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Heat transfer in direct-fired rotary kilns Gorog, John Peter 1982

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HEAT TRANSFER IN DIRECT-FIRED ROTARY KILNS by JOHN PETER GOROG .S. (Hons.)* Michigan Technological U n i v e r s i t y , 1975 M.S., Michigan Technological U n i v e r s i t y , 1977  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of M e t a l l u r g i c a l Engineering  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1982 ©John Peter Gorog, 1982  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree t h a t for  Library  shall  for reference  and  study.  I  f o r extensive copying of  that  h i s or  be  her  g r a n t e d by  f i n a n c i a l gain  shall  not  the  be  of  Department o f  Metallurgical  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 October  27,  1982  further this  Engineering  Columbia  thesis  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  make  head o f  representatives.  copying or p u b l i c a t i o n  the  University  the  s c h o l a r l y p u r p o s e s may  understood  the  I agree that  permission by  f u l f i l m e n t of  advanced degree a t  Columbia,  department or for  in partial  written  ABSTRACT The overall heat-transfer mechanism within a d i r e c t - f i r e d rotary k i l n has been examined theoretically. To accomplish this task, the work has been divided into three parts:  (1) the characterization of radiative heat  transfer within the freeboard area; (2) the overall heat transfer mechanism in the absence of freeboard flames; and, (3) the overall heat transfer mechanism in the presence of freeboard flames. The radiative heat transfer between a nongray freeboard gas and the i n t e r i o r surface of a rotary k i l n has been studied by evaluating the fundamental radiative exchange integrals using numerical methods. Direct gas-to-surface exchange, reflection of the gas radiation by the k i l n w a l l , and k i l n wall-to-solids exchange have been considered.  Graphical repre-  sentations of the results have been developed which f a c i l i t a t e the determination of the gas mean beam!ength and the total heat f l u x to the wall and to the solids.  These charts can be used to account f o r both k i l n size  and solids f i l l ratio as well as composition and temperature of the gas. Calculations using these charts and an equimolal CO^-h^O mixture at 1110 K indicate that gas-to-surface exchange i s a very localized phenomenon. Radiation to a surface element from gas more than half a k i l n i n diameter away i s quite small and, as a r e s u l t , even large axial gas temperature gradients have a negligible effect on total heat f l u x .  Results are also  presented which show that the radiant energy either reflected or emitted by a surface element i s limited to regions less than 0.75 k i l n diameters away. The radiative exchange integrals have been used, together with a  ii  modified reflection method, to develop a model f o r the net heat flux to the solids and to the k i l n wall from a nongray gas.  This model i s compared  to a simple r e s i s t i v e network/gray-gas model and i t i s shown that substantial errors may be incurred by the use of the simple models. To examine the overall heat-transfer mechanism i n the absence of freeboard flames a mathematical model has been developed to determine the termperature distribution in the wall of a rotary k i l n .  The model, which  incorporates a detailed formulation of the radiative and convective heattransfer coefficients i n a k i l n , has been employed to examine the effect of different k i l n variables on both the regenerative and the overall heat transfer to the solids.  The variables include rotational speed, per cent  loading, temperature of gas and solids, emissivity of wall and s o l i d s , convective heat-transfer coefficients at the exposed and covered w a l l , and thermal d i f f u s i v i t y of the w a l l .  The model shows that the regenerative  heat flow i s most important in the cold end of a rotary k i l n , but that generally the temperature distribution and heat flows are largely independent of these variables.  Owing to this i n s e n s i t i v i t y i t has been pos-  s i b l e to simplify the model with the aid of a r e s i s t i v e analog.  Calcula-  tions are presented indicating that both the shell loss and t o t a l heat flow to the bed may be estimated within 5 per cent using t h i s simplified model. F i n a l l y , to examine the overall heat-transfer mechanism i n the presence of freeboard flames a mathematical model has been developed to determine both the temperature and heat flux distributions within the flame zone of a rotary k i l n .  The model, which i s based on the one-  dimensional furnace approximation, has been employed to examine the iii  effects of fuel type, f i r i n g rate, primary a i r , oxygen enrichment and secondary a i r temperature on the flame temperature, solids heat flux shell losses, and overall flame length.  iv  TABLE OF CONTENTS Page ii v vi i i ix xv xix  Abstract Table of Contents List of Tables ... L i s t of Figures .. L i s t of Symbols ., Acknowledgement ., Chapter 1 . 2  1  INTRODUCTION .. RADIATIVE HEAT TRANSFER WITHIN THE KILN FREEBOARD 2.1  Introduction  6  • ••  2.2 Representation of the emissive characteristics o f the gas, solids and k i l n wall 2.2.1  Emissive characteristics of the freeboard gas...  2.2.2 Emissive characteristics of the k i l n wall and solids 2.3 Radiative exchange between freeboard gas and k i l n wall. 2.3.1  2.4  Total radiative heat flux from an isothermal gas  6  7 7 11 11 12  2.3.2 Radiative heatflux from a non-isothermal freeboard gas  16  2.3.3  22  Reflected gas radiation  Radiant exchange between the freeboard gas and k i l n solids  28  2.5 Radiant exchange between the k i l n solids and wall  35  2.6 Mathematical model of the total radiative exchange in rotary kilns  38  2.6.1  Real gas model development  2.6.2 Comparison between real and gray gas radiative models i n rotary kilns  v  38 46  Page  Chapter OVERALL HEAT TRANSFER IN THE ABSENCE OF FREEBOARD FLAMES.  50  3.1  Introduction  50  3.2  Previous work  <  51  3.3 Model development  •  53  3.4  Heat-transfer coefficients in rotary kilns 3:4.1  Radiative heat-transfer coefficients  3.4.2 Convective heat-transfer coefficients  60 68 72  3.5 Model predictions  3.6  60  3.5.1  Convective heat-transfer coefficient at the covered wall •  79  3.5.2  Kiln speed  82  3.5.3 Convective heat-transfer coefficient at the exposed wall  84  3.5.4  Kiln loading  86  3.5.5  Emissivities of solids and wall  88  3.5.6 Thermophysical properties of wall  91  Simplified r e s i s t i v e network  94  OVERALL HEAT TRANSFER IN THE PRESENCE OF A FREEBOARD FLAME.  100  4.1  Introduction  100  4.2  Previous work  101  4.3 Model development  103  4.3.1  Selection of modelling technique  103  4.3.2 Model assumptions  113  4.3.3 Model formulation and solution  118  vi  Chapter  Page 4.4 Model predictions.. 4.4.1  5  123  Fuel type  129  4.4.2 Firing rate  134  4.4.3  136  Secondary a i r temperature...  4.4.4 Use of primary a i r  141  4.4.5  146  Oxygen enrichment.  CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK..  151  5.1  151  Conclusions  5.2 Suggestions for future work  154  REFERENCES  155  Appendix Al  Radiative properties for equimolal C0 -H 0 mixtures  158  A2  Derivation of equation (2.9)  161  A3  Finite difference equations for rotary k i l n wall  167  A4  Determination of radiative heat flows using the network method  170  2  2  A5  Solution f o r the r e s i s t i v e network used to predict heat flows and temperatures i n the absence of a freeboard flame.. 189  A6  Solution f o r the r e s i s t i v e network used to predict heat flows and temperature in the presence of a freeboard flame.. 192  A7  FORTRAN source l i s t i n g and sample output for k i l n wall model  195  FORTRAN source l i s t i n g and sample output f o r k i l n flame model  246  A8  vii  LIST OF TABLES Table  Page  3-1  Constants for use with Eq. (3.18)  71  3-2  Summary of input data used for computer simulations  7 4  3- 3  Influence of process variables on regenerative action of the wall and total heat flow to solids near feed end of a rotary k i l n .  °1  4- 1  Combustion properties of natural gas  125  4-2  Combustion properties of No. 6 fuel o i l . .  126  4-3  Combustion properties producer gas (Lurgi-Air Blown)  127  4-4  Summary of input data used for computer simulations  128  Al-1  Summary of emissivity data for equimolal C0„-H 0 gas mixtures at 830, 110 and 1390 K ...t.... Summary of absorptivity data for equimolal CO2-H2O. gas mixtures at 1110 K for blackbody radiation at 277, 555 and 833 K  Al-2  A4-1 A4-2  9  159 160  Summary of view factors needed to evaluate the 4-zone k i l n wall model  174  Summary of view factors needed to evaluate the 16-zone r e s i s t i v e analog of Fig. 4-3  180  viii  LIST OF FIGURES Figure 1- 1  Schematic diagram of rotary k i l n showing; (a) axial cross section; (b) major heat flow paths i n non-flame zone I I ; and; (c) major heat flow paths i n flame zone I  2  2- 1  Emissivity, e , and emissive character, de  2-2  equimolal C0 - H 0 gas mixture at 1390, 1710 and 830 K... Absorptivity of an equimolal CO2 - H2O gas mixture at 1110 K for blackbody radiation at 277, 555 and 833 K  10  Exchange integral for gas-to-kiln wall radiation versus k.pD  15  Top: Schematic cross-section of an empty rotary k i l n showing s l i c e s used i n evaluating the non-isothermal heat transfer from the freeboard gas to a d i f f e r e n t i a l element, dA, on the k i l n wall. Bottom: Assumed temperature prof i l e in freeboard gas  17  q  2  2-3 2-4  2-5  2  2  Heat-flux d i s t r i b u t i o n densities f o r gas-to-wall radiative heat flux versus axial position, z/D, f o r both an isothermal and a non-isothermal equimolal C0 - H^O freeboard gas mixture  21  Schematic diagram of the cross-section of an empty rotary k i l n showing d i f f e r e n t i a l elements, dA-j and dA2» used to evaluate the nature of reflected gas radiation from the k i l n wall  23  Exchange integral f o r reflected gas radiation-to-kiln wall versus k.pD  26  2  2-7  2-8  9  Normalized cumulative gas-to-wall radiative heat f l u x versus axial position, z/D, for both an isothermal and non-isothermal equimolal C0 - H 0 freeboard gas mixture 2  2-6  ( r ) / d r , for an  g  i  2-9  Flux surface of reflected gas radiation incident on k i l n wall for a small p i l o t k i l n , D = 0.4m, f i l l e d with an isothermal equimolal C0 - H 0 freeboard gas mixture (right). Iso-flux lines for the flux surface shown at right ( l e f t )  27  Schematic diagram of the cross-section of a rotary k i l n showing d i f f e r e n t i a l elements, dA, and dA , used to evaluate either the radiative exchange between the freeboard gas and the solids or the radiative exchange between the solids and k i l n wall  29  ?  2-10  ?  2  ix  Figure 2-11 2-12  2-13  2-14  2-15  2-16  2- 17  3- 1  3-2  Page Exchange integral f o r gas-to-sol ids radiation versus k.pD as a function of f i l l r a t i o , F/D. Heat flux d i s t r i b u t i o n from gas-to-sol ids surface as a function of both position across the solids surface and solids f i l l r a t i o , F/D Modified solids surface-to-wall view factor versus axial position, z/D, f o r a p i l o t k i l n , D = 0.4m, f i l l e d with an isothermal equimolal freeboard gas as a function of solids f i l l r a t i o , F/D.... Schematic diagram of the cross-section of a rotary k i l n showing the path of radiant energy leaving the solids surface.. Schematic diagram of the cross-section of a rotary k i l n showing the path of radiant energy emitted from the freeboard gas  3 3  3 4  3 7  4 0  4  '  Radiative resistance network for a rotary k i l n where the freeboard gas, k i l n wall and solids surface are assumed to be gray  47  Radiant exchange, for a r o t a r y - k i l n , D = 6m, from an isothermal freeboard gas at 1110 K to an isothermal black solids surface at 833 K for both a real and a gray gas at a f i l l r a t i o , F/D, of 0.30  48  Schematic diagram of rotary kiln showing nodal configuration used to determine temperature distributions i n the k i l n wall  5 6  Computer flow-diagram used to determine temperature distributions i n the rotary-kiln wall  68  3-3  1-zone  62  3-4  4-zone radiation analog of rotary-kiln wall  53  3-5  Circumferential k i l n wall temperature p r o f i l e s used i n the 1- and 4-zone radiation analogs Comparison of 1- and 4-zone radiant heat flows incident on the inner k i l n wall  55  3-6  radiation analog of rotary-kiln wall  x  56  Page  Figure 3-7  3-8  3-9  3-10  3-11  3-12  3-13  3-14  3-15  3-16  3-17  Axial temperature profiles f o r a d i r e c t - f i r e d , lime k i l n with no preheater. Sankey diagrams show the r e l a t i v e contribution of freeboard and regenerative heating of the solids within the calcination, preheat and drying zones of the rotary k i l n  73  (a) Circumferential inner and (b) radial wall temperature profiles for the calcination or high temperature zone of a rotary k i l n  76  (a) Circumferential inner and (b) radial wall temperature profiles for the drying or low temperature zone of a rotary k i l n  77  The influence of the convective heat transfer c o e f f i c i e n t at the covered wall on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n  80  The influence of k i l n speed on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n  83  The influence of the convective heat-transfer c o e f f i c i e n t at the exposed wall on the inner wall circumferential temperature within the low temperature region of the rotary k i l n . ..  85  The influence of solids f i l l ratio on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n . . . .  87  The influence of solids emissivity on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n  89  The influence of wall emissivity on the inner wall c i r cumferential temperature p r o f i l e within the high temperature region of the k i l n  90  The influence of the thermal conductivity of the wall on the radial temperature p r o f i l e within the low temperature region of the k i l n  92  The influence of the specific heat of the wall on the radial temperature p r o f i l e within the low termperature region of the k i l n .  93  xi  Figure 3-18  Page Simplified r e s i s t i v e network used to predict heat flows within the rotary k i l n  95  Exposed wall temperatures predicted by r e s i s t i v e analog versus the integrated exposed wall temperature within the c a l c i n a t i o n , preheat and drying zones  96  Heat received by the solids predicted using the r e s i s tive analog versus the heat received by the solids based on the integrated average wall temperatures within the c a l c i n a t i o n , preheat and drying zones..  98  Predicted heat loss from the k i l n shell using the r e s i s tive analog versus the integrated heat loss from the shell within the c a l c i n a t i o n , preheat and drying zones..,  99  Zonal configurations and temperature d i s t r i b u t i o n s used for both the (a) one-dimensional and .(b) zone models  106  4-2  Resistive analog of one-dimensional model  108  4-3  Resistive analog of multi-zone model  109  4-4  The influence of axial temperature gradients on the onedimensional flame model  110  4-5  The influence of relative flame size on the onedimensional flame model.  HI  4-6  The influence of k i l n wall r e f l e c t i v i t y on onedimensional flame model  112  4-7  Variation of flame emissivity with distance from burner...  114  4-8  Simplified r e s i s t i v e network used to predict heat flows within the flame zone of a rotary k i l n  117  Schematic diagram of the cross-section of a rotary k i l n showing heat flow paths and r e s i s t i v e elements f o r (a) non-flame zone and (b) flame zone  119  Schematic diagrams of rotary k i l n showing (a) zonal configuration f o r one-dimensional flame model and (b) the major heat flows within each s l i c e  120  Computer flow-diagram used to determine temperatures and heat flows within the flame zone of a rotary k i l n  124  3-19  3-20  3-21  4-1  4-9  4-10  4-11  xii  Figure 4-12 4-13 4-14  4-15 4-16 4-17 4-18 4-19 4-20  4-21 4-22 4-23  4-24 4-25 4-26  Page The influence of fuel type on flame temperatures within the rotary k i l n  130  The influence of fuel type on the solids heat f l u x within the flame zone of a rotary k i l n  131  ....  Bar diagrams showing the influence of fuel type on the energy distribution within the flame zone of a rotary kiln  133  Schematic diagram of natural gas burner used i n flame model calculations  135  The influence of f i r i n g rate on flame temperatures within the rotary k i l n  137  The influence of f i r i n g rate on the solids heat f l u x within the flame zone of a rotary k i l n  138  The influence of secondary a i r temperature on flame temperatures within the rotary k i l n  139  The influence of secondary a i r temperature on the solids heat flux within the flame zone of a rotary k i l n  140  Bar diagrams showing the influence of secondary a i r temperature on the energy distribution within the flame zone of a rotary k i l n  142  The influence of primary a i r on flame temperatures within the rotary k i l n  143  The influence of primary a i r on the solids heat f l u x within the flame zone of the rotary k i l n  144  Bar diagrams showing the influence of primary a i r on the energy distribution within the flame zone of a rotary kiln...  145  The influence of oxygen enrichment on flame temperatures within the rotary k i l n  147  The influence of oxygen enrichment on the solids heat flux within the flame zone of a rotary k i l n  148  Bar diagrams showing the influence of oxygen enrichment on the energy distribution within the flame zone of a rotary k i l n  149  xiii  Figure A2-1 A2-2  A3-3  A4-1  A4-2  Page Orthogonal views of cylinder and hemisphere; (a) elevation; (b) end; and; (c) plane  163  Schematic diagram of cylinder of diameter D and hemisphere of radius r showing the l i m i t s of integration used i n Eq. (A4.5)  164  Projected area of intersection for a cylinder of diameter D and hemispheres with r a d i i of r/D = 0.5, 1, 1.5 and 2  I  Schematic diagram of the cross-section of a rotary k i l n used to evaluate view factors for (a) 1-zone and (b) 4-zone analogs  172  6  6  Schematic diagram of concentric cylinders showing zonal configuration used i n 16-zone analog  176  A4-3  View factor, F^, curves for concentric cylinders  177  A4-4  View factor, F^, curves for concentric cylinders  178  A4-5  Schematic diagram of the cross-section of a rotary k i l n used to evaluate view factors within the flame zone  187  xiv  LIST OF SYMBOLS a  weighting factor for gray-plus-clear gas model, emissivity  A  area (m )  A  xs  2 cross-sectional area of solids (m )  A  xt  AF  2 cross-sectional area of empty k i l n (m ) stoichiometric air-to-fuel r a t i o (kg/kg)  b  weighting factor for gray-plus-clear gas model, absorptivity specific heat (J/kg)  P  C  d d  o  distance between two elements (m) equivalent burner diameter (m)  D  k i l n diameter (m)  E  2 emissive power (W/m )  E  2 emissive power, l o c a l l y defined (W/m )  f  absorption c o e f f i c i e n t , absorptivity (nf atm )  F  view factor  F  L  9 G 9 Gr  G  1  1  flame length (m) function, l o c a l l y defined 2 momentum flowrate (kg m/s ) 2 gas mass flux (kg/m hr) Grashof number 2  h  cv  h  R  H  F  H  loss  convective heat-transfer coefficient (W/m K) 2 radiative heat-transfer coefficient (W/m K) gross heating value of fuel (J/kg) hydrogen loss  XV  exchange integral for gas-to-kiln wall radiation  h  exchange integral for gas-to-kiln wall radiation,  1  T  2  exchange integral for gas-to-kiln wall s l i c e radiation  l  exchange integral for reflected gas radiation-to-kiln wall  3  l  *4  exchange integral for gas-to-sol ids radiation  J  radiosity (W/m )  J  r a d i o s i t y , l o c a l l y defined (W/m )  k  absorption c o e f f i c i e n t , emissivity  k  thermal conductivity (W/m K) - chapters 3 and 4  K  extinction coefficient ( n f )  o  ( n f atm" ) - chapter 2 1  1  1  L  average mean beamlength (m)  m  mass flowrate  N  unit vector normal to element, l o c a l l y defined  P  partial pressure (atm)  PA  per cent stoichiometric a i r as primary  PO  per cent primary a i r as oxygen  Pr  Prandtl number  q  heat flow (W/m)  Q'  heat flow (W)  r  radial position (m)  m  R  I  R  f  o Re  R  (kg/s)  inner radius (m) radius separating steady and unsteady state regions (m) outer radius (m) transverse flow Reynold's number  Re  rotational Reynold's number  R  vector between two elements, l o c a l l y defined xv i  s  width of s o l i d s  t  time  T  temperature  ^cov ex  T  surface  (m)  (s) (K)  i n t e g r a t e d average c o v e r e d w a l l  temperature  i n t e g r a t e d average exposed w a l l  temperature  V  volume  w  d i s t a n c e from c e n t e r o f s o l i d s  W  halfwidth of solids  z  d i s t a n c e along  a  absorptivity  a  g  (m )  absorptivity  B  dimensionless  e  emissivity  0  angle,  P p  e •  P  kiln axis  of real  (m)  for its  distance along  defined l o c a l l y  reflectivity  or dimensionless  equivalent fuel density  z/r  kiln axis,  z/D  density  d i s t a n c e , r/D, (kg/m  defined  )  3  slope  a  Stefan-Boltzman  T  transmissivity  diffusivity  transmissivity  (m  2  /s)  constant  of real  (5.67 x 10  gas  for its  -8  own  (rad)  h a l f angle solid  kiln axis,  (kg/m )  thermal  angle  gas  own r a d i a t i o n  (rad)  d i s t a n c e along  K  n  gas  angle  wall  (m)  dimensionless  kiln  *  surface  s u r f a c e t o element on s o l i d s  subtended by s o l i d s (sr)  xvii  (rad)  2 4 W/m K )  radiation  locally  (m)  Subscripts a  ambient  cov  covered  cp  combustion products  en  entrained  ex  exposed  f  f l ame  F  fuel  9 .  