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Computer simulation of the push-type slab reheating furnace Li, Zongyu 1986

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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.  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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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