COMPUTER SIMULATION OF THE PUSH-TYPE SLAB REHEATING FURNACE by ZONGYU LI B.Sc. (Eng), Northeast Institute of Technology, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metallurgical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITSH COLUMBIA DECEMBER 1986 Zongyu Li, 1986 In presenting degree at this the thesis in University of partial fulfilment of British freely available for reference and copying of department publication this or of thesis by this for his thesis study. scholarly or her Columbia, of purposes l The University of British 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6(3/81). J*» . r/j f / Columbia H, , I agree may representatives. for financial gain hrtifa //i<rj \cJ requirements that the I further agree be It shall not permission. Department the & *J j is an advanced Library shall make that permission for granted head by understood be for the that allowed without it extensive of my copying or my written ii ABSTRACT A mathematical heat-transfer model for the slab reheating furnace has been developed. Radiation in the furnace chamber was calculated using the zone method, with the gas temperature distribution being determined using a finite-difference assumed, and heat transfer in the slab was approximation of two-dimensional transient conduction. These individual calculations were coupled to allow prediction of the temperature profiles in, and heat flux to, refractory walls and slabs at any point inside the furnace. The emissive/absorptive characteristics of the gas mixture within the furnace chamber were simulated with a clear-plus-two-gray-gas model which simulated the real gas behaviour to within 5%. For the calculation of radiative exchanges, the furnace chamber was subdivided into 432 isothermal zones, and radiative exchange factors to slab surfaces were evaluated rather than relying on empirical or experimental estimations as in previous studies. An iterative technique was devised in order to combine the radiative and slab heat conduction calculations. For the purpose of identifying the mechanism of skidmark formation, the region of skidrail/slab contact was examined in detail by introducing a radiation shielding factor to account for the presence of the skid structure. The gas temperature distribution inside the furnace chamber was found to have a significant influence on the heat flux to the slab surface. Nonuniform gas temperature transverse to the push direction causes an uneven transverse slab temperature distribution and subsequent rolling problems. Higher gas temperatures near the sidewall refractory were iii shown to cause serious distortion of the transverse heat-flux distribution. The heating practice for the hot charging of slabs was simulated by the model in order to improve the process from the standpoint of energy conservation and slab temperature uniformity. Model predictions have shown that the fuel input could be reduced substantially near the slab entrance where the port to the chimney is located, thus maximizing the residence time of the combustion products. Alternatively the throughput of the furnace can be increased if the fuel input remains the same as for charging cold slabs. The extent of increase in production rate can be determined by the off-line computer model. The model was used to predict the thermal behaviour of slabs for various thicknesses, steel grades and push rates. The results consistently indicated that the selection of an appropriate push rate is crucial to the final temperature distribution. The study of the mechanism of skidmark formation showed' that the radiation shielding effect of the skidrail was the dominant factor, accounting for 90% of the heating deficit around the slab/skidrail contact region. Computer simulation of the possible measures that could be taken to alleviate the skidmark height Coating and width of the skidrail improved highly reflective materials on the formation has indicated that reducing the radiative heat transfer in the contact region. exterior surface of the skidrail to increase reflectivity from 0.3 to 0.8, could enhance heat transfer locally around the the skidrail by about 25% - 30% when the skidrail temperature is lower than the slab bottom temperature. iv TABLE OF CONTENTS Page Abstract ii Table of Contents iv List of Tables vii List of Figures viii Nomenclature xii Acknowledgement xvi Chapter 1. Introduction 1 2. Literature Review 8 2.1 General Modelling of a Reheating Furnace 8 2.1.1 Slab Temperature as an Input Parameter 9 2.1.2 Global Furnace Temperature as an Input Parameter 10 2.1.3 Application of the Zone Method to the Reheating Furnace 17 2.2 Hot Charging of Slabs 20 2.3 Previous Studies on the Formation of Skidmarks 21 V 3. Objective of the Present Work 26 4. Real Gas Treatment For the Zone Method 28 4.1 Method of Treatment of Real Gas Emissivity and Absorptivity 28 4.2 Application of the Method to the Real Gas in the Reheating Furnace Chamber 5. 30 Methodology 39 5.1 Zone Method for the Radiation inside a Furnace Chamber 40 5.1.1 Direct Exchange Coefficient 40 5.1.2 Total Exchange Coefficient 43 5.1.3 Energy Balance for a Surface Zone 46 5.2 Zoning Technique in the Reheating Furnace 5.3 Direct Exchange Coefficient and Total Exchange ; 48 Coefficient Calculation 52 5.3.1 Calculation of Direct Exchange Coefficient 55 5.3.2 Calculation of Total Exchange Coefficient 59 5.3.3 Energy Balance on the Slab Surface and Refractory Wall 61 5.4 The Slab Conduction Model 66 5.4.1 Heat Conduction inside the Slab 73 5.4.2 Coupling of the Zone Method with the Heat Conduction Model 75 5.5 Sensitivity Analysis of the Nodal Division in the Conduction Model 6. Results and Discussion .. 78 83 c vi 6.1 General Thermal Behaviour of Reheating Furnace 6.1.1 Slab Temperature and Refractory Wall 83 Temperature Response 83 6.1.2 Characteristics of Heat-Flux Distribution 87 6.1.3 The Effect of Push Rate on the Slab Temperature Distribution 91 6.1.4 The Effect of Slab Dimension on the Slab Temperature Distribution 95 6.1.5 The Effect of Steel Grades on the Slab Heating Process 95 6.1.6 Off-line Computer Control Model 97 6.1.7 Influence of Gas Temperature on the Slab Heat Flux and Temperature Profiles 7. > 101 6.2 Skidmark Effect 107 6.3 Hot Charging Practice 125 Conclusions 134 References 137 APPENDIX I: Computer FLow Chart for the Real Gas Treatment 142 APPENDIX II: Calculation of View Factor From Furnace Chamber to the Slab Bottom Surface 144 APPENDIX III: Flow Chart of the Code "ENERGY" 145 v/7 LIST OF TABLES Page 4.1 Typical Composition of the 4.2 The Change of Coke Oven Gas Composition of and CO2 31 with H2O the Mixing Ratio 33 4.3 Coefficient 4.4 Linear 5.1 Summary 6.1 Computer of A^ and B^ for 1 j Correlation for the Weighting 37 Coefficient a ^- 37 g of S u b - C h a m b e r Zoning Prediction of the Shielding 55 Effect of Skidrail as a Function of its Width W 6.2 Computer Prediction of Function of its Height H 113 the Shielding Effect of Skidrail as a 114 viii LIST OF FIGURES Page 1.1. Basic heat transfer paths and processes in the reheating furnace 2 1.2. Heat-transfer enviroment inside the reheating furnace 4 1.3. The relationship between the reheating furnace and the other relevant processes 2.1. Temperature response 5 and heat-flux distribution inside the reheating furnace (from Hollander) 11 2.2. An experimental approach to evaluate 4>CQ (from Yoshisuka) 2.3. A nodal system of a furnace zone (from Veslocki and Smith) 19 2.4. Common defects in the continuously cast slabs. 22 2.5. Temperature distribution in the skidmark region (from Roth et al) 24 4.1. Change of r j 35 4.2. Linear interpolation of a^- 4.3. Comparison between the real gas emissivity and the predicted g with temperature 15 36 emissivity 38 5.1. Geometry for radiative exchange between two surfaces 41 5.2. Radiative energy balance for a surface zone 44 ix 5.3. Reheating furnace showing division into sub-chambers (Schematic) 49 5.4. Typical sub-chamber showing the fictitious surface 51 5.5(a). Zone numbering sequence for the slab surface 53 5.5(b). Zone numbering sequence for the furnace side wall 53 5.5(c). Zone numbering sequence for the gas zone 54 5.5(d). Zone numbering sequence for the furnace roof 54 5.6. Subdivision of a surface zone pair 57 5.7. Non-uniform subdivision of gas zones used for the calculation of g l . 58 Ji 5.8. Structure of furnace refractory walls 63 5.9. Heat transfer for the slab in the reheating furnace 64 5.10. Contact region between the skidrail and the slab (detail) 68 5.11. Radiative network among skidpipe, slab surface and the fictitious surface 5.12. Radiative network 70 for the slab end surface and the furnace side wall 5.13. Nodal 74 division of the slab (transverse to the furnace longitudinal direction) 5.14. Sensitivity of internal 76 slab temperature to the node spacing in the thickness direction 5.15. 79 Sensitivity of slab surface temperature to the node spacing in the longitudinal direction 80 5.16. Isotherm contours with respect to three nodal divisions 81 6.1. Predicted slab reheating furnace temperature profiles longitudinally in the 84 X 6.2. Predicted slab temperature contours at three longitudinal positions 86 6.3. Predicted refractory wall temperatures 88 6.4. Predicted heat flux to top slab surface as a function of axial position in reheating furnace 6.5. 89 Prediction of heat flux to slab across width of the reheating furnace 6.6. 90 A comparison of direct radiation to the slab surface from the chamber gas and refractory 6.7. Comparision of the 92 slab centreline temperature for two slab push rates 93 6.8. Slab temperature contours at furnace exit for two push rates 6.9. The effect of slab thickness on the slab temperature 6.10. Thermal diffusivities of two steels 98 6.11. Predicted longitudinal slab temperature profiles for two steels 99 6.12. Predicted longitudinal slab surface heat flux 94 at furnace exit profiles for two gas temperature profiles 6.13(a). Effect of non-uniform 102 transverse gas temperature on the transverse irradiation flux to the slab surface 6.13(b). Effect of non-uniform transverse gas temperature 103 on the transverse irradiation flux to the slab surface 6.14. Effect of temperature 6.15. Irradiation non-uniform transverse gas temperature 104 on slab contours at the furnace exit flux position=24.2 m) 96 across the slab bottom 106 surface (furnace axial 109 xi 6.16. Longitudinal slab bottom temperature distribution at the furnace exit 6.17. Comparison between the conductive heat loss and the radiative heat loss at the skid/slab contact region 6.18. Comparison of the skidmark Ill and centreline temperature of a slab 6.19. ;. Radiative network Effect of reflectivity 118 of skidrail on the net heat flux to the bottom surface 6.21. 120 Effect of view factor from furnace chamber to skidmark area around the skidmark region 6.22. A proposed design of 121 the skidrail system to reduce the skidmark 123 6.23. Possible alternative 6.24. Predicted slab rail design to reduce skidmark formation top surface temperature response for hot 124 and cold charging 6.25. 127 Predicted top and centreline temperature response for hot and cold charging 6.26. 129 Comparison of longitudinal heat flux to the slab surface for cold and hot charging 6.27. 130 Predicted slab temperature contours at the furnace exit for hot and cold charging 6.28. 116 among the slab bottom surface, the skidrail and the furnace bottom chamber 6.20. 110 Comparison of two 131 predicted furnace exit for two push rates slab temperature contours at 132 NOMENCLATURE A area of an element a thermal diffusivity (m ) 2 (mVs) gas absorptivity weighting coefficient C shielding factor of the skidrail. CR contact resistance (J/(m K) S specific heat ( J / k g ° C ) D diameter of the skidpipe (m). d width of the skidrail (m). E radiative emission from a blackbody (W/m ) e natural logarithm 2.7183 e net radiative exchange coefficient F view factor G.G. i J total exchange coefficient between two gas zones (m ). G.S. i J total exchange coefficient between a gas zone i and a surface zone j (m ). US- directed exchange coefficient for a gas zone i to a a surface zone j (m ) 2 2 2 » j ¥ j direct exchange coefficient for a gas zone i to a surface zone j H total incident heat flux to a surface(W/m ) h height of the skidrail (m) h convective heat transfer coefficient (W/m K) k extinction coefficient of a gray gas (kPa.ni)" L path length M number of node divisions in the slab thickness direction. ND number of node divisions in the slab transverse direction. N number of zones N number of gray gas components 2 2 1 (m) (m ). 2 2 2 xiii n unit normal vector P Pressure (Pa) p partial pressure (Pa) Pr Prandtl number Q quantity of heat flow q heat flux (W/m ) R thermal resistance (KmVW) Re Reynolds number r . gas emissivity weighting coefficient S.(jj directed exchange coefficient between surface zone i and gas zone j s.Sj direct exchange coefficient between surface i and j S.S. total exchange coefficient between surface i and j g»i ( j>/a ) (W) 2 ( UD/v ) (m ) J (m ). J (m ). directed exchange coefficient between surface i and j 2 (m ). 2 s distance between two skidrails (m) T temperature ( ° C ) t time (s) U over-all heat transfer coefficient^W/m K.) V volume of a gas element W total leaving radiative flux from a surface (W/m ) x fraction of the coke oven gas in the fuel mixture for the reheating furnace x coordinate for the slab width direction (m) y coordinate for the slab thickness direction (m) 2 (m ) 3 2 Subscript b bottom furnace chamber c centre of an element xiv CG subscript pertaining to the experimental approach to determine en subscript pertaining to enviroment F furnace chamber g gas in the furnace chamber H slab bottom surface i subscript pertaining to an element i j subscript pertaining to an element j n subscript pertaining to gray gas component r refractory side wall s slab surface t top half of furnace chamber to subscript pertaining to the total amount T the exterior surface of the skidrail w subscript pertaining to the refractory wall wa subscript pertaining to the cooling water 0 Greek Alphabet a absorptivity. B relative dimension of slab/skidrail contact zone 6 thickness of the slab (m) 6 relative error. v kinematic viscosity (mVs) c emissivity. p reflectivity of a surface. 8 angle between the unit normal vector of an element and a line connecting it to another spatial element (degree) thermal conductivity (W/mK) effective radiative exchange coefficients, transmissivity. XV/ ACKNOWLEDGEMENT I would like to express my sincere gratitude to Drs. J.K. Brimacombe and P.V. Barr for their guidance and understanding throughout the course of this study. I am also grateful to the National Science and Engineering Research Council and the People's Republic of China for providing financial support for this research. It from my would be a mistake not to mention the friendship and encouragement I received friends, fellow students, the Faculty and technical staff. Their enthusiasm will become the memory I cherish in my future. Last but not least, I would like to express my appreciation to Ms. Savithri for her patience figures. in typing my thesis and Hemaguptha's excellent engineering drawing of most INTRODUCTION 1. Despite furnace considerable efforts essential to the is still presence of defects from the to direct contemporary roll continuously cast iron and steel plant continuous casting process, which slabs, Due must the reheating primarily be to the removed before subsequent processing, no hot strip mill in the world has yet achieved one hundred percent direct hot rolling and thus eliminated the reheating process. At present, the detection and removal of these defects require that the slab be cooled, thus creating a major barrier to n the achievement of direct rolling. Currently, two approaches toward eliminating this obstacle are being pursued. One is to develop mechanical automatic systems to enable detection and removal of defects while the while the second method defect formation relies upon mathematical and thus identify inspection and slab is in the models to define the conditioning hot state^, mechanisms of corrective measures. Both methods will take time to be fully successful and will involve large capital expenditure. Therefore, the reheating furnace is likely to remain an indispensible part of the steel hot working process for decades to come. A typical three-zone push-type reheating mechanically pushed through the reheating furnace is shown in Fig 1.1. Slabs are furnace and are elevated to a temperature level suitable for rolling. Skidpipes are constructed at the centreline of the furnace to support the slabs during the heating process. The furnace chamber is heated by burners located in various positions within the furnace, thus forming a high temperature field in the. enclosure. Heat transfer convection within the reheating and conduction, furnace chamber is very complicated, involving radiation, with radiation being the dominant mode in the operating (2) temperature CO2 range and H2O of the furnace v . Since all reheating furnaces burn hydrocarbon fuels, in the products of combustion are the primary sources of gas radiation. The basic paths of heat transfer inside the reheating furnace are shown in Fig. 1.1 arid the FURNACE LONGITUDINAL DIRECTION Primary zone Heating zone Soaking zone TANGENTIAL BURNER ROOF BURNERS ENTRANCE SOLID HEARTH SKIDRAIL TANGENTIAL BURNER TANGENTIAL BURNER Note: 1 Gas to Slab Surface (Radiation + Convection) 2 Refractory Wall to Slab Surface (Radiation) 3 Gas to Refractory Wall (Radiation + Convection) 4 Refractory Wall to Enviroment (Convection + Radiation ) 5 Slab to Refractory ( Radiation ) Fig. 1.1 Basic heat transfer paths and processes in the reheating furnace. 3 heating enviroment for a slab being heated inside the furnace is shown in Fig. 1.2. Primary radiative transfer occurs from furnace gases to slab surfaces and to the bounding refractory wall while secondary exchanges occur between the slab and refractory wall surfaces as well as between areas on the refractory. Convective heat transfer occurs from furnace gas to the refractory wall and furnace gas to the slab surfaces. Heat transferred to the slab surface is then conducted depressions to on the the interior slab of bottom the slab. surface due The to water-cooled conduction skids between cause the temperature slab and the contacting skids and radiative shadowing by the skids resulting in visible skidmarks. The operation of the processes in an continuous integrated casting the reheating rolling machine or furnace is closely linked to that of the mill. For delays in instance the the rolling hot charging of schedule influence on the reheating process . Fig. 1.3 shows the reheating other processes in a typical hot strip mill where the exert slabs from a very other the significant furnace in relationship to slabs may originate either from a slabbing mill or a continuous caster. The latter is providing a progressively larger proportion of slabs foT hot rolling. The reheating process received little attention only a few papers until the early 1970s, prior to which were published on the mechanism of reheating slabs in a furnace chamber. The reheating furnace was regarded as an essential, but not critical, process in the steel hot working plant As long as the furnace could provide sufficiently hot slabs so as to meet the rolling schedule, the furnace operation was considered acceptable. The operation of reheating experience furnaces of was operators. mainly With increasing cost of energy, the based on progressively reheating furnace more heat stringent balances quality furnace was identified combined with the requirements and the as a key process affecting product quality and fuel consumption in the hot strip mill. Since most mechanical properties of the control slabs are in the strongly roughing dependent upon operation or temperature, thermal stress rolling problems, such as gauge build-up due to non-uniform SIDE BURNER REFRACTORY WALL SYMMETRICAL PLANE OF THE FURNACE RADIATION+CONVECTION WEAR BAR CONDUCTION LOSS SKIDRAIL RADIATION +CONVECTION Fig. 1.2 Heat transfer enviroment inside the reheating furnace CONTINUOUS SLAB CASTER REHEATING FURNACE */////> SLABBING MILL ROUGH ROLLING Fig. 1.3 The >////?