gas  gen  generated  i  index  j  index  max  maximum  min  minimum  0  at burner  pa  primary a i r  s  sol ids  sa  secondary a i r  sh  outer shell  T  total  w  wall  A  wave length (um)  xviii  ACKNOWLEDGEMENTS To a l l the people who helped throughout my stay at The University of British Columbia I would l i k e to extend my sincere gratitude. I would expecially l i k e to thank my co-advisors Dr. J . K. Brimacombe and Dr. T. N. Adams who allowed me to work in a very independent manner while s t i l l  offer-  ing the guidance and advice needed to complete this project. The d i s cussions and assistance of Dr. Alec Mitchell, Dr. Ernest Peters, Mr. Peter Madderom, Dr. A. M. Johnson and Dr. Stewart Ballantyne have also proven to be invaluable. I would also l i k e to thank Stelco Inc. and NSERC for t h e i r assistance in the form of financial support of this project. Special thanks must also be given to Mrs. M. Curtis for her assistance i n helping me to obtain U.S. student loans during my stay in Canada. F i n a l l y , I would l i k e to take.this opportunity to formally thank my parents, John and Elizabeth, for the endless encouragement and patience they have given during the long years of study.  Without t h e i r help i t  would have been impossible for me to complete my education.  In response  to the love they have shown, I would l i k e to dedicate my thesis to both of them.  1 Chapter 1  INTRODUCTION  Rotary kilns are primarily heat exchangers in which s o l i d s are heated to reaction temperature by hot gas flowing in the freeboard above the bed.  Processes in which rotary kilns are used as reaction vessels  include: 1)  the roasting of p y r i t i c ores  2)  the direct reduction of iron bearing ores  3)  the induration of iron-ore pellets  4)  the incineration of industrial waste material  5)  the calcination of limestone, alumina or dolomite  As shown i n Fig. 1-1(a),  a rotary k i l n i s simply a long refractory-  lined cylinder inclined at a small angle to the horizontal. Material to be processed i s fed into the upper or back end while the vessel i s slowly rotated about i t s axis.  In response to this rotation the material moves  along the k i l n length where i t i s exposed to hot combustion gases produced by a burner located at the lower or front end of the k i l n .  A k i l n with  this heating configuration i s referred to as being d i r e c t - f i r e d in that a l l the fuel i s burnt within the confines of the vessel.  This i s i n  contrast to an indirect-fired k i l n in which the fuel i s burnt externally to the vessel.  In this study, only d i r e c t - f i r e d kilns w i l l be  considered.  As with any high-temperature, energy intensive reactor, soaring energy costs coupled with an increasingly competitive market place have  2  Figure 1-1  Schematic diagram of rotary k i l n showing; (a) axial cross section; (b) major heat flow paths i n non-flame zone I I ; and; (c) major heat flow paths i n flame zone I .  provided incentives to more c l e a r l y define the fundamental p r i n c i p l e s 2-11 that govern k i l n operation.  This has led some investigators  mathematically model rotary k i l n s .  to  In these studies attempts have been  made to predict or examine operating conditions which w i l l enable the k i l n industry to better understand t h e i r operating practices while simultaneously  improving both heat u t i l i z a t i o n and product q u a l i t y .  While  a l l of these investigators have c l e a r l y identified these objectives as an overall goal, in general, the success that any of these models has achieved has been rather limited in that none of the models are able to predict accurately k i l n performance for more than a very l i m i t e d range of operating conditions.  Inasmuch as rotary k i l n s are primarily  heat exchangers, the ultimate success of any k i l n model i n predicting either temperature p r o f i l e s or product quality i s c l e a r l y dependent on i t s a b i l i t y to characterize accurately the various heat flows within the kiln.  For this reason, the emphasis of this thesis i s placed on  how  to best characterize the mechanisms of heat transfer within a d i r e c t fired rotary k i l n . The heat transfer within the rotary k i l n i s complex because i t i n volves radiative exchange in the freeboard space as well as gas and solids convection, and conduction within the bed.  However owing to the  high temperatures attained in the freeboard, radiation i s believed to be the dominant heat-transfer mechanism.  1  Examination of F i g . 1-1 (a)  reveals that two d i s t i n c t regions of heat transfer within the k i l n be e a s i l y i d e n t i f i e d :  the flame zone I and the non-flame zone I I .  In the l a t t e r region, shown schematically i n Fig. 1—1(b), heat i s  may  4 transferred to the solids burden by two paths, across the exposed upper surface and the covered lower surface of the bed.  The mechanisms of  heat flow to the two surfaces are very different.  At the upper surface  the solids receive heat d i r e c t l y by radiation and convection from the freeboard gas whereas at the lower surface, heat flows by a combination of radiation and conduction from wall to solids.  The l a t t e r heat-  transfer path i s part of the regenerative cycle.of the k i l n wall which, as i t rotates through the freeboard, also receives thermal energy v i a radiation and convection from the hot combustion gas.  Of course a portion  of this heat i s lost to the surroundings as depicted i n Fig. l - l ( b ) .  For  region I of the k i l n , in which flames are present in the freeboard, the overall mechanism of heat transfer i s similar to that described for region I I , however, the radiative heat transfer to the burden surface and exposed wall i s further complicated by the presence of a v i s i b l e flame, shown schematically i n Fig. 1-1(c). The primary objective of this study i s to describe mathematically the overall heat-transfer process within a rotary k i l n .  To accomplish  this task, the work has been divided into three parts: 1)  The characterization of radiative heat transfer within The freeboard area.  2)  The overall heat transfer mechanism in the absence of freeboard flames.  3)  The overall heat transfer mechanism in the presence of a freeboard flame.  In this way, a framework i s developed which, in the future, may be used  to predict more accurately heat flows at any position along the enti k i l n axis.  6  Chapter 2  RADIATIVE HEAT TRANSFER WITHIN THE KILN FREEBOARD  2.1  Introduction Radiation amongst the gas, solids and wall has been incorporated 2-11  into a number of mathematical models  of rotary k i l n processes based  on simplifying assumptions such as gray gas behavior. Moreover many of the models have assumed simplified geometries and ignored the effects of axial temperature gradients on radiation.  However the v a l i d i t y of  these assumptions has not been checked, largely because a detailed analysis of radiative exchange involving a real gas i n rotary kilns has not been developed. Thus the primary objectives of the work described in the present chapter are to examine theoretically the radiative interchange amongst the gas, solids and wall of a rotary k i l n , and to develop a radiation model based on real-gas behavior. The chapter may be divided into three sections: 1)  The radiative characterization of the freeboard gas, k i l n wall and sol ids.  2)  The determination of the radiative exchange between freeboard gas and k i l n w a l l , gas and solids,and  k i l n wall and  solids. 3)  The prediction of total radiative exchange amongst the freeboard gas, k i l n wall and solids based on a real-gas radiative model, and comparison of simpler gray-gas models.  7 In this way, the importance of real-gas behavior in rotary kilns i s demonstrated. 2.2  Representation of the emissive characteristics of the gas, solids and k i l n wall 2.2.1  Emissive characteristics of the freeboard gas The freeboard gases common to rotary k i l n operation are  composed mainly of CC^ - h^O mixtures in nitrogen generated by the combustion of hydrocarbon fuels with a i r .  Because these gases emit and  absorb radiation in d i s t i n c t bands, the use of gray-gas approximation, i.e. the emissivity and absorptivity are equal and constant at a given gas temperature, i s not v a l i d .  Rather, these mixtures should be treated  as real gases in which the emissivity and absorptivity need not be equal because absorptivity, due to the banded characteristics of the CO^ and h^O, i s a function of both the gas and the emitting surface temperatures. 12 For these mixtures Hottel and Sarofim  suggest that gas radiation can  be visualized as that due to the weighted sum of a s u f f i c i e n t number of gray and clear gas components to approximate the banded characteristics of a real gas.  Thus according to t h e i r approach the gas emissivity may  be represented by e  = Ea (1 - e " i ) i k  y  p r  (2.1)  subject to the restrictions that the a^ are a l l positive and za  i  = 1  since emissivity approaches 1 with increasing pr.  (2.2)  8 In a similar manner, the absorptivity of a real gas at temperature Tg f o r blackbody radiation from a surface at T" can be represented by s  a  = Zb ( l - e i P ) i _ f  9  (2.3)  r  1  where again the b. are a l l positive and Eb.  = 1  (2.4)  In the present study, equimolal C0 - H 0 gas mixtures have been 2  2  used to represent the freeboard gas i n a rotary k i l n .  Fig. 2-1 shows the  emissivity of these gas mixtures at 830, 1.1.1.0 and 1390K (data points) taken from the experimentally based charts of Hottel and Sarofim.  The  solid lines i n the same plot represent a five-term f i t of Eq. (2.1) to the experimental points, achieved by the method of least squares using 13 a pseudo Gauss-Newton algorithm.  I t may be noted that a simpler  method outlined by Hottel and Sarofim could be employed to f i t the emiss i v i t y data i f computer time i s unavailable. Fig. 2-2 shows the absorptivity of an equimolal C0 - HgO gas mixture at 111 OK f o r black2  body radiation emitted from a surface at 277, 555 and 833K where the solid lines represent a three-term f i t of Eq. (2.3) to the experimental points.  A complete summary of the radiative properties f o r the equimolal  COg * H^O gas mixtures used in this study i s presented i n Appendix A l . Owing to the diversity of k i l n operations, the gas temperatures  arbi-  t r a r i l y chosen here are typical of those found i n a small p i l o t k i l n  1  while the gas compositions represent the products of combustion of (CH ) 2  n  where the ratio of C0 to H 0 i s one to one. Although the use 2  2  of an equimolal gas at the selected temperatures w i l l affect the numerical values reported below, the results of this work are general in  P { C O  0  2  0  (ft-atm)  6  .  2  + H  8  2  1.5 P ( C O  Figure 2-1  + H ^ ) D  4  I  0.5  2  2  O ) D  (m  -  10  2.5  3  atm)  Emissivity, £ , and emissive character, d e ( r ) / d r , for an equimolal g  C0  9  g  - H„0 gas mixture at 1390, 1110 and 830 K.  p(co +H o)D(ft - atm) 2  2  _0LO  Tg = 1110 K  a  0.6  0.75  1.00  p(co +H o)D ( m - a t m ) 2  Figure 2-2  2  Absorptivity of an equimolal C0 - HgO gas mixture at 1110 K 2  for blackbody radiation at 277, 555 and 833 K.  11  that the techniques employed may be applied to kilns operating under a wide variety of conditions. 2.2.2  Emissive characteristics of the k i l n wall and solids In the current work the k i l n wall and solids are assumed, at  different times, to be either gray (e = a = constant) or black (e = a = 1). The gray assumption has been used because the spectral emissivities of the solids and wall refractories common to rotary kilns are not known. This assumption i s thought to introduce only a small error provided the emissivity of both the solids and wall are high.  The blackbody assumption  was made in some examples only for purposes of s i m p l i f i c a t i o n , and i s not necessary to the f i n a l solution. 2.3  Radiative exchange between freeboard gas and k i l n wall The characterization of radiative exchange between the freeboard  gas and k i l n wall has been studied by considering 1)  three cases.  A black cylinder f i l l e d with an isothermal gas f o r which the total radiative flux incident on the k i l n wall has been estimated using a d i f f e r e n t i a l element located on the wall.  2)  A black cylinder f i l l e d with a non-isothermal gas to establish the influence of temperature gradients i n the freeboard gas on the total heat f l u x .  3)  A gray cylinder f i l l e d with an isothermal gas t o determine the nature of reflected gas radiation.  In each of these cases, an empty cylinder has been assumed, but as w i l l be shown l a t e r , this has a minor effect on the f i n a l solution.  12 2.3.1  Total r a d i a t i v e heat flux from an isothermal  gas  For the evaluation o f the r a d i a t i v e exchange between an isothermal freeboard gas and k i l n wall consider an empty k i l n where the k i l n wall i s assumed black to eliminate surface r e f l e c t i o n s .  Then the  r a d i a t i v e flux from the freeboard gas to a d i f f e r e n t i a l element, dA, located on the k i l n wall is given by  14*  cosedA  j— e  K,E„ dV A  o  A  -K r A  dx  (2.5)  irr  V  where dV = 2Trr sin<J>d4dr = r d t t d r 2  (2.6)  2  Integrating Eq. (2.5) over a l l wavelengths and rearranging y i e l d s  -v H A  dA  r  =  r  y y J J  d  £  n (  r  ^aui  )  dr  c  o  s  e  ni^dndr TT  (2.7)  9  fi  In this form, the heat flux from the gas to the wall may be considered to originate from a number of concentric d i f f e r e n t i a l hemispherical  shells  of radius r and thickness dr centered above the d i f f e r e n t i a l element dA. The contribution to the total emissivity that each of these s h e l l s would make i f they were f u l l  hemispheres is  ( d e ( r ) / d r ) d r . However, as only a g  f r a c t i o n of these s h e l l s l i e e n t i r e l y within the k i l n the emissive contributions from each must be m u l t i p l i e d by a c o r r e c t i o n f a c t o r ,  g(r),  equal to the area projected onto the base of the hemisphere by the section of the hemisphere within the k i l n w a l l , divided by the projection of the e n t i r e hemisphere.  For volume-surface r a d i a t i v e exchange Hottel  *see l i s t of symbols f o r d e f i n i t i o n of terms used  13 and Sarofim have shown that g(r) = /cosedft/V, therefore Eq. (2.7) may be rewritten as de  %+M_f  (r) dr E  9 g ( r )  JJ  dA  r|. geometric correction factor  I [  ^ 9  Av dr  emissive character of the gas  emissive power  For the present configuration g(r) i s equal to the projected area of intersection between a hemisphere of radius r and k i l n or cylinder wall of diameter D, divided by the total projected area of the hemisphere. By use of symmetry, g(r) takes the following dimensionless form for t h i s case r D 2 /*' 2 2 2 2 2 % g ( r ) = ^ 2 J ' [ ( J ) •-(§)• - ( -^- ") ] d ( f ) r  V  9  0  for £  <_ 1  "  - 1 for £  > 1  (2.9)  L  where for hemispheres with r a d i i larger than the k i l n diameter the lower l i m i t i s set so as to eliminate those portions of the gas or hemisphere which l i e outside the k i l n wall.  A complete derivation of Eq. (2.9) i s  presented in Appendix A2. Having considered the geometry, the contribution to the total emissivity from each of the hemispherical shells may be evaluated by d i f f e r e n t i a t i n g Eq. (2.1) as follows (r)  de  -r dr 9  -k.pr = za.k.pe . l v  1  (2.10)  14 Substituting Eqs. (2.9) and (2.10) into Eq. (2.8) and l e t t e r i n g p = r/D and  5 = z/D P  CO  VvdA .  4 TTP  E dA  g  Ea k.pDe"S / pDp  /* 2  [P -S -(P -C ) J 2  i  •/  2  2  %  d? dp  (2:11)  ^  1  0  0  for p < 1  VP -]  for p > 1  2  Finally, letting 6 =  2  2  5/P the total radiative flux from an isothermal  freeboard gas to a d i f f e r e n t i a l element located on a black k i l n wall may be evaluated as follows  dA _ Ea.I,.  g  }  E: (dA  1  • . i•  (2.12)  11  g where -1 1 I = i = | -k Dy* ' i Mi 2i k  iP  e  p D p  y [l-B -p (l-B ) f 2  2  2  2  2  dg d  P  (2.13)  x  2  for p < 1 Vl - (1/P ) f o r P > 1 2  Due to symmetry along the k i l n axis Eq. (2.13) represents h a l f the t o t a l heat flux incident on dA. Eq. (2.13) was integrated numerically as a function of k^D using 15 a modified Gauss-Lengendre algorithm  where the upper l i m i t on p was  set equal to 5 k i l n diameters since contributions to the t o t a l flux beyond this point were found to be i n s i g n i f i c a n t .  Results of these  integrations are presented in Fig. 2.3 where 1 ^ i s plotted as a  1—I—I  Figure 2-3  I  I  f  T  T  T  Exchange integral for gas-to-kiln wall radiation versus  k.pD.  16 function of k^pD. Using this graph and Eq. (2.12), the t o t a l radiative flux from the freeboard gas to wall may be evaluated f o r any gas composition whose emissivity can be described by Eq. (2.1). To v e r i f y t h i s approach, using the method outlined by Hottel and Sarofim, ^ the mean beam1  length f o r an i n f i n i t e cylinder f i l l e d with a gray gas was calculated by use of Eq.(2.12).  A value of 0.97D was obtained which, considering the  judgement involved i n determining mean beamlengths, compares favorably with the value of 0.94D reported by Hottel and Sarofim thereby confirming the v a l i d i t y of this approach. 2.3.2  Radiative heat flux from a non-isothermal freeboard gas The influence of temperature gradients along the k i l n axis  on the radiant flux received by the wall may be established by considering a black cylinder divided into s l i c e s as shown i n Fig. 2-4. For this configuration the contribution to the radiant flux from each s l i c e may be evaluated using Eq. (2.8) where the influence of temperature gradients on both the emissive character and the emissive power of the gas must now be considered.  The influence of gas temperature on the emissive  character, d e (r)/dr, for a real gas has been shown i n F i g . 2-1 where g  the derivative i s seen to remain nearly constant over the range of temperatures shown.  This indicates that the influence of temperature  gradients on the emissive character of the gas i s negligible i n comparison with their effect (^Tg)  o n  the emissive power. Assuming  the emissive character of the gas to be constant, the contribution to the total heat flux from the j  s l i c e may be evaluated using Eq.  (2.13) where the l i m i t s of integration, shown i n Fig. 2-4, are now  \  \  Z/D  Figure 2-4  Top: Schematic cross-section of an empty rotary k i l n showing s l i c e s used i n evaluating the non-isothermal heat transfer from the freeboard gas to a d i f ferential element, dA, on the k i l n wall. Bottom: Assumed temperature prof i l e i n freeboard gas.  18 chosen i n such a way as to eliminate radiant contributions from freeboard gas lying outside the s l i c e of interest. radiant contribution from the j  w  Ea.Ii. i i 2i  s l i c e i s given by  a  where  Rewriting Eq. (2.13) the  1  i 2i !  (2.14)  (M)  for p < 1 j for p > 1  2i  2  0  Vl and E, a  (2.15)  [l-g -p (l-3) ] d3dp 2  !  2  2  for p i l - (VP ) 2  for p > 1  ..th j i s the emissive power of the gas for the j " ' s l i c e . 1  For a non-  isothermal gas with a temperature gradient shown i n Fig. 2-4 the emissive power of each individual s l i c e i s evaluated at the midpoint  temperature.  As an example, the radiant exchange between the freeboard gas and a d i f f e r e n t i a l element on a black k i l n wall was evaluated f o r a small p i l o t k i l n , (D = 0.4m), using Eq. (2.14) for both an isothermal (T =1110K) g  and non-isothermal  (250K/m)  equimolal gas  (P^Q  =  Q  =  0-12 atm).  In  the non-isothermal calculation, the temperature of the gas d i r e c t l y above dA was set equal to the isothermal gas temperature.  Thus a direct com-  parison of the isothermal and non-isothermal cases can be made. Results of these calculations can be seen i n Fig. 2 - 5 which shows the normalized cumulative heat flux from the freeboard gas to a d i f f e r e n t i a l element  Axial  position , Z / D  Normalized cumulative gas-to-wall radiative heat flux versus axial position, z/D, for both an isothermal and a non-isothermal equimolal C0  9  - H 0 freeboard gas mixture. 9  20 located on the k i l n wall plotted as a function of position along the k i l n axis.  Clearly the influence of temperature gradients i s small.  