 0^=449 .5 - 0.45017 Cpsteel = FaCpa + FpeTCPper + (l-Fa- F^Cp^ Enthalpies of Transformation: Hy^ = 221656.4-864.47+1.979572 - 0.00147873; (7<720) Hy^ = -2.917;ri07 + l 145907-148.872 + 0.0639973; (720<7<780) Hy^ = 3277373-105757+11.54572 - 0.0042473; (7 > 780) H^per =70651 +225.237-0.346972 + 6.755JC10-573 (*) Medina F., MEng. Essay, Thermal and Microstructural Evolution of a Hot Strip on a Runout Table, UBC, September, 1992, pp.70-75. APPENDIX 122 Initial Conditions for the Runout Table Model The initial conditions to solve Eq. [5.1.6] (T0(y)) are obtained from regression analysis of the predictions of a Finishing Rolling Mill model developed at UBC (82), and are given by: Q = r 0(y)-r,(0) r 0 ( V 2 ) - r 0 ( 0 ) 6 r e / = 1.5988/ -0.5988(/) 2 AO — = -0.01869/ +0.01869(y*)2 Ay / = JL ; Ay = y-yref; y r e / = 5mm T0 (Lc 12) = 0.998970 (0) + 1700.0LC where Lc and70(0) are the measured thickness and initial top surface temperature respectively. 


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