£> relationship processes. FINISH ROLLING between the reheating 7ZZ$> furnaces and PRODUCT the other relevant 6 temperature hence distribution inside the slab are related to the thermal to the reheating furnace. The reheating furnace is history of the slab, and also the non-electrical energy in the hot strip mill. Consequently, attention reheating operation and various attempts were made (^^"^ major consumer of began to focus on the to better understand the mechanism of reheating. Numerous mathematical models have been developed since the early 1970s, and some plants have adopted these models to monitor and control the furnace (2),(9-lo0 However, these models tend to oversimplify operation of their the reheating complex heat exchange processes inside the furnace chamber, particularly, gas radiation to the slab surface and heat transfer in the radiation contact region between to the slab surface was coefficients and good agreement adjusting effective temperature coefficients. the skidrail and the often represented by slab. Gas and refractory ambiguous 'effective' with the measured slab temperature Most of these measurements and a detailed picture of heat transfer heat-transfer was achieved only by referred inside the wall to the average slab furnace chamber other than conduction in the slab remained unclear. Many questions, such as the influence of the gas temperature process distribution could temperature not transverse to the be answered such simplistic mathematical on the models. slab heating Uneven distribution, induced by fuel firing conditions and burner arrangement, depression of slab temperature quality by direction of slab movement adjacent to the skids are thought to be detrimental gas and the to the of the rolled product Therefore, the necessity to develop more sophisticated models becomes apparent The present study attempts to fill the gaps in previous works by developing a more sophisticated mathematical model. The model utilizes the geometry and operational parameters from the Stelco Lake Erie Works reheating furnace. In order to cope with the complexity of the radiative exchange in the furnace chamber filled with an absorptive-emissive gas, the zone method (^)'^O) w a s employed to calculate the radiative exchanges among gas, 7 refractory wall surfaces two-dimensional transient slab. The model and slabs. The results heat-conduction model significantly increases predictive to provide predict boundary temperature capacities, facilitates conditions profiles for a within the and analysis of the process and provides guidance for design of the reheating furnace. The inputs to the current model are the production rate and the gas temperature profile inside the chamber. In order to proceed beyond this point, gas flow patterns would have to be obtained, which is beyond the scope of this project. However the model could eventually be extended so as to require only the input of two operational parameters: firing rate and the slab push rate, both of which are readily available. fuel 8 2. L I T E R A T U R E R E V I E W The purpose of the reheating furnace is to supply slabs which are adequately heated for deformation in a rolling mill. Important operational objectives are the maximization of the tonnage through the furnace/mill system and efficiency of heating. Yet these objectives are difficult to achieve because of the many operating variables involved(/.e., fuel firing rate, slab push rate, furnace chamber, and their gas temperature etc.). These relationships cannot be known factors until interact inside the furnace the mechanism of heating is well understood. Therefore, the need to develop a heat-transfer model becomes obvious. As was mentioned published in Chapter 1, numerous papers have been on the mathematical modelling of reheating furnace mostly in the last fifteen years. Owing to progressively more stringent requirements for slab quality, a number of studies also have appeared on the skidmark phenomenon (21-26) 2.1 General Modelling A furnace model basic objective of a Reheating Furnace in the development of is to better understand the temperature can computer be control adopted , subsequently energy for saving, and any mathematical on. purposes Despite different purposes, existing models can be classified into input parameters required. reheating response of a slab being heated. Such a various so model of the such as diversified on-line forms monitoring, developed for three categories according to the 9 2.1.1 Slab Surface Temperature as an Input Parameter Avoiding the Hollander complicated heat and Fitzgerald ^ exchanges occuring within the furnace enclosure, proceeded directly to the slab conduction problem by assuming the relationship between surface temperature and time of a slab being heated in a furnace. Neglecting edge effects in the slab, the problem was reduced to that of semi-infinite conduction in one dimension which is governed by the following equation^ Ji= <-> 0 J2 The initial condition depended upon the preceding slab process and was taken to be T(y,0) = T (y) q (2.3) The boundary conditions conditions before the solid hearth were assumed to be : (T) = y = Q fjCt), (T) = y = g f (t) while at the solid hearth (refer to Fig 1.1), (-oTWo where y denotes the represent explicit the = the adiabatic condition was imposed 0 ( 1 5 ) coordinate in the temperature-time finite-difference (2.4) 2 slab thickness direction and while relationship method was for utilized both ^® to top and obtain bottom f-^(t) and slab solutions for f2(t) surfaces. The Eq. (2.1). The discretization steps, Ay and At, were restricted by the stability criterion A. y)V (a At)> 0.5 (27).(28) Because fj(t) dependent upon the and f2(t) situation were difficult to express in analytical form and were entirely to be simulated, Hollander argued that they could be 10 determined iteratively depending on furnace length, throughput and the required total heat content of the slab at the exit. Once f^(t) and f2<t) had been defined, the temperature distribution through the slab thickness was obtained using the finite-difference approximation. The net heat flux to the slab surface at each axial point was calculated as follows T W e where X " < J7 >y=0 = X s < - "V^ X T 1 } ( is the thermal conductivity of the slab, T. is the surface temperature the temperature estimate 1 and T 6 ) ^ is of the node adjacent to the slab surface. An empirical model was used to the required fuel firing rate and the fuel distribution at the different positions inside the furnace. Typical model results are shown in Fig 2 . 1 ^ These features relatively simple, semi-empirical of the slab temperature models have predicted response, such as the centreline the fundamental temperature. The method was also applied to on-line computer control of a reheating f u r n a c e ^ ' ^ * ^ ^ . Since these models relied upon knowledge of the slab surface temperature response and ignored the heat transfer inside the furnace chamber, it was impossible to link the slab temperature field distribution. In which to the operational addition, are believed edge effects to be mainly parameters, such as the gas temperature and the transverset responsible for many slab temperature rolling problems and its distribution (25),(29) > w e , r neglected. 2.1.2 Global Furnace Temperature as an Input Parameter t Transverse refers to the direction in the plane of the slab normal to the motion of the slabs in the furnace. Since the slabs move through the furnace sideways, the transverse direction in the furnace is actually the rolling direction after the slabs are discharged. e 11 Fig. 2.1 Temperature response and furnace (from Hollander) heat flux distribution inside the reheating 12 This is the most commonly encountered reheating furnace model in the literature. has been widely used for on-line monitoring of slab temperature control of the furnace Rather than ( 5 ) , ( 3 1 ) , ( 3 2 ) requiring ^ 0 p u r m z a u the surface ( H ) « ( 1 2 ) , ( 1 4 ) , ( 3 0 ^ o n of furnace performance temperature of the slab c o m p It U t e r (5),(9)_ to be specified, heat transfer from furnace gases and refractory walls was utilized for the boundary conditions at the top and bottom slab regardless of whether surfaces. All the radiative heat transfer to the slab surface, from the furnace gases or refractory walls, was expressed in a single parameter , which is determined by furnace geometry and operational variables. C o l l i n ^ ^ introduced a configuration factory and calculated the heat transfer to the slab surface as follows q st = 4> . o( T\ st gt q, sb = <p , o( T I - M M where T' ) s v sb T s gb + ' 4 ') h.( T gt t + vh ( - T , gb T ) s ' - T ) s ' the top and and bottom slab surface. <f> and 0 ^ ' were ( 2 . 8 ) ' v the subscripts s and g represent slab surface and gases respectively, while represent ( 2 . 7 ) v assumed t and b to vary longitudinally but to be constant at each axial position in the furnace. The methods used to estimate <t> were not reported by Collin^ ), but reportedly the results given in Hottel and 30 MacAdams^) the were applied. Other gas-to-slab surface radiative papers^) (^ (^ , , also cited Hottel and McAdams for exchange. According to them, the net heat flux from a gray gas at T^ to a gray heat sink at T^ is % - 1 = ^ T g " T l) ( Z 9 ) 13 where <p is <t> = —T — + e g *T -j 7=;— ~ C e. 1 (2.10) 1 and A^. C A^ A = = 1 A 1 I A + A (2.11) R (2.12) R is the area of the sink( in a reheating furnace it refers to the slab surface); A R is the area of the refractory surface and C is the ratio of the area of the heat sink to the area of refractory surface, and e are the emissivities of the sink and the gas respectively. Although Eq. (2.10) provides a simple formulation, the net heat flux is based on very restrictive assumptions: CO The gas and flame in the furnace chamber can be assigned a single mean temperature T . (u) The gas is gray. (iii) The surface of the heat sink is gray and can be assigned a single temperature. (iv) External losses through the furnace walls are negligible and internal convection to refractory walls of area A R is negligible. (v) The disposition of sink surface and refractory wall is such that from any point on the walls, the view-factor to sink surface is the same as from any other point This is referred to as the 'speckled' wall condition. 14 Yoshisuke ' v and Ishida defined the radiative heat transfer inside the furnace chamber with an equation similar to E q . (2.8) q = 0 C a G where $ ^ was determined temperature of into a slab ( Tj, - experimentally passing through and <t> QQ was determined (2.13) using a heat-resistant the reheating the slab surface was calculated from surface ) TJ from was proposed originate by from (or emitting has been shown the significantly transverse F w g sources: to radiation heat from T h e net heat = h A c + a length of with axial surface s s ( of the transfer T g A s the less - v T a( g flux + T ) ' e g 4 reheating surface. A 5% of a A - a furnace, position. Therefore slab across the slab experiment to the of slab surface wall, radiation the furnace the total was considered from the to flames gas. However, convection heat at typical furnace rate to the slab was expressed by: ) geometrical transfer exchanges inside the furnace chamber the refractory than S and gas 'zones' gradient represented in the the radiative The contribute was based on purely absorptivity ^ \ to record the E q . (2.13). The values of 4>QQ thus calculated were combustion products), and convection from q to Fitzgerald three temperatures^). Due approach to characterize logger (Fig. 2.2). T h e heat the measured temperature only valid for the particular conditions and geometry Another furnace data similar furnace chamber was adopted by F o r d ^ ) . F ( T WS ) w 4 A e ( T )« s s s ' v the v gas and surface the furnace uniform view W will vary was considered to consist of several temperature (not factors (in the clear treatment temperatures (2.14) ' of the equal). The evaluation medium) radiative while a exchange g of was the inside the After removal a n d Instrument Container recovery, It i s sent (Data collecting Capsule + to the data recorder Insulation device) Fig. 2.2 An experimental approach to evaluate* C G (from Yoshisuka) 16 Models average in temperature have adequately this class have provided predictions that agree of the fulfilled slab. Because of their tasks such as temperature simplicity well with the and flexibility, measured these models monitoring, or computer control of the furnace. In these applications, only the prediction of the rising trend of slab temperature is required. However, the absorptive-emissive gases within the reheating quite different furnace chamber exhibit behaviour than that of the assumed gray gases. An emissive-absorptive real gas has discontinuous bands of emission and absorption while the behaviour of a gray gas is just the opposite. Moreover, the gas temperatures within the furnace vary dimensions while for the models it has been assumed that the gas temperature in three changes only in the longitudinal direction of the furnace. The most difficult problem with these models has been the effective heat-transfer coefficient, which is a function of many furnace parameters , such as the geometry of the furnace, gas temperature this key parameter and, worse, it determined operational distribution, slab dimensions and so on. A theoretical to different is not feasible experimentally basis to link operating conditions of the furnace has not been developed to (5),(15) characterize ^ all o n conditions under which the j the y experiments f situations ^ or empirically. pa^ui^ Values furnace and of 4> the were conducted. Owing to complicating factors such as the skidrails in contact with the bottom surface of the slab or the influence of the furnace side wall, the transverse heat-flux particularly when side burners are installed distribution is likely to be quite uneven, (29) -rh e transverse temperature distribution (down the length of the slab) has been shown to be critical to the final product quality. (29). One-dimensional heat-transfer models (the variables depend only on the dimension of the furnace) are incapable of addressing these problems. longitudinal 17 Ambiguity also exists for the definition of the furnace temperature the radiative temperature exchanges inside the furnace chamber. The value balanced among gas, refractory gas temperature these temperature or it could be the positions on the roof of the sensors were subjected to radiation and slabs. The temperatures to the were normally obtained from the readings of thermocouples or radiative pyrometers mounted in different furnace; chosen might refer wall and slab temperatures alone. The 'furnace temperatures' used to evaluate from gases, refractory thus measured represent a balance among refractory wall wall, gas and slabs. 2.1.3 Application of the Zone Method to the Reheating Furnace Hottel et al published a series of pioneering works (19),(20),(35),(36) o n ^ development of the zone method for the computer modelling of the heat transfer inside an enclosure. The zone method allows the presence of an emissive-absorptive medium in the enclosure and a temperature gradient in the medium. However, owing to the relatively slow calculation speed of the computers at that time, the new method until and Sarofim (35) again summarized the the late 1960s and the early application of the zone method while Pieri (37) extended flames into Hottel to the calculation of radiation the zone medium. Johnson and B e e r ^ ) 1970s. method to allow developed a mathematical the zone method. These studies have from was not very popular non-luminous flames for concentration gradients in the model for incorporating luminous demonstrated that the zone method is mathematically a reliable technique in spite of its complexity. Patankar^) critically summarized and compared the different methodologies for the modelling of furnaces. He reviewed different stages in the development of the mathematical modelling of furnaces and concluded that before the 1970s, a furnace was normally treated as a uniform-temperature enclosure (so-called zero-dimensional model); then one-dimensional analysis was employed for long furnaces of modest width. Recent developments in computers 18 and numerical methods have made it possible to make two- or three-dimensional analyses of furnaces, with steadily increasing realism and refinement Modelling of reheating furnaces has, to the author's knowledge, remained at the stage of one-dimensional analysis of radiative exchanges within the furnace chamber. Despite the demonstrated advantages of the zone method, remarkably few studies using this technique for the reheating furnace are available in the literature. The model developed by Veslocki and Smith reheating equivalent has been the most thorough application of the zone method to the furnace. The to furnace chamber " zones") but the was discretized into a series of nodes (they are slab was considered to be only one dimensional in the thickness direction, thus posing a one-dimensional conduction problem. Fig. 2.3 shows the nodal from system balances. For in a given example, the furnace top zone. surface Node of temperatures each slab was were calculated assumed to receive heat heat by radiation from the roof surface nodes , from nodes in the circulating gas, and from nodes in the flame. In addition, the flowing directly over it top slab surface receives heat by convection from the gas The heat received from the furnace chamber was transferred to the interior of the slab by conduction. The main results from this model are : CO- The longitudinal gas temperature profile has a significant effect on required furnace energy input, and on the temperature (ii). In gradients in the slab. response to a delay, the firing strategy fuel firing rate during the delay, but to should be not only to reduce continue to use reduced firing rates following the delay. (iii). A slab temperature developed from the model. estimator suitable for on-line computer control was 19 FIVE ROOF NODES EIGHT TOP FLAME NODES TWENTY SLAB NODES I 1 i I EIGHT BOTTOM FLAME NODES FIVE NODES Fig. 2.3 A nodal system of a furnace zone (from Veslocki and Smith) FLOOR 20 However, quantitative reheating furnace application of details were the of not model radiative given, nor would be exchanges between was the limited by detail the of each component gas and failure to flame account within radiation. for the the The furnace sidewalls and the skid system. Because the effects of refractory side wall and skidrails were not considered in the model, the heat flux variation in the furnace width direction was neglected. As pointed out before, the assumption of uniform transverse gas temperature is likely to be contrary to resultant reality and the resulting nonuniform heating. Fitzgerald^) and Fontana^) also applied the zone method to predict radiative exchanges inside the reheating furnace. Since both studies adopted a one-dimensional model of the furnace chamber and neglected the influence of the refractory skidrail, they could not amplify the results of Veslocki and Smith( ). In 14 mean beam length method was applied to charaterize side wall and the both studies, the gas radiation to the slab surface . However, the mean beam length approach is only valid for gas of uniform temperature in an enclosure. 