Comparison, for the small p i l o t k i l n , between the total heat f l u x received by a d i f f e r e n t i a l element on the k i l n wall for a non-isothermal gas and that for an isothermal gas indicate that less than a 4 per cent error i s introduced by ignoring temperature gradients in the freeboard gas.  Hence the use of Fig. 2-3 may be extended to include  conditions  where temperature gradients along the k i l n axis are present. The negligible influence of temperature gradients on the total heat flux received by a d i f f e r e n t i a l spot located on the k i l n wall may be explained by considering the distance along the k i l n axis over which radiant exchange between the freeboard gas and wall occurs.  Fig. 2-6 shows the  heat flux d i s t r i b u t i o n densitites for both the isothermal and non-isothermal cumulative heat flux curves plotted in Fig. 2-5.  As can be seen i n  Fig. 2-6, 86 per cent of the radiant energy incident on the k i l n wall originates from freeboard gas within an axial s l i c e only 0.6 k i l n diameters long.  Because there i s no long range radiant exchange between the  freeboard gas and k i l n w a l l , the gas seen by a spot on the wall appears to be roughly isothermal even though an axial temperature gradient e x i s t s . The localized nature of gas radiation results from the low transmissivity of a real gas for i t s own radiation.  For example, the transmissivity of  a real gas for i t s own radiation under the p i l o t - k i l n conditions described above i s t y p i c a l l y less than 0.2 while the transmissivity f o r gray or blackbody radiation may be as high as 0.6, Fig. 2-2.  -1,5  -0.9  ~0.3  0.3 Axial  Figure 2-6  position,  0.9  1.5  Z/D  Heat-flux distribution densities for gas-to-wall radiative heat flux versus axial position, z/D, for both an isothermal and a non-isothermal equimolal C0  9  - H 0 freeboard gas mixture.. 9  ro  ~*  22 2.3.3  Reflected gas radiation To examine the nature of reflected gas radiation from the  k i l n wall consider a gray cylinder f i l l e d with a real gas. For t h i s configuration the total flux of gas radiation incident on a d i f f e r ential element, dA^, located on the k i l n wall may be evaluated using Eq. (2.12).  Since the k i l n wall i s now assumed to be gray, only a por-  tion of this radiative flux w i l l be absorbed with the remainder being diffusely reflected away from the wall.  For a gray cylinder of re-  f l e c t i v i t y P the total flux of reflected gas radiation leaving dA^, i s s  given by ^reflected —dA"  _  y c -Vii^g  o iz\  aT  p s  ( 2 , 1 6 )  of which a portion w i l l be re-absorbed by the gas with the remainder being transmitted to the surrounding wall.  To determine the amount of  reflected energy received by the wall consider the two d i f f e r e n t i a l elements shown i n Fig. 2-7. For t h i s configuration the amount of reflected energy leaving dA-j, which i s transmitted to the surrounding wall i s given by the following equation ^reflected (dA -> w) 1  _ Y /  A, 2  p  cr  sTVli g reflected radiant energy leaving dA^ E  «-k^pd e  1  transmissivity  cose, cose ud  2  dA view factor dA-j •*• wall 1  ,„  0  2  ( 2  2  J  7  )  23  Figure 2-7  Schematic diagram of the cross-section of an empty rotary k i l n showing d i f f e r e n t i a l elements, dA-j and dA , used to evaluate the nature of reflected gas 2  radiation from the k i l n wall.  24 Eq.  (2.17) may be r e w r i t t e n i n terms o f c y l i n d r i c a l coordinates <j> and z  using the f o l l o w i n g transformations dA d  = | d<j,dz  2  2  = D  (2.18)  2 l yos^ (  +  Z  R  cose  = 1  _ _'- = — I R || N , |  (2/|9)  |(l-COS*)  d  12  =y  cos9 2  (2.20)  x c  "*2 -  f  ( 1  I R 111 \ \  -  C O S  ^  (2.21)  d  2  S u b s t i t u t i n g Eqs. (2.18-2.21) i n t o Eq. (2.17) and l e t t i n g £ = z/D the r a d i a n t energy r e f l e c t e d by dA^, which s t r i k e s the k i l n wall may be evaluated as f o l l o w s  R e f l e c t e d (dA  •=>- w)  —wrr-^ i  -"srVii'si  ( 2  -  2 2 )  g  where oo  Tf  .If f  j 3 1  2*J  n-cos^) H-cos*  J  0  0  L  2  +  ?2  2  e  -k D[(!f5i) l P  +  C  2  ] \  D  ,  (2.23)  "12  J  Note t h a t the l i m i t s of i n t e g r a t i o n used in Eq.  (2.23) take  i n t o account  symmetry both about and along the k i l n a x i s . Eq. the  (2.23) was  integrated numerically as a f u n c t i o n of k^.pD where  upper l i m i t of £ was set equal to 5 since c o n t r i b u t i o n s beyond t h i s  point were found to be i n s i g n i f i c a n t .  Results  of these integrations  are presented in Fig. 2-8 where I_. i s plotted as a function of  k.pD.  This graph may be used together with Fig. 2-3 and Eq. (2.22) to calculate the total reflected flux received by the surrounding k i l n wall from a d i f f e r e n t i a l spot on the wall for any gas composition whose emissivity can be described by Eq. (2.1).  For the small p i l o t k i l n  previously mentioned, evaluation of Eq. (2.22) for an isothermal gas (T  = niOK) indicates that only 20 per cent of the radiant energy  reflected by a d i f f e r e n t i a l spot i s directed toward and i s received by the surrounding wall.  In other words, for a small p i l o t k i l n 80  per cent of reflected gas radiation from the wall i s reabsorbed by the freeboard gas.  Note that for larger k i l n s , where the absorption paths  are much greater than the 0.4 m of the p i l o t k i l n nearly a l l the reflected gas radiation to the overall heat transfer i n rotary kilns i s expected to be negligible. The distribution of reflected freeboard gas radiation from a d i f ferential spot located on the k i l n wall to the surrounding wall may evaluated using Eq. (2.23). isothermal gas (T  be  For the small p i l o t k i l n f i l l e d with an  = 111 OK) Fig. 2-9 shows the normalized value of I^.  as a function of both angular wall position, <j>, and axial position, z/D. Where the k i l n wall has been unfolded into the plane of the paper.  1^,  the reflected energy, i s represented by the surface shown on the right side of Fig. 2-9, which w i l l be referred to as a "flux surface".  Due to  symmetry about dA^, only half of the total flux surface has been presented.  The flux surface may be taken to represent the d i s t r i b u t i o n of  ro  =5  I.Ofc"  O  ^ reflected gas- -wall  c o  Eg dA  T3  O  o cn  kjpD  I3i  0 0.5 1.0  1.0 0.623 0.404 0.186 0.095 0.054 0.023 0.012 0.008  ZD  -o o 0}  J >2 ajlli^i  3.0 4.0 6J0 8.0 10.0  s  CTl <U  O c a  JC  o  X  UJ  8 k|PD  Figure 2-8  Exchange integral for reflected gas radiation-tok i l n wall versus k..pD.  TT/2 T3  O  C  o to o  Q.  O  5 3  —  C <  -TT/2r—  -25  Figure 2-9  -2.0  -1.5  -1.0  -0.5 Axial  0 Q5 position,Z/D  1.0  1.5  2.0  25  Flux surface of reflected gas radiation incident on k i l n wall for a small p i l o t k i l n , D = 0.4m, f i l l e d with an isothermal equimolal COg-HpO freeboard gas mixture (right). Iso-flux lines for the flux surface shown at right ( l e f t ) .  28 reflected freeboard radiation incident on the k i l n wall where the volume bounded by the flux surface represents the total flux of reflected energy received by the wall.  For the flux surface shown here  the iso-flux l i n e s , as a function of both angular and axial wall  posi-  t i o n , are drawn on the l e f t side of Fig. 2-9 and may  map  be used to  the regions on the k i l n wall where the flux of reflected energy i s greatest.  Thus for the small p i l o t k i l n , the reflected radiant gas  energy incident on the wall i s limited to an area within one diameter of dA^,  and as expected, the areas receiving the greatest  flux are d i r e c t l y adjacent to dA^.  The localized nature of  gas radiation again results from the low transmissivity for i t s own 2.4  kiln  reflected  of a real  gas  radiation.  Radiant exchange between the freeboard gas and k i l n solids The radiative exchange between the freeboard gas and solids  has  been characterized as a function of f i l l ratio for the case of an isothermal real gas and black solids.  The assumption of an isothermal  gas  does not introduce significant error for reasons stated in the previous section.  For the configuration shown in Fig. 2-10,  the radiant exchange  between the freeboard gas and a d i f f e r e n t i a l element, dA^, the s o l i d surface, may  be calculated  located on  using Eq. (2.24)  (2.24)  a r where by definition (2.25)  w h "Figure 2 - 1 0  1  3  Schematic diagram of the cross-section  of a rotary k i l n  showing d i f f e r e n t i a l elements, dA-j and dA » used to 2  evaluate either the radiative exchange between the freeboard gas and the solids or the radiative exchange between the solids and k i l n wall.  30  C o m p a r i s o n o f E q s . (2.7) a n d (2.24) i n d i c a t e t h a t t h e s a m e b a s i c e q u a t i o n can be again used t o c a l c u l a t e t h e radiant energy received by t h e solids.  Asrepresented  g a s may be c o n s i d e r e d  b y E q . (2.24) t h e h e a t f l u x f r o m t h e f r e e b o a r d t o come f r o m c o n i c a l g a s e l e m e n t s c o n v e r g i n g  d i f f e r e n t i a l element l o c a t e d on t h e s o l i d s s u r f a c e .  on a  At this point, i t  m u s t b e e m p h a s i z e d t h a t b o t h E q s . (2.7) a n d (2.24) r e p r e s e n t t h e t o t a l v o l u m e - s u r f a c e e x c h a n g e a s g i v e n b y E q . (2.5).  However, b y r e v e r s i n g  t h e o r d e r o f i n t e g r a t i o n , E q . (2.24) may be u s e d m o r e c o n v e n i e n t l y  to  e v a l u a t e t h e r a d i a n t f l u x as a f u n c t i o n o f p o s i t i o n on t h e s o l i d s s u r face.  I n t e g r a t i n g E q . (2.24) f i r s t w i t h r e s p e c t t o r , / ( d e ; ( r ) / d r ) d r g  = e ( d ) , t h e radiant energy from the freeboard gas i n c i d e n t on the g  e n t i r e s o l i d s s u r f a c e may be e v a l u a t e d  as f o l l o w s  (2.26)  R e f e r r i n g t o F i g . 2-10, E q . (2.26) may be r e w r i t t e n i n t e r m s o f c y l i n d r i cal coordinates  dA,  <j> a n d z by a p p l y i n g t h e f o l l o w i n g d<}>,dz  cos <f>  (z, = a x i a l p o s i t i o n s o l i d s )  (2.27)  (z =axial  (2.28)  2  9  position wall)  [§cos<j> - ' ( § - F ) ] + [ | s i n * - ( § 2  2  2  (| - F - \ cos<i> ) 2  COS0  transformations.  N  =  d  FHan^+fZg  z j  2  (2.29)  (2.30)  31 cose  2  cos^L"!" cos<(. -  =  - F)].+  2  sin$ [|- s i n ^ 2  (| - F j t a n ^ )  (2.31)  d Substituting  Eqs.  ( 2 . 2 7 - 2 . 3 1 ) i n t o Eq.  (2.26) and  letting  5 =  z^/D,  Eq. ( 2 . 3 2 ) i s o b t a i n e d which may be employed t o e v a l u a t e t h e r a d i a t i v e f l u x r e c e i v e d by a d i f f e r e n t i a l  ^ "* Sdz  s l i c e across  the s o l i d s  surface  = za.I,. i  5  1  l E g  4  (2.32)  1  where cf>  «  (2TT-<J>) L  L 41  J  J  .  =  \  2  J  W l T  o o * A  DAE(Bsiny+ Ccos* ) B  +  C  2  p  D  ^  t  f  + ^2)2  ^  ( 2  .  3 3 )  1  2  L  \  ' U "  ( 2  _^  C 0 S  h  (2.34)  B = z\ sin<j) - (]r - ]j) tan<j>  (2.35)  C = \  - (1 - £ )  (2.36)  / cos ^  (2.37)  2  cos^  E = (1 - § )  1  2  Note t h a t t h e r a d i a n t f l u x r e c e i v e d by a d i f f e r e n t i a l surface w i l l  e l e m e n t on t h e  change o n l y as a f u n c t i o n o f a n g u l a r p o s i t i o n , <J>, s i n c e  gas t e m p e r a t u r e has been assumed t o be i s o t h e r m a l a l o n g t h e k i l n  solids the  axis.  T h e r e f o r e i n d e r i v i n g Eq. ( 2 . 3 2 ) , t h e v a l u e o f z ^ , t h e p o s i t i o n o f dA^ measured a l o n g t h e s o l i d s  s u r f a c e , has been s e t equal t o 0.  l i m i t s o f i n t e g r a t i o n i n Eq. and a l o n g t h e k i l n  Also,  ( 2 . 3 2 ) t a k e i n t o a c c o u n t symmetry both  the about  axis.  Eq. (2.33)was i n t e g r a t e d n u m e r i c a l l y as a f u n c t i o n o f b o t h t h e  fill  32 r a t i o , F/D, and k.pD where the upper l i m i t of z, was again set equal t o 5 since contributions beyond this point were found to be i n s i g n i f i c a n t . Results of these integrations are presented i n Fig. 2-11 where for a given f i l l r a t i o , 1^. i s plotted as a function of k^D.  Using t h i s graph t o -  gether with Eq. (2.32) the total radial flux received by the k i l n solids may be evaluated for any gas composition described by Eq. (2.1). To verify the results plotted i n Fig. 2-11 the mean beamlength for an i n f i n i t e cylinder (F/D - 0.0) f i l l e d with a gray gas was calculated using Eq. (2.32). A value of 0.95D was obtained which agrees favorably with the value of 0.97D previously calculated using Eq. (2.12). Referring again to Fig. 2-10, the flux of radiant energy from the freeboard gas to a d i f f e r e n t i a l element, dA^, on the solids surface may be evaluated as a function of angular position ^ by rewriting Eq. (2.31) as fol1ows % + d A  d  l  A  l g E  • (27r-<f>) L  C  y  "J  f a-A(Bsin<j) + Ccos<j>J ?  J  Tr(B +C H ) 2  2  2  _  n/R2 2+r2\ s. J  k  n  + r  2  0 <fr  L  Eq. (2.38) was numerically integrated both with respect t o f i l l  ratio,  F/D, and angular position, ^ , for the case of the small p i l o t k i l n f i l l e d with an isothermal gas at 111 OK.  Results of these integrations are shown  in Fig. 2-12 where the modified view factor of the gas f o r dA-j i s plotted against position across the s o l i d surface for three f i l l r a t i o s .  Thus at  1.0  - 0.8  c o o  -5 o  tt  2 •a o  0.6 0  0.10  KiPD/ (A O Cn  c o» e: o <J  I  0 . 3 0  4 i  0.0  0.0  0.0  Q 2 5 0  0 2 0 0  Q I 8 4  0 1 6 5  0.75  0.0  0 5 2 0  0 4 7 3  0 4 4 3  0 4 0 4  1.0  0 5 9 6  0 5 6 6  Q 5 3 5  0.491  2J0  Q 8 I 4  0.787  0.760  0.713  3.0  0 . 9 0 5  Q 8 8 3  Q 8 6 5  Q 8 2 3  4.0  Q 9 4 6  0.929  0.918  0.883  5.0  0 . 9 6 2  0 - 9 5 2  0 9 4 7  0.916  I Q O  0 . 9 9 2  0-995  0 9 9 0  Q 9 6 7  0.15  0 . 2 5  0  X  UJ  0  0 2 5  0  O  0.2  0.05  AxS / A T X  8  10  kjPD  Figure 2-11  Exchange integral for gas-to-sol ids radiation versus k.pD as a function of f i l l r a t i o , F/D.  o o  -0.05  F/D  n JO  0.08  CO  •o X  3  O  -0.8  1 -0.6  1 -0.4  1 -0.2  0  10.4  Q2  1 0.6  1 0.8  X Radial  Figure 2-12  position,  w/W  Heat-flux distribution from gas-to-solids surface as a function of both positions across the solids surface and solids f i l l r a t i o , F/D.  35 constant f i l l r a t i o , the radiant energy received by the s o l i d s i s greatest at the center of the surface and decreases continuously as the k i l n wall i s approached.  Also at a fixed radial position, w/W, the radiant energy re-  ceived from the gas i s seen to decrease with increasing f i l l r a t i o . In either case the decrease in radiant energy results from there being less radiating gas d i r e c t l y above the point of interest on the solids surface. 2.5  Radiant exchange between the k i l n solids and wall The radiant exchange between the solids surface and k i l n wall has  been characterized as a function of f i l l ratio for the case of a black k i l n w a l l , gray solids and a real gas.  To simplify the mathematics the  wall and solids are assumed to be isothermal.  Referring t o Fig. 2-10, the  radiant exchange between the solids surface and k i l n wall i s given by  q . [ f g +s / / r  J  J  A, A  T (d) 9  eD ss  M  ,  I I  C 0 S 6  1  C O S 9  2 dA A 2 c  TTO  I I  A  (2.39)  1  I  2  energy emitted by solids  transmissivity  view factor solids -+ wall  where from Eq. (2.3). x (d) = 1 - ( d ) = 1 - Eb.(l - e - i ) f  g  p d  0 g  (2.40)  As before Eq. (2.39) may be rewritten i n terms of c y l i n d r i c a l coordinates. Substituting Eqs. (2.27-2.31) and Eq. (2.40) into Eq. (2.39) and l e t t i n g £ = z/D, Eq. (2.41) i s obtained which gives the radiant f l u x received by the k i l n wall from a s l i c e on the solids surface.  36 »  r  q  (2TT-<p. )  r  r  Sdz E J 1 s s  J  J  0  0  *i_  l£  • DAE(Bsin<j>, • + Ccos<j>) 0  Wir •  L  1  '  <J>^ =° (2ir - <j>^) -Z /* f f b.DAECBsin*, + Ccos<j>J _ , i J J J ~ — rMl-e i f  f  n n  p  R 2 D (  + ri + B  +  C  +  ^  Jd^d^,  (2.41)  Again the l i m i t s of integration in Eq. (2.41) r e f l e c t symmetry both about and along the axis.  By inspection the f i r s t term of Eq. (2.41) i s re-  cognized to be, i n the absence of a freeboard gas, the view factor between a s l i c e on the s o l i d surface and the k i l n wall which has a value of 1 since the  solids cannot see themselves. To verify Eq. (2.41) the f i r s t term was  integrated as a function of f i l l ratio and was found to be 1 f o r a l l cases. Comparison of Eq. (2.33) with the second term of Eq. (2.41) shows them to be equivalent where now the constants b.. and f.. from Eq. (2.3) may be used together with Fig. 2-11 to evaluate the second integral of Eq. (2.41). By substitution, Eq. (2.41) may be rewritten as follows  q -5-^SdzE s  =  e(l s  - Eb.I,.) . 1  (2.42)  4 1  1  Using Eq. (2.42) together with Fig. 2-11 the total radiant f l u x received by the wall from the solids surface may be evaluated for any gas composition whose absorptivity i s described by Eq. (2.3).  Eq. (2.42) has been  employed to calculate the solids-to-wall radiant flux f o r the small p i l o t k i l n containing an isothermal gas at 1110K and solids with an emissivity of 0.8 and a temperature of 833K.  F i g . 2-13 shows the modified view  Figure 2-13  Modified solids surface-to-wall view factor versus axial position, z/D, for a p i l o t k i l n , D = 0.4m,  f i l l e d with an isothermal equimolal freeboard gas  as a function of solids f i l l r a t i o ,  F/D.  CO  ^1  38 view f a c t o r of the s o l i d s surface f o r the wall of t h i s k i l n , as determined by Eq. (2.42), p l o t t e d as a f u n c t i o n of a x i a l p o s i t i o n .  Thus the r a d i a n t  f l u x received by the w a l l i s seen t o increase with i n c r e a s i n g f i l l r e s u l t i n g from a decrease i n the o p t i c a l thickness of the gas.  ratio  A l s o , 89  per cent of the r a d i a t i v e exchange between the s o l i d s l i c e and k i l n w a l l i s l i m i t e d to a region 1.5 k i l n diameters long.  This region i s s l i g h t l y  longer than t h a t predicted f o r the r a d i a t i v e exchange between the freeboard gas and k i l n w a l l .  As mentioned e a r l i e r t h i s r e s u l t s from the increased  t r a n s m i s s i v i t y of a r e a l gas f o r gray-body r a d i a t i o n as compared t o t h a t f o r i t s own r a d i a t i o n .  The l o c a l i z a t i o n of r a d i a t i v e t r a n s f e r again would  have the e f f e c t of minimizing the i n f l u e n c e o f temperature gradients  in  both the s o l i d s and k i l n w a l l . 2.6  Mathematical model of the t o t a l r a d i a t i v e exchange i n r o t a r y k i l n s 2.6.1  Real gas model development In t h i s s e c t i o n , a mathematical model i s presented which,  based on previous d i s c u s s i o n , may be used to describe the t o t a l r a d i a t i v e interchange amongst the freeboard gas, k i l n w a l l and s o l i d s . ing assumptions  The f o l l o w -  have been made i n the model:  1)  The freeboard gas i s r a d i a t i v e l y a r e a l gas.  2)  The freeboard gas, k i l n w a l l and s o l i d s are taken t o be isothermal since the i n f l u e n c e of temperature gradients  along  the k i l n a x i s have been shown to be i n s i g n i f i c a n t . 3)  The k i l n wall and s o l i d s are r a d i a t i v e l y gray s u r f a c e s .  Based on these assumptions  a modified r e f l e c t i o n method  17  was used to  describe the net r a d i a n t l o s s f o r the freeboard gas, k i l n w a l l and s o l i d s .  39  The essence of the reflection method of calculating the net radiant loss is to follow the emission and subsequent reflections from each surface or gas in order to determine how much of these emissions and reflections are absorbed by the surface or gas of interest.  Having traced the emission  and absorption from the freeboard gas, k i l n wall and s o l i d s , the net radiant loss may then be calculated f o r each by taking the difference between the total amount of radiant energy emitted and the t o t a l amount absorbed. For the k i l n s o l i d s , the path of emitted energy i s shown i n F i g . 2-14. As seen here radiant energy leaves the solids surface, travels through the gas where a portion i s absorbed, and strikes the k i l n w a l l .  At the wall a  portion of the energy i s absorbed with the remainder being reflected back through the gas to either the solids or the remaining w a l l .  This process  is repeated f o r each reflected ray until a l l the energy leaving the solids has been absorbed. Two reflections have been considered, shown by s o l i d lines in Fig. 2-14, where the energy remaining after the second r e f l e c t i o n i s distributed between the solids and wall as i f they were black, as described below.  The amount of radiant energy attenuated by the gas f o r  each ray i s equal to the difference between the energy i n i t i a l l y emitted or reflected and that which successfully strikes either the wall or solids surface.  For a real gas, absorption occurs only within bands bounded by  discrete wavelengths. Therefore, the amount of energy attenuated by successive reflections i s decreased since f o r each r e f l e c t i o n there i s less radiant energy lying within the wavelength regions where absorption • occurs.  