2.2 Hot It Charging of Slabs is to be expected that the hot charging of slabs would decrease the furnace fuel consumption per ton of slab, because the energy content of each slab from the preceding process is delivered to the furnace. However, this simple and effective process of conserving energy has not been widely slab defects as well as adopted yet owing to technical problems management problems in the overall W^M^O^ mill, eg, the s u c n as prompt transportation of hot slabs from the primary rolling process to the entrance of the reheating furnace. In required order normally to maintain before they slab are quality, inspection and transferred for conditioning subsequent hot of the working. slabs are Forrest and 21 Wilson^) have summarized the surface defects usually found in continuously cast slabs (Fig 2.4). presence The of longitudinal midface, or longitudinal corner cracks, can be visibly detected, but entrapped slag and transverse broadface cracks are more difficult to find. The problem conveyance arising in of hot hot slabs to charging. An the entrance insulated of vehicle a reheating is being furnace is another developed in Japan^) in order to retain the energy of the hot slabs during transit The potential benefits offered by hot charging can only be realized if proper heating strategies are available. However, heating strategies for hot charging have yet to be established owing to the newness of the process. 2.3 Previous Studies on the Formation of Skidmarks As previously mentioned, conduction around the slab/skidpipe contact surface and the shadowing effect of skidpipes beneath heat flux, generation thus of forming skidmarks regions in a of the slab bottom surface cause distortion of the local temperature reheating furnace depression produces known as unfavourable subsequent rolling process. This region gives rise to different deformation final rolling skidpipes process and therefore are an indispensable makes supporting accurate structure gauge for control the difficult current skidmarks. effects the properties in the However, push-type furnaces, the generation of skidmarks is irrevocably associated with the process. Ford et al in The suggest that the mechanism of skidmark formation involves: CO Conduction across the slab/skid interface. Cu) Radiative shadowing of the slab by the supporting skid structure. since reheating 22 Fig. 2.4 1. Longitudinal Midface Crack 2. Star Cracks 3. Longitudinal Corner Crack 4. Transverse Crack ( 4 a ) 5. Slag Patches Corner Crack ( 4 b) Surface Defects Common defects in the continuously cast slabs. 23 It was concluded that the radiation shadowing. In most significant factor distribution explicit in finite-difference the of skidmarks was the flux to the slab and skidpipe in the reheat zone. In the soaking zone of the furnace (Fig 1.1), An formation the model, carbon dioxide and water vapour were approximated by a gray gas in the calculation of radiative surface. in the slab and blackbody radiation was assumed for the slab method skidpipe. was Although assumptions were not realistic, some important employed to the gas' 'gray solve the and temperature the 'blackbody' and useful results have been generated by the model. CO Thickers slabs require greater soaking time to even out the temperature difference in the slab and reduce the skidmarks. (ii) The inlet temperature little effect influenced on by of cooling-water skidmarks, radiation thus shielding in the range considered (25-100 ° C)had indicating of the that skidmark skidpipe and formation very little was by the strongly direct conductive heat loss to the cooling water. Howell et al (22),(23) between the -2 kW/m skidpipe caTr and the i } ec o u t investigation of the thermal contact conductance slab and concluded that its value varies from 0.1 to 5.0 -1 C according to the contact pressure and temperature and 150-1000° C respectively. Weaver and Barraclough (2^) of skidrail geometry with experiments c o m in the ranges 0.1-1.0 MPa bined to study the configuration the computer modelling of skidrails and concluded that a teardrop-shaped design would be the best for minimizing skidmarks. Roth et al (24) employed radiative through a one-dimensional model to shadowing to be predominant the wear bar simulate skidmark in the formation and rail having only a minor the region of the skids are shown in Fig. 2.5. . temperatures. They also showed of skidmarks, with heat conduction influence. Temperature predictions in 24 AT , = 26°C 1050 AT 1100 '38°C 1150 Temperoture, 6 Transfer bar leaving roughing mill Fig. 2.5 Temperature distribution in the skidmark region (from Roth el d) 25 In general, past studies have concentrated on the effect of skidpipe design on the generation of skidmarks and have achieved useful results {eg , the identification of optimal geometrical skidrail shapes). However, the formation of skidmarks will later be shown to be closely related to the operation of the furnace as well as to skidrail design. Moreover, the one-dimensional description of radiative exchanges inside the furnace chamber cannot accurately characterize heat transfer around the skidmark regions, which includes the effects of refractory side wall and nonuniform transverse gas temperatures. The radiative exchanges among the exterior surface of the skidrail, the slab bottom surface and the furnace chamber must also be studied in detail furnace operating parameters. in order to link the temperature of the skidmark to the 26 3. O B J E C T I V E S Although have fulfilled temperature, some previous tasks mathematical such as THE PRESENT heat-transfer computer control WORK models and of the prediction reheating of trends furnace in slab many inadequacies have been identified in the previous chapter. More stringent requirements be the OF for slab heating quality and further achieved quantitative by a one-dimensional description of the model of heat-transfer improvements in furnace operation cannot the furnace processes chamber. occurring A inside more the detailed, furnace is necessary. The objective heat-transfer of the present study is to formulate a sophisticated mathematical model of the the reheating furnace, capable of providing detailed predictions of slab and refractory wall temperatures. Since the influence of the furnace refractory side wall and the nonuniform transverse gas temperature distribution in the process are to be taken into model account, adopted which to are simulated three-dimensional calculate major as characterized model a geometry heat-transfer radiative participants gray by the gases in by zone-type exchanges. The the radiative previous in model order to identify was combined with order to predict the temperature Gas temperature, a furnace chamber combustion gases inside have been bands. Skidrails their the exchanges and have often models, discontinuous absorption of effects treated were also on slab heating. two-dimensional unsteady as has the been chamber, been inaccurately 'real' gases and incorporated into the The chamber furnace conduction model in of slabs being heated in the furnace. slab push rate , slab initial temperature distribution and steel grade are the required inputs to the model. To proceed beyond this point would require complete knowledge of the gas flow pattern inside the furnace chamber, due to the coupling of the heat-transfer problem to the flow field. However, the gas flow field is currently unavailable 27 and would be exceedingly difficult to obtain; Therefore the gas temperature profiles had to be assumed as input to the model. Any extension of the present study require knowledge of the gas flow pattern inside the chamber so that the assumption of the gas temperature profile can be removed. The geometry and operational parameters for the model calculations are taken from the reheating furnace at the Lake Erie Works ( L E W ) of Stelco. Although the model is applied to simulate the thermal behaviour of the Stelco furnace, the modelling approach is general and the results generated from the model can be applied to other slab reheating furnaces, since the Stelco furnace is typical of a three-zone type reheating furnace. The model was applied to investigate the following: CO The effects of varying furnace operating parameters, such as both longitudinal and transverse gas temperature distributions, steel slab grade and slab push rate etc, on slab exit temperatures and skidmark formation. Cu) The mechanism of skidmark formation and identification of measures to eliminate or reduce iL (iii) The effects of various heating strategies for hot-charged slabs so as to minimize fuel consumption and improve slab temperature uniformity. 28 4. R E A L GAS T R E A T M E N T F O R T H E Z O N E M E T H O D Gas radiation plays a dominant role in the thermal behaviour (20),(36),(43) Q f furnaces. Since most reheating furnaces are fired with hydrocarbon fuels, carbon dioxide and water vapour in the gaseous combustion products constitute the major source of radiation within the furnace chamber. In the Stelco LEW reheating furnace, which is fired by natural gas and a mixture of natural gas and coke-oven gas under pre-mix conditions, the contribution of particulates to the radiative emission is unlikely to be significant and therefore has been ignored in the present study. Since the furnace gases are mixed by high momentum jets from combustion burners, a well-stirred chamber assumption has been adopted. In order to characterize the emissive and absorptive behaviour of the combustion gas in the Stelco reheating furnace, a mixture of three gray gases was assumed to represent the radiative gas, which exhibits discontinuous emission and absorption over behaviour of the real specific wavelength bands. The approach described below has been applied in a three-dimensional heat-transfer model of the reheating furnace chamber. 4.1 Method of Treatment of Real Gas Emissivity and Absorptivity Absorption and emission of radiation by a real gas occurs only over specific bands pf the wavelength extremely difficult. spectrum. Thus the calculation of gas radiative exchange is rendered In order to overcome this complication real gas emission/absorption can be simulated by the weighted summation of a sufficient number of gray gases, so that the mathematical formulation which characterizes a gray gas can be applied to the real gas. As described by Hottel and Sarofim^^, the total gas emissivity then can be approximated by 29 the following equation: N e == g with the where in are the extinction practice it has been dependence found can values r of (4.1) 1 for T g j and E q . (4.1). kj The N and r kj are weak carried by the be change the L which can for theory that be characteristic gas temperature, form (l-exp(-k.pQ) coefficients coefficients. Although in temperature of . 8 , 1 restriction that k. weighting r L i= 0 of real . are gas components both functions weighting lies in the obtained gray range by fitting gas temperature coefficients^^. the gas emissivity functions of gas of of the furnace real with and total r . are temperature, so that After gas temperature all selecting gas temperature, emissivity the data a a set to is reflected the in .. g.i A similar treatment can be applied to the absorptivity depends on relationship between absorptivity given by the following both gas temperature and g ( T s • T and emissivity, of a real emitting according to gas, except that the surface temperature Hottel and T . g Sarofim The j s equation: 0.65 T a absorptivity g ' p L ) ( = T T ) s [ % ( T s ' ' T pL )] ( 4 J ) g and a g is also expressible as follows N a = Z j= 0 e The determination of agj absorbing gas temperature, a ( ^ is similar T T ) [ l - e x p ( - k pL)] (4.4) ° to that of T g , but also of the rgj, the except emitting agj is a function not surface temperature, T . g only of 30 4.2 Application of the Method to the Real Gas in a Reheating Furnace Chamber At the Lake Erie Works of Stelco, the fuel fired in the reheating furnace consist of the following: Soak zone Natural gas Heating zone Natural gas or mixture of natural of natural ( private gas and coke oven gas. Primary zone Natural gas or mixture gas and coke oven gas The composition of typical coke oven communication with Stelco). Because CO2 emissivity and absorptivity of furnace molar composition of stoichiometry. this purpose, complete Stelco and H2O gas, their depend upon the For gas at the is shown in Table 4.1 are the major components affecting the respective partial pressures, which, in turn, combustion products, must be calculated combustion has been assumed inside the from fumace chamber since the excess air is 10%. For the stoichiometric calculations, the fraction of coke oven gas was taken to be x and the natural gas to be (1-x). Assuming complete combustion and excess air of 10%, the stoichiometric equation for the natural gas ( CH4 ( l - x ) ( C H + 2.20 + 8.272N ) — > 4 2 2 ) combustion is ( l - x ) ( C O + 2 H O + 8.272N + 0.2O ) 2 2 2 2 (4.5) 31 Table 4.1 Typical Composition of the Coke Oven Gas Gas Molar H 0.513 2 CH 4 0.234 H 0 2 0.055 N 2 0.100 0.052 0.018 CO co 2 C H 2 o C fraction 0.019 4 0.007 2 0.002 2 6 H Similar stoichiometric equations apply to the coke-oven gas combustion : H + 0.55O + 2.068N 2 2 2 C H + 2.20 + 8.272N 4 2 C O + 0.55O + 1.88N 2 4 2 6 2 — > C 0 + 2 H 0 + 8.272N + 0.2O 2 2 2 2 2 —> 2 2 (4.6) 2 2 C O + 1.88N + 0.05O —> 2 C H + 3.850 + 14.5N 2 H O + 2.068N + 0.05O —> 2 C H + 3 . 3 0 + 12.4N 2 2 —> (4.8) 2 2 C 0 + 2 H 0 + 1 2 . 4 N + 0.3O 2 2 2 2 2 (4.9) 2 2 C O + 3 H O + 14.5N + 0.35O 2 (4.7) 2 2 (4.10) The total moles in the final combustion product are : 0.055x+O.lOOx + 0.018x + 0.007x + 1.025x + 0.328x+3.359x + 0.08145x+(1- x) + 2(1- x)+8.272(1- x) + 0.2(1- x) = 11.472- 6.5x (4.11) 32 The moles of CO2 are (l-x)+0.0018x+0.328x The moles number of the final l-0.654x (4.12) = 2-0.92x (4.13) are 2(l-x) + 0.055x+1.025x Therefore, = molar fraction of the two emissive-absorptive gases in the combustion products can be expressed as: rn 2 H O 2 n (1.0 11.472 = C 0 = u Both the - 0.654x) 6.5x ( -° 11.472 2 volume ..... <- > 4 (415) ^- °- ) 6.4986x 92x fraction of CO2 i : > ) and H2O are functions of the 4.2 shows the change of molar composition of CO2 and H2O Because furnace the mixing parameter in the 14 reheating fraction x. Table with the mixing parameter seldom goes higher than x. 0.4, %H20/%C02 was found basically to be 2.0. This is the ratio the present calculations are based upon. The distribution partial pressure inside the of furnace CO2 and H 0 2 can be determined if the total pressure chamber is given. According to previous measurements ^ \ the variation in total pressure inside the pusher-type reheating furnace chamber is less than 1% and the total pressure of the chamber can be assumed to be atmospheric. Therefore, the ratio of partial pressure of CO2 as to H2O in the final combustion product can be taken 33 Table 4.2 The Change of Composition of C O 2 and with the Mixing Ratio. Coke oven gas proportion co % H 0% 0.0(pure natural gas) 0.1 0.3 0.4 0.6 0.9 1.0(pure coke oven gas) 8.72 8.64 8.44 8.32 8.02 7.32 6.96 17.44 17.63 17.63 18.40 19.12 20.84 21.72 2 2 throughout the furnace chamber. Based on the above calculated partial pressure and the known gas temperature range of the furnace, the emissivities of both carbon dioxide and water vapour can be obtained from Fig. 6.9 and Fig. 6.11 in Hottel & Sarofim( °). However, because the atmosphere of 2 the reheating furnace is a mixture of not only C O 2 and H 2 O but also radiatively inert gas components, the latter interfere with the emissivity of this real gas. In addition, some of the CC"2 and H 2 O absorption bands overlap. For these reasons, the real emissivity of the combustion products in a furnace' is actually smaller than the direct sum of emissivities of CO2 and A H2O. correction factor has been introduced as suggested by Hottel and SarofW ) : 20 V Three reheating C = gray furnace C0 + 2 e H 0 2 * A gases were assumed to simulate as mentioned previously. <' > e One 4 17 the of emissivity of the the three gases real is a gas in the clear gas characterized by an extinction coefficient kg=0.0. A computer program was developed to fit 34 Eqs. (4.1) and (4.4) to within 5% using a typical gas temperature calculation of the extinction coefficient kj. The output T c of 1300° C for the of the program was the desired extinction coefficient kj and the temperature dependent r computer program is shown in Appendix I. The program was written for general purposes g i and a g j . The flowchart of the and could be used to fit any number of gray gases. The extinction coefficients of the three gray gases calculated from the computer program are : ICQ = 0. ^ An exponential of r g i = 1.499x10' form, r ^ = g 2 (kPa.m) , k _1 2 = 2.168x10" HkPa.m) -1 AjExpt-Bj-T), was utilized to correlate the variation with gas temperature, as shown in Fig. 4.1. The values of Aj and Bj are listed in Table 4.3. As indicated in Eqs. (4.3) and (4.4), the weighting coefficients for the absorptivity a g i T . g are not only a function of the gas temperature T , but also of the surface temperature g For the sake proposed to of convenience, in using the zone correlate the change of the weighting method coefficient , a linear a o i with equation was the surface temperature at a specified gas temperature as follows: a . g.i where Cj = and Dj C T i s are the + D. i slope and intercept (4.18) ' v respectively, and T g is the surface temperature. The slopes and intercepts for these linear equations are expressed in Table 4.4. Fig 4.2 shows a comparison between the linear interpolation and the real values of absorptivity at a gas temperature of 1000 ° C . Excellent agreement was found. From the results shown in Table 4.3 and Table 4.4, the following relationship was Legend n 900 1 1000 Gas Fig. 4.1 1 1 1100 1200 temperature 1 1300 (C) r„j change with temperature r 1400 r I Gas temperature V Legend 1000 (C) X Correlated a X Reala , f Correlated a G3 Real a i 900 I WOO 1100 Surface Fig. 4.2 i 1200 temperature r 1300 1400 (C) Linear interpolation of a 1500 Table 4.3 Coefficients of and B^- for i Correlation coefficient 0 1 2 0.3601 0.4338 0.5001 4.06x10"* -2.79x10"* -1.349x10"* 0.995 0.919 0.995 Table 4.4 Linear Correlation for the Weighting Coefficient a Gas Temp. a °C C 900 1000 1100 1200 1300 1400 0.00038 0.00039 0.00041 0.00042 0.00043 0.00043 g,0 a 0 D 0 C 0.18126 0.14458 0.11224 0.08337 0.05727 0.03339 a g.l l D -0.00026 -0.00028 -0.00029 -0.00031 -0.00032 -0.00033 g i g,2 D l -0.00012 -0.00012 -0.00011 -0.00011 -0.00010 -0.00010 0.56178 0.60801 0.64934 0.68666 0.72074 0.75212 2 0.2569 0.2474 0.2384 0.22994 0.22990 0.21448 found r .( T ) g.r g ' = a .( T , T ) when T g,r s g ' g = T s (4.19) which indicates that Kirchoffs Law was satisfied by the above treatment of the real gas model. When the emissivity simulated by the three-grey-gas model is compared with the real gas emissivity (Fig. 4.3), less than 5% error is observed. The results generated by the current gas model were incorporated into the heat-transfer model to calculate radiative exchanges between the furnace gas and its bounding surfaces within the furnace chamber. 0.30-\ W A A A Legend A /A REAL 1ST. APPRO. 2N_D.^P_PRO. PREDICTED 0.00 0 50 Optical Fig. 4.1 100 150 length Change of r (kPa.m) g i with temperature 200 39 5. Heat transfer inside a reheating parts, the first gases, the bounding refractory heat being the conduction convective heat METHODOLOGY inside radiative the transfer furnace chamber can be divided into two and convective exchanges occuring among the wall and the slab surface (Fig. slab and provide through the the boundary 1.2), refractory conditions furnace and the second being wall. for distinct slab Since and radiative and refractory wall conduction, the two heat-transfer processes are interrelated. The radiative chamber because the exchange is the temperature dominant inside the furnace level is well above 900° C. The convective heat transfer accounts for less than 5% of the total^). In furnace chamber, a zone-type^) the zone method furnace chamber, irregular geometry inside a reheating data, energy model was developed. In its ultimate form, of the thus formidable of the furnace furnace is likely and the combustion and gas flow obstacle complicated to be complex balances on gas zones are transfer exchanges in the requires knowledge a of heat order to calculate radiative heat-transfer presenting mode to burner and data impossible and its for patterns application. orientation, Due the is unavailable. this inside the reason gas to gas the flow Lacking this temperature profiles have been assumed for the current study. Four points distinguish this investigation from previous studies of reheating furnace heat transfer: CO Gas temperature gradients, in both longitudinal and transverse the furnace chamber, can be accomodated. Cu) The influence of refractory side walls was taken into account directions of 40 The (Iii) ernissive/absorptive approximated characteristics of using a clear-plus-two-gray-gas the furnace model rather gas were closely than a single gray gas. (iv) The energy interchanges near the slab/skidrail contact region were calculated in detail using a separate conduction heat-transfer model which was integrated into the furnace chamber heat-transfer model. 5.1 Radiative Heat Transfer Inside the Furnace Chamber An introduction to the zone method is provided below. one is directed to reference (20). For a detailed description, The initial step in the application of the method to a furnace chamber is to subdivide the furnace enclosure and emitting-absorbing gas mixture into a sufficient number of surface zones and volume zones, respectively, so that each may be assumed to be isothermal with uniform radiative properties. The next step is to evaluate the radiative exchanges between each zone pair for each gray gas component. The radiative flux between two zones in an enclosure containing a clear gas plus two gray gases is the summation of coefficients k.), temperature the independent weighted contributions from each gray gas i (with absorption in proportion to the absorptivity coefficients a_.- evaluated at the of the emitting zone. To simplify the nomenclature it is to be understood that the following formulation is based on a single gray emission band (k=constant). 5.1.1 Direct For the Exchange Coefficients pair of surface zones Aj and A shown in Fig. 5.1 the radiative energy 41 Fig. 5.1 Geometry for radiative exchange between two surfaces 42 emitted by surface element dAj which impinges upon dAj directly is Q A i A = A "j E. cos©, ; — • y ; X. A A . i "y j cosfl. J T. 1 1 = where sX is the direct exchange coefficient between ^ .dA.dA. 1 J iTsT E. (5.1) surface i and surface j. For a gray gas sjsT = sjsT (reciprocity) (5.2) From Eq. 4.1 the transmissivity r j -» j is given by T m . = exp(- /jkpdl) (5.3) The net direct radiative exchange between Aj and Aj is therefore = Q ^ A . «*A. l j A A IX ij v ( E. - I J E. ) j ' K J (5.4) ' A similar result can be obtained for gas-surface direct exchange, % * A j = *fi < gi - E E j W > where the direct exchange coefficient between a gas zone i and a surface zone j is defined as 7? ij 8 = , i ir f f J J v i The direct f e" . • l f k p d l dVjdA 2 j exchange coefficients pressure distribution of the gas. 4kcos0.. H r (5.6) J will be functions of both chamber geometry and partial 43 Energy conservation requires that for surface zones N s _ Z j=l and s.s. 1 + _ sT *J J I j=l IX = A. where N g values 1 evaluating the multiple i n . analytical approximate zone + 8 J 1 4kAV = J direct exchange integrations format are coefficients extremely (except for a few to very were further subdivided into subzones obtain simple approach has been taken involving the uniform pairs E q . (5.1) and E q . (5.6), and are impossible to geometries).Therefore subdivision m e t h o d ^ ) 2 and o f multiple the the summation an in which over these integrations. Total Exchange Coefficients The which from difficult subzones was performed according to the definition 5.1.2 (5.8) 1 denote the total number of surface and gas zones respectively. o f the express g.s. and In both (5.7) 1 for gas zones Z j=l the g Z j=l N direct directly exchange coefficients consider only the radiation originating from a zone i impinges upon another the direct radiation through single or multiple from i zone j . Total to j and the radiation A. W i i = = originating from i account for and reaching j reflections in the enclosure, are required. As shown in F i g . 5.2, a radiative A. W. i i exchange coefficients, which A . V( I e. E. I A . e. E. i l l balance on a surface zone A j requires that : + I + p.H.) *i r p.( i v N Z j= 1 g _ g.s. E . ' ' j s Z j i N + = (5.9a) ' _ ' s.s. W.)(5.9b) J J 1 44 Fig. 5.2 Radiative energy balance for a surface zone. 45 where W. is the total leaving radiative flux from a surface, H . is the total radiative flux and p . and e j are respectively the reflectivity incident and emissivity of the surface i. When applied to each surface zone, Eq. (5.9b) provides a set of N simultaneous equations which may be rearranged into the more useful form where 6 .. is defined by ij ... • \ " X If all zone temperatures solved for the N i = j 1 0 i*j are known, the set of simultaneous linear leaving flux equations (5.10) can be densities, from which the net zone radiative flux follows readily. The net heat flux between j and i can be evaluated from Q. . J * i = V Q. . J •» i Q. V 1 . J = S.S. ( E. i J J E. ) i ' K (5.11) ' where SjSj is the total exchange coefficient term having the dimensions of area. The total exchange coefficients can be derived from Eq. (5.10) resulting in S.S. i J = A. e. - J — J (.W. Pj i J 5.. e. ) y J (5.12) A. e. i j = ¥ j = G S ^ ¥ j i j < W + ^ Vi i k w 513 > <- > 5 14 where ^Wj is the leaving flux density at the surface j per unit emissive power of zone i. 46 Again, since a gray component is being considered, reciprocity applies and s x " 1 J = J Since the net radiative s x " , G . S . 1 1 radiative energy exchanges with = s i J r j , 1 C J T J . * = j 1 (5.15) G T J . j 1 ' emitting from a zone must equal the summation of its all the other zones in the enclosure, the total exchange coefficient for a surface zone must satisfy the condition N N f=i i j jLi 'ft s s + (i = l. = A ii e (516) N) 2, g while, for a gas zone N N g j=l 2 S . G . + j=l I 0= S.1.3 1. 1O G . = 4k 1gA V . e ( 5 . 1 7 ) N ) 2 g Energy Balance F o r a Surface Zone Energy conservation for requires that the a surface zone summation of radiation i in an enclosure filled with a gray gas received from gas zones and surface zones plus convection from the adjacent gas must equal its rate of radiative emission plus the rate of 47 conduction away from the surface. This result can be expressed as Z j where Q ^j (5.18) E . ) gj + = V wall Z j i ( S.S. E. ) Ji J + heat e loss i A i E + h. A. ( T i i g v T. ) i ' i and ( 5 J 8 heat enthalpy changes with respect > to term). For the steady-state condition, the unsteady term can be dropped. Eq. yields temperatures. Ji includes n e time(unsteady UX ( as many The flux simultaneous non-linear distribution follows directly equations as there are unknown zone from the solution of Eq. (5.18) for the unknown zone temperatures. However, all real emitting/absorbing gases exhibit a variation in absorption coefficient with wavelength. To account for this, the directed exchange coefficients the direction of arrow energy exchange in the direction of the radiation between two surface zones using STsT and S.Gj (with flux) are introduced and the net the clear-plus-two-gray-gas emissivity model is Q. A V i . ** A. J E . S.S. W l j S.S. E . j l sj (5.19) ' v with SX~ i J = Z n a ( T. )( s,n I v 3X") , l j 'VL (5.20) S.S. J i = Z n a ( T. s,n j v X ^JS i~ )n 48 Similarly, the net heat exchange between a gas zone and a surface zone is Qr ~c Gj * S. = ^ 1 j E . g,i S^"E j 1 i S ( 5 J - 2 1 > with G i i S " I S j°i = J g.n < g X i j >»• r T G S \ „ < j »< j ° i 'n T <- ' S 5 The subscript n denotes the three components of gray 22 gases being used to simulate the radiative behaviour of the real reheating furnace gas. The energy balance on a surface zone Aj of unknown temperature is given by L j S.S. E . ) J i sj + = where T g k E j Q„ . ^net,i a t is the temperature ( G.S. E . ) J i gj + + h. A. ( T , i i gk T . ) s,i e. A. E . I I s,i (5.23) ' v of the gas contiguous to A. , and Q ^[ nt includes useful flux, such as heat loss through conduction or a transient term if any. The solution to the set of nonlinear simultaneous equations Eqs. (5.23) results in the final temperature 5.2 field. Zoning of the Reheating Furnace As shown in Fig. 5.3, the Stelco L E W furnace is about 32 m by 11 m by 2-3 m. If the entire furnace was to be considered simultaneously, a computationally number of zones would result Fortunately, due to the characteristics . of unmanageable the reheating furnace geometry, the opening (Fig. 5.3) between furnace sub-chambers is sufficiently narrow TRANSITIONAL ZONE SUB-CHAMBER 1 SLAB SUB-CHAMBER 3 SUB-CHAMBER 2 SKIDRAIL SUB-CHAMBER 5 SUB-CHAMBER 4 Fig. 5.3 Reheating furnace showing division into sub-chambers (Schematic) VO 50 to provide a barrier to the radiative interaction between sub-chamber which passes through the transitional both because the area of opening between them. Radiation originating in one section is unlikely furnace to be reflected back, sub-chambers is small compared with the whole chamber and because the gas in the chamber is highly absorptive. Thus for the evaluation of direct considered as 5 connection area separate (Fig. exchange between passes through exchange and exchange enclosed sub-chambers 5.3). A typical each adjacent the total opening. order by sub-chamber sub-chamber In coefficients, to introducing is shown is assumed to examine the the fictitious in be above reheating Fig. only furnace was surfaces at the 5.4. that assumption The radiative which directly (that radiation passing through the opening has a rare chance of being reflected back), an estimation the equivalent effective emissivity emissivity approaching unity of the fictitious surface was calculated. implies that the majority of the radiation opening will be absorbed by the furnace at Stelco, the effective next sub-chamber. From the An of effective impinging upon this geometry of the emissivity e ^ was calculated to be approximated reheating 0.984 based on the following equation e f = "j TC <- > 5 (^X-Vl 24 c where e A c is the emissivity of the interior wall (0.5), A^. is the area of fictitious surface and is the total area of the interior emissivity indicates that the assumption that the radiative refractory fictitious interaction wall of the sub-chamber. The magnitude surface between is close to a blackbody. It justifies of the each adjacent sub-chamber is only in the form of direct radiation through the opening. The advantages: division of the entire furnace chamber into different sub-chambers has two 51 i 52 (i) It simplifes the calculation of direct and total exchange coefficients in such a large furnace, since their calculations are being restricted to a small sub-chamber. (ii) since furnace sub-chambers resemble each other in geometrical configuration, a general computer program could be employed for the calculation of these coefficients. Each furnace sub-chamber (including gas) was discretized into a number of zones. In order to characterize the heat flux and temperature variation in both the longitudinal and transverse directions of the furnace, the gas volume, slab surface and roof refractory walls were all divided into a series of zones in these two directions. To investigate the effect of the side walls, both left and right refractory side walls were divided into a series of zones along the furnace longitudinal direction. In the transverse direction, four gas zones and four slab zones were created to account for any nonuniformity of temperature. For zone longitudinal numbering, furnace sub-chamber axis. A maximum typical use was numbering made of symmetry sequence on a with slab respect surface in the is shown in Fig. 5.5a. Each pair of symmetrical surface zones differ Exchange coefficients calculated for one half of the the first by 20. furnace chamber were applied to the other half. The zone numbering sequences for the side refractory (Fig. 5.5c), and the furnace roof zones (Fig. 5.5d) to wall (Fig. 5.5b), the gas are also shown. The number of zones for each furnace sub-chamber is listed in Table 5.1. 5.3 Direct Exchange Coefficient and Total Exchange Coefficient Calculation One of the major tasks in the application of the zone method is the evaluation of the these direct exchange coefficients heat-flux is coefficients between vital to distribution in the the zones. The ultimate prediction furnace enclosure. A importance of the of accurate temperature unique nonuniform calculation of fields and the division method was \ Fig. 5.5(a) Zone numbering sequence for the slab surface FURNACE LONGITUDINAL DIRECTION 49 Fig. 5.5(b) 50 51 52 53 54 Zone numbering sequence for the furnace side wall 54 FURNACE LONGITUDINAL DIRECTION FURNACE SYMMETRICAL UNE TRANSVERSE DIRECTION OF THE FURNACE Fig. 5.5(c) FURNACE LONGITUDINAL DIRECTION Zone numbering sequence for the gas zone 50 42 49 41 ii Fig. 5.5(d) Zone numbering sequence for the furnace roof 55 Table 5.1 Summary of Sub-Chamber Zoning Subchamber Gas zone Surface Zone Total 1 2 3 4 5 32 24 32 12 16 88 60 88 36 44 120 84 120 48 60 introduced to calculate the gas-to-surface direct exchange coefficients. The evaluation of total exchange coefficients between zones in a medium having a large extinction coefficient was simulated by a pure diffusion process^)' ^ \ 5.3.1 Calculation of Direct Exchange Coefficients Since coefficients analytical for the solutions are not available, furnace chamber using the evaluation Eq. 5.1 of the direct exchange and Eq. 5.6 requires the use of approximate numerical methods. Owing to the large computational effort required for even a coarsely zoned furnace chamber, it is imperative to carry out all calculations efficiently. The direct radiative small due to the intervening absorptive medium. Since the angles 6 j and 6 j be relatively (Fig 5.1) exchange between two zones separated by a large distance will will vary slightly within the geometrical domain of both zones, the direct exchange coefficients for widely separated zones were approximated by s.s. ii = ( cos©. ) ( COS0. ) — J — - — i r c 2 k( cos0. ). g. Sj s V r exp(-kp r ^ V K ) AA.AA. c ' i j (5.25) ) (5.26) J Av. A A. 3 exp(-kp r c 56 where r is the centre to centre distance between zones, c Radiative exchange between zones increases rapidly with decreasing zone separation. Thus closely spaced zones were discretized into a series of subzones, each sufficiently small that 6 j and 0 j could be considered constant over the subzones. According to the definition of multiple integration, the direct exchange coefficients could be approximated by ( s.s. s ij s 1 J A criterion COS0. )..( Z Z 1 i j »( Z i Z j for COS0. , 'J r .. J i l exp(-kp( r ) .. 2 c k( cosfi. ).. 1\ F V A V . A A. , i c ij n zone ).. )..) AA.AA. V c H V l j exp(-kp( r ; c subdivision was developed . (5.27) > v )..) (5.28) 1J An estimate of the relative importance of radiative exchange between any two zones can be determined by the ratio of their approximate direct exchange coefficient (calculated from Eq.(5.25) or Eq.(5.26)) to the summation of all direct exchange coefficients to the zone (equal to the area of the surface zone as per Eq. (5.7)). If further the ratio of a zone pair was above a prescribed value (1%), subdivision was carried out Fig. 5.6 illustrates the method of the subdivision of a surface zone pair. Since gas elements closest to any surface will contribute disproportionately to the surface irradiation, a non-uniform division method was devised to account for this effect Thus instead of uniform divisions in the whole gas zone, those gas elements closest to the surface were more finely divided . Fig 5.7 illustrates the method utilized. The accuracy of the direct exchange calculation was established from Eq. (5.7) and by defining the error 5 according to N ( 5 = Z s N g s.s. + Z l=f s.g. ) - A. X 100% (5.29) i For 8 > 5% additional subdivision was carried out However, for the second gray gas ( 57 Fig. 5.6 Subdivision of a surface zone pair 58 Fig. 5.7 Non-uniform subdivision of gas zones used for the calcualtion of gjSj 59 k = 2.168x10" 1 (kPa.m)~*, 2 the diffusive method (20),(48) w a s uge£ j t Q s j m u j ^ a t e xadiative exchange (Section 5.3.2). The code T l . C H A M B E R written in F O R T R A N IV was employed to calculate the direct exchange coefficients. Calculation of the Total Radiative Exchange Coefficient. 5.3.2 Total exchange coefficients were calculated with Eqs. (5.10) to (5.14). From the left hand side of Eq. (5.10), a general transfer matrix can be written as S 11 S " S D unique l / p l S 21 S S 12 S 22 " S A 2 / p 2 = (5.30) s A A property was nl found s s n n n 2 s in the transfer matrix, which can lead A lp n ^n to savings in computational time. Since any convex surface zone i ( typical of the furnace zones ) cannot 60 directly irradiate itself iX = (5.31) 0.0 (i = l, 2, N ) 3, the diagonal elements in the transfer matrix become si. - A./p. = - (5.32) A./p. 1M 11 1M K-'-^J Utilizing Eq. (5.7) and noting that p. = 1- e. . 0< e. <1. |- A./ P i | > F. yield the following result N I s _ i j S.S. I _ < I SjSj - A./p. | (5.33) Thus the transfer matrix is a typical diagonal-priority matrix, which is very stable. For this reason a decomposition method was employed to solve Eq. (5.10) for the Wj, and iterative improvement of the result was unnecessary. Using a similar technique as described by Eq. (5.29), the reliability of the calculation of total exchange coefficients was checked and the results found to be accurate to better than 5% . When the medium is highly absorptive, the subdivision method becomes excessively demanding of computation time. For the second gray gas ( k = 2.168x10" ^(kPa.m)~^), when 2 a beam of radiation impinges upon it and one measures the radiative intensity at 1 m from the source, only 0.3247% of the original radiation will be transmitted through the gas. 61 It has been shown ' that when the product of the centre-to-centre v zones and extinction coefficient k is greater distance between the than 3.0, radiation can be approximated as a diffusion process and adjacent gas zones will interact radiatively with the surface. All other radiation will be blocked by the absorbing medium. Applying the diffusion method developed by Sarofim and Hotted ) 20 to the second gray component, the net radiative transfer between an adjacent gas zone and a surface zone was calculated from the following (GS). %et = — < g " «net = 3Y E ^ E W s > <- > 5 3kT < g " E = E s > (5 34 " > 35 Comparing Eq. (5.34) to Eq. (5.35), (GS) where and 2 = 8A/3kD A is the interfacial the surface zone, k area, D (5.36) is the centre-to-centre is the extinction coefficient, distance between and (GS) 2 the gas zone is the total exchange coefficient between the gas zone and its contiguous surface zones. The total exchange coefficients were evaluated with the code C O T E C . 5.3.3 Energy Balance on the Slab Surface and Refractory Since the model assigns the gas temperature gas flow pattern is not required. The interior slab surface temperature Net heat transfer Wall distribution in the furnace chamber, the refractory wall surface temperature and the were obtained by applying an energy balance at each surface zone. at a typical refractory wall zone i is the radiation resulting from the 62 zones of gas and the other refractory wall and the slab surface, convection from the adjacent gas and conduction through the refractory wall (assumed to be one dimensional). In steady state this can be expressed by A. I = U. A. ( T. l l l v i=number e. E s,i. I - T + A. h.