For t h i s reason, the gas absorbs almost no radiation after the  Figure 2-14  Schematic diagram of the cross-section of a rotary k i l n showing the path of radiant energy leaving the solids surface.  41  F i g u r e 2-15  Schematic diagram of the c r o s s - s e c t i o n o f a r o t a r y  kiln  showing the path o f r a d i a n t energy e m i t t e d f r o m the f r e e board gas.  42 second r e f l e c t i o n since a l l the radiant energy within the banded regions has previously been absorbed, i.e. the gas becomes transparent a f t e r only two r e f l e c t i o n s .  Also, after two reflections a large percentage of the  radiant energy emitted by the solids has either been absorbed by the wall or reabsorbed by the solids.  For these reasons the radiant energy remain-  ing after two reflections was distributed between the k i l n wall and s o l i d as i f they were black as indicated on Fig. 2-14 by dashed l i n e s . The error introduced by t h i s approximation i s small, especially i f the s o l i d and wall emissivities are high.  In a similar manner the path of radiant  energy from the k i l n wall may also be traced. For the freeboard gas the path of emitted energy i s shown i n Fig. 2-15. As seen here radiant energy emitted by the gas i s received by both the k i l n wall and solids.  At the wall or solids a portion of t h i s incident  tion i s absorbed with the remainder being reflected.  radia-  Since the absorp-  ;  t i v i t y of a real gas for i t s own radiation i s high only one r e f l e c t i o n need be considered.  The energy remaining after the second r e f l e c t i o n i s com-  pletely absorbed by the freeboard gas. Using t h i s development the net radiant loss for the freeboard gas, k i l n wall and solids was determined as follows  -ws<V™swws¥s -ws<Vssw<'s^sss F  F  T  F F  T  FF  W  p e  AE  energy emitted by solids and reabsorbed by solids "(2.4,3)  43 V m ws w w w (L  )F  e  A  E  V V ws ww w w w w (  F  F  p  E  A  E  V V ws L w w w w (  F  F  p  e  A  E  energy emitted by wall and absorbed by solids  V V ws sw s wS w w (  F  F  ws  F  ws  F  T(3L  p  p  A  E  m ww w w w w )F  p  e  A  E  2 T(3L  m ws ww sw s w w w w )F  F  F  p  p  e  A  Vg< rA g L  E  energy emitted by gas and absorbed by solids  energy emitted by wall  E A E  www  V  ( L  m  V  (2L  sw s s s  ) F  e  A  E  m ww sw w s s s )F  V' m 3L  V  E  ( 3 L  F  )F2  p  e  A  E  ww sw w s s s F  p  e  A  E  m ws sw s w s s s ) F  F  p  p  £  A  E  F  ww  T(3L  m  F  sw  T(3L  m ws ww s w s s s  F  ww  T(3L  m ws sw s w s s s  )F2  )F  )F  ww sw w s s s F  F  F  p  p  p  e  p  p  A  e  e  E  A  A  E  E  energy emitted by solids and absorbed by wall  44 V m ww w w w (L  )F  e  A  E  V m Lwww )F  (2L  V  (2L  e  A  E  m ws w s w w w )F  F  p  e  A  E  S  energy emitted by wall and reabsorbed by wall  V V ww w w w w (  2  F  p  e  A  E  V V ws ww sw s w w w w 3 3 (  F  F  ww  F  p  p  £  A  E  m ww w w w w  x(3L  2F  F  ww  F  sw  F  sw  )F  T(3L  E  A  E  m ws ww sw s w w w w  T(3L  T(3L  p  )F  F  F  p  p  e  A  E  m L ws s w w w w )F  F  p  p  e  A  E  m ws ww sw s w w w w )F  F  F  p  p  e  A  E  Vg m w g ( L  ) A  E  % wwVg^ g F  £  ( L  m  ) A  w g E  energy emitted by gas and absorbed by wall (2  Vsw s g< m> g m s g p  T  ) A  g g  L  £  )A  (L  E  energy emitted by gas e  g  H  ( L m  E  " ^ m^ sw s s s L  [x(g  e  F  A  E  - T(2L )]F P s A E M  s w  w  S  S  energy emitted by solids and absorbed by gas  S  • [t(2L )  - T Om' L Jw IwFs^wFw^s^sE s  "  - ^ m  m  J  3L  )]F  ws sw s w s s s F  p  p  £  A  E  45  -  [T(2L ) - T ( 3 L ) ] F m  m  -  "  <  T  3 L  m  w s  F  s w  P P 5  w E w  A E w  energy emitted by wall and absorbed by gas w  » ws ww s,AVs w w F  F  F  A  E  (2.45)  "  <"gVg< n,> w g L  A  E  - tVww ' yvi .i , .v, +  |i  £  s  "  ["5  Vg' n,'  +  L  ] F  1L  energy emitted by gas and reabsorbed by gas  ws'Vg< ,A g L  E  where, referring to Fig. 10 sw  (2.46)  = 1 sin<f).  ws  WW  (2.47)  TT - (f>,  =1  (2.48)  WS  The transmissivity, t by  g  a.k.e" i k  , of a real gas f o r i t s own radiation i s given p r  (2.49)  __ k. J J J  The average mean beamlength for a rotary-kiln was evaluated, as outlined by Hottel and Sarofim,  16  using the charts developed i n the previous  sections. Based on these calculations the average mean beamlength f o r a rotary-kiln may be determined as follows ^  = 0.95(1 -  (2.50)  where F/D represents the f i l l r a t i o as defined by F i g . 2-10. Eqs.2.43 - 2.50 may be used to estimate the total radiant exchange between the freeboard gas, k i l n  wall and s o l i d s . 2.6.2  4  g  Comparison between real and gray gas radiative models i n rotary k i l n s To check the gray gas assumption which has been used in e a r l i e r  models, the radiative exchange amongst the freeboard gas, k i l n wall and solids was calculated for a gray gas using the radiative network shown i n Fig. 2-16  and compared to that obtained using the real gas model developed  here, Eqs. (2.44-2.5D).  The freeboard gas used i n these calculations  again taken to be an equimolal ( P Q C  = P^  Q  = 0.12  was  atm) isothermal gas  mixture at 111 OK where for s i m p l i c i t y the solids were assumed to be black and the k i l n wall to be gray at temperatures of 833 and 944K, respectively. Results showing the radiative flux received by the solids from the gas f o r a k i l n 6m in diameter with a f i l l r a t i o F/D, Fig. 2-17.  of 0.30  are presented i n  As can be seen the radiant energy received by the s o l i d s from  either a gray or real freeboard gas increases with the r e f l e c t i v i t y of the k i l n wall because a larger fraction of the gas radiation incident on the wall i s reflected back to the s o l i d s .  Comparing the radiant energy  received by the solids from a gray gas to that of a real gas, the former is seen to increase more rapidly as the r e f l e c t i v i t y of the wall  increases.  For a gray gas the increase in radiant energy received by the solids results from the transmissivity of the gas for i t s own radiation being too high.  Recall from previous discussion the transmissivity of a real gas  for i t s own radiation i s very low, less than 0.2 for the small p i l o t k i l n , thereby allowing only a small fraction of the reflected energy to be transmitted  back to the s o l i d s .  However, for a gray gas, the trans-  missivity i s equal to 1 minus the emissivity, shown in Fig. 2-1 to be as  AsFsv/Tg Figure 2-16  A € w  w  Radiative resistance network for a rotary k i l n where the freeboard gas, k i l n wall and solids surface are assumed to be gray.  48  T~T  220  diameter = 6 m  Kiln  \E  F/D =0.30 A / A = 0.25  JSC  x s  to T3  x t  200  O  / /  /  O  /  </»  Co7>  /  180  /  cr> c  —  Gray g a s  /  o J=  /  o X 0)  / /  «_  c D  Real g a s  160  /  o or  0  0.2  Reflectivity  Figure  2-17  0.6  0.4  o f kiln wall,y0  w  Radiant exchange, for a r o t a r y - k i l n , D = 6m, from an isothermal freeboard gas at 1110 K to an isothermal black solids surface at 833 K for both a real and a gray gas at a f i l l r a t i o , F/D, of  0.30.  49 high as 0.8, thereby allowing too much reflected energy to arrive at the solids surface.  For this reason the use of the gray gas assumption i n  predicting the radiative exchange i n rotary-kilns may lead to s i g n i f i c a n t error, greater than 20 per cent, i f the r e f l e c t i v i t y of the wall exceeds 0.2.  Based on calculations of this type i t i s suggested the real gas  model be used to evaluate the total radiative exchange i n rotary k i l n s where i f the emissivities of the k i l n wall and solids are high (e,, and w e  s  > 0.8) the gray-gas assumption may be applied with no more than a  20 per cent error.  50 Chapter 3  OVERALL HEAT TRANSFER IN THE ABSENCE OF FREEBOARD FLAMES  3.1  Introduction The purpose of the work described i n t h i s chapter i s t o develop a  fundamental understanding of the o v e r a l l h e a t - t r a n s f e r mechanism f o r t h a t region of the k i l n i n which there are no flames present w i t h i n the f r e e board area.  Toward t h i s g o a l , a d e t a i l e d mathematical model has been  developed which takes i n t o account a l l of the h e a t - t r a n s f e r steps p r e v i o u s l y described and shown schematically i n F i g . 1-1(b).  Inasmuch as  r a d i a t i o n was the c e n t r a l t o p i c of the previous chapter and t h a t the convective heat t r a n s f e r at the upper surface o f the bed i s described 118 elsewhere, '  in analyzing the o v e r a l l heat flow i n t h i s r e g i o n ,  emphasis i s placed here on c h a r a c t e r i z i n g the regenerative heat t r a n s f e r . Therefore, the model developed i n t h i s chapter, has l a r g e l y been used to explore the regenerative a c t i o n of the k i l n wall and i t s importance r e l a t i v e to the other h e a t - t r a n s f e r steps, the e f f e c t of d i f f e r e n t k i l n v a r i a b l e s on regenerative as well as o v e r a l l heat flow to the bed, and the p o s s i b i l i t y of employing a s i m p l i f i e d model to p r e d i c t the i n s i d e w a l l temperature, heat l o s s through the r e f r a c t o r y wall and o v e r a l l heat t r a n s f e r to the s o l i d s . The approach taken i n t h i s part of the work i s again t h e o r e t i c a l . The chapter may be d i v i d e d i n t o four major s e c t i o n s : 1)  The development of a mathematical model to p r e d i c t the temperature f i e l d i n a k i l n w a l l .  51 2)  The determination of heat-transfer coefficients f o r use in the model.  3)  Application of the model to predict the  regenerative  action of the wall and overall heat transfer to the solids as a function of k i l n variables. 4)  Development of a simplified model to predict average inside wall temperature, heat transfer to the s o l i d s and heat loss through the w a l l .  3.2  Previous work The e a r l i e s t attempts to predict the regenerative action of the k i l n 19  wall were r e l a t i v e l y crude.  Heilegenstaedt  calculated the circumferen-  t i a l temperature in the wall by assuming i t to be a slab of i n f i n i t e thermal conductivity.  The slab was insulated on one side and alternately  heated and cooled by a hot gas and a well-mixed charge respectively on 20 21 the other side to simulate conditions at the k i l n w a l l .  Other studies  *  followed in which similar calculations were performed for a k i l n wall of f i n i t e thermal conductivity.  A detailed review of these early investiga22  tions has been presented by V a i l l a n t  who went on to consider the more  refined case of a rotating wall of f i n i t e thermal conductivity, which i n cluded shell losses and wall radiation to the solids surface. dimensionless equations for a slab being alternately heated and  A set of cooled,  with constant heat-transfer coefficients at the slab surfaces, was and solved using an analog simulator. the amount of regenerative  derived,  Results of his study indicate that  heating of the solids increases with both  52 rotational  speed and l i n i n g thermal conductivity, whereas the f i l l  ratio  and l i n i n g thickness have no e f f e c t . 3  More r e c e n t l y , Cross and Young  have developed a one-dimensional  heat-flow model to predict the temperature v a r i a t i o n i n the wall of an induration k i l n .  Their findings are in agreement with V a i l l a n t in that  higher rotational speeds r e s u l t in more e f f i c i e n t heat t r a n s f e r to the underside of the burden.  The heat transfer in a rotary heat exchanger 23  has been studied by Kern  f o r the case of a non-radiating freeboard  gas.  Results of t h i s study also indicate improved regenerative e f f i c i e n c y at higher rotational speeds with the influence being more pronounced at greater flow rates.  Attempts to predict the wall temperatures of both a 24 11 limestone-calcination and d i r e c t - r e d u c t i o n k i l n reveal the v a r i a t i o n of the inner-wall temperature to be t y p i c a l l y 50-100 K. In a l l of these t h e o r e t i c a l s t u d i e s , accurate predictions of wall temperatures and heat flows depend c r i t i c a l l y on values of both convective and r a d i a t i v e heat-transfer c o e f f i c i e n t s which must be s p e c i f i e d . f o r t u n a t e l y , although c r i t i c a l to model success, previous  Un-  investigators  o f f e r l i t t l e j u s t i f i c a t i o n f o r the heat-transfer c o e f f i c i e n t s used i n t h e i r work.  In these studies, convective heat-transfer c o e f f i c i e n t s have  e i t h e r been calculated from untested equations or chosen on a r u l e - o f thumb b a s i s , while the equations needed to determine the r a d i a t i v e heatt r a n s f e r c o e f f i c i e n t s are e i t h e r u n s p e c i f i e d , incomplete or i n a form which i s not e a s i l y understood.  Therefore, i t becomes d i f f i c u l t to apply,  or assess the accuracy of these e a r l i e r models.  For t h i s reason, a de-  t a i l e d evaluation of a l l heat-transfer c o e f f i c i e n t s needed f o r the c a l c u l a tions in the present work has been included in t h i s chapter.  53 3.3  Model  development  Heat c o n d u c t i o n i n a r o t a t i n g  k i l n wall  i s governed by t h e  following  equation  <L (  ar  K  k  8T k 3T __w^ + J«L _ w w ar..' r ' ar  '  1 r  3_ 3$  2  - »;  c P  3T _Wv a<j> '  f k V l  w  i r  ( 3  -  l )  w The problem can be s i m p l i f i e d c o n s i d e r a b l y by n o t i n g t h a t t h e k i l n may be d i v i d e d i n t o two r e g i o n s : s u r f a c e which undergoes rotates (II)  (I)  a thin active layer at the  a r e g u l a r c y c l i c temperature change as  inner  the wall  through the f r e e b o a r d and then passes beneath t h e b u r d e n ; and  a steady-state  l a y e r e x t e n d i n g from the a c t i v e r e g i o n t o t h e o u t e r  s u r f a c e which does not e x p e r i e n c e any temperature v a r i a t i o n as of k i l n r o t a t i o n , i . e . ally  wall  i n F i g . 3-1  between them i s  3 T / 3 t = 0. w  These two r e g i o n s are shown s c h e m a t i c -  where the c y l i n d r i c a l l o c a t e d a t R^.  by making t h e f o l l o w i n g  i n t e r f a c e ( t h i r d dimension  The problem can be f u r t h e r  The t h e r m o p h y s i c a l  2)  C o n d u c t i o n o f heat i n the l o n g i t u d i n a l  3)  simplified  i.e.  p r o p e r t i e s o f the k i l n w a l l  are  direction  i.e.  constant.  is  a T / 3 z = 0. w  C o n d u c t i o n o f heat i n t h e c i r c u m f e r e n t i a l d i r e c t i o n negligible,  infinite)  assumptions:  1)  negligible,  a function  is  aT /3(}> = 0. w  22 Vaillant  has shown t h e e r r o r i n t r o d u c e d by n e g l e c t i n g  both t h e l o n g i t u d i n a l  conduction  in  and c i r c u m f e r e n t i a l d i r e c t i o n s t o be s m a l l  ( < 2 pet.). A p p l y i n g the a p p r o p r i a t e s i m p l i f i c a t i o n s  first  to the a c t i v e  region  54 of the w a l l , the governing partial d i f f e r e n t i a l equation, Eq. (3.1), i s reduced as follows: (I)  For Rj <_ r <_ R : f  3T w ar 2  2  , 3T , 3T, . 1 w _ J_ w w r  8  r  K  , f-i o\  3 t  This equation can be solved to y i e l d the radial temperature d i s t r i b u t i o n in a s l i c e of the wall as i t rotates with respect to time, subject to the following boundary conditions: 1)  At the exposed inner w a l l , for 0 <_ 4> <_ 2  -*»Fl  „  • <cv h  r=Rj 2)  w  3 r  I  H  g-*w  At the covered inner w a l l , for 3T - k -r-^ [ 'r=R  3)  +  = h V s c  (IT -  L  R  -  <  3  <f>  L  <  2TT  (T - T ) w  and t > 0:  NW  g,s->w  2(TT  <j>)  and t > 0: (3.2b)  s  At the interface separating the two regions, r = R^., f o r 0 < . <> j < 2TT and t > 0: T  w  = T  f  (3.2c)  An i n i t i a l condition i s defined as follows for t = 0: T w  = T (r) w '  where T (r) i s given later i n the text.  (3.2d)  55 For the steady-state region the governing partial d i f f e r e n t i a l equation, Eq. (3.1), i s reduced as follows: (II) For R.f <— r <— R o : v / dT  , dT 1 ' w  2  w dr  2  + r  =  o  (3.3)  d r  which can be solved subject to the following boundary conditions;  1)  At the outer k i l n surface, for 0 <_ <f> <_ 2TT:  -^W 2)  Q  = '^/Va'^-^  At the interface separating the two regions,  "  (3  3a)  r = R^, f o r  0 <_(()<_ 2 rr T  w  =  T  (3.3b)  f  For the active layer Eq. (3.2) was solved numerically using the 25 e x p l i c i t finite-difference method schematically in Fig. 3-1.  and the nodal configuration shown  The finite-difference equations f o r each  node type are summarized in Appendix A3. Having evaluated the temperature p r o f i l e within the active region, the heat loss per meter of k i l n length, P i  o s s  >  i s taken to be the d i f -  ference between the energy received by the exposed wall as i t passes through the freeboard and that transferred to the solids by the covered wall as i t moves beneath the burden.  Expressed mathematically  Figure 3-1  Schematic diagram of rotary kiln showing nodal configuration used to determine temperature distributions in the k i l n wall. cn  57  << _ h  q  '°ss=  V »  +  V  where T  and T  V - ^ v _ °  » L W  (3.4)  represent the integrated average temperatures of the CO V  GX  exposed and covered w a l l , respectively. For the steady-state region Eq. (3.3) can be solved a n a l y t i c a l l y to y i e l d the radial temperature p r o f i l e i n the wall as shown below T - T T(r) = T ln(~). R k R,„/ 0\ , w f f out o f  a  f T  R  where h  . =  o u t  h  (3.5)  R  h ^ + h sh->a sh+a D  R  cv  The heat loss per meter of k i l n length i n the steady-state region, '"''loss'  c a n  b e  0 D , t a i n e  d y differentiating Eq. ( 3 . 5 ) with respect to r b  and substituting into Fourier's law of conduction as follows 2TT  ^'loss '  y  k (T R*  - T )  '  k  ,  ^  <ir> < h - V  n l  o  +  out  o  The flow diagram for the computer algorithm i s shown i n Fig. 3-2.  As  can be seen, at the start of any calculation, for the nodal configuration of Fig. 3-1, the total number of nodes and t h e i r positions within the unsteady-state region are calculated using the input data.  Thus, the  position of the interface separating the two regions, R^, i s taken to be an input parameter.  The i n i t i a l temperature d i s t r i b u t i o n within the  k i l n wall i s then a r b i t r a r i l y determined for both regions by extension of  58 Read  input  data  Set up nodes  Set initial temperature distribution for entire wall using E q 3 . 7  Solve for cyclic temperature distribution  Adjust No  L  Print results  temperature  at R using E q 3 . 8 and reset s t e a d y state temperatures f  7  C°p ) , s  Figure 3-2  Computer flow-diagram used to determine temperature distributions in the rotary-kiln wall.  59  Eq. (3.5)  as  follows T - T  T(r) =  T  k  •,„/ 0\ R  where T = (T  r  1n(^-) o  a R  + T )/2.  w  i  I  "out  K  (3.7)  o  Commencing w i t h t h i s t e m p e r a t u r e d i s t r i b u t i o n t h e  $  f i n i t e - d i f f e r e n c e t e c h n i q u e t h e n i s employed t o c a l c u l a t e t h e t e m p e r a t u r e f i e l d in the a c t i v e l a y e r . s e p a r a t i n g t h e two r e g i o n s  Heat f l o w s i n t o and o u t o f t h e i n t e r f a c e a r e t h e n compared u s i n g  Eqs.  (3.4)  and ( 3 . 6 ) .  I f t h e heat f l o w s a r e not e q u a l , t h e i n t e r f a c e t e m p e r a t u r e , T^, i s  cor-  r e c t e d as shown below  ^ ^V °=  fln( T  f  -  )+i  ,q,i  s  °  °  +  T  a  < - > 3  8  w and t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e w a l l cess continues u n t i l  the f i n a l  The main advantages  solution is  is recalculated.  This  reached.  o f t h i s a l g o r i t h m a r e t h a t both t h e number  nodes and t i m e s t e p s needed t o c a l c u l a t e t h e f i n a l  temperature  i s assumed t o be t r a n s i e n t .  This r e s u l t s  kiln  in a ten-fold reduction  i n computing t i m e r e l a t i v e t o t h e complete t r a n s i e n t s o l u t i o n ; and a k i l n t y p i c a l l y may t a k e up t o 20 hours t o r e a c h s t e a d y - s t a t e t h e s a v i n g s i n computer t i m e a r e c o n s i d e r a b l e .  of  distri-  b u t i o n are g r e a t l y reduced compared t o t h a t r e q u i r e d i f t h e e n t i r e wall  pro-  because  operation,  The main d i s a d v a n t a g e  of  t h i s approach i s t h a t i t cannot be used t o examine s t a r t - u p p r o c e d u r e s . complete FORTRAN s o u r c e l i s t i n g o f t h e computer a l g o r i t h m t o g e t h e r w i t h an example o f t h e programs  o u t p u t are p r e s e n t e d i n Appendix A 7 .  A  60 3.4  Heat-transfer coefficients in rotary  kilns  In o r d e r t o p r e d i c t t e m p e r a t u r e p r o f i l e s i n the k i l n w a l l , r e l i a b l e v a l u e s f o r a l l h e a t - t r a n s f e r c o e f f i c i e n t s used i n t h e model must be d e t e r mined as mentioned e a r l i e r .  In t h i s d i s c u s s i o n  d i v i d e d i n t o two m a j o r g r o u p s :  t h e c o e f f i c i e n t s have  the r a d i a t i v e c o e f f i c i e n t s a t the  w a l l , burden s u r f a c e and o u t e r s h e l l , and t h e c o r r e s p o n d i n g  been  inner  convective  coefficients.  