( v I T , gk T . ) si ' ) en ' (5.37) of refractory wall zone where h. is the local convective heat transfer coefficient, U. is the thermal conductance of 1 1 the refractory wall and T depends the on conductivities. g n structure The design is the temperature of of the the of the surroundings. The calculation of Uj refractory refractory walls wall and together properties for Stelco LEW furnace are shown in Fig. 5.8 conductances were U =1.207 W / m K from Glinkov™. transfer^ ), an 2 with corresponding thermal the material relevant . The calculated values for the (top wall) and U =0.725 W/m K(bottom 2 2 T Convection to the their fi refractory wall was calculated using 1^ = 7.80 Since convection is likely established estimation of wall). W/(K.m ) 2 obtained to account for less than 5% of the total heat the convective heat transfer coefficient h| should result in negligible error. The are in the furnace chamber was subdivided into 234 refractory top half), zone temperature Zones refractory on resulting in 234 non-linear wall zones (of which 116 simultaneous equations to be solved for and, subsequently, zone heat flux. the slab wall zones (Fig. surface 5.9). are The subjected net energy to the incident same heating on the conditions as the exposed slab surface is 63 PLICAST VERIUTE MORJNT 1 s t QUALITY FIREBRICK PLIBRICO Top Half Furnace Unit of dimension: mm MOFUNT 1 s t QUALITY FIREBRICK PLICAST VERILITE Bottom Half Furnace Thermal Conductivity (W/ m.K) 0.198 0.44 1.31 0.1784 Fig. 5.8 Structure of furnace refractory walls 64 TOP FURNACE CHAMBER TRANSVERSE DIRECTION THICKNESS DIRECTION RADIATIVE+CONVECTIVE HEAT CONDUCTION ////// CONDUCTION LOSS RADIATION+CONVECTION BOTTOM FURNACE CHAMBER Fig. 5.9 Heat transfer for the slab in the reheating furnace 65 conducted into the slab and the skidrail system. For an exposed slab surface zone G"£" E . + L Ji gj j Z j Ji - i=number E . sj A. e. E . + I I s,i X. A. ( I I 3y V / A. h.( T , i r gk T . ) si ) , y=surface v (5.38) ' of slab surface zone The right hand side of Eq. (5.38) represents the heat being conducted into the slab. Xj is the local slab thermal conductivity. The situation at the slab/skidrail contact region is more complex and will be discussed in Section 5.4. Since the temperature from distribution radiative transfer the furnace transfers are coupled heat-transfer solved in conjunction with in chamber the slabs is unknown and related by Eq. (5.38), the radiative phenomena. the two-dimensional Thus the radiative to the and conductive equations have to be conduction problem. The coupling of the equations from the zone method and from the two-dimensional conduction model will be presented in Section 5.4. In performing an energy balance over the refractory wall and slab surface zones, the emissivity of the the refractory emissivity wall and the slab, of the refractory respectively, particularly required. its However, information on temperature, is rather s c a r c e ^ ^ ^ . Therefore a value of 0.5, which is commonly adopted for industrial application^), wall, are dependence on 4 was selected. According to Masashi< ), not vary much with temperature and 0.8 is a good estimate. 46 the slab emissivity does 66 5.4 The Slab Conduction Model To describe heat flow within the slab as it moves through the furnace, an unsteady state two-dimensional conduction model was applied. Heat conduction was considered in the through-thickness ( y ) and width ( x ) directions only, since conduction in the slab longitudinal direction can be reasonably ignored. The governing equation is *S< I f ) = The initial temperature previous process ( K < distribution eg. hot > depends on the charging from h < *tr > + thermal history Three types of boundary of the a continuous casting machine). inspected and conditioned at room temperature, its initial temperature value. -> ( 5 3 9 conditions have been identified slab from the If the slab is will be at the ambient for this slab conduction problem : (i) The Exposed Slab Surface The top slab surface and the exposed area of the slab bottom surface exchange energy directly with the furnace chamber, with the slab surface temperature by In Eq. 5.38. order to link the slab conduction equation to the gradient given furnace radiative exchange , a radiative exchange coefficient I j e i OS? E . + L J gj j 1 ~ st^E J i A. ( T *1 s, k v (i = l, 2, sj . - A. e . E . 1 1 s,i T. «) 1 ( 5 ' 3 Substituting Eq. (5.40) into Eq. (5.38), the 4 0 ) n) was introduced into the model. Ts^ is then the 'characteristic' temperature gas. - boundary of the chamber condition can be written in a 67 familiar form: - 1^ 9y M The radiative ) u A y=boundary heat-transfer = e.( T r T*)+ v 4 h.( T i g T. ) i v (5.41) ' coefficient serves to connect the chamber zone model to the slab conduction model. Eq. 5.41 can be employed to determine the minimum time step for the explicit Although radiative finite-difference e^ was flux at evaluated used to calculate the for the geometrical any other points on the temperature centre of each zone was obtained between ej and the neighbouring zones. Thus the radiative distribution in the slab. slab surface using a linear zone, the interpolation flux for any point on the slab surface could be calculated and the boundary conditions established. Under coefficients position e^ are in examined given the by operating influenced furnace introducing conditions in the by zone the chamber. The a temperature slab reheating furnace, temperature sensitivity to perturbation as the slab 50 °C. of the radiative well as by surface The exchange geometrical temperature variation of ej was was typically found to be less than 1% , thus indicating little sensitivity to changes in the slab surface temperature. Since the variation of surface temperature between adjacent zones is small, linear interpolation between zones will be unlikely to introduce significant error. (ii) The slab/skidrail contact region In this vicinity, radiation and convection from the furnace enviroment are partially blocked by the skid structure, thus reducing heat transfer from the chamber. This effect is compounded by conduction at the skid/slab contact area, resulting in localized depression of slab temperature. Fig. 5.10 depicts the situation in detail. The problem was analyzed by introducing the fictitious surface AB, through which all radiation coming from the gas and the refractory wall to the slab bottom must pass. The slab bottom surface, the exterior 68 H: Bottom slab surface T: Exterior surface of skidrail F: Fictitious surface representing furnace chamber s: distance between centreline of two adjacent skidrails h: height of the skidrail d: width of skidrail Fig. 5.10 Contact region between the skidrail and the slab (detail) 69 surface of the skidrail and the fictitious surface form a radiative infinite in the direction perpendicular to the page). If diffuse and to have uniform properties enclosure (assumed to be all the surfaces are assumed to be and • temepratures, and furthermore, gas emission within this small region is ignored, the view factor from the chamber to the slab can be expressed analytically (the derivation is shown in Appendix II) (5.42) Based on values of s=1700(mm), d=301(mm), h=475(mm) from the Stelco LEW furnace the view factor was calculated to be ^ =0.65. The slab surface around the skidmark region receives heat directly from the furnace chamber as well as heat reflected and radiated from the exterior surface of the skidrails. If the exterior surface of the skidrail is assumed to be in radiative equilibrium, an electrical analogue shown in Fig. 5.11, can be established. The assumption of radiative equilibrium for the skidrail is reasonable because the convective heat transfer from the furnace gas to surfaces of the skidrail is of the same magnitude as that from the interior skidpipe wall to the cooling water. Introducing subscripts F, T, H to indicate the fictitious surface, the exterior surface of the skidrail and the slab bottom surface respectively, it can be seen that F. F 1.0 ->T - F. F -* H (5.43) The total resistance expression R, to 1.0/( A F P F. F •• H + 0.5 A ^ F F F -»T ) 70 Fig. 5.11 Radiative network among skidpipe, slab surface and the fictitious surface 71 can be rearranged in the form A p R = t o 1.0/( F , p + H 0.5 F p ^ j ) (5.44) The shielding factor of the skidrail can be expressed by C = F _^ p The first term of Eq. 5.45 to the slab surface, + (1.0 - F p ^ H )/2.0 (5.45) gives the direct radiation from the gas and the refractory wall while components from the exterior the second term represents the reflected and reradiated wall of the skidrail. The shielding factor of the skidrail for the Stelco LEW reheating furnace was calculated to be C=0.83. As previously noted, the slab will also lose energy to the cooler skidpipe due to conduction at the contact area. Values for contact resistance between skidrail and slab were obtained from Ford ^ \ .1.4 CR = 0< ) 1.4 + 3.2/20.0( T 4.4-4.2/300.0( T g l a b g l a b -120.0) -200.0) 1 2 0 T . , <120°C slab ^ T slab ^ 2 0 ° (5.46) 200< T . , < 5 0 0 ° C slab 0.2 The Dittus-Boelter equation^) was utilized to determine the convective heat-transfer coefficient between the cooling water and the interior tube. h = 0.023Re£ Pr* 4 X/D and the average conductivity of the pipe blanket was taken to be 0.623 W/mK. (5.47) 72 The overall heat transfer coefficient to the skid-pipes was calculated assuming steady state u c = <rii -r * > 1/ + ^ + where A x is the thickness of the insulating jacket, and X is its average conductivity. In characterizing the total conductance from the furnace chamber to the slab bottom surface in Eq. (5.45), uniform irradiation has been assumed for all the surfaces. In reality, this assumption does not hold. The slab zone around the contact region will be subjected to the most severe shadowing effect of the skidpipe. The shadowing of the skid structure will decrease as the distance from the surface zone to the slab/skidrail interface increases. Therefore, the shielding factor suggested in Eq. (5.45) was applied only for the zone near the slab/skidrail interface. Coupling Eq. (5.42) with Eq. (5.48), the boundary condition for a zone in the contact region is 9y ' C* " « V U " ' = 1 where 0 , the parameter B U c be ^radiative — from ^conductive c < - wa > T <- > T 5 49 characterizing the dimension of the contacting area, is given by width of contacting zone/node size Ax is the overall heat-transfer evaluated U ^convective Eq. (5.46) coefficient to (5.50) for the heat lost to the cooling water and can Eq. (5.48), and includes the contact resistance, the conductive resistance of the pipe blanket and the convective resistance of the cooling water. Eq. (5.38) can be used to evaluate q presence of the skidpipe. n e t ^- The term C is the shielding factor due to the 73 (Ui) The side surface of the slab Since the slab thickness is typically less than 6 % of the slab length, heat transfer to the end surface will not be a significant factor in determining the overall temperature distribution. Heat transfer to the end surface of the slab can be expressed by « = v where v is the radiative its proximity, the mr - a [ { y + h exchange factor ando < - T g T s > • <- > 5 51 is the Stefan-Boltzman constant Owing to end of the slab was assumed to interact radiatively with the refractory side wall, resulting in the analogous electrical circuit shown in Fig. 5.12, from which ^s ->r e r A r s e s AB=0.24 m ; DC=3.8 m and the perpendicular distance between AB and D C is 1.15 m (refer to Fig. 5.12). According to the cross-string method F — ~ AB-+CD F AC + B D - ( A D + B C ) 2AB — ~ S-*R _ ~ „ u o { : > ' 0 i ) The boundary condition for the ends of the slab is therefore found to be " where T r 5.4.1 X ( Il } x=L °- 6 6 5 a ( T r" T 4 ) + h ( T g " T s } ( 5 ' 5 4 ) represents the average interior refractory wall temperature. Heat Conduction Inside the Slab By assembling Eq. (5.39) and Eq. (5.54), the complete slab conduction equation and its corresponding boundary was = employed to compute conductions can be the temperature defined. An explicit distribution in the finite-difference slab. The model longitudinal 74 \ SIDE REFRACTORY WALL s SLAB SIDE SURFACE * tk. REFRACTORY WALL E R J -WW iii* Fig. 5.12 R •— Radiative wall. s J \*vW _1_ network • for the slab vVWV- E S Izli end surface and the furnace side 75 cross-section of two-dimensional the slab push in Fig. 5.13, into a boundary conditions for the slab surface, and the slab conduction equation, required different minimum slab longitudinal direction. 7 divided different the with was Since the nodes in shown direction) the 30 as the through and mesh, to nodes thickness nodal (transverse time steps, a compromise time step was selected in order to maintain slab the stability of the explicit finite-difference calculation. 5.4.2 Coupling of the Zone Method with the Heat-Conduction Model. As seen in Eq. (5.38), the slab conduction problem and the radiative chamber problem are phenomena coupled by the shared boundary condition. Thus the two individual models have to be solved together in order to obtain the temperature field and heat-flux distribution. As mentioned earlier, Eq. (5.41) and Eq. (5.49) provide the connection between the zone method conduction, model. in the The chamber common model parameter and ej the finite-difference determined for the scheme chamber in the slab model was incorporated into the conduction model. Since in the zone method, the directed exchange coefficients are functions of the zone temperatures, which are to be obtained from the solution procedure, an initial assumed temperature profile is required in order to start the chamber calculation . Energy balances are not required for the gas zones, since the gas temperature solution of the energy balance (Eq. 5.37) for the refractory distribution was assigned. The wall zones and slab surface zones (Eq. 5.38), when combined with the conduction model of the slab , provide a new .estimate for the temperature distribution. This new previous distribution and the procedure repeated until From a guess of the initial temperature distribution was compared with the satisfactory convergence was obtained. profile, generally 8 to 10 iterations were required. A detailed outlined of the procedures is as follows: 76 Y SYMMETRICAL LINE OF SLAB N D NODES > M NODES E o Fig. 5.13 o o Nodal division of the slab (transverse to the furnace longitudinal direction) 77 (/) Calculate the direct exchange coefficients for the furnace chamber, (/i). Assume temperatures for all refractory wall and slab surface zones. (iii) . Evaluate total exchange coefficients in the. furnace chamber.(Eq. (5.10) to Eq. (5.14) and Eq. (5.36)) (iv) Evaluate directed exchange coefficients from the known zone temperatures. (v) Solve energy balance ( Eq. (5.37) ) for the refractory distribution using U B C library routine Q N E W T wall temperature ( \ 5 9 (vi) Calculate the radiative coefficients and obtain heat flux on the slab surface zone. (vii) Using the calculated boundary conditions, initiate the two-dimensional unsteady-state conduction model for the slab. (viii) Compare the calculated temperature if the maximum temperature profile with the assumed profile and, deviation exceeds 10° C, adopt the new profile and return to (iv). If not, calculation procedure is stopped. The flow chart of the procedure is shown in Appendix As the has been described, the heat-transfer clear-plus-two-gray gas emissive/absorptive chamber and the two dimensional transient III. model consists of three sub-models, i.e., model, of the zone model the furnace conduction model of the slab. The interaction among the sub-models is illustrated by the flow chart in Appendix III. 78 5.5 Sensitivity As mentioned finite-difference resultant earlier, method temperature computing unsteady cost. number i.e. to A general Fig. 5.13. sample selected. convective simulate ( heat transfer. symmetrical heating. Evidently, variables: M temperature calculated the and constant M=5 then ND results temperature distribution (Fig. 5.15); the when both indicate and have been nodes, the the AD An to are Nodes ND to plotted predicted of and The in simulate multi-mode thickness slab and F explicit sensitivity of order to the minimize the two-dimensional heat transfer nodes their when With (Y axis); at the ND the length of ND=40 is corresponding temperature fine to the imposed constant M differ three has modes of the side on boundary been radiative BC conditions. values was first 5.14 shows 5 5.16, of to from found is depend to 25 to The than in which and 3%. predicted The even for 50 and the satisfactorily The three division. ND and to to two (ND=25). converge is relatively' insensitive grids being due to the the 3% less nodal upon varied 10 less than varied was Fig. will from by was ND=50 in Fig varies slab and distribution combined temperature ND depicted was AB=4.35(m), dependent. M=10 M=6, have relationships, M thickness and and conductive predicted held varied to are M M=8 the AD=0.24(m) temperature the varied smoother contours for of condition reveal between are number considered E of for along written system with adiabatic satisfactory. difference M that the was profile are was dimensions order across temperature investigated the distribution. using an axis)). sensitivity In calculated spacing was of the slab are ND. profile the contours and was grid rectangular denotes slab temperature in a the slab COND2 with CD thermophysical properties the within code nodes (X AB, the to M rectangle Surfaces transfer obtain predictions of longitudinal A held heat conduction problem boundary, and Analysis of the Nodal Division in the Conduction Model situation temperature These the results number of improved curve fitting possible 79 o Fig. 5.14 Sensitivity of internal thickness direction. slab temperature to the node spacing in the 80 1300-1 1200 - 1 2 S a E o 1100- 1000 Legend -Q ND=25 ... 900- ND=40 a ND=50 800- M=6 700 0 1 Slab Fig. 5.15 2 longitudinal 3 distance Sensitivity of slab surface temperature longitudinal direction. 4 (m) to the node spacing in the 81 Note: Isotherm Values are TxlO* CO 4.35(m) (b) M =7 ND = 30 0.97 -1 01 | 4.35(m) (c) Fig. 5J6 M = 9 N D = 40 Isotherm contours with respect to three nodal divisions. 82 with additional M=7 points. In order to obtain by ND=30 node points was selected. satisfactory temperature contours, a network of 83 6. The heat-transfer RESULTS AND DISCUSSION model developed for the reheating furnace was used to investigate three aspects of furnace behaviour: (/) To predict the effects of different operational parameters, such as slab size, gas temperature distribution , push rate and steel grade , on the temperature distribution in the slab and refractory walls. (ii) To calculate the nonuniform heat-flux the slab/skidrail contact region in alleviate the skidmark (iii) detail and temperature and to identify distributions around possible measures to effect To develop improved heating strategies for the hot charging of slabs. The effect of varying each of the identified furnace operating parameters was established by holding the remaining parameters constant in the model. Unless otherwise specified, the standard slab selected for the calculation was medium-carbon (0.23%) steel and 4.35 m wide by 0.24 m thick. Although the 0.23% carbon steel is choosen as a sample grade of steel for the calculation, common carbon steels are expected to exhibit similar heating behaviour under the same heating conditions, since they have very similar thermal 6.1 General Thermal 6.1.1 Slab Temperature Behaviour of the Reheating Furnace and Refractory By asssuming gas temperature refractory obtained walls were properties^), calculated by Wall Temperature Response profiles, the temperature the heat-transfer for a pushing rate of 0.0034m/sec. ( 200 response of the slab and the model. T/hr. Figure 6.1 shows results for slabs charged in tandem ). 84 Bottom 0 5 10 15 Longitudinal Fig. 6.1 Predicted furnace slab temperature gas 20 25 position (m) profiles longitudinally 30 in the reheating 85 Since the level of gas temperatures in most reheating furnaces is similar, the range of the assumed distribution reports^)- (16) a n ( j m The gas temperature transverse for direction e the current calculations is based on measurements of suction temperature the values thermocouples t (drop-off) temperature for and to vary temperature the surface temperature difference ( 230 previous from Stelco. of both top and bottom chambers was assumed to be uniform in the longitudinally, as shown in Fig. 6.1, from 1400° C. Under these conditions, the slabs are heated from room temperature exit of of 1180 - 1200° C, steel. As expected, the a value slab centreline well above temperature 900° C to to a predicted the austenizing is lower until the slab reaches the soaking zone, with the maximum than the temperature ° C ) occuring at the centre of the heating zone ( about 20 m into the furnace ). The lag in centreline $ temperature is mainly due to the low conductivity of the steel. Isotherm temperature contours distribution in in a slab the offer slab a cross useful section. visual In form particular for a a two-dimensional detailed temperature distribution around the slab/skidrail contact region will be clearly depicted. Figure 6.2 shows temperature Although contours at three different at the entrance to the axial furnace positions ( 25.2 m , 27.5 m, 32.0 m ). soaking zone(25.25 m), the slab surface temperature is significantly higher than at the centreline, the influence of the skids is relatively minor. As the slab progresses through the soaking zone adjacent to the skids becomes more apparent the At localized 27.5 slab temperature depression m, the average temperature of the skidmarks is 25° C lower than the unaffected surface, which increases to 50° C at the exit In the soaking zone, the centreline temperature quickly approaches the surface temperature. These results are indicative of the fact that, before the soaking zone, the major heat sink in the slab is its center, but once the slab enters the soaking zone, the slab/skidrail contact t As was pointed out before, these values are not gas temperatures but are "balanced" temperatures among gas, refractory walls and slabs in the furnace chamber. t Centreline refers to the geometric centre of the slab cross-section Note: Isotherm Values are TxlO CO 4.35(m) (a) x=25.19(m) 4.35(m) (b) x=27.11(m) 4.35(m) (c) Fig. 6.2 x = 32.00(m) (furnace exit) Predicted slab temperature contours at three longitudinal positions 87 region becomes the main heat sink and the skidmark effect grows quickly until the slab is discharged. Predictions for the top and side furnace refractory shown in Fig. 6.3. The refractory wall surface temperature wall surface tmeperatures stabilizes at 250-270 ° C than the local gas temperature. The drop in the refractory side wall temperature are lower is due to the fact that the furnace geometry is relatively unfavourable for the radiative heat transfer to the side wall compared with that to the roof surface. In addition, the refractory side wall near the exit end of the furnace loses heat to the surroundings through the furnace discharge door. 6.1.2 Heat-Flux Distribution to the Slab Surface Model predictions for the net surface heat flux (Fig. 6.4) slab heating furnace. In rate occurs around the midpoint of the the soaking zone, the radiative locally high slab temperature. heat flux heat-flux distribution can heating zone, about 20 m into is much reduced, partly due to the the These results are in agreement with other s t u d i e s ^ ^ ^ . O f 2 more significance, however, are the transverse heat-flux these are believed to exert indicate that the maximum distributions shown in Fig. 6.5, since a significant impact on the rolling process^ ). The transverse 29 be seen to exhibit nonuniformity in contrast to the gas temperature which was specified to be uniform in the transverse direction. Near the furnace entrance (10.67m), the heat-flux side slab surface wall where approaches the the profile receives more furnace discharge (28 convex, since the maximum heat is concave due to the influence of the m into the flux occurs in radiative energy. However, furnace), the heat the middle of the refractory as the slab flux profiles becomes furnace. This effect results from the slab end temperatures becoming greater than those in the central part of the furnace so that the net radiative heat received is proportionally decreased. 1400- 1200- 1000- 800- ouu Legend ? . s .1 E M P • (A S_SIJ_M E D) A 400 ROOF_TEMP._ _ SIDE WALL T E M P . ?nn - i i 10 15 Longitudinal Fig. 6.3 i i 20 25 position (m) Predicted refractory wall 1 30 temperatures 35 89 1600 Fig. 6.4 Predicted heat flux position in reheating to top furnace slab surface as a function of axial 90 20 -r i 15- c o 3 r Legend io- Slab l o n g i t u d i n a l position 10.67(m) Slab l o n g i t u d i n a l position 28.01(m) X .3 O 3: 5- -6 •4 -2 Furnace Fig. 6.5 Predictions furnace 0 transverse of heat flux 2 4 position to slab (m) across width of the reheating 91 The importance of direct radiation to the slab surface from the refractory wall, relative to direct radiation from the gas, is shown in Fig. 6.6. At the furnace entrance, gas radiation provides the dominant contribution to the slab surface heat flux, while near to the furnace exit, the refractory wall provides about 58% of the total radiation. This efficiency of surface/surface radiative exchange, relative moving to open radiative tube (ORT) to gas/surface exchange, is the basic reason for furnace design^ ). These furnaces rely 29 heavily on radiation from the refractory wall. The Effect of Push Rate on the Slab Temperature Distribution 6.1.3 Varying the rate of slab throughput (push rate) is a common plant practice. Fig. 6.7 shows a comparison of the centreline temperature and 0.004m/s temperature (235T/hr.); the remaining parameters were rates although the contours (Fig 6.8) surface for rolling. These results demonstrate the constant The centreline in the discharged slabs under the two temperatures do production rate also exacerbates the process of skidmark of held (at furnace exit) is predicted to be 80° C lower for the higher push rate. The same effect is seen in temperature production for two push rates: 0.0034m/s (200T/hr.) furnace is the poor that the major conductivity of the not differ formation, much. The higher which is not desirable obstacle to increasing the productivity slab being heated. Since increasing the productivity implies reducing the residence time of the slab inside the reheating furnace, the energy received at the surface does not have adequate time to conduct into the centre of the slab, resulting in a large temperature slab. An alternative difference between the centre and surface of the process, electrical induction heating, might the magnetic field used induces eddy current inside the slab. alleviate this situation since 92 Fig. 6.6 A comparison of direct chamber gas and refractory radiation to the slab surface from the 93 Fig. 6.7 Comparision of the slab centreline temperature for two slab push rates 94 Note: Isotherm Values are TxlO (a) (b) Fig. 6.8 CC) Push rate: 0.004 m/sec. Push rate: 0.0034 m/sec. Slab temperature contours at furnace exit for two push rates 95 6.1.4 The Effect of Slab Dimension on the Temperature Distribution Increasing the slab thickness, while holding the other operational parameters constant, has a similar effect to increasing the push rate. contours at the furnace exit, for two different centreline temperature than that of the Figure 6.9 shows predicted temperature thickness slabs, 240 mm and 300 mm. The of the 240 mm slab at exit can be seen to be about 130° C higher 300 mm slab, while the temperature difference between centreline surface also tends to be slightly less. This could reduce subsequent problems in the gauge control of the rolled products. One solution for the push-type reheating and final furnace to heat thick slabs is to increase the residence time of the slabs in the furnace chamber by lowering the push rate of the furnace. Another approach is to increase the gas temperature of the furnace chamber by increasing fuel firing rates. However, latter since the temperature poor conductivity of the slabs being heated is likely model discussed later thick slabs be arranged in groups to facilitate slabs of different to is a limit to the to lead to a larger difference between the surface and centre of the slabs. Both approaches can be simulated by the off-line made there ensure of thin Sec. 6.1.6. It If furnace chamber, a compromise has to be slabs and occur. This heating strategy is very difficult is strongly advisable that easier control of the the reheating furnace. thicknesses are mixed in the that overheating in underheating of thick slabs do not to achieve and has to be carefully simulated with the off-line model. 6.1.5 The Effect of Steel Grades on the Slab Heating Process It is to be expected that different temperature response curves as a Since common most carbon steels result have grades of steel might exhibit of differences very similar in their somewhat different thermophysical properties. thermophysical properties^), their responses under the same heating conditions will show little difference. However, alloy steels 96 Note: Isotherm Values are TxlO (°C) 4.35(m) (a) (b) Fig. 6.9 The Slab thickness = 0.30(m) Slab thickness = 0.24(m) effect of slab thickness on the slab temperature at furnace exit 97 can have quite significantly shown in Fig. 6.10. different thermophysical properties from mild carbon steel, as The thermal diffusivity of the alloy steel( 3.5% Ni, 1.0% C r - M o ) is almost 35-40% lower than the common carbon steel. A comparison of temperature prediction for carbon steel (0.23%C) and alloy steel( 3.5% Ni.1% Cr-Mo) is given in Fig. 6.11. At the furnace exit, the surface temperature than that of the carbon steel. While even greater, roughly about 152° C. of alloy steel was found to be about 100° C lower the disparity between the centreline temperatures was These discrepancies cannot be eliminated entirely by altering the heating condition {eg. increased firing) since the phenomenon originates from the properties thermal have the diffusivity more temperature alloy of material heated. values (Fig. 6.10) tendency uniformity steel. The being to accumulate Since the alloy steel exhibits (generally) lower than carbon steel, energy entering the alloy slab will near the surface. In order to obtain satisfactory slab at the furnace exit, additional soaking time will be required for the computer model can be used to study the heating of alloy steel. The calculated slab exit temperature different push rates for the can be compared and a proper push rate can be selected based on the discharge temperature requirement set by the rolling process. 6.1.6 Off-line Computer Control Model As the previous sections have demonstrated, the model is capable of simulating the effects of furnace operating parameters on the slab heating process. Clearly the model could be used off-line for computer control because it is capable of predicting desired operating parameters in the form of a data bank for different slab conditions. An example is given below to illustrate the steps in using an off-line model. The requirement from the rolling mill and information from the slabbing mill consist of the following: 98 0.04 H 0 1 1 1 1 200 400 600 800 Temperature Fig. 6.10 1 1000 1 1200 (C) Thermal diffusivities of two steels (from reference(50)). 1400 99 Fig. 6.11 Predicted longitudinal slab temperature profiles for two steels. 100 Grade of slab : 0.23% C, carbon steel Slab dimension: 4.35x0.24x1.0 m 3 Slab charging temperature: Room temperature 25-30° C. Required slab exit temperature(average): 1200 ° G . Maximum the allowable difference between centre temperature and the average slab temperature: 50° C Maximum allowable difference between skidmark temperatures and the average slab bottom surface temperature: 50° C. The of desired operating parameters, which are controlled variables in the terminology process control, are the required gas temperature variables were obtained iteratively running the program Appendix conditions were III). Suitable operating profile and the "ENERGY" push (flow selected based on rate. These chart shown in satisfactory exit slab temperature. The desired operating parameters are push rate (0.0034m/s) and the desired top and bottom gas temperature profiles shown in Fig. 6.1. It must be emphasized that the model predictions are conditional upon control of fuel firing rates so that the specified gas temperature profiles are obtained. A data bank of results from the off-line simulation (for various sets of operating variables) would then be stored for later retrieval by the on-line process computer. In on-line control, as long as the requirements from the rolling process and the from slabbing automatically fit mill the are input into the process computer, the process the information computer will situation on, hand into a certain category and choose the appropriate operational parameters from the memory and execute the necessary commands. 101 6.1.7 Influence of Gas Temperature on the Slab Heat Flux and Temperature Profiles Within the furnace chamber, the primary source of radiation is the furnace gas and therefore, the heat-flux distribution. Figure 6.12 distributions will be strongly influenced provides a comparison of net radiative the top slab surface for two specified gas temperature by the gas heat flux temperature distributions for profiles. Lower gas temperatures at the furnace entrance result in significantly lower net heat fluxes to the the slab surface. It is also interesting to note that gas temperature effects on heat flux are localized, the reason being that the emitting gas is highly absorptive of its own radiation. Thus nonuniform gas emission will not penetrate more than a metre or so. At the slab exit end, the heat flux for the lower gas profile is higher since the slab surface temperature is lower. All the above calculations are for uniform gas temperatures in the transverse direction of the furnace. However, this is unlikely to be the case in reality, especially when side burners are installed^). Then severely nonuniform transverse gas temperature profiles may result in correspondingly nonuniform transverse slab temperatures. The model was used to investigate this problem. At the furnace longitudinal position of 15.22 gas temperature higher gas difference temperature irradiation-flux of 100° C region is was assigned in located near the the m, the maximum cross-furnace direction. If refractory side wall, the the transverse distribution at this longitudinal position is shown in Fig. 6.13(a). The flux is severely distorted with the maximum value close to the refractory side wall. As was shown in Fig. 6.5, the transverse net-flux distribution to the slab surface is not uniform across the furnace, even under transverse isothermal conditions. This is due to the presence of the side wall, which contributes more radiative heat to the slab surface closest to it This effect would be aggravated in the presence of high gas temperature near the refractory side wall because again the slab area adjacent to the refractory side wall would receive more heat than that in the central part of the furnace. Figure 6.14 discharged slab heated under two gas temperature shows temperature fields: one with contours of a uniform transverse 102 Fig. 6.12 Predicted temperature longitudinal profiles. slab surface heat flux profiles for two gas 103 1500-1 1200- 2? O v. q> 80 900- a -2 70 to O cr> to X 3 600-60 10 c to 30050 g 5 c: o-f-6 40 —T— -4 0 -2 Furnace transverse Fig. 6.13(a) Effect of the 2 4 distance(m) non-uniform transverse gas transverse irradiation flux to the slab surface. temperature on the 104 1500 1200 <X> v. Q) 80 900 -2? 70 o Q> co 600 3 H h60 to C o 300 q> to A 50 -4 -6 -2 Furnace Fig. 6.13(b). -c Effect of the I 0 transverse non-uniform % .2 40 2 4 direction transverse (m) gas transverse irradiation flux to the slab surface. temperature on the 105 temperature distribution temperature case of of the the and the other with a locally nonuniform distribution. The skidmark near the centre of the furnace is about 10° C lower in the nonuniform transverse gas temperature than its counterpart heated under a uniform gas temperature. When higher temperature gas is located near the central part of the furnace, at the same longitudinal position, the effects of the side wall on the cross-furnace irradiation flux is somewhat suppressed, since the gas radiation makes up for the reduced radiation coming from the refractory when the side wall. Figure 6.13(b) shows the transverse irradiation flux variation gas temperature at the furnace centre is 100° C higher than that near the refractory side wall at the longitudinal position x=15.22(m). Though the flux still exhibits a slight central minimum, a more even irradiation flux distribution is found, compared with the case in which high gas temperatures are concentrated near the refractory side wall. It should be pointed out that the above calculations are nonuniformity in the transverse gas temperature. If limited the gas temperature to a small scale shifts towards either the centre or the side of the furnace on a large scale, the consequences for slab heating would be more visible and severe. Based on the above results, it is understood that precautions have to be taken to prevent hot gas from circulating near the refractory side unfavourable heating of the slab, such as a large temperature overheating near the wall, because this causes drop along the slab length, slab end and a more severe skidmark effect In the design of side burners for diffusion flames, the centre of the fuel heat release should be in the centre of the furnace. o 106 Note: Isotherm Values are TxlO -3 (°C) 107 6.2 SKIDMARK The to the EFFECT occurence of rolling process, particularly nonuniform temperature deformation in the nonuniform edging in 1% yield problem loss. that skidmarks and the in There also is is the also a believed in roughing aprons and supporting structure, positive roughing process. This deterioration computer which include the simulation gauge in the to the impact which extra width at each in of is the is a results in effects region of nonuniform nonuniform spread and allowance that results in skidmark. The immediate transfer bar "head-end the rapid undesirable skidmark results from skidmarks results The slab variation from Turn-down owing operation. and requires result mill. slabs brings many width variation mills to of roughing ductility rolling process. Thus, rougher/edger heating the nonuniform turn-down" The in deterioration turned-down of table end in rolls, the early will, in turn, result in costly repairs. model considers mechanisms of formation of skidmarks, following: (/) The the contact area between (if) The (Hi) T h e shielding effect of the the skidrail, slab bottom conductive heat loss to the radiative which exchange between prevents radiation from reaching surface and the skidrail. water-cooling skidpipes. the exterior surface of the skidrail and the slab bottom surface. The computer reheating model process at was the applied Stelco to Lake analyze Erie contact was described previously in Chapter the Works. 