3.4.1  Radiative, heat-transfer c o e f f i c i e n t s The r a d i a t i v e c o n d i t i o n s t h a t e x i s t w i t h i n t h e f r e e b o a r d a r e a  were s i m p l i f i e d by making t h e f o l l o w i n g 1)  Both t h e k i l n s o l i d s  assumptions:  and w a l l a r e taken t o be r a d i a t i v e l y g r a y  because t h e s p e c t r a l e m i s s i v i t i e s o f the s o l i d m a t e r i a l s and wall  r e f r a c t o r i e s are not w e l l  known.  T h i s assumption  thought t o i n t r o d u c e o n l y a small e r r o r provided t h e o f both t h e s o l i d s  2)  The f r e e b o a r d gas contains  and w a l l a r e  is emissivity  high.  i s t a k e n t o be r a d i a t i v e l y g r a y even though  it  CO^ and H^O w h i c h e m i t and absorb r a d i a t i o n i n d i s t i n c t  bands, and t h e r e f o r e s h o u l d s t r i c t l y be t r e a t e d as a r e a l  gas.  The e r r o r i n t r o d u c e d by t h e g r a y - g a s assumption may be g r e a t e r 20 per c e n t ,  f o r the reasons d e s c r i b e d i n Chapter 2 , but i n  than  this  work where t h e o b j e c t i v e i s t o examine t h e n a t u r e o f t h e o v e r a l l h e a t - t r a n s f e r mechanism, g r a y - g a s b e h a v i o r i s adequate t o  properly  c h a r a c t e r i z e t h e r a d i a t i v e i n t e r a c t i o n o f t h e g a s , w a l l and I f t h e o b j e c t i v e was e i t h e r t h e p r e d i c t i o n o f d e s i g n o r control  then the r e a l - g a s  treatment is  preferable.  solids.  process  61 3)  The presence of flames within the freehoard has been ignored; and hence the model i s valid only for regions remote from flames.  4)  There are no radial temperature gradients in the solids bed or freeboard gas so that either phase can be characterized by a unique temperature at a given point i n the k i l n .  5)  The influence of axial temperature gradients i n the s o l i d s , wall and freeboard gas i s negligible (< 15 pet.).  A detailed discussion  of; downstream radiation effects has already been presented in chapter 2. Based on these assumptions, the radiant heat transfer within the freeboard may be estimated using a 1-zohe wall model where, i n the presence of circumferential temperature gradient, the wall temperature i s taken to be an integrated average over the entire exposed w a l l . A l t e r natively, the radiant heat flows may be calculated using a multi-zone analog to more accurately characterize the effect of circumferential temperature gradients at the exposed inner w a l l .  In the multi-zone model  the wall i s divided up into zones or sections of equal area each at an isothermal temperature so as to approximate the circumferential temperature gradient.  In this study, the 1-zone model has been used to determine  the radiant heat flows, and hence the radiative heat-transfer c o e f f i c i e n t s , within the freeboard area.  To estimate the error associated with t h i s  approach, the radiant heat flows were calculated using both a 1- and 4-zone model and compared.  The resistance analogs of the two models  are shown in Figs. 3-3 and 3-4, respectively, while the view factors  €  <J  F  w  A  g  w  €  g  F  e„F A  A gw"g  g sg c  s c  6  g  F  gs  EX  €  As F swT g  A  H  WW  Figure 3-3  L  A  'Fws Tg c  1-zone radiation analog of rotary-kiln wall.  £  A s  A  Figure 3-4  4-zone radiation analog of rotary-kiln w a l l .  64  are given in Appendix A4.  To make the 1- and 4-zone predictions d i r e c t l y  comparable, as shown i n Fig. 3-5, the integrated wall temperature,  T,  employed i n the 1-zone model was obtained using the 4-zohe temperature distribution.  In this way the influence of temperature gradients can be  precisely established.  For these calculations the emissivities of the  solids and wall were both taken to be 0.90 while that of the gas was 0.24. The gas and solids temperatures were a r b i t r a r i l y fixed at 1773 and 773 K, respectively, with a k i l n diameter of 3 m. temperature gradients used  Using these temperatures the  in the 4-zone model were a r b i t r a r i l y selected  such that the integrated average was 1273 K f o r each calculation. Results of the calculations are presented i n Fig. 3-6 where the percentage d i f ference between the 4- and 1-zone heat flows, { Q _ 4  - °^-zone^4-zone °' }10  z o n e  are plotted as a function of the circumferential 4-zone wall temperature gradient, A Tw where A T  w  =  w,max " w,min  T  T  =  T  4 " l T  (3  -> 9  As can be seen the error introduced in using the 1-zone approximation in estimating the exposed wall heat flow i s less than 30 per cent f o r circumferential gradients as large as 600 K.  However, because the  circumferential gradients common to rotary k i l n operations are not expected to exceed 100 K, the use of the 1-zone approximation i n predicting radiative heat transfer at the exposed wall should be accurate to within 5 per cent. For circumferential gradients of TOO K, F i g . 3-6 also shows the error introduced in using the 1-zone approximation i n predicting the solids heat flow to be less than 3 per cent. 1-zone model shown in Fig. 3-3, has been adopted  Thus the  to characterize the  4 - Zone model  I-Zone model  OJ k_  - T  -T„  n  O OJ CL  e  OJ  V  ,  T  .  +  T  2  +  T  3  +  T  4  )  /  4  T  EX  =  T  W  OJ  c c 0  05  10 Fraction of exposed  Figure 3-5  0  05  10  wall  Circumferential k i l n wall temperature p r o f i l e s used i n the 1- and 4-zone r a d i a t i o n analogs.  Figure 3-6  Comparison of 1- and 4-zone radiant heat flows incident on the inner k i l n wall.  67 radiative  heat-transfer  Exposed i n n e r The  c o e f f i c i e n t s as d e s c r i b e d  below.  wall  radiant  heat-transfer  c o e f f i c i e n t a t the exposed  wall,  h  ,  R  g,s->w is  given  by (J  E  :  _ R  where  the temperature  average  over  determined  g,s-w  Although heat-transfer  ex'  .  .  (  T  . i s t a k e n t o be t h e i n t e g r a t e d  wall  and the r a d i o s i t y ,  m a t r i x methods  power o f t h e w a l l ,  temperature o f t h e exposed  Exposed s o l i d s  ex  v  )  w VW  of the wall,  standard  Note t h e e m i s s i v e average  p  t h e e n t i r e exposed  using  w  e  - E  E  y  as d e s c r i b e d ,  is evaluated  J  g  x  , may b e  i n Appendix for the  A4.  integrated  wall.  surface n o t employed  d i r e c t l y i n the wall  coefficient at the solids  surface,  model, the r a d i a t i v e h  , i s given  R  by  g,w->s e  n  R R  where  the radiosity,  J  s  s  (  J  s  "  E  s  )  n tT - T )  =  g,w-s  V ' g  , may a l s o  (3.11)  V  be d e t e r m i n e d u s i n g  standard  matrix  methods.  Outer  Shell  22 Vail!ant at the outer  h a s shown t h a t shell, h  the radiative  , may b e e v a l u a t e d R  heat-transfer as  coefficients  follows  sh+a  <3.,2,  68 where = {1 t ' J — + 'sh  C  3.4.2  3  (^-) 'sh  + (^-) > 'sh  2  (3.13)  3  Convective heat-transfer coefficierits For completeness, the convective heat-transfer coefficients  used in the present study are summarized below. Covered inner wall Tscheng and Watkinson  have reported that the convective heat-  transfer coefficient at the covered w a l l , *  H  11.6 k  «  ID  Rj  , i s given by w*s  2 co.R,  n  • H I T (-30V )  < -™>  1  w-*s for  h  <}i/30 K  $  L  3  s  < 10 . Eq. (3.14) was employed to evaluate the convec-  t i v e heat-transfer coefficient for a lime k i l n where the solids thermal conductivity, k  and thermal d i f f u s i v i t y , < were taken to be 0.692 -7 2 18 W/m K and 5.1 x 10" m /s, respectively. Results of these calculations s>  s>  for a 3.5 m I.D. k i l n indicate that the heat-transfer coefficient at the 2 covered wall t y p i c a l l y l i e s in the range of 50 to 100 W/m K.  *Note that heat transfer beneath the solids occurs simultaneously via radiation, convection and conduction; however, to be consistent with ref. 18 in the remainder of this text i t w i l l be referred to as a convective heat flow.  69 Exposed inner wall Under the freeboard conditions found in rotary k i l n s , the convective heat-transfer c o e f f i c i e n t at the exposed wall can be determined from the 18 following correlation. „  cv  .  0  .  0 3 6  ^ 0 . 8 , 0 . 3 3 ,0,0.055 R e  p i  ( 3 J 5 )  This equation applies to the case of turbulent flow, which i s not f u l l y developed, in a cylinder with 10 < L/D < 400.  These conditions 5  apply. 5  in a rotary k i l n where Reynolds numbers are t y p i c a l l y 2(10 ) to 5(10 ) in the freeboard and the length to diameter ratio i s usually less than 50/1.  Thus for a 3.5 m ID x 135 m long k i l n with Re = 2.5 ( 1 0 ) , Eq. 2 (3.15) yields a heat-transfer coefficient of about 12 W/m K. In the 2 5  ensuing calculations, a range of 10-30 W/m  K has been employed, the  upper end of which would apply close to the entrance region of the k i l n . Exposed solids surface Although i t does not figure directly in the regenerative action of the k i l n w a l l , convective heat transfer from the freeboard gas to the solids i s important in the calculation of the overall heat flow to the bed.  Of al1 the heat-transfer coefficients, i t i s the least known.  Experimental studies indicate that the equation h  cv ^.  =  0.4 G' g  (3.16) '  0 , 6 2  f i t s measurements obtained with kilns of 0.19^  v  and 0.40 m ID. ^ 2  How-  ever, Eq. (3.16) has not been tested with data from a large production  70 k i l n ; and thus i t s usefulness i n scale-up i s uncertain. Nonetheless i f 2  a 3.5 m ID k i l n i s again considered, with a gas mass flux of 8275 kg/m h r , 2 Eq. (3.16) gives a value of 107 W/m K for h  . This i s considerably  larger than the value that was calculated for the exposed inner w a l l , TO:  oc  and conforms broadly with experimental measurements reported previously. ' The high value i s believed to result from the large surface area presented by the solids to the freeboard gas as compared to the chord length normally used to characterize the exposed area of the bed.  The action of  the solids particles tumbling down the upper surface of the bed may also contribute to the enhanced solids heat transfer r e l a t i v e to the exposed inner wall.  In this work h  has been allowed values i n the range of g+s  o  50 to 100 W/rri K. The convective heat-transfer coefficient at the outer s h e l l , h , 27 sh-*a is given by Outer shell 0.36 0.11 k Pr"-"" o o n « h c v  h a  =  ( 0  '  "  5  +  +  ( 3  "  1 ? )  for values of Re / /Gr greater than 0.2. For operating conditions where this ratio i s less than 0.2 the influence of k i l n rotation can be ignored 31 and the convective coefficient may be determined from k Pr ' 0  h  rv  sh->a  cv  =  ~Si  3  C  (> Re  (* > 3  18  u  where the values of the coefficient C and the exponent N are presented in Table on h c  sh-^a  3-1.  For most  may be ignored.  k i l n operations the influence of k i l n speed  TABLE  3-1.  Constants f o r use with Eg. (3.18)  Re  C  N  4-40  0.911  0.385  40-4000  0.683  0.466  4000-40000  0.193  0.618  40000-400000  0.027  0.805  72 3.5  Model predictions To examine the regenerative interaction of the k i l n wall with the  freeboard gas and solids a number of computer simulations, using the algorithm previously described, were performed.  In these simulations  the process variables of interest were: 1)  gas and solids temperature;  2)  wall and solids emissivities;  3)  convective heat transfer coefficients at the exposed; and covered walls ;  4)  k i l n speed;  5)  solids f i l l r a t i o ; and  6)  thermal d i f f u s i v i t y of the wall.  Gas and solids temperatures, used in this study, were taken from the temperature p r o f i l e shown in Fig. 3-7, which i s thought to be typical 28 of that f o r a d i r e c t - f i r e d lime k i l n with no preheater.  For these  calculations, pairs of gas and solids temperatures were taken along the k i l n axis at the positions labeled I, II and III which correspond to thermal conditions within the calcination, preheat and drying zones respectively.  Adopting these temperatures, for each zone, the model was  employed to predict the temperature distribution i n the k i l n wall over the range of operating conditions summarized in Table using these temperature  3-2.  Then  d i s t r i b u t i o n s , both the regenerative and free-  board heat transfer to the solids together with heat losses at the outer surface were calculated f o r each simulation.  In t h i s way, the  [  I Radiation to burden surface Convection to burden underside  |jg3 Convection to burden surface  CO  F i g u r e 3-7  A x i a l temperature p r o f i l e s f o r a d i r e c t - f i r e d , l i m e k i l n w i t h no p r e h e a t e r . Sankey diagrams show the r e l a t i v e c o n t r i b u t i o n o f f r e e b o a r d and r e g e n e r a t i v e h e a t i n g o f the s o l i d s w i t h i n the c a l c i n a t i o n , preheat and d r y i n g zones o f the r o t a r y k i l n .  74  TABLE 3-2.  Summary of input data used for computer simulations  Location  ]  o n  e  tyK)  T (K) $  I  Calcination  1920  1200  II  Preheat  1350  950  III  Drying  1060  420  Range of process variables studied: T  298 K  a  % e  (0.5 + 0.9) (0.5 + 0.9)  s  h  (10 + 30) W/m K 2  r v  h  (50 100) W/mK 2  Vs  c  h  (5 + 15) W/mK 2  c v cv  sh+a  h...  (50 100) W/m K 2  (1 + 3) rpm F/D  (0.1 + 0.3)  k  (1 + 3) W/m K  w  Rj  1.75  m  R.  1.98  m  75  role each process variable plays i n determining the amount of heat transferred to the solids by the regenerative action of the wall may be established.  Furthermore, by comparing the magnitudes of both the regenera-  tive and freeboard heat-transfer rates, the r e l a t i v e importance of regenerative heating may be established over a wide range of operating conditions. At this point i t must be emphasized that the purpose of the present investigation i s to examine general trends i n k i l n operation.  For t h i s  reason, no attempt has been made to compare model predictions to s p e c i f i c operating data. A typical pair of circumferential and radial wall temperature prof i l e s , calculated using the model, are plotted in Figs. 3-8 and 3-9 f o r both the calcination and drying zones, respectively.  As seen i n F i g .  3-8, f o r the calcination zone the circumferential inner wall temperature variation i s 55 K with an active layer thickness in the wall of 9 mm. For the drying zone, Fig. 3-9 indicates that the temperature variation at the inner wall has been reduced to 40 K while the active layer t h i c k ness remains nearly constant at 9 mm.  Examination of the 100 simula-  tions performed in this study reveals that the circumferential temperature variations at any point along the k i l n axis are t y p i c a l l y in the range of 30 to 90 K with an active layer thickness that rarely exceeds a depth of 15 mm.  Comparison of inner wall circumferential temperature  variations, under similar operating conditions, for d i f f e r e n t zones reveals that the amplitude of temperature cycling at the inner wall i s d i r e c t l y proportional to both the gas temperature, T , and the  76  2000 1900 Colcinotion zone — high temperolura  eoo 1700  -  1600 ^ •50V*'m*K  W 2 R P M €,•€^•0^75  1500  O,«2-6*l0"'tnVi  WOO  F/0«0-IS  BOO 1200  025  0 5 0 Fraction  of  told  0 7 5  IO  cyete  (a)  0  f75  160  » 0  HJ5  Radiol  wall  position  r95  2 0  (m)  (b)  F i g u r e 3-8  (a) C i r c u m f e r e n t i a l i n n e r and (b) r a d i a l w a l l temperature p r o f i l e s f o r the c a l c i n a t i o n o r h i g h t e m p e r a t u r e zone o f a r o t a r y  kiln.  77  1000 *  Drying zone — low temperature  800  IOW/rn"K  OJ-ZRPM |  €,=€.= 0-75  600  a *  ru, "cv,  \-\WmK  400 F  w  =2-8X10 mVs  F/D = 015  200  =50  0—*  n  cv -w  = 50  "cv  0-25  =20  5  s  050  075  1-0  Fraction of total cycle (O  ~  a> 5 o  800 600  a > a. J  400 200 0 1-75  1-80  1-85 Radial wall  Figure  3-9  (a)  Circumferential  temperature  profiles  1-90 1-95 position (m) (b)  20  i n n e r and (b) r a d i a l f o r the d r y i n g  t e m p e r a t u r e zone o f a r o t a r y  kiln.  o r low  wall  78 difference between the gas and solids temperatures, AT  . On the other g»  s  hand comparison of the active layer thicknesses indicates that neither Tg nor ATg  s  s i g n i f i c a n t l y influences the depth of the active wall region.  The above i s i l l u s t r a t e d by comparing the temperature p r o f i l e s of Figs. 3-8 and 3.9 where increasing T  and AT  by 860 and 80 K, respectively,  increases the amplitude of the temperature cycle by 15 K, a change of 33 per cent, while the active layer thickness remains nearly constant at 9 mm.  Thus, the temperature cycling i n the wall i s expected to be  greatest in the hotter regions of the k i l n or i n areas where there exists a large difference between the gas and solids temperatures. In order to determine the role of regenerative heating i n the overall heat flow to the s o l i d s , temperature p r o f i l e s s i m i l a r to those shown in Figs. 3-8 and 3-9 were used to calculate the total heat transferred to the solids for each simulation.  Based on these calculated  heat flows, Sankey Diagrams were constructed for each computer run.  The  results of these calculations are summarized in F i g . 3-7 where an "average" Sankey Diagram has been constructed  for each zone.  Thus the  relative amounts of energy transferred to the solids d i r e c t l y from the freeboard and i n d i r e c t l y by the regenerative function of k i l n position.  action of the wall i s a  In the calcination zone, where high gas and  s o l i d temperatures e x i s t , the dominant path of heat transfer to the solids i s freeboard  radiation.  In this region roughly 84 per cent of  the total energy received by the solids results from t h e i r radiative interaction with the freeboard gas and exposed w a l l .  Moving along the  k i l n axis toward the feed end this fraction i s seen to decrease such that, in the drying zone the radiative contribution i s only 34 per cent. It follows that changes i n either the regenerative or convective heat transfer to the solids w i l l influcence k i l n operation more strongly within the low temperature regions of the k i l n .  For this reason, the  discussion that follows i s centered mainly on this zone. 3.5.1  Convective heat-transfer coefficient at the covered wall Fig. 3-10 shows the influence of convective heat transfer  at the covered wall on the inner wall temperature. Thus i t i s seen 2 that an increase in h from 50 to 100 W/m K causes a decrease i n cv  both the exposed and covered wall temperatures, and hence also a decrease in the temperature driving force for heat transfer to the solids. At the covered wall this effect i s overcome by the larger value of h  such that the quantity of heat transferred regeneratively to the c  w+s  solids i s increased, in this case by 53 per cent. This translates into an 18 per cent increase in the total heat flow received by the solids as shown i n Table 3-3 because regenerative heating accounts f o r only 34 per cent of the t o t a l .  At the exposed w a l l , the lower temperature  causes the net radiative exchange between the exposed wall and solids to be decreased by 13 per cent which corresponds to a 4.5 per cent decrease in the total energy received by the solids.  Thus the net  effect of increasing the convective heat-transfer coefficient at the covered wall by a factor of 2 i s only a 13.5 per cent increase in the total heat received by the solids as summarized i n Table 3-3.  On this  1100 T  'g  Drying zone — low temperature 900  . A v —s  emperatui  o  700  h  OJ= 2 RPM  € =V0-75  cc vv  g—s  = 50W/m*K  = 50 W/m K  T  h  2  T  c v  w,i = 100  w  k =IW/mK w  h-  a =2-8xl0" mVs F/D = 015 7  500  -  w  T  's  i  300 0  1  025 Fraction  Figure 3-10  »  0-50 of  total  075  10  cycle  The influence of the convective heat-transfer coefficient at the covered Wall on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n .  „  TABLE 3-3.  Influence of process v a r i a b l e s on regenerative a c t i o n of the w a l l and t o t a l heat flow to s o l i d s near feed end of a rotary k i l n Change (per cent) V a r i a b l e Range  1 -* 2  (rpm)  L  h  cv  •30  0.08 •* 0.18  F/D h  Difference between average exposed and covered-wall temperature  w+s  Heat flow exposed wall to s o l i d s  Regenerative heat flow to solids  Total heat flow to solids  -18  +38  + 10  +18  +4.0  + 10  (W/nf K)  50 -* 100  +36  -13  +53  + 14  (W/nT K)  10 •> 30  +10  +6.0  +15  + 10  0.75 -> 0.9  nil  +7.5  nil  + 5  0.75  nil  nil  nil  nil  1 + 3  -65  -4.1  -6.7  - 5  5.3 -> 8.04  +19  +1.0  -1.0  < +2  g+W  W  k(W/m K)  a(m /s) x IO 2  7  0.9  oo  82 basis, changes i n h c v  are not expected to strongly influence heat w*s  transfer within rotary kilns.  In other words, s i g n i f i c a n t changes i n  regenerative heating w i l l not always lead to markedly improved k i l n performance.  As w i l l be seen i n the remaining discussion there i s a  recurrent "trade-off" between the freeboard and regenerative heating in that an increase i n one i s normally accompanied by a decrease i n the other. 