5. formation Heat of transfer skidmarks around the during the slab/skidrail 108 The skidmark is a region of the slab having a locally low the skidmark not only affects the temperature it also causes distortion of the temperature in Fig. 6.2(c). It The temperature skidpipe, temperature. However, distribution of a slab at the contact area, but distribution of the slab as a whole, as was seen is seen that the skidmark effect extends to the top surface of the slab. of the central region of the slab is also low, due to the presence of the since the heat received on the slab bottom surface is distributed between the slab/skidrail contact region and the central part of the slab because both are heat sinks. As shown in Fig. 6.15, the irradiation flux distribution across the length of the slab bottom is very uneven, and in particular, deep troughs are observed around the skidmark region. This serious distortion indicates that the skidmark region receives much less heat than the normal exposed slab surface, which has great impact on the transverse temperature distribution (slab rolling the direction). Figure 6.16 shows direction of a discharged slab. The the temperature troughs represent distribution skidmark along temperatures slab length and they are determine the roughly 50° C lower than the average slab temperature. An relative investigation importance of with the the skidrail computer radiative model has been shielding effect from the slab to the skidpipe, as shown in Fig. 6.17. slab surface radiation skidmark and is far due to more the presence of important than the the It performed and the to conductive heat loss was found that the reduction of skidrail conductive is the loss primary to the cause of skidpipe, the which accounts for less than 1% of the total surface radiation. However, the conductive heat loss does assume more importance Thus, reduce in order to the at the slab discharge due severity to the of skidmarks, emphasis has to radiative shielding of the skidrail using highly insulating material the geometrical between skidrail size and of the slab. skidpipe Use of by a higher narrowing highly conductive heat loss, but more importantly, the insulating reduces the slab temperature. be placed on the and consideration given to dimension material of not the only contact reduces area the necessary size of the skidrail and 109 Fig. 6.15 Irradiation flux position=24.2 m) across the slab bottom surface (furnace axial 110 1400 300Slab Fig. 6.16 longitudinal direction (m) Longitudinal slab bottom temperature distribution at the furnace exit Ill Fig. 6.17 Comparison between the conductive heat loss and the at the skid/slab contact region. radiative heat loss 112 increases its exterior surface temperature , which consitutes a major part of the radiative shielding loss. This will become clearer when coating with reflective materials is discussed. Based on the computer predictions, it is recommended that the following measures be considered to alleviate the skidmark effect: (1) Reduce the geometrical size of the skidrail The shadowing geometrical size of the surface the of effect of the skidpipe can skidrail is reduced, allowing slab. The current structure and be improved more radiation geometrical dramatically to if reach the dimensions of the the bottom contacting region between the longitudinal skidrail and the slab are shown in Fig. 5.10. The width of the longitudinal skidrail is 301 mm and the two skidrails (s) is 1700 the skidrail, where height (h) is 475 mm; the distance between mm. About 18% of the bottom area of the slab is covered by it does not have access to the outside radiation. The area bounded by the slab/skidrail contact line is also affected by the presence of the skidrail. All the outside radiation has to pass through the surface AB in Fig. 5.10 to reach the bottom surface of the slab. reaching Ignoring the slab the influence bottom of surface cross-furnace relative to skidrails, the incident the proportion outside of radiation radiation has been calculated as a function of the width of skidrail, W, and is listed in Table 6.1. From Table 6.1, the proportion of incident radiative heat transfer from the furnace chamber to the slab bottom surface can be increased noticeably by narrowing the skidrail width W. Based on the geometry of the skidrail in the Stelco LEW Reheating Furnace, the computer predicted ratio, PR, is 66.7%. If 170 (mm), However, the width of the current ski'drail is reduced to the values of PR will be increased by 3.3%, from the original 66.7% to 70.0%. the requirements. reduction Better of insulation the width and high of the thermal skidrail capacity is often materials limited are by insulation necessary if the 113 Table 6.1 Computer Prediction of the Shielding Effect of Skidrail as a Function of Its Width W W/AB 0.44 0.40 0.35 0.20 0.18 0.15 0.10 0.05 Width of skidrail (mm) % PR 748.00 678.00 595.00 340.00 305.00 255.00 170.00 85.00 50.4 52.6 55.4 63.5 65.2 67.0 70.0 72.9 AB: Distance between skidrails PR: percentage of radiation incident on plane AB which reaches bottom surface of slab. dimensions of the current skidrail are to be reduced. Ceramic materials offer extremely good insulation properties and durability and might be suitable for this purpose. According shielding effect to the of the computer predictions, another skidrail is to reduce its height. potential Table 6.2 way to alleviate lists the • proportion the of incident heat reaching the slab bottom surface as a function of the height of the skidrail. It is seen that reducing the height of the shielding effect than changing its width. In is not as restricted by the skidrail offers more potential to reduce the addition, variation of the height of the skidrail thickness of insulation layer. The current skidrail uses double tubes inside the skidrail and the height is 475 mm. From the computer results, about 30% of radiation is directly blocked by this height Since the current system uses double tubes ( each of diameter skidrail, potential exists for converting the double-tube skidrail 145 mm ) inside the to single-tube which will increase direct radiation from the furnace atmosphere significantly. This change would reduce 114 Table 6.2 Computer Prediction of the Shielding Effect of Skidrail as a Function of its Height H the Height (mm) PR 500.0 475.0 450.0 425.0 400.0 375.0 300.0 250.0 200.0 175.0 150.0 100.0 50.0 65.6 66.7 67.7 68.8 69.9 71.1 74.7 77.2 80.0 81.1 82.5 85.3 88.2 % height of skidrail by almost half which implies that the net gain of direct radiation would be increased from the current 66.7% to 77.2%. This is a significant improvement in the heat-transfer conditions. Changing from the double-tube since it only requires the to single-tube skidrail system is technically feasible modification of the exit and entrance of the cooling water. However, better insulation is required for the single-tube skidpipe to eliminate the case in which constant vaporization of cooling water could lead to a "burn-down" accident of the skidrail system. Moreover, stronger materials support the load of slabs in the furnace. are prefered for the skidpipes in order to 115 (2) Provide adequate heat to the bottom surface of the slab Since the transfer from skidmark the bottom originates at the bottom furnace chamber (to surface of phenomenon temperature the described the beginning of contours of a slab moving toward Section 6.2. insufficient As heat rise between top Increasing surface top and bottom of the slab results in the at slab, ensure equal temperature and bottom of the slab) will aggravate the skidmark effect difference between the temperature "head-end turn down" was observed from the exit of the furnace, Fig. 6.2, the the low temperature region near the skidmark propagates to the top surface. Figure 6.18 compares the skidmark temperature and the centreline temperature of a slab. The results clearly indicate that the slab centreline is cooler during reheating until the slab is near the furnace exit The reason for the difference is that heat received on the bottom surface not only has to reach the skidmark region, but also has to conduct to the cooler central part of the slab. Therefore, more heat should be input to the bottom furnace chamber. (3) Provide a more even transverse heat-flux Most transverse rolling direction problems in the arise from furnace(rolling a distribution nonuniform direction of temperature slabs). distribution However, the in variation the of transverse gas temperature and heat flux has been neglected by previous models which have been based on a one-dimensional gas temperature distribution. From the computer simulation results shown in Fig. 6.13(a), the transverse irradiation severely if the higher gas temperature flux distribution will be distorted concentrates near the refractory side wall. Then the slab end will be subjected to a much higher intensity of radiation than the central area. The temperature of the central area is inevitably more severely depressed, and this in turn, 116 •9 1 1 0 5 1 1 1 1 1 10 15 20 25 30 Furnace Fig. 6.18 longitudinal position I 35 (m) Comparison of the skidmark and centreline temperature of a slab. 117 enhances the spread of the low adverse heating situation. In higher gas temperatures temperature region near the the contour shown in Fig. 6.14, concentrating near the refractory skidmark and leads to an corresponding to the case of side wall, the temperature of the skidmark near the centre of the slab is found to be more depressed than near the end. Therefore, a long flame ( shifting the main heat release to the central part of the furnace ) is preferred if side burners are used. This will have significant impact on the skidmark effect (4) Coat the skidrail with highly reflective materials Because the efficacy of coating highly reflective material on the skidpipes to enhance radiation to the slab bottom surface is debatable, a computer simulation was performed to investigate the the exterior effect. A surface of radiative network was established among the the skidrail and the surface AB, as was shown in Fig. 5.10. components is shown in Fig. 6.19, slab bottom surface, furnace chamber represented by the A new radiative fictitious electical analogue for the three which removes tha assumption made in Chapter 5 that the skidrail is in radiative equilibrium. Of primary interest is net radiative heat flux to the slab bottom surface. Performing an energy balance on each node in the electrical analogue: For the slab bottom surface E -W, R W - 4 W. K E. - 5 W, R 2 For the exterior surface of the skidrail E -W K l Solving Eqs. (6.1) W, E' K and (6.2) W. - W, 3 W K t and Wjj 5 in terms of E , Ejj and Ej., the net heat flux to s 118 SKIDRAIL —r Fig. 6.19 Radiative network : VIEW FACTOR among furnace bottom chamber. the slab bottom surface, the skidrail and the 119 the slab surface is: Q v - n e t to slab R (6.3) 2 Two cases have been studied to determine the effect of modifying the reflectivity of the skidrail exterior are shown in bottom surface on the net heat flux Fig. 6.20. If surface temperature, the temperature to the slab bottom surface. The results of the skidrail, T , t T^, no substantial increase in the is higher net heat than the slab flux to the slab bottom surface is observed, which indicates that there is no necessity to coat new on to the skidrail surface. However, if T the skidrail from 0.1 to 0.9, raises the Often, due to insufficient insulation, the that of the slab bottom t is lower net heat surface. Therefore it than T^, increasing the reflectivity flux temperature material from 32.4 of the kW/m to 46.9 2 kW/m . 2 skidrail surface is lower is recommended to apply of than high-reflectivity coating materials. (5) Improve the design of the skidrail The essence of improving skidrail design is to alter its shapes and configuraton to increase the view factor from the furnace chamber to the area of the slab bottom adjacent to the skidrail. When the reflectivity of the exterior modifying the shape of the skidrail to increase this view > T f l bottom or T^ > surface of the skidrail factor, either is in the case of T T , will significantly enhance the radiative heat flux impinging on the slab t surface, as shown in Fig. 6.21. Thus, this is a very effective way to reduce the skidmark and confirms the results of previous workers. In a recent study by Lee( ), 25 been confirmed fixed, experimentally that a teardrop-shaped skidrail is the best it has design for minimizing skidmarks. The present configuration of skidrail in the Stelco reheating furnace is 120 i O co E o -»— ~o -Q <D X 0.2 0.4 Reflectivity Fig. 6.20 Effect surface. of reflectivity of 0.6 0.8 of the skidrail skidrail on the net heat flux to the bottom 121 20 T I 0.5 0 0.55 View Fig. 6.21 I 0.60 1 1 0.65 0.70 factor Effect of view skidmark region. from I 0.75 furnace r 0.80 0.85 0.90 chamber to 0.95 slab factor from furnace chamber to skidmark area around the 122 rather unfavourable with respect to the radiative heat reaching the slab bottom surface, since the skidrail is too wide and too high (double tubes) and has a rectangular shape. Clearly there is considerable scope for improvement in the configuration of skidrail reheating furnace. A proposed design of configuration is shown in Fig. 6.22 It should be emphasized that the temperature in the LEW (sketch). of the exterior surface of the skidrail, T , plays a very significant role in the net heat flux to the slab bottom surface. When T > heat (bottom slab surface temperature), flux is observed than in the under case where any conditions, a > The substantially temperature T t higher depends mainly upon the insulation of the skidrail and increases with better insulation. Because the direct conductive heat loss to the skidpipe is minor compared with the radiative shielding effect, the main function of improving insulation of the skidrail is to increase its surface temperature. Based on the model prediction, a possible design of an alternative skidrail is shown in Fig. 6.23. The skidrail is offset slightly in the reheating furnace, allowing the area which originally contacts temperature the offset the skidrail to distribution in the slab bottom is recommended at the be exposed end of to the radiative enviroment so that surface will be more even. Implementation the skidmarks becomes most apparent (as was reported heating zone, in Fig. 6.2) where the formation the of of and the heat flux to the slab surface remains high. In attention conclusion, to alleviate the skidmark effect, it is equally important to both the design of the skidrail and the appropriate operation of the to pay reheating furnace. A good design of skidrail but improper operation of furnace will also bring severe skidmark problems. 123 I i Fig. 6.22 A proposed design of the skidrail system to reduce the skidmark Slab motion Fig. 6.23 Possible alternative rail design to reduce skidmark formation. 125 6.3 Hot Charging Practice Many advantages accrue from the hot charging of slabs: (/) Increase the the reheating the throughput of the reheating furnace. It is quite common that furnace is the production-constraint process in the rolling mill. production rate of the reheating furnace can be increased, the If production rate of the whole mill will be increased as well. (ii) Reduce the residence time of the slab in the reheating furnace and therefore reduce oxidation and scale thickness on the slab surface. (iii) Reduce energy consumption per ton of slab in the reheating furnaces. (iv) Reduce the skidmark effect and thermal stress build up in the slab due to a shallower temperature However, hot involved, as corresponding remain charging was discussed heating unclear. is not in strategies a profile. straightforward Chapter and 2. their This section focuses on In practice the effect since there implementation are of many hot charging, on the thermal behaviour of investigating the thermal effect hot of problems the the slab charging practice using the computer model. Compared with cold charging slabs, hot charging brings more the previous process into the reheating furnace. The direct energy content consequence is that the from hot charged slab does not require as much heat from the furnace to reach the required rolling temperature. The effective utilization of this physical heat Evidently, there are two alternatives available. One is to lower furnace chamber, so that less fuel input becomes a crucial the gas temperature problem. of the is required and energy conservation is achieved. 126 The other approach is to increase the production rate, so that the total fuel consumed per unit ton of slab is decreased. Figure 6.24 shows the top surface temperature compared with a cold charged slab under two response of a hot charged slab different gas temperature profiles. The slab hot charging temperature has been taken as 600 ° C , as is commonly encountered in the Stelco set of hot reheating furnace. charged slabs, the The heating strategy is gas temperature is substantially such that, lowered at slightly reduced at the slab discharge end. The justifications for the the heating slab entrance side, and for such a heating strategy are as follows: (/') In order furnace to miximize gases) are expected exchange heat with discharge end will entrance the the to stock have longer furnace have efficiency, as long a being heated. residence the residence The time combustion fuel than products (or time as possible to input that at the fired at side, where the port of the chimney is located. Therefore, furnace the slab maintaining higher gas temperature at the slab discharge end will increase the time for the thermal exchange between hot gas and the slab. (ii) Since the direct heat is hot charged slab required at the has virtually furnace been entrance . "preheated", The fuel no immediate input could be proportionally decreased. Figure 6.24 shows charging slab is substantially that, even though decreased (200° C the lower furnace gas than the temperature for the hot corresponding cold charged slab at the furnace entrance), the exit top slab surface temperature of the hot charged slab remains 20-30 ° C higher than that of the cold charged slab. 127 1600 1400- 1200- (S 1000- 3 5 800- <b 600 / / 400- Gas temp, p r o f i l e for hot charging^ / Top j m r j a c ^ t e m p ^ o f a_ cold slab / 200- / Top s u r f a c e temp, of a hot slab -r T" Furnace 6.24 15 10 5 Fig. Gas temp, p r o f i l e for cold c h a r g i n g Predicted charging. slab longitudinal top surface 20 25 position temperature 30 35 (m) response for hot and cold 128 As was discussed in Section 6.1, for the cold charged slab, one of the main heat sinks is the centre of the slab; and heat conduction from the surface to the centre is slow, which directly limits the production rate of the furnace. However, for the hot charged slab, the centre is no longer an important heat sink. From Fig. 6.25, the disparity between the centreline and the surface temperature less severe than its counterpart rate for a remains of the hot charged slab is and smoother, because it is already basically the same, but shows that the the area under heating in a high or range. Figure 6.26 compares the net heat flux along a furnace hot charging and a cold charging case. It variation is observed that for the cold charged slab. The reason is that the the hot charged slab is lower medium temperature it between shape of the net heat flux the hot charging curve is significantly less than for cold charging. This is consistent with the fact that a hot charged slab requires less heat than a cold charged slab inside the furnace to reach the same temperature. Temperature Fig. 6.24. 6.27. contours in a discharged cold and hot charged slab are provided in The hot and cold slab heating strategies are the same as those shown in Fig. The temperature homogeneous temperature than distribution that in the in the hot cold charged and the average temperature charged slab. The slab tends difference to be much between the more skidmark of the rest of the slab is less than 30° C, which is an improvement for supplying a slab of nearly uniform ductility for rolling. The mentioned production rate was kept constant before, temperature the production rate could in the above be increased calculation. However, for hot charging compares the discharged slab temperature gas contours in a cold charged slab and in a hot charged slab under the same gas temperature productivities, temperature the level is the same as for cold charging. A rise in production rate is equivalent to a fuel saving. Figure 6.28 different if as was level is 200 slightly T/hr lower and for 300 the T/hr hot respectively. charged slab The but profile overall slab the severity but with discharge of the 129 1400 1200- 1000- 800 600- 400 Top s u r f a c e temp, of a cold slab C e n t r e l i n e t e m p . of a cold slab Top j>urJ^ace_temp^of a_hot slab 200- Centreline temp, of a hot slab 0 r 0 5 10 Furnace Fig. 6.25 -r— 15 15 longitudinal 20 25 position Predicted top and centreline temperature charging 30 35 (m) response for hot and cold 130 25 H OH O 1 5 1—! 10 Furnace Fig. 6.26 1 15 longitudinal 1 1 1 20 25 30 position 1 35 (m) Comparison of longitudinal heat flux to the slab surface for cold and hot charging. in Note: Isotherm values are TxlO -3 (°C) 4.35(m) (a) Charging temp.= 600 °C, productivity = 200T/hr. under improved heating strategy (shown in Fig. 6.24 ) 4.35(m) (b) 6.27 Charging temp. = 25° C, productivity = 200T/hr. Predicted slab temperature contours at the furnace exit for hot and cold charging 132 Note: Isotherm Values are TxlO Charging temperature = 600 °C, productivity = 300 T/hr. (a) (b) Fig. 6.28 CO Charging temperature = 25 ° C , productivity = 200T/hr. Comparison two o f two predicted productivity. slab temperature contours at furnace exit for 133 skidmark effect is basically unchanged. Thus the off-line computer model is capable of simulating a range of throughputs and hot charging temperatures, for a given set of conditions. and finally of providing the most desirable production rate 134 7. CONCLUSIONS In striving towards the objective of understanding the thermal behaviour of reheating furnaces and the formation of skidmarks in slabs, a comprehensive computer model has been developed capable of predicting the temperature simulation and heat-flux distribution to the slab and the refractory wall for various operating conditions. The computer simulation model predicts the following: (1) The characteristic heat-flux the gas temperature distribution to the slab surface is determined largely by profile in the furnace. The investigation of transverse heat flux, which could not be undertaken by previous one-dimensional furnace chamber models, shows that a nonuniform gas temperature mainly when responsible for the severe the distortion high temperature distortion of of gases are transverse heat distribution across the furnace chamber is heat flux in this concentrated near flux is likely to direction. the refractory occur, which In particular, side wall, a will lead to adverse heating of the slab. Therefore, the centre of heat release should be shifted toward the central part of the furnace. (2) The formation of skidmarks has many important characteristics: (a) The skidmark penetrates more or skidrails contact to the effect top a region. not surface of less affected. is is The the a local slab and the phenomenon. entire study clearly shows that the dominant factor In to order merely causing the alleviate the heat deficit skidmark Its influence slab temperature radiative near effect, the is shielding of slab/skidrail emphasis should be directed to reducing the skidrail size and improving the configuration. (b) The effectiveness of coating reflective materials on the exterior surface of 135 the skidrail to reduce skidmarks depends on the temperature the local bottom slab surface, Tjj, and the If > T , an increase of reflectivity enhance the skidmark effect the heat transfer slab bottom If The benefit t > surface temperature, from surface of the skidrail, T . t , increasing the in this regioa would alleviate the skidmark (c) T between of the skidrail would be favourable to t reduce the exterior difference Generally the reflectivity skidrail thus coating of high does not is cooler than reflectivity materials system is primarily effect improved insulation for the due to increase the exterior surface temperature skidrail of the skidrail. This reduces the absorption of heat originating from the slab bottom surface by the skidrail. The reduction in heat loss across the slab/skidrail contact is of secondary importance. (d) The modification of the current skidrail geometry to reduce the shielding effect to a minimum is crucial to alleviate the present severe skidmark problem. An alternative skidrail design and a single-pipe skidrail system have been proposed based on the results of the computer simulation. A comparison refractory role wall to the in the uniform between heat flux transverse the radiative contributions from slab surface has indicated that the to the heat-flux use of open radiation tube the gas significantly from the latter plays a significant slab surface. The model results suggest that a more distribution across the furnace should result from the . For the case of hot charging slabs, the required gas temperature can be lowered and compared with that for at the charge end cold charging slabs; and fuel savings can be achieved by this heating strategy. From the slab heating profiles, it has been found that the heating rate, or the 136 production rate, is limited by heat conduction from the surface to the centre of the slab. The computer simulation provides a flexible behaviour of a reheating furnace. Further work off-line model to monitor is needed to combine the the thermal current model with a model of gas flow and combustion inside the furnace chamber. Optimization of the model, and provision of on-line computer control of the reheating processes are other areas for future work. 137 REFERENCES 1. Ishihara, S., Hachiya, S., Fujisawa, F., Ono, S. and Yamaguchi, T. of Direct-Link Continuous Caster Hot Rolling in Japan Awarded the Okochi Memorial Prize, 1984, 2. Hollander, F. , Huisman, R.L. , "Computer , "Development Process", Research and Development pplO-19. Controlled Reheating Furnaces Optimize Hot Strip Mill Performance", Iron and Steel Engineer Year Book, 1972, pp.427-439. 3. Syogo Matsunaga, Bunsho Hiraoka, "Simulation Experiment a Reheating Furnace", Tetsu-to-Hagane, 4. Laws, W.R. and Salter,F.M., Furnaces for Hot Working", ISI 5. Ishida, R., Reheating on Gas Row Vol. 56, 1970, pl575 (In "Future Trends in Design of Patterns in Japanese). Continuous Strip Mill", Proceeding of the conference on "Reheating Pusher for Hot Control for publication, London, 1968, ppl32-140 Matsuura, Furnace Y., Sawae, of a Hot M. Strip and Mill", Ohtomo, R&D, A, "Computer Research and Development (Kobe Steel Ltd.) Vol. 33, No.4, 1983, pp81-84. 6. Fontana, P.G., Boggiano, A., Furinghetti, Distribution in Continuous Furnace A. Using and Pastorino, a Dynamic B., " Optimizing Mathematical Heat Model", BTF-special issue 1985, pp9-14. 7. Hills, A.W.D. and Sellars, C M . , "Observation in the Conference". Proceeding of the Conference on "Reheating for Hot Working", ISI Publication, London, 1968, ppl86-187. 8. Matsukawa, T. and Yoshibe, Y. "Recent development in Reheating Furnace", Nippon Steel Technical Report No.20, Engineering Division, December 1982, ppll9-130. 9. Gauvrit, Optimale M., Dolpert, D., Maitre, J.F. and des fours de rechauffage Gosse, metallurgique". G. , "Optimisation et synthese Rev. Gen. Therm., Vol. 138, Fr. Juin 1973 pp535-543. 10. Report "For Saving, Measure Steel Temperature not Furnace Temperature", Metal Producing, March 1985, pp58-60. 11. Strack, Hubert und Kohne Heinrich, "A Mathematical Model for Radiation Exchange of Fuel-Fired Furnace Chamber and its Apllication to Calculate Temperature Fields in Pusher Type Furnace", Stahl. u. Eisen 100 ,1980, Nr. 16. August 11, pp887-893 12. Davies, EM. , Lucas, D.M., Performance of Rapid-heating Master, S.J. , "Use of Models for Predicting Furnace", Proceeding of the Conference on the "Reheating 138 for Hot Working", ISI 13. Price, J.C. publication, London, 1968, pp. 29-45 ."Temperature Control for Slab Reheating Furnace", Iron and Steel and Verification of a Model", Iron and Steel Engineer, April 1982, Engineer, September 1980, pp59-64 14. Veslocki, Timothy Slab Reheating A. and Smith, Clifford, C. " Development Furnace Mathematical Vol. 57, No.4, pp46-51. 15. Misaka, Y., Takahashi, R., Shinjo, A., Nariai, Y., and Kooriki, Control of a Reheating Furnace at Kashima Steel works' Hot M. , "Computer Strip Mill", Iron and Steel Engineer, May 1982, Vol. 59, No.5, pp51-55. 16. Fitzgerald, F. and Sheridan, A.T. , "Heating of Slab in a Furnace", Journal of the Iron and Steel Institute, London, 1970, Vol.208, ppl8-28. 17. Hollander, F. , "Reheating Processes and Modifications to Rolling Mill Operations for Energy Savings", Iron and Steel Engineer, June 1983, Vol. 60, No. 6, pp55-62, 18. Glatt, R.D. and Macedo, F.X. "Computer Control of Reheating Furnaces", Iron and Steel International, December, 1977, Vol. 50, pp381-396. 19. Hottel, H.C. and Cohen, E.S. , "Radiant Allowance for Nonuniformity of Heat Exchange in a Gas Filled Enclosure: Gas Temperature", A.I.Ch.E. Journal, 1958, Vol.4, pp3-14. 20. Hottel, H.C. and Sarofim, A.F. , "Radiative Transfer", McGraw-Hill, N.Y., 21. Ford, R., Suryanarayana, R. and Johnson, J.H., "Heat-transfer water-cooled Skid Pipe in Reheating Furnace", Ironmaking 1967. Model for Solid Slab and Steelmaking, 1980, Vol. 7,No.3, ppl40-146. 22. Howells, R.L., Probert, S.D. and Ward, J. ," Influence of Skid Design on Skid-mark Formation", Journal of the Iron and Steel Institute , January 1972, Vol. 210, pp 10-20. 23. Howells, R.L. High , Ward, J. and Probert, Temperatures", S.D., "Thermal Conductances of Contacts at Journal of the Iron and Steel Institute, March 1973, Vol. 211, ppl93-196. 24. Roth, J., Sierpinski, H., Chabanier, J. and Germe, J., "Computer Control of Slab Furnaces Based on Physical Models", Iron and Steel Engineer, August 1986, Vol. 63, No.8, pp41-47. 25. Weaver, A. Lee and Improves Strip Quality", Barraclough, William F. "Application Iron and Steel Engineer, September of 1986, Hot Alloy Skids Vol. 63, No. 10, 139 pp 25-28. 26. Compell, Frank and Bonner, B.D., "Skid System Redesign at Great Lakes' 80-in Hot Strip Mill", Iron and Steel Engineer, Oct 1986, Vol. 63, No. 10, pp. 275-276. 27. Hopkirk, R.J. "Analysis of Heat Transfer in Complex Geometries Due to Combined Conductance, Radiation Transfer, Vol. II, & 28. and Natural Convection", Numerical Methods in Heat Edited by R.W. Lewis, K. Morgan and B.A. Schrefler, John Wiley Sons Ltd., 1983, pp319-342. Jaluria, Y. , "Numerical Study of the Thermal Processes in a Furnace", Numerical Heat Transfer, Vol. 7, Hemisphere Publishing Corporation, 1984, pp 211-224. 29. Kozo, S., Hoshine, Y., Takamori, O., Kawabata, Matsukawa, T., "Development of Open A., Sannomiya, K., Radiant Tube Type Goto, K. Reheating amd Furnace", Technical Report, Transaction ISIJ, Vol. 25, 1985, pp972-976 30. Collin, R. , "A Flexible Mathematical Model for the Simulation of Reheating Furnace", source unclear. 31. Kay, H. ."Design of Furnaces for Reheating", Metal Technology, October 1975, pp450-462. 32. Fontana, P., Model of Boggiano, A., and Furinghetti, Slab Reheating Furnace: A., "A Theory Two and Dimensional Practical Mathematical Utilization", private communication. 33. McAdams, W.H. and Hottel, H.C., "Heat Transmission". 3rd. Ed., McGraw-Hill, York, 34. New 1954. Fitzgerald, F., Conference "Aspects of on Furnace "Reheating for Design Hot for Hot WOrking", Working", ISI Proceeding of publication, the London, 1968, Radiation from ppl23-131. 35. Hottel, H.C. and Sarofim, A.F., "The Status of Calculations of Non-luminous Flames", Journal of the Institute of Fuel, Sept 1973, pp295-300. 36. Hottel. H.C. , "The Melchett Lecture for 1960 " Radiative Transfer In Combustion Chambers", Journal of the Fuel, June 1961, pp 220-234. 37. Fieri, G . , Sarofim, A.F. and Hottel, H . C , " Radiative Heat Transfer in Enclosures: extension to of Hotell-Cohen Zone Method Allow for Concentration Gradients", Journal of the Inst, of Fuel, Sept 1973, pp 321-330. 38. Johnson and Beer, "The Zone Method Analysis of radiant Heat Transfer: A for Luminous Radiation", Journal of the Institute of Fuel, September Model 1973, pp 140 301-309. 39. Patankar, S.V. and Spalding, D.B. Transfer in Furnaces: A Review", ," Mathematical Models of Fluid Flow and Heat Journal of the Institute of fuel, Sept 1973, pp. 279-283. 40. Shinji Hori, Shoji Nishitomo and Shinya Tanifuji, " Reheating Furnace Combustion Control System For Hot Charge Rolling", Hitachi Review, Vol. 32, 1983, No. 1. 41. Steve, Forrest and Keith Wilson, "Requirements For the Hot Charging of Cast Slabs at Stelco's Lake Erie Works", private communication. 42. Tamura, Y., "Temperature Measurement of Steel in the Furnace", TFJ4PERATURE: 1982 American Institute of Physics, pp. 505-512. 43. Siegei, Robert and Howell, J.R., "Thermal Radiation Heat Transfer", 2nd. Edition, Hemisphere Publishing Corporation, 1981, Washington, D.C. 44. Chen Kuangyu, "Heat Transfer in Furnaces", Mechanical Industrial Publication, Beijing 1980 (In Chinese). 45. Eckert E.R.G. and McGraw-Hill, N.Y., 46. Masashi M., Thermometry", Drake, R.M. , "Analysis of Heat and Mass Transfer", for Radiation 1972. Kon, K. and Presented at the Utsuno, M., "Emissivity of 109th. ISIJ meeting, April Metals 1985, Lecture No. S390, 1985 ISIJ. 47 Barr, P.V. and Brimacombe, J.K., Department of Metallurgical Engineering, University of British Columbia, private communication. 48. Afgan, N.H. and Beer, J.M., Editor, "Heat Transfer in Flames", Scripta Book Company, division of Hemisphere Publishing Corporation, Washington D . C , 1974, pp. 3-5, pp. 47-73. 49. Samsonov, G.V. and Viniski, I.M. , "Handbook of Refractory Compounds", IFI/Plenum Data Company, N.Y., 1980, pp. 275-276. 50. "Physical Constants of Some Commercial Steels at Elevated Temperatures", Ed. B.I.S.R.A., Butterworths, London, 1953. 51. Weitrteich, J. G., "Control of Strip Shape During Cold Rolling", Journal of the Iron and Steel Institute, December, 1968, pp.1203-1206. 52. Barr, P.V., Rechard, J. and Brimacombe, J.K., "A Heat Transfer Model of the Tall Coke-Oven Flue", Centre of Process Metallurgy, Dept of Metallurgical Engineering, 141 University of British Columbia, Paper to be published. (Private communication) 53. Salter, F.M., "Improve Pusher Furnace Skid Systems", Proceeding of ISI 'Slab Reheating', ISI 54. publication, ' London, 1972, Conference pp83-95. Lowers, T.M., Heap, M.P. , Michelfelder, S., "Mathematical Modelling of Combustion Chamber Performance", Journal of Institute of Fuel, Dec. 1973. pp. 343-351. 55. Chigier, N.A,, "Application of Models Results to Design of Industrial Flames", Journal of the Institute of Fuel, December, 1973. pp 271-278. 56. Massey,R. and Sheridan, S., "Mathematical Ingots and their and Isothermal Modelling the Soaking of Subsequent Rolling", Journal of the Institute of Fuel, Dec. 1973, pp 373-382. 57. Zuber, I. and Konecny, V. , "Mathematical Model of Combustion Chamber for Technical Applications", Journal of the Inst of Fuel, Sept 1973, pp 285-294. 58. Glinkov, M.A. ."General Theory of Furnaces", Journal of the Iron and Steel Institute, June 1968, pp584-594. 59. Moore, Carolyn, (internal "UBC publication), NLE: Zeros University of of Nonlinear British Equations", Columbia, Vancouver, Computing March, Centre 1984, pp 17-20. 60. Peacock, G.R. ," Noncontact pyrometry in Hot Strip Mills and Reheat Furnaces", Iron and Steel Engineer, May 1982, pp30-38. 61. Barnes, Donald C. , "Tips on Industrial Heating, December, 1982, 62. Lu, Yong-Zai and Williams, Computers A - Case Industry, No. 4, 1983, 63. Improving Process Furnace Energy pp28-31. T.J., "Energy Savings and Productivity Study of Efficiency", the Steel Ingot Handling Process", Increases With Computer in ppl-18. Hollander, F.,"Design, Development and Performance of On-line Computer Control in a 3-zone Reheating Furnace", Iron and Steel Engineer, January, 1982, pp44-52. 64. Koinis, F.J. and Meyer, W. Robert, "Improve Thermal Efficiency in Reheating Furnace Through Analysis and Correlation", Iron and Steel Engineer, February 1982, pp. 43-46. 65. Birks, N. and Jackson, W. , "Simultaneous Scaling and Decarbonization Journal of the Iron and Steel Institute, January 1970, pp81-84. of Steels", 142 Appendix I COMPUTER FLOW CHART FOR THE REAL GAS TREATMENT Select INPUT i=l, T . e k .(pL^ 2. . . . . I n linear regression o f ln( e .-Li. )=ln r , -^Plj Note. I iteration of linear a r i | I e 1 ? K > i Repeat are ] INPUT (pL) . T « j = l , 2, . . . . m. k«>Z K an optimum I i < g j " gj " l " gj ~ e is an iterative process to select I—* regression ta It n n times terms of form if there exponential r . 8' 1 > ± J = At r I T. , solve r ^j ) T g.2 M 1. b v j ( T, equation I COmTNUE linear regression to fit ( r_, I I = J = a < T 1, 1. . T T. ). J N M .pL^ ) linear algebra Equation ) ebra 311 1 Appendix n Calculation of the View Factor from the Furnace Chamber to the Slab Bottom Surface Arc AC CD = EB JT/4 /(lWd/2)) = AD = + 2 (s-d) 2 AD = ( rr /4)d + C D According to symmetry: EB = AE AD = BD = ( jr/4)d + (h - d/2) AE According surface is to string method, the F AB-ED = view factor ( AD + (s - d) from EB - the AE - bottom B D )/( finally = (Ah-(d/2)) 2 + 2 - (h - chamber to d/2))/s 2xAB) the slab bottom 145 Appendix Dl FLOW CHART OF THE CODE "ENERGY" GAS EMISSrVTTY AND ABSORPTIVITY TREATMENT INPUT THE GEOMETRY OF THE FURNACE CHAMBER AND PHYSICAL PROPERTIES I DIRECT EXCHANGE COEFFiaENTS CALCULATION SELECT DIFFERENT k. AND a. AT VARIOUS TEMPERATURE TOTAL EXCHANGE COEFFICIENTS CALCULATION INPUT GAS TEMPERATURE AND ASSUMED REFRACTORY WALL AND SLAB TEMPERATURE CALCULATE Ji 1 J 146 PERFORM ON ENERGY BALANCE REFRACTORY SOLVE T H E WALL TEMP. DISTRIBUTION O F REFRACTORY WALL THE USING QUASI-NEWTONIAN TECHNIQUE OBTAIN H E A T F L U X ON T H E SLAB S U R F A C E Z O N E 2-D UNSTEADY CONDUCTION MODEL
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Computer simulation of the push-type slab reheating furnace Li, Zongyu 1986
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Title | Computer simulation of the push-type slab reheating furnace |
Creator |
Li, Zongyu |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | A mathematical heat-transfer model for the slab reheating furnace has been developed. Radiation in the furnace chamber was calculated using the zone method, with the gas temperature distribution being assumed, and heat transfer in the slab was determined using a finite-difference approximation of two-dimensional transient conduction. These individual calculations were coupled to allow prediction of the temperature profiles in, and heat flux to, refractory walls and slabs at any point inside the furnace. The emissive/absorptive characteristics of the gas mixture within the furnace chamber were simulated with a clear-plus-two-gray-gas model which simulated the real gas behaviour to within 5%. For the calculation of radiative exchanges, the furnace chamber was subdivided into 432 isothermal zones, and radiative exchange factors to slab surfaces were evaluated rather than relying on empirical or experimental estimations as in previous studies. An iterative technique was devised in order to combine the radiative and slab heat conduction calculations. For the purpose of identifying the mechanism of skidmark formation, the region of skidrail/slab contact was examined in detail by introducing a radiation shielding factor to account for the presence of the skid structure. The gas temperature distribution inside the furnace chamber was found to have a significant influence on the heat flux to the slab surface. Nonuniform gas temperature transverse to the push direction causes an uneven transverse slab temperature distribution and subsequent rolling problems. Higher gas temperatures near the sidewall refractory were shown to cause serious distortion of the transverse heat-flux distribution. The heating practice for the hot charging of slabs was simulated by the model in order to improve the process from the standpoint of energy conservation and slab temperature uniformity. Model predictions have shown that the fuel input could be reduced substantially near the slab entrance where the port to the chimney is located, thus maximizing the residence time of the combustion products. Alternatively the throughput of the furnace can be increased if the fuel input remains the same as for charging cold slabs. The extent of increase in production rate can be determined by the off-line computer model. The model was used to predict the thermal behaviour of slabs for various thicknesses, steel grades and push rates. The results consistently indicated that the selection of an appropriate push rate is crucial to the final temperature distribution. The study of the mechanism of skidmark formation showed' that the radiation shielding effect of the skidrail was the dominant factor, accounting for 90% of the heating deficit around the slab/skidrail contact region. Computer simulation of the possible measures that could be taken to alleviate the skidmark formation has indicated that reducing the height and width of the skidrail improved radiative heat transfer in the contact region. Coating highly reflective materials on the exterior surface of the skidrail to increase reflectivity from 0.3 to 0.8, could enhance heat transfer locally around the the skidrail by about 25% - 30% when the skidrail temperature is lower than the slab bottom temperature. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0078627 |
URI | http://hdl.handle.net/2429/26308 |
Degree |
Master of Applied Science - MASc |
Program |
Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
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