3.5.2  Kiln speed The influence of k i l n speed, co, on the inner wall tempera-  ture i s shown i n Fig. 3-11. I t i s evident that, for higher rotational speeds, both the inner wall temperature and the difference between the integrated exposed and covered wall temperatures are decreased. For this case, increasing co from 1 to 2 rpm increases the t o t a l heat flow received by the solids by less than 10 percent, Table 3-3; and thus, changes in k i l n speed are not expected to s i g n i f i c a n t l y influence the solids heat flow at any point along the k i l n axis. The effect of k i l n speed can be seen i n terms of i t s influence on the heat-transfer coefficient at the covered w a l l , Eq. (3.14), and on the heating cycle during rotation.  With increases i n k i l n speed, the  heating cycle changes because the time available f o r the wall to heat and cool, as i t moves through the freeboard and beneath the solids respectively, i s decreased. Therefore, at higher k i l n speeds, the difference between the integrated exposed and covered wall temperatures is reduced.  For the case shown i n Fig. 3-11, increasing co from 1 to  1100 r  Drying  zone — low  temperature  900  r-0J=IRPM h_ =IOOW/m*K 0) k_  700  D  O  s  w.  =122  «•*  a = 2 8xiO~V/s  QL  F/D =015 500  h  r v  h  cv  h  CV c v  300  Figure 3-11  h  w  w  a>  6 <u I—  £ =e =075 k =IW/mK  =50 W/m*K =20 S  fc  '°  sh—a  025  050 Fraction of total  0-75 cycle  10  The influence of k i l n speed on the inner wall circumferential temperature profile within the low temperature region of the rotary k i l n . .  CO CO  84  2 rpm decreases this difference by 10 K, a change of 30 per cent. At the increases with co raised to  same time, according to Eq. (3.14), h c  Vs  the 0.3 power which, as discussed in the previous section, decreases both the exposed and covered wall temperatures. reduced cycle time and increased h  The two effects -  - thus account f o r the observed cv  w+s influcence of k i l n speed seen in Fig. 3-11. The small influence of k i l n speed on total heat transfer to the solids can be explained again by the opposing effects of regenerative and freeboard heating. Heat transfer from the covered wall i s increased because the increase of h cv  with rotation speed more than offsets w+s  the decrease in the integrated covered-wall temperature.  However the  net radiant exchange between the exposed wall and solids surface i s decreased due to the decrease in the exposed-wall temperature so that the net heat flow to the solids remains essentially unchanged. While changing the k i l n speed does not alter the heat-transfer conditions s i g n i f i c a n t l y , i t must be mentioned that co does strongly affect the residence time of the solids and hence overall k i l n performance.  This aspect of rotary-kiln behaviour i s beyond the scope of  this work. 3.5.3  Convective heat-transfer coefficient at the exposed wall The influence of altering the convective heat-transfer  coefficient at the exposed w a l l , h  , i s shown in Fig. 3-12.  In-  g-*w increases both the exposed and covered wall temperatures; c  creasing h  g->w and thus the temperature driving force beneath the solids and that CV  0  Figure 3 - 1 2  025  050 07f Fraction of total cyclr  10  The influence of the convective heat transfer coefficient at the exposed wall on the inner wall circumferential temperature within the low temperature region of the rotary k i l n .  co  86  between the exposed wall and solids surface are both increased which results in improved freeboard and regenerative heat flows.  The e f f e c t , 2  however, is r e l a t i v e l y small since increasing h c  g^w  :f rom 10 Jto 30 W/m  K  increases the total heat flow to the solids by only 10 percent, Table 3-3.  Although this change w i l l improve k i l n performance a problem may  arise i f the increased freeboard gas velocities required to improve convective heating cause an increase in dust loading in the freeboard. Thus, altering the convective heat transfer within the freeboard may prove to be impossible i f increased dust loading cannot be tolerated. 3.5.4  Kiln loading Fig.  3-13 shows the influence of solids f i l l r a t i o , F/D on  the inner wall temperature.  It is evident that, for larger f i l l  ratios,  both the inner wall temperature and the difference between the integrated exposed and covered wall temperatures are again smaller.  For this case,  increasing F/D by a factor of 2 increases the total solids heat flow by only 10 per cent, Table 3-3, which is a r e l a t i v e l y minor e f f e c t . Like k i l n speed, the effect of solids f i l l r a t i o can be seen i n terms of changes to the convective heat-transfer c o e f f i c i e n t at the covered wall and the heating cycle during wall rotation.  By increasing  F/D the time available to heat any point on the exposed wall is decreased while the time spent beneath the solids i s increased.  Thus  both the maximum and minimum inner wall temperatures are reduced during rotation.  The covered-wall heat transfer c o e f f i c i e n t is also  reduced during rotation.  The covered-wall heat transfer c o e f f i c i e n t  is also reduced by increasing F/D since according to Eq. (3.14),  1100  r  Drying zone—low temperolure 900 =65W/m*K  OJ  700  h  cv cv  0J=2RPM  k_  Z3  € =€ =0 75 k =IW/mK  o  s  w.  0> Q.  a =28xiO"V/s  CU  w  500  J00  w—s  =50  w  CY  w  E  r-  F/D =008 T  cv,  h  cv  c v  = 50W/m*K  sh—o g  —-w  =20  j_  025  0-50  075  10  Fraction of total cycle Figure 3-13  The influence of solids f i l l ratio on the inner wall circumferential temperature profile within the low temperature region of the rotary k i l n .  00 •^1  .4  88 h c  V>s  depends on the half angle subtended by the solids raised to the  -0.7 power. Thus the heat flux to the exposed and covered surfaces of the bed i s reduced.  However this effect i s overcome by the fact  that the area available for heat transfer at both the covered wall and exposed solids surface i s increased with larger f i l l r a t i o s .  The  increase in heat transfer rate to the solids i s most pronounced on the freeboard side of the bed where, f o r the case shown i n Fig. 3-13 i n creasing F/D from 0.08 to 0.18, improves freeboard heat transfer by 30 percent (12 percentof the total heat flow to the s o l i d s ) .  Regenera-  tive heating, on the other hand, remains e f f e c t i v e l y unchanged because the combination of a reduced temperature driving force and covered-wall heat-transfer coefficient i s s u f f i c i e n t to roughly balance the i n creased contact area. Again i t must be noted that although the k i l n loading does not strongly influence the total heat flow received by the s o l i d s , i t does s i g n i f i c a n t l y a l t e r the thermal load and residence time of the solids.  For these reasons F/D has a major effect on k i l n operation but  is also beyond the scope of the present work. 3.5.5  Emissivities of solids and wall Figs. 3-14 and 3-15 show that neither the emissivity of the  solids nor that of the wall have a significant influence on the inner wall temperature. This i s reflected in a small effect also on the total heat flow to the solids.  Increasing the solids emissivity from  0.75 to 0.9 increases the total heat received by the solids by only 5 per cent, Table 3-3.  This small improvement results primarily from an  1100  Drying zone — l o w temperature 900  700 OJ  Z3  w  D  k.  T  h  cv  h  cv  =50 W/m* K =20  w  OJ  CL  E  CO  OJ =2RPM k = IW/mK a =2-8x|0" m*/^ F/D =015  500  300  h,.. cv, wc v  sh—a  JL  JL  0-25  050 Fraction  Figure 3-14  =100  075  . of  total  cycle  The influence of solids emissivity on the inner wall circumferential temperature p r o f i l e within the low temperature region of the rotary k i l n .  CO  U3  2000 r  Calcinalion zone—high  temperature  1800  £  Z3  1600  ?  0J=2RPM k = 20W/mK  QL  6  w  OJ  H  a =6-7x|0" m^s  h  w—s cv  7  1400  w  €  w  =0-75  T w , x  50 W/m* K = 50 = 10  •cv. 'sh—o  cy.g — w  = 10  1200  1000  025  X  050 Fraction of  Figure 3-15  075 total cycle  The influence of wall emissivity on the inner wall circumferential temperature profile within the high temperature region of the k i l n .  VD O  increase i n the fraction of radiant energy absorbed at the s o l i d s surface.  The same can be said f o r increasing the emissivity of the w a l l . 3.5.6  Thermophysical  properties of wall  The influence of the thermal conductivity, k , and s p e c i f i c w  heat, C  , of the wall on the wall temperatures are shown i n Figs. 3-16 w  p  and 3-17, respectively.  In F i g . 3-17 the radial temperature p r o f i l e s  within the k i l n wall are plotted for two refractories having thermal conductivities of 1 and 3 W/m K respectively, and a constant thermal -7 2 d i f f u s i v i t y of 6.674(10  ) m /s. As can be seen, an increase i n k  w  causes both the exposed and covered temperatures to be decreased while the active layer thickness remains constant at 6.5 mm.  A concomitant  effect i s that the total heat received by the solids i s reduced by 5 per cent, Table 3-3.  This i s clearly an unwanted result but even more  serious i s the fact that increasing k tures and heat losses.  w  increases both the s h e l l tempera-  In this case, the heat loss i s increased by  nearly a factor of 2.5. With respect to wall specific heat, Fig. 3-17, increasing C  by  w 30 per cent with k held constant, i . e . a 30 per cent reduction i n w p  thermal d i f f u s i v i t y , causes the active layer to be reduced by 1.5 mm or 25 per cent and the amplitude of the temperature cycle to be decreased by 8 K or 23 per cent.  However, the total heat flow received  by the solids i s lowered by less than 2 per cent, Table 3-3. Based on these calculations, the k i l n l i n i n g should be constructed using a refractory with low thermal conductivity and high thermal i n e r t i a (low thermal d i f f u s i v i t y ) .  The f i r s t property results i n  Figure 3-16  The influence of the thermal conductivity of the wall on the radial temperature profile within the low temperature region of the k i l n .  to ro  0  Figure 3-17  175  180  185 190 195 Radial wall position (m)  20  The influence of the specific heat of the wall on the radial temperature profile within the low temperature region of the k i l n . CO  94 lower heat losses, an obvious finding, while the second gives r i s e to a small amplitude of the temperature cycle and shallow depth of the active layer thickness.  This could be important i f thermal spalling  i s s i g n i f i c a n t l y influenced by wall-temperature cycling. 3.6  Simplified r e s i s t i v e network As shown in Table 3-3, the overall heat flow received by the  solids i s not s i g n i f i c a n t l y altered by any of the process variables examined in this study.  This i n s e n s i t i v i t y suggests that the mathemati-  cal model used to predict k i l n heat flows and temperature p r o f i l e s may be greatly simplified. In pursuit of this goal, the network shown in Fig. 3-18 was developed by extension of the standard radiative analog to include both convective and conductive heat flows.  Using the method outlined i n Appen-  dix A5, this resistive network was used to estimate the exposed inner wall temperature, the overall heat flow to the solids and the heat loss through the refractory wall. The results of these calculations are presented in Figs. 3-19 to 3-21 and compared to the predictions of the model formulated e a r l i e r in this chapter.  In Fig. 3-19 the exposed wall temperature predicted using  the modified network i s plotted against the integrated exposed wall temperature, T  . Thus i t i s seen that along the entire k i l n axis the  6X  modified analog of Fig. 3-18 accurately predicts the exposed wall temperature.  For any of the cases considered in this investigation the error  introduced by the use of this approximation i s less than 5 per cent.  I  Figure 3-18  Simplified resistive network used to predict heat flows within the rotary k i l n .  vo cn  _  1800  -  o  o c o  CJ _>  1 m ©  1400  A  1  1  Calcination Preheat Drying  1  —  0>  /  —  1000  /  600  & X  UJ  Figure 3-19  /  —  /  O  w  —  —  4  O.  CO  /  —  f —  o  6  /  /  /  /  /  /  JO  o.  1  zone  "«)  •a  1  l  —  /  /  _  /  /  f 200  200  1  1  l  600  Integrated  l 1000  exposed  wall  1  1  1400  temperature  1  1800  (K )  Exposed wall temperatures predicted by r e s i s t i v e analog versus the integrated exposed wall temperature within the calcination, preheat and drying zones.  vo cn  97 For the same operating conditions F i g . 3-20 shows the t o t a l heat r e ceived by the s o l i d s p r e d i c t e d using the modified network.plotted against the integrated heat flows c a l c u l a t e d using the 1-zone model. Again the s o l i d s heat flow i s seen to be a c c u r a t e l y p r e d i c t e d using the analog s o l u t i o n .  F i n a l l y , as shown i n F i g . 3-21, p r e d i c t i o n s o f the  heat losses through the k i l n w a l l based on the analog c i r c u i t are w i t h i n 5 per cent of the more complex f i n i t e - d i f f e r e n c e c a l c u l a t i o n s .  There-  f o r e , i n the absence o f a freeboard flame, the modified r a d i a t i v e n e t work o f F i g . 3-18 may be used to a c c u r a t e l y p r e d i c t both w a l l temperatures and heat flows a t any point along the k i l n a x i s .  !200r  0  400  800  1200  Heat received by the solids based on integrated average wall temperatures (kW/m) Figure  3-20  Heat received by the solids predicted using the r e s i s t i v e analog versus the heat received by the solids based on the integrated average wall temperatures within the c a l c i n a t i o n , preheat and drying zones.  vo CO  120  4> JZ V)  80 E cn o o c O  Q) >  - ""E.  Calcination zone  40  Preheat Drying -O 0)  3  XL  0  1  i  40  1  Integrated hoat loss from kiln Figure 3-21  i  80  i  i  1210  (kW/m)  Predicted heat loss from the k i l n shell using the r e s i s t i v e analog versus the integrated heat loss from the shell within the calcination, preheat and drying zones.  vo vo  TOO  Chapter 4  OVERALL HEAT TRANSFER IN THE PRESENCE OF A FREEBOARD FLAME  4.1  Introduction The purpose of the work described i n this chapter i s t o develop a  fundamental understanding of the overall heat-transfer mechanism for that region of the k i l n in which flames are present i n the freeboard area.  Toward this goal, a detailed mathematical model has been developed  which takes into account a l l of the heat-transfer steps shown schematic a l l y i n Fig. 1-1(c). Comparison of Figs. 1-1(b) and T-l(c) reveals that the major heat-flow paths are similar for both the flame zone (I) and the non-flame zone (II) of the k i l n , the major difference being that in the flame zone the radiating gases i n the freeboard are largely found within the confines of the v i s i b l e flame as opposed to the entire freeboard volume.  Therefore in the flame zone, the solids and exposed wall  receive heat primarily from a well defined flame where convection, due to the high flame temperatures, plays only a minor role i n the overall mechanism. The regenerative heating i s again present; however i t s role is also reduced due to the high flame temperatures.  For these reasons  the model developed in this chapter i s very similar to that developed i n the preceeding chapter for zone II-type heat transfer.  Thus, the study  of flames i s r e a l l y only an extension of the concepts previously developed in this text.  101 The approach taken i n developing a flame model i s again theoretical. The chapter may be divided into two sections: 1)  The development of a model to predict temperatures and heat flows in the presence of a freeboard flame.  2)  Application of the model to examine the flame characteristics and heat flows as a function of k i l n variables.  4.2  Previous work The number of studies dealing with the general areas of f l u i d flow*  mixing and heat transfer i n furnaces are too numerous to be considered within the context of this study.  Therefore, i n writing t h i s review  only that work relating d i r e c t l y to flames within rotary k i l n s has been included.  The interested reader i s referred to a variety of other  39-42 sources for a more complete review of flames and furnace systems. As i s the case with other aspects of rotary k i l n s , few studies dealing solely with flame characteristics i n kilns have been undertaken. 34 Rhuland  has studied flame length using a small cold-flow model together  with a f u l l - s i z e cement k i l n .  In his cold-flow experiments a small p l e x i -  glass vessel was designed i n which the mixing and combustion processes of the k i l n were simulated by reacting dilute acid and a l k a l i solutions with thymolphthalein as an indicator.  For these conditions, a blue colored  zone was produced which had the essential appearance and characteristics of a flame.  Based on these laboratory experiments and measurements i n a  f u l l - s i z e operating k i l n , Rhuland was able to deduce a general equation for flame length in a rotary k i l n as a function of burner and k i l n  102 dimensions and burner momentum. 35 Pearce,  in a similar study, developed a heat-transfer model based  on a short, well s t i r r e d , constant-temperature, constant-emissivity flame of length equal to 2.5 to 3 k i l n diameters.  In his work, average values  for both flame temperature and emissivity were based on direct measurements of kiln-type flames. The success of this approach has never been f u l l y tested i n that the author was unable to collect a comprehensive set of temperature or heat flux distributions from a production k i l n against which to test his model.  However, the model was p a r t i a l l y  validated using a small laboratory simulator. 36 Using the techniques of partial modelling Moles,  in a manner  similar to Rhuland, used both isothermal a i r and water models to examine the flow patterns at the hot end of rotary k i l n s .  In an attempt to  verify their work the authors collected and analyzed a considerable amount of industrial k i l n data. The results of their study indicate that the flame characteristics are s i g n i f i c a n t l y influenced by the secondary a i r path which is largely determined by the shape or design of the k i l n f i r i n g hood through which the secondary a i r i s introduced. On this basis, the authors have shown that accurate model predictions can only be obtained from a test carried out on a correctly scaled geometric model of the particular system  under investigation.  Hence, the generalized  use of the equations derived by Rhuland may under some operating conditions y i e l d incorrect results as no allowance was made to compensate for the f i r i n g hood configuration. 37 More recently, based on the observations of Moles, Jenkins  has  developed a heat-transfer model f o r a large enclosed flame i n a rotary  103  kiln.  The model d e v e l o p e d i n t h i s s t u d y i s based on t h e zone method o f 38  Hottel  modified t o take i n t o account the s p e c i f i c f i r i n g c o n d i t i o n s  common t o a l a r g e c e m e n t k i l n .  Using t h i s model, both temperature and  h e a t - f l u x d i s t r i b u t i o n s a l o n g t h e k i l n w a l l were p r e d i c t e d a n d t h e n compared t o those measured i n a f u l l - s i z e o p e r a t i n g k i l n w i t h r e a s o n able success.  The major drawback t o t h i s approach r e s u l t s from t h e  a p p r o x i m a t i o n s used i n d e f i n i n g t h e zone s t r u c t u r e . To a v o i d complex f l u x geometry  c a l c u l a t i o n s Jenkins approximated  t h e k i l n s y s t e m by u s e  o f an a n n u l a r g r i d t h e r e b y e l i m i n a t i n g t h e p r e s e n c e o f t h e s o l i d s  burden.  H e n c e , t h e m o d e l may n o t b e u s e d t o e s t i m a t e t h e h e a t - f l u x d i s t r i b u t i o n of the solids. In d e v e l o p i n g a m a t h e m a t i c a l  model f o r an i n d u r a t i o n  k i l n , Young  3  and C r o s s heat flow.  used a one-dimensional  f l a m e model t o e s t i m a t e t h e s o l i d s  D i v i d i n g t h e f l a m e z o n e i n t o a number o f d i s c r e t e a x i a l  s l i c e s , t h e a u t h o r s e s t i m a t e d t h e heat r e l e a s e p a t t e r n o f t h e flame by u s i n g a gamma d i s t r i b u t i o n t h a t was b a s e d o n e x p e r i m e n t a l o b s e r v a t i o n s of a s i m i l a r k i l n type flame.  Unfortunately, the authors gave very  l i t t l e information concerning the exact nature o f t h e i r heat release d i s t r i b u t i o n s and i t i s t h e r e f o r e d i f f i c u l t t o a s s e s s t h e v a l i d i t y o f the o v e r a l l model. 4.3 M o d e l d e v e l o p m e n t 4.3.1 Selection o f modelling technique The a p p r o a c h t a k e n i n d e v e l o p i n g a m a t h e m a t i c a l  model o f t h e  f l a m e z o n e was b a s e d o n c o n s i d e r a t i o n s o f b o t h t h e n u m e r i c a l a c c u r a c y o f the parameters  i n v o l v e d a n d t h e t i m e r e q u i r e d t o make t h e c a l c u l a t i o n s .  104 Examination of the preceding section reveals that three basic techniques have been previously employed to study the overall heat-transfer mechanism within the flame zone of a rotary k i l n .  These techniques,  are summarized below. 1)  Well-stirred furnace: Using this technique the gases within the flame zone are assumed to be perfectly mixed resulting in a flame of constant temperature.  The solids heat flow  is then calculated using this temperature. 2)  One-dimensional furnace:  In this technique the flame zone  is divided into a .number of axial slices each at some uniform temperature.  A heat balance i s then written for each s l i c e .  Using an assumed heat release pattern f o r the flame the heat balances are then solved consecutively for each s l i c e t o y i e l d both flame temperatures and solids heat flows. 3)  Hottel zone method: Using this technique the gases and bounding surfaces of the flame zone are divided into an arbitrary number of gas and solids zones such that each i s isothermal. Heat and energy balances are then written for each zone.  The  resulting set of simultaneous nonlinear equations are then solved to y i e l d temperatures for each zone.  Based on these  temperatures the heat flows of interest may then be calculated. For this study, the well-stirred furnace approximation was rejected because i t oversimplifies the problem and therefore would not s i g n i f i cantly contribute to a better understanding of the overall heat-transfer  105 mechanism in the flame zone.  Of the two remaining techniques the zone  method of Hottel, i s more robust in i t s a b i l i t y to r a d i a t i v e l y account for both axial and radial temperature gradients but, the f l u x geometry calculations needed to include the solids burden are both complex and tedious thereby diminishing the overall appeal of t h i s approach. Further, i t should be noted that i n the absence of any temperature gradients, the two methods are equivalent in their a b i l i t y to estimate heat flows i n the flame region.  Thus, for the present study, i f the influence of axial  temperature gradients within the flame region can be neglected, as was the case i n zone II-type heat transfer, the one-dimensional approach is preferred because i t provides the desired level of accuracy i n a more straight-forward manner. To check the accuracy of the one-dimensional model r e l a t i v e to the zone method, a number of calculations were performed i n which the flame and surrounding wall were approximated by two concentric cylinders where f o r s i m p l i c i t y , the solids burden was ignored. The configurations used f o r both the one-dimensional and zone models are shown schematically in Figs. 4-1(a) and (b), respectively.  For both of these configurations,  the inner cylinder, used to approximate a flame of constant diameter, was assumed to be black (e = a= 1), while the outer cylinder or wall was taken to be gray (e = a= constant).  In these calculations, the wall  temperature used for both models was 1173 K and to make the predictions d i r e c t l y comparable the flame temperature of the one-dimensional model and that of zone-9,  shown in Fig. 4-1(b), were assumed equal at 1773 K.  The remaining flame-zone temperatures were then evaluated, r e l a t i v e to the zone-9  temperature, using a linear temperature gradient, AT/Ax,  I - Zone  T  1  9  «I773 K  o.  E  ©  T, -1173 K  Distance (o)  16—Zone  !  @ ®  1  ©  1  ©  1  ®i® 1 ; 1  -—I^TPTTTVK  0) w  © : © i • i  O k. 4)  a.  E  A Z  ^T,  - 1173 K  Distance (b)  Figure 4^-1  Zonal configuration and temperature distributions used for both the (a) one-dimensional and (b) zone models. o CTl  107 such as that shown i n Fig. 4-l(b).  Based on these temperatures, the  radiative heat flows received by the wall were then calculated for both the one-dimensional and zone models using the r e s i s t i v e networks shown i n Figs. 4-2 and 4-3, respectively, where the view factors needed to solve these analogs are summarized i n Appendix A4. Aside from considering the influence of axial temperature gradients, the influence of both flame diameter and wall r e f l e c t i v i t y on the overall heat transfer were also considered. In this way, the error associated with the use of the one-dimensional approximation was determined. The results of these calculations are shown i n Figs. 4-4 to 4-6 i n which the ratio ( q  z o n e  - ^i-D^zone  f o r  z o n e  ^' ^  a D e l  ^  e d  i n  F  ig « s  4-1 (a) and (b), are plotted as a function of axial temperature gradient, AT/AX, relative flame s i z e , r , / r , and wall r e f l e c t i v i t y ,  p , respec-  9  tively.  As shown i n Fig. 4-4, for axial temperature gradients of 300 K/m  the use of the one-dimensional approximation introduces only a 20 per cent error.  In considering the ratio of flame diameter to that of the k i l n ,  Fig. 4-5 shows that f o r a r e l a t i v e l y long thin flame, r - j / r  = 2  °- » 25  the error introduced by the one-dimensional model does not exceed 25 per cent. P  w  F i n a l l y , Fig. 4-6 shows that for a highly r e f l e c t i v e w a l l ,  = 0.75, the error i n using the one-dimensional approach i s again  less than 25 per cent.  Based on these calculations, for an operating  k i l n where t y p i c a l l y the maximum temperature gradients rarely exceed 300 K/m, the flame to k i l n diameter ratios are greater than 0.5 and the wall r e f l e c t i v i t i e s range between 0.2 and 0.5, the use of the onedimensional approach in modelling the flame should be accurate to  til-  — A W  I €,  Figure 4-2  AW  A,  F A, |9  Resistive analog of one-dimensional model  Figure 4-3  Resistive analog of multi-zone model.  o vo  no  a>  c o si cr Q I  cr 1  CD C  o  N CT  200  0  Axial  Figure 4-4  400  temperature gradient, A T / A Z  The influence of axial temperature gradients on the one-dimensional flame model.  (K/m)  Figure 4-5  The influence of relative flame size on the one-dimensional flame model.  112  Wall  F i g u r e 4-6  reflectivity, p  The i n f l u e n c e o f k i l n w a l l dimensional  flame model.  w  r e f l e c t i v i t y on o n e -  113 within 20 per cent. Thus, i n the present study, the one-dimensional approach was chosen over the more complex zone method with very l i t t l e loss of accuracy. 4.3.2 Model assumptions In developing the one-dimensional flame model the following assumptions have been made. 1)  Both the k i l n solids and wall are taken to be r a d i a t i v e l y gray because the spectral emissivities of the s o l i d materials and wall refractories are not well known. This assumption i s thought to introduce only a small error.  2)  The flame i s taken to be radiatively gray and of constant emissivity at any position along the k i l n axis.  Depending on the operating  conditions, the emissivity of the flame l i e s between the emissivity of clear gases  and a high value of 0.95.  The peak emissivity  exists over a short section on the flame axis as shown i n Fig. 4-7 which represents a typical relationship between distance from the 39  burner and flame emissivity.  Trinks  has suggested t h i s r e l a t i o n -  ship may be approximated by taking the flame emissivity to be a constant value of 2/3 times the maximum emissivity based on the C/H r a t i o of the f u e l .  This i s shown by a dashed l i n e i n Fig. 4-7.  Aside from fuel composition, the emissivity of a flame i s also a function of other variables, the most important of which are: fuel-toa i r r a t i o , temperature of fuel and a i r , rate of mixing of the fuel  114  Figure 4-7  V a r i a t i o n of flame e m i s s i v i t y with distance from burner.  115 and a i r and the thickness or shape of the flame.  Thus, f o r a  given set of operating conditions, the f i n a l shape of the emiss i v i t y curve i s best estimated with the advice of the burner manufacturer.  However, within the context of this study, the  flame emissivity i s chosen solely on the basis of composition using the method outlined by Trinks.  The error associated with  an approach of this type i s thought to be small and may be easily changed as more information becomes available or to s u i t a specif i c set of operating conditions. The flame i s taken to be of constant diameter and i t s shape may therefore be approximated by use of a cylinder. The level of recirculation within the flame region i s thought to be small  and therefore the gas surrounding the flame i s  composed primarily of air.  and ^  from the supply of secondary  The presence of gases which are given o f f by the reacting  solids into the freeboard volume has been ignored. The overall flame length, F  L>  i s calculated  using the equation  45 of Bee*r as follows P . Pp.*! e • hv (-^) / F. = 6d (1 + AF )(-^-) Pp sa P  L  e  (4.1)  p  0  C  where f o r a double coaxial-type burner 0  = _f p  F  PA . p  pa  (4.2)  116 (  "F  +  %a>  (4.3)  and ( A F ) m  AF* =  F- pa  (4.4).  m  V In developing Eq. (4-1) the term (p /p e sa 43 work of Ricou and Spaulding  i s taken from the  and takes i n t o account the i n -  fluence of secondary a i r temperature on the o v e r a l l flame l e n g t h . The.flame lengths predicted by Eq. (4.1) are i n broad agreement "57  w i t h those measured by previous i n v e s t i g a t o r s  '  '  and a l l  thought to be accurate to w i t h i n 20 per cent of the actual length. A i r entrainment by the f u e l gas j e t i s instantaneously mixed and burns a s t o i c h i o m e t r i c  amount of f u e l .  By extension of the work described i n Chapter 3 the heat flows w i t h i n the flame zone may be approximated by the r e s i s t i v e n e t work shown i n F i g . 4-8.  A d e t a i l e d comparison between t h i s n e t -  work and that previously described and seen i n F i g . 3-18, shows them t o be very s i m i l a r where the gas node of F i g . 3-18,  Eg,  has been replaced by a s i m i l a r node E^, which i s used t o represent the flame.  The remaining d i f f e r e n c e between these  networks, r e s u l t s from the r a d i a t i v e i n t e r a c t i o n of the exposed w a l l and s o l i d s surface.  For zone II-type heat t r a n s f e r ,  when no flame i s present i n the freeboard, r a d i a t i v e exchange  n  ' f_w cv  A  w  h  Figure  4-8  cvf,  Simplified resistive rotary  kiln.  $  A  s  network used t o p r e d i c t heat f l o w s w i t h i n t h e f l a m e zone o f a  118 between the wall and solids occurs via the single path shown i n Fig. 4-9(a); and hence a single resistor i s used to j o i n the radiosities.  However f o r zone I , where there i s now a flame i n  the freeboard, radiative exchange between the solids and wall occurs v i a the two  paths shown schematically i n Fig. 4-9(b).  As a r e s u l t , f o r the flame zone, the resistance between the wall and solids radiosities must be altered as shown i n Fig. 4-9(b).  These same changes may also be seen i n Figs. 4-8 and  3-18. 8)  The specific heat of the fuel j e t mixture i s taken to be a constant at any axial position.  The value used i n the present  study was that of the combustion products at the adiabatic flame temperature.  The error associated with an approximation  of this type i s assumed to be less than 10 per cent. 4.3.3  Model formulation and solution Based on the preceding discussion, Eq. (4.1) i s f i r s t used  to calculate the overall flame length for a particular set of operating conditions.  Then, as shown i n Fig. 4-10(a), the flame i s divided into  n slices of equal size.  In this study n was chosen such that each  s l i c e was approximately 0.5 m thick; dividing the flame into thinner s l i c e s offered no significant improvement i n the f i n a l solution. Having divided the flame into s l i c e s , a heat balance may then be constructed around each s l i c e .  For the general s l i c e  , Fig. 4-10(b)  shows a l l of the heat flows which must be included within the heat  119  Figure  4-9  Schematic diagram o f the c r o s s - s e c t i o n o f a r o t a r y k i l n showing h e a t f l o w paths and r e s i s t i v e e l e m e n t s (a) n o n - f l a m e zone and (b) f l a m e zone.  for  Secondary air  d.ecA>"*(*»"  8  •a  (a)  Flame ^ Qq«n r — * — — **-  1  Solids (b)  Figure 4-10  Schematic diagram of rotary k i l n showing (a) zonal configuration for one-dimensional flame model and (b) the major heat flows within each s l i c e . ro  121 balance as  follows:  Q' y  i  +  z  heat  v  i  v  Q'  +  s  I  i  T  =  m T,z  +  Q'L.  sh  V  (4.5)  Q' .  z+Az  V  H  ,  o  ;  _L  heat  out  z  f  J  T  y  :  heat generated within slice  where  w  =  gen  i  in  Q' z  Q'  p  C  n  c  p  (4.6)  dT  298  Qgen  =  ( m  en,z Az  "  +  %n  -  A  W  sh  and  ( T  m  • \ en,z>  sh  "  T  F  H  (  "  1  H  loss  /* A '  )  7  (  4  7  )  (4.9)  a>  T  z+Az ^  A z  +  ™T,z Az/  =  C  +  P  d c  ( '  T  4  p  1 0  >  298 Combining  Eqs.  (4.5  - 4.10)  and r e a r r a n g i n g y i e l d s  w h i c h may be u s e d t o c a l c u l a t e T  2  +  A  given  z  the value of T  Ml  K,z P C  W  ( c p  V  2  9  8  »  +  " sh out < sh " V A  •8,..  "T,z+  _ C . Z  cp  S  e n , z  "X  T  h  (  298}  {m I  v  1  (  i  - % , z >  Z  -  £ f l F  -  H  f o r any  7  l o s s  equation slice.  )  »  °s " s '  T > z + z  '  +  the following  E  C  }  p r  cp  (4.11)  122 To solve Eq. (4.11) both the rate of entrainment, m , and the total en  mass flowrate of the fuel gas j e t , m-p must be determined.  Based on  Eq. (4.1), the entrainment of a i r by the fuel gas j e t i s taken to be a linear function of axial distance as follows (4.12)  men where  (4.13)  nu + mpa and  (4.14)  z> 6  The actual flame lengths and rates of entrainment are, as i s the case for flame emissivities, also influenced by operating conditions. Within the context of the present study, Eqs. (4.1) and (4.12) are adequate; however, they may be easily altered to more closely match a s p e c i f i c kiln/burner configuration as the need dictates.  F i n a l l y , by extension  of Eq. (4.12), the total mass flow of the fuel gas j e t i s given by (4.15)  Based on the i n i t i a l fuel and a i r temperatures at the k i l n discharge together with Eqs. (4.12) and (4.15), Eq. (4.11) may be used to determine the flame temperature consecutively for each s l i c e . of the s o l i d s , J  s  >  The radiosity  needed to solve Eq. (4.11) together with the solids  and outer shell heat flows may be determined using the r e s i s t i v e network  123 of Fig. 4-8 as outlined i n Appendix A6. Following these steps a computer algorithm was written which may be u t i l i z e d to solve for both flame temperature and the heat flux d i s t r i bution of the solids bed. i s shown i n Fig. 4-11.  The flow diagram f o r the computer algorithm  A FORTRAN source l i s t i n g together with a sample  of the program output are given in Appendix A8. 4.4  Model predictions To examine the overall heat-transfer mechanism within the flame  zone of a rotary k i l n a number of computer simulations, using the algorithm previously described, were performed.  In these simulations  the process variables of interest were: 1)  fuel type  2)  f i r i n g rate of fuel  3)  temperature of secondary a i r  4)  the amount of primary a i r  5)  oxygen enrichment.  The solids temperature, adopted in this study, was again taken from the temperature profile shown i n Fig. 3-7.  For a l l of these calcula-  tions, the solids temperature was taken at the position labeled I, 1200 K, and held constant. The fuel types considered were natural gas, No. 6 fuel o i l and producer gas where the combustion properties are summarized in Tables 4.1 to 4.3, respectively.  In that convection  plays only a minor role within the flame zone, contributing less than 10 per cent of the total solids heat flow as shown i n the preceding  124  Read  input data  Calculate  flame  using  length  Eq (4.1)  Divide flame into axial slices of  equal size  Consecutively solve for flame temperatures using Eqs (4.11) (A6.I - A6.5) (4.15) and (4.12)  Solve Eqs  for heat flows ( A 6.5  - A 6.7)  Print  c F i g u r e 4-11  using  results  Stop  Computer f l o w - d i a g r a m used to determine t e m p e r a t u r e s and heat flows w i t h i n the flame zone o f a r o t a r y  kiln.  125  TABLE 4-1  Combustion properties of natural  gas  Composition (% by volume):  CH  97.38  4  C H 2  g  2.17  C H  8  0.15  3  N  2  0.30  Gross heating value  =  5.521 x 10  =  992.9 B T U / f t  =  16.97 kg/kg  Hydrogen loss  =  9.82%  Average flame emissivity  =  0.25  7  Stoichiometric a i r to fuel r a t i o  Average s p e c i f i c heat  •=  1550 J/kg K  J/kg 3  126  TABLE 4-2  Combustion properties of No: 6 fuel o i l  Composition (% by weight): C  86.20  H  9.70  0  1.58  S  0.60  N  0.72  Gross heating value  = 4.221 x 10 J/kg 7  =  18145  BTU/lb  Stoichiometric a i r to fuel ratio  = 13.26 kg/kg  Hydrogen loss  = 5.06%  Average flame emissivity  =  Average specific heat  = 1400 J/kg K  0.85  127  TABLE 4-3  Combustion properties producer gas (Lurgi-Air Blown)  Composition (% by volume): CH  5.0  4  CO  16.0  H  25.0  2  C0 N  2  2  Gross heating value  14.0 40.0 =  6.974 x 10 J/kg  =  177  6  BTU/ft  Stoichiometric a i r to fuel r a t i o  =1.81  kg/kg  Hydrogen loss  =  6.61%  Average flame emissivity  =  0.25  Average specific heat  =  1490 J/kg K  3  128 TABLE 4-4  Summary of input data used for computer simulations  Fuel type:  natural gas No. 6 fuel o i l producer gas  T s  =  1200 K  T a  =  298 K  h  =  50 W/mK  =  20 W/m K  f+s  cv  h cv  h cv  f+w sh+a  2  2  =10  W/m K 2  Rj  =  1.52 m  T  =  298K  =  0.8  p  0.8  w T sa  =  298 to 773 K  PA  =  20 to 40 per cent of stoichiometric  FR  =  9.2 to 16 MW = 36 to 55 (x 10 ) BTU/hr 6  Oxygen enrichment*— 21 to 39 volume per cent of Or, i n primary a i r  *In t h i s study, a i r i s taken to be 21 per cent 0 and 79 per cent N on a volume basis. 2  2  129 chapter, the convective heat-transfer coefficients at the exposed walls, 2 solids surface and outer shell were fixed at 20, 50 and 10 W/m respectively.  K,  Adopting these conditions within the flame zone the  model was employed to examine the flame characteristics over the range of operating conditions summarized in Table 4.4.  The results of these  simulations are reported below. 4.4.1  Fuel type In order to compare the overall heat-transfer character-  i s t i c s of natural gas, No. 6 fuel o i l and producer gas, both the equivalent diameter, d , and the gross f i r i n g rate of the fuel were fixed Q  at 0.16 m and 14.5 MW, respectively. In this way, any difference in behavior may be attributed solely to the fuel type rather than the burner/kiln configuration or operating conditions under consideration. The r e s u l t s of these calculations are presented in Figs. 4-12 and  4-13  where the flame temperature and solids heat flux are plotted against the axial distance from the k i l n discharge, respectively. As plotted in Fig. 4-12, the maximum flame temperatures for a l l three fuels com40 pare favorably  with those previously reported i n the l i t e r a t u r e .  Furthermore, the predicted flame lengths for both the natural gas and No. 6 fuel o i l are in broad agreement with those observed in a lime44 sludge k i l n operating under similar conditions.  On this basis, the  overall model generally conforms with experimental observation.  In  comparing the predicted flame lengths of natural gas and fuel o i l , the l a t t e r i s seen to be longer as a result of an increase i n the  130  ?  0  l  I  I  I  I  i  I  2  4  6  8  10  12  Axial  Figure 4-12  distance  from  kiln  discharge  L 14  (m)  The influence of fuel type on flame temperatures within the rotary k i l n .  131  i  300  r "#G Fuel oil  Natural  CM  E  gas  /  150 Firing rate = 14.5 MW =(49.5 X 10 B.T.U./hr)  JSC  T = 298 K  X  f  3  T  P A  =298 K  T = 298 K % PA = 20 deq =0.160 m S A  o JC  •o o cn  -150'  1  0  Figure 4-13  2 Axial  ^ 8 10 12 14 4 6 distance from kiln discharge (m)  1  The influence o f fuel type on the s o l i d s heat f l u x w i t h i n the flame zone o f a r o t a r y k i l n .  132 effective fuel gas density, p . g  In other words, the denser fuel o i l  jet i s less able to entrain the secondary a i r needed for combustion. It i s t h i s increase in flame length coupled with a high value of flame emissivity which results in the lower flame temperatures seen in Fig. 4-12.  In comparing the producer gas to the other fuels, i t s short  flame length results from s i g n i f i c a n t l y reduced stoichiometric a i r requirements.  Because no kilns are currently using producer gas as a  fuel i t i s impossible to compare these results to any real operating data.  As shown i n Fig. 4-13, the solids heat flux i s greatest for the 2 No. 6 fuel o i l reaching a maximum value of 280 KW/m at the flame t i p 45 which compares favorably to that measured in a cement k i l n type flame 2 where values were found to range between 200 and 300 KW/m  .  For each  fuel type, Fig. 4-14 shows the relative distribution of energy within the flame zone.  For No. 6 fuel o i l , 18.2 per cent of the total energy  input i s transferred to the solids within the flame zone as compared to 9.8 per cent for natural gas and 0.4 per cent for producer gas. Thus, i t would appear that fuel o i l and natural gas may be interchanged easily as k i l n fuel with very l i t t l e modification to the operating conditions.  However, because the solids heat flux i s s i g n i f i c a n t l y  higher using fuel o i l , care must be taken to avoid overburning of the k i l n product.  Since v i r t u a l l y no heat i s transferred to the solids  within the flame zone using producer gas, i t i s doubtful that this fuel i s completely interchangeable with either No. 6 fuel o i l or natural gas.  Because very l i t t l e information regarding the use of  producer gas as a k i l n fuel i s available, further studies must be performed to evaluate the overall effect of t h i s fuel on k i l n  5.89-  70.82  # 6 Fuel  Natural  oil  gas  9.96^9.82^ 92.26  0.75y Lurgi  producer g  0.38-^6£(  I  1  Sensible Heat  £3^^^  heat  of combustion  transferred  Heat  loss  to  products  solids  from shell  H IOSS 2  Figure 4-14  Bar diagrams showing the influence of fuel type the energy distribution within the flame zone of rotary k i l n .  134 performance. 4.4.2  Firing rate The burner shown schematically in Fig. 4-15 has been used  to examine the effect of f i r i n g rate on the heat transfer characteri s t i c s within the flame zone.  The burner basically consists of two  concentric annuli which are 8.7 (11/32) and 101.6 mm (4 in.) wide. Gas passes through the inner annulus at an angle of 15° to the k i l n axis.  Primary a i r i s introduced through the outer annulus. For this  configuration using natural gas as the f u e l , Eqs. (4.2) to (4.4) take the following form. 2{1 + PA(AF*)} d  o  P  =  <(V  + 6 P  aK  <  4  J  6  )  } 2  where G  P  + G  na  P  = 1414 +  (PA(AF*)}  a  (  2  (4.17)  ^ 0 _ ) o.065 1.308  l_I_P0 +  1.224  1 + PA(AF*) P6  _ L _ + (1 - P0)PA(AF*) 0.67 1.224  (4.18) +  P0(PA)(AF*) 1.308  and AP* = •  {AF - (1 - PO)PA(AF) - 4.31(P0)PA(AF)} c  1 0.67  (  H - PO)PA(AF) 1.224  +  (PO)PA(AF) 1.308  }  (4.19)  Figure 4-15  Schematic diagram of natural gas burner used in flame model calculations.  co cn  136 Using Eqs. (4.16 - 4.19) a number of computer simulations were performed f o r natural gas where the f i r i n g rate was varied from 9.2 to 16 MW.  The results of these calculations are presented in Figs. 4-16  and 4-17 where the flame temperature and solids heat flux are plotted against the axial distance from the discharge end, respectively.  As  seen here both the flame temperature and solids heat flux increase with f i r i n g rate while the flame lengths remain constant. These pre- 42 dictions are consistent with the observations of Beer.  Therefore,  beyond a certain point, increased f i r i n g rates w i l l tend to overburn the product while s i g n i f i c a n t l y increasing the backend temperatures of the k i l n .  Clearly the f i r i n g rate must be carefully adjusted to  suit each operation. 4.4.3  Secondary a i r temperature The influence of secondary a i r temperature on the heat-  transfer characteristics was again studied using the burner of Fig. 4-15 and natural gas. The results of these calculations are shown in Figs. 4-18 and 4-19 where the flame temperatures and solids heat flux are plotted against the axial distance from the discharge end. As seen here, increasing the secondary a i r temperature s i g n i f i c a n t l y increases the flame length while having only a minor effect on both the maximum flame temperature and solids heat flux.  The increase i n  flame length results from a decrease in the secondary a i r density since the fuel gas j e t i s less able to entrain a lower density gas. Because the fuel burns over a longer distance, for a constant energy input, the flame temperature decreases as more energy i s transferred to the  Axial  Figure 4-16  distance  from  discharge  end (m)  The influence of f i r i n g rate on flame temperature within the rotary k i l n .  Axial Figure 4-17  distance  from discharge  end (m)  The influence of f i r i n g rate on the solids heat flux within the flame zone of a rotary k i l n .  139  Figure 4-18  The influence of secondary a i r temperature on flame temperatures within the rotary k i l n .  Figure 4-19  The influence of secondary a i r temperature on the solids heat flux within the flame zone of a rotary kiln.  141 heat sinks.  However, because the secondary a i r adds additional energy  in the form of sensible heat, this effect i s offset and the maximum flame temperatures and solids heat flux remain nearly constant. For secondary a i r temperatures of 298, 523 and 773 K Fig. 4-20 shows the r e l a t i v e energy distribution within the flame zone.  As seen here, by  increasing the secondary a i r temperature from 298 to 773K the total heat received by the solids i s increased by nearly a factor of two. Since the maximum heat flux at any position i s not s i g n i f i c a n t l y i n creased the use of preheated secondary a i r appears to be a reasonably safe method f o r improving k i l n productivity without overburning the product. 4.4.4 Use of primary a i r The influence of primary a i r on the heat-transfer characteri s t i c s was again studied using the burner of Fig. 4-15 and natural gas. The results of these calculations are plotted i n Figs. 4-21 and 4-22 where the flame temperatures and solids heat flux are plotted against the axial distance from the discharge end. As seen here, increasing the amount of primary a i r decreases the overall flame length which causes both the maximum flame temperature and solids heat flux to. increase.  The total energy distribution for primary a i r rates of 20,  30 and 40 per cent of stoichiometric i s shown i n Fig. 4-23. Thus, by increasing the primary a i r from 20 to 40 per cent of stoichiometric the total amount of heat received by the solids i s decreased from 9.8 to 5.0 per cent of the total energy input.  This results from the cor-  responding decrease i n flame length because the distance available f o r  142  Natural gas firing rate = I4.5MW(49.5 X I 0 b t u / h r ) 6  % PA = 2 0 T =2SGK T =298K p  s a  76.31  T. =778K fl  8.69' Sensible  heat of  Heat transferred Heat H  Figure 4-20  2  combustion  products  to solids  lost by outer  shell  loss  Bar diagrams showing the influence of secondary a i r temperature on the energy distribution within the flame zone of a rotary k i l n .  Figure 4-21  The influence of primary a i r on flame temperatu within the rotary k i l n .  Axial  Figure 4-22  distance  from discharge  end (m)  The influence of primary a i r on the solids heat flux within the flame zone of the rotary k i l n .  Natural Firing  gas rate = 14.5 MW (49.5 X 10  btu/hr)  Tp=298 K T  P A  T  S A  *298K = 298K 76.31 % P A =20 9.83 '  9.82  79.69 %PA=30  7.3r 82.7!  9.82  2.39 %PA=40 5 . 0 8 ^ 9.8 2  C  J  w//m  Sensible  of  Heat  transferred  Heat  lost  H  Figure 4-23  heat  2  /  combustion  products  to solids  from outer  shell  lost  Bar diagrams showing the influence of primary a i r on the energy distribution within the flame zone of a rotary k i l n .  heat transfer i s reduced.  Even though the maximum flame temperature  and solids heat flux are increased with additional primary a i r i t i s not enough to offset the decrease in heat transfer length.  Hence the  total amount of energy received by the solids i s decreased.  On this  basis, the primary a i r should be set at the lowest possible level necessary to maintain the flame. 4.4.5  Oxygen enrichment The influence of oxygen enrichment on the heat-transfer  characteristics was again studied using the burner of Fig. 4-15 and natural gas.  In this study oxygen was substituted for primary a i r on  a mass basis.  That i s , i f 20 per cent of the stoichiometric a i r re-  quirement was used as primary a i r , a level of 0 per cent oxygen enrichment would correspond to 21 per cent oxygen i n the primary a i r while 20 per cent enrichment corresponds to 39 per cent oxygen i n the primary stream.  The results of these calculations are presented i n  Fig. 4-24 and 4-25 where the flame temperatures and solids heat flux are plotted as a function of axial distance from the k i l n discharge. It i s seen that, by increasing the level of oxygen i n the primary stream both the maximum flame temperature and solids heat f l u x are s i g n i f i c a n t l y increased while the flame length i s simultaneously decreased.  For oxygen enrichment levels of 0, 10 and 20 per cent of  the primary stream Fig. 4-26 shows the distribution of energy within the flame zone.  As seen here, by increasing the level of oxygen in the  primary a i r from 21 to 39 per cent the energy received by the solids increases from 9.8 to 12.2 per cent of the total energy input. In  Axial  Figure 4-24  distance  from  discharge end (m)  The influence of oxygen enrichment on flame temperatures within the rotary k i l n .  Axial  Figure 4-25  distance  from discharge  end (m)  The influence of oxygen enrichment on the solids heat flux within the flame zone of a rotary k i l n .  Natural gas F i r i n g rate = 14.5 MW (49.5  X l O btu/hr)  % PA = 2 0 T =298 K T =298K T =298K 76.3 F  P A  S A  % PA as 0 = 0 2  9.8 5 ^  9.8 2 ^  75.31 % P A as 0 = 10 2  11.0' 2 ^ 9. 8 2 ^ 3.(5 5 ^  74.2 5 ^  1  12.28  Sensible  heat  Heat transferred Heat H  Figure 4-26  2  lost  k  95 2  of  % P A as 0 = 2 0 2  /  combustion products  to solids  by outer shell  loss  Bar diagrams showing the influence of oxygen enrichment on the energy distribution within the flame zone of a rotary k i l n .  150 effect, this temperature. the a c t u a l  behavior results By u s i n g  from an i n c r e a s e i n t h e a d i a b a t i c  oxygen  flame  i n t h e p r i m a r y a i r stream the i n c r e a s e  f l a m e temperature i s  in  g r e a t enough t o overcome the reduced  h e a t - t r a n s f e r l e n g t h o f the f l a m e .  This  is  i n c o n t r a s t t o the use  of  i n c r e a s e d p r i m a r y a i r where because t h e a d i a b a t i c t e m p e r a t u r e remained constant  this  i n c r e a s e i n flame temperature was not l a r g e enough  overcome t h e r e d u c e d h e a t - t r a n s f e r l e n g t h o f t h e f l a m e . oxygen may be used t o i n c r e a s e s o l i d s maximum v a l u e of  the s o l i d s  Therefore  o u t p u t ; however because the  heat f l u x i s  t o a v o i d o v e r b u r n i n g o f the p r o d u c t .  to  i n c r e a s e d , c a r e must be used  Chapter 5  CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK  5.1  Conclusions The following conclusions concerning heat transfer in a direct-  f i r e d rotary k i l n may be drawn from t h i s study. 1)  Because the freeboard gases common to rotary-kiln operation contain CO^ and r^O which emit and absorb radiation i n d i s t i n c t bands, the gray-gas approximation i s not v a l i d ; and these mixtures should be treated as real gases.  2)  The majority, 86 per cent, of radiant energy received by the k i l n wall or the solids surface originates from gas within an axial s l i c e +0.3 k i l n diameters along the k i l n axis.  The  localized nature of gas-wall or gas-solids radiative exchange results from the transmissivity of a real gas for i t s own radiation being very small. 3)  The influence of temperature gradients in the freeboard gas on the radiative exchange between the gas and the k i l n wall or the solids surface i s negligible.  This i s again due t o the  transmissivity of a real gas for i t s own radiation being very low.  4)  Greater than 80 per cent of the freeboard gas radiation reflected from either the k i l n wall or the solids surface i s reabsorbed by the gas on the f i r s t r e f l e c t i o n . Hence, the i n fluence of reflected gas radiation on the t o t a l radiant heat transfer in rotary kilns i s minimal and i s a local phenomenon.  5)  The radiant exchange between the solids surface and the k i l n wall occurs over an axial s l i c e + 0.75 the k i l n axis.  k i l n diameters along  Hence, the influence of temperature gradients  along the k i l n wall or the solids surface has a negligible e f f e c t on the radiative exchange between the k i l n wall solids 6)  and  surface.  Exchange integrals for gas-to-wall, gas-to-sol ids, reflected gas-to-wall and wall-to-sol ids radiative exchange have been evaluated.  Based on these integrals, charts have been developed  which may be used to evaluate the radiative heat transfer in rotary kilns for any combination of k i l n diameter, gas composit i o n , temperature and solids f i l l r a t i o . 7)  Using the results shown here, a modified r e f l e c t i o n method has been used to develop a model which describes the total radiat i v e exchange in rotary k i l n s for the case of a real freeboard gas.  8)  Predictions of the radiative exchange in rotary kilns using the real-gas model have been compared to those using a simple gray-gas radiative thermal c i r c u i t .  Based on calculations of  t h i s type the use of gray gas may introduce significant error, greater than 20 per cent, when the r e f l e c t i v i t i e s of the k i l n wall and solids exceed 0.2. 9)  During a single k i l n rotation, the unsteady state or c y c l i c temperature region i s limited to a thin layer at the inner wall that rarely exceeds a depth of 15 mm.  10) At any point along the k i l n  axis, the cyclic temperature  variation at the inner surface of the k i l n wall l i e s i n the range of 30 to 90 K. 11) In the high temperature regions of the k i l n , 60 to 80 per cent of the heat received by the solids results from their radiative interaction with the freeboard gas and exposed wall.  12) Within the low temperature regions of the k i l n , 70 per cent of heat received by the solids results from the combination 7  of freeboard convection and the regenerative heating of the wall.  13) The temperature distribution within the wall at any point, along the k i l n axis and the total heat flows received by the solids and l o s t to the surroundings, are largely i n dependent of the process variables considered i n this investigation.  154 14)  Since the temperature distribution of the k i l n wall and heat flows are largely independent of the operating conditions, the model used to predict the amount of heat received by the solids and shell losses within the non-flame zone may be greatly s i m p l i f i e d by using the r e s i s t i v e network of Fig. 3-18 which incorporates radiative, convective and conductive heat flows. Using t h i s simplified model, both the wall temperatures and heat flows may be estimated to within 5 per cent at any point along the k i l n axis.  15)  The one-dimensional furnace approximation may be used to model freeboard flames within the rotary k i l n .  The error associated  with t h i s approach i s less than 20 per cent. 16)  The r e s i s t i v e network of Fig.' 4-8 which incorporates radiation, conduction and convection may be used to estimate heat flows within the flame zone of a rotary k i l n .  5.2  Suggestions f o r future work Having analyzed the various heat flows within the d i r e c t - f i r e d  rotary k i l n an obvious extension of the work would be to develop an overall steady-state and/or dynamic model which u t i l i z e s the r e s i s t i v e networks developed in t h i s text.  Based on the success of these  models a series of studies might then be undertaken to better characterize the radiative properties of the various k i l n materials, the convective heat-transfer c o e f f i c i e n t s , the internal heat transfer within the solids bed flame zone.  or the mixing and f l u i d flow conditions within the  155  REFERENCES 1.  J . K. Brimacombe and A. P. Watkinson: Met. Trans., 1978, Vol. 9B, pp. 201 - 219.  2.  V. Venkateswaran and J. K. Brimacombe: Met. Trans., 1977, Vol. 8B, pp. 387 - 398.  3.  R. W. Young and M. Cross: pp. 129 - 137.  4.  S. L. Wingfield, A. Prothers and J. B. Auld: Vol. 47, pp. 64 - 72.  5.  A. Saas: Ind. Eng. Chem., Process Des. Dev., 1967, Vol. 6, pp. 532 - 535.  6.  J . W. Lyons, H. S. Min, P. E. Parisot and J. F. Paul: Ind. Eng. Chem., Process Des. Dev., 1962, Vol. 1, pp. 29 - 33.  7.  A. Manitius, E. Kurcyusz and W. Kawecki: Ind. Eng. Chem., Process Des. Dev., 1974, Vol. 13, pp. 132-142.  8.  J . B. Riffaud, B. Koehret and B. Coupal:  Iron and Steel Making, 1976, Vol. 3, J . Inst. Fuel, 1974,  B r i t . Chem. Eng. and  Proc. Tech., 1972, Vol. 17, pp. 413 - 418. 9.  H. A. Spring: Automatica, 1972, Vol. 8, pp. 309 - 323.  10.  V. A. Kaiser and J. W. Lane: Ind. Eng. Chem., Process Des. Dev., 1968, Vol. 7, pp. 319 - 320. W. Schnabel: Ph.D. Thesis, Faculty of Mining and Metallurgy, Rheinisch-Westfalischen Technical University, Aachen, 1977.  11. 12.  H. Hottel and A. Sarofim: Radiative Transfer, McGraw-Hill, New York, 1967, pp. 199 - 256.  13.  W. J . Dixon, Ed.: Biomedical Computer Programs P-Series, University of California Press, Berkeley, 1977, pp. 461 - 520.  14.  Op. c i t . , H. Hottel and A. Sarofim:  15.  P. Madderom: Quadrature Integration Routines, University of  pp. 279 - 287.  B r i t i s h Columbia, Vancouver, 1978, pp. 27 - 44. 16.  Op. c i t . , H. Hottel and A. Sarofim:  pp. 273 - 299.  17.  J . Sucec: Heat Transfer, Simon and Schuster, New York, 1975, pp. 233 - 307.  156 18.  S. H. Tscheng and A. P. Watkinson: Vol. 57, pp. 433 - 443.  Can. J . Chem. Eng., 1979,  19.. W. Heiligenstaedt: "Warmetechnische Rechnungen fur Industrieofen" 4. Auflage, Verlag Stahleisen M.B.H., Dusseldorf, 1966. 20.  M. Imber and V. Paschkis: Drehrohrofen a l s Warmeaustauscher, 1960, Vol. 4, pp. 183-197.  21.  H- S. Carslaw and J. C. Jaeger: Conduction of Heat in Solids, Oxford University Press,' London j 1947.  22.  A. V a i l l a n t : Ph.D. Thesis, Faculty of Mechanical Engineering, Columbia University, 1965.  23.  J . Kern:  24.  W. Zimmer: Private Communication, Kennedy Van Saun Corp., Danville, Pa.  25.  F. Kreith and W. Z. Black: Basic Heat Transfer, Harper and Row, New York, 1980, pp. 165 - 190.  26.  A. P. Watkinson and J. K. Brimacombe: unpublished research, University of B r i t i s h Columbia.  27.  D. A. Kaminski, Ed.: Heat Transfer Data Book, General Electric Co., New York, 1977, pps. 511.2 (1-6).  28.  F. Schwarzkopf: Lime Burning Technology, Kennedy Van Saun Corp., Danville, Pa., 1974, pp. 29-33.  29.  A. K. Oppenheim: Trans. A.S.M.E., 1956, Vol. 78, pp. 725 - 735.  30.  Op. c i t . , F. Kreith and W. Black: pp. 333 - 337.  31.  Op. c i t . , F. Kreith and W. Black: pp. 249 - 253.  32.  C. M. Lee: Zeros of Nonlinear Equations, University of B r i t i s h Columbia, Vancouver, 1977, pp. 16 - 48.  33.  D. C. Hamilton and W. R. Morgan: Radiant Interchange Configura-  Int. J . Heat Mass Transfer, 1974, Vol. 17, pp. 981 - 990.  tion Factors, MACA TM 2836, 1952. 34.  W. Ruhland:  J . Inst, of Fuel, 1973, Vol. 40, pp. 69 - 75.  35.  K. W. Pearce:  36.  F. D. Moles, D. Watson.and P. B. Lain: Vol. 46, pp. 353 - 362.  J . Inst, of Fuel, 1973, Vol. 46, pp. 363 - 371. J . Inst, of Fuel, 1973,  37.  B. G. Jenkins and F. D. Moles: Trans. Inst. Chem. Eng., 1981, Vol. 59, pp. 17 - 25.  38. Op. c i t . , H. Hottel and A. Sarofim:  pp. 470-487.  39. W. Trinks and M. H. Mawhinney: Industrial Furnaces, Vols. I and I I . John Wiley and Sons, New York, 1961. 40. R. T. Read, Ed.: North American Combustion Handbook, North American Mfg. Co., Cleveland, 1978. 41. M. W. Thring: The Science of Flames and Furnaces, Chapman and . H a l l , London, 1962. 42. J. M. Beer and M. A. Chigier: Combustion Aerodynamics, Applied Science Publishers, London, 1972. 43.  F. P. Ricou and D. B. Spaulding: pp. 21 - 32.  J . Fluid Mech., 1961, Vol. 11,  44. T. N. Adams: Private Communication, Weyerhaeuser Co., Tacoma, Wa. 45.  R. Graf and R. Payne:  I FRF, 1981, Doc. No. F 32/a/42.  Appendix Al  RADIATIVE PROPERTIES FOR EQUIMOLAL C0 -H 0 GAS MIXTURES 2  2  159 TABLE A l - T . .  Summary o f e m i s s i v i t y d a t a f o r equimolal  C0 -H 0_gas ?  o  m i x t u r e s a t 830, 1110 and 1390 K  P  (C0  2  + H O)  T L  2  (atm - m)  e  g  (C0  = 830  2  (K)  + H 0) 2  T  e  g  = 1110  (C0  2  (K)  T  e  + H 0) 2  g  = 1390  (C0  2  + H 0) 2  0.0  0.0  0.0  0.0  0.06  0.10  0.13  0.16  0.09  0.12  0.16  0.18  0,12  0.14  0.18  0.21  0.18  0.16  0.20  0.25  0.24  0.18  0.23  0.28  0.30  0.20  0.25  0.30  0.37  0.22  0.27  0.32  0.49  0.24  0.30  0.36  0.61  0.28  0.33  0.38  0.91  0.32  0.38  0.44  1.22  0.36  0.43  0.43  1.83  0.42  0.50  0.54  3.05  0.49  0.57  0.64  Coefficients:  l  a  2  a  a  3  a  4  a  5  k  l  k  3  k  4  k  5  4 gray + 1 c l e a r gas model -  Eq.  (K)  (2.1)  0.40  0.43  0.48  0.04  0.09  0.06  0.04  0.04  0.O5  0.06  0.08  0.10  0.46  0.36  0.31  0.66  0.59  0.69  3.67  3.58  10.17  13.71  14.57  6.73  59.38  43.18  26.90  0.0  0.0  0.0  160  TABLE A 1 - 2 : .  Summary of a b s o r p t i v i t y data f o r eguimolal C0 -H P_gas 2  2  mixtures at 1110 K f o r blackbody r a d i a t i o n at 277, 555 and 833 K  p  ( C 0 + H 0) 2  L  2  (at m-m)  T  s  = 277 (K)  T  g  = 1110 (K)  a  ( C 0 + H 0) 2  2  T = 555 (K) s T = 1110 (K) g ( C 0 + H 0) a  2  2  T  s  = 833 (K)  T a  = 1110 (K) g ( C 0 + H 0) 2  2  0.0  0.0  0.0  0.0  0.03  0.16  0.11  0.10  0.09  0.30  0.21  0.18  0.18  0.39  0.26  0.23  0.30  0.45  0.32  0.28  0.46  0.57  0.37  0.32  0.61  0.63  0.41  0.36  0.91  0.70  0.48  0.42  1.52  0.85  0.57  0.52  Coefficients:  b  l  b  2  b  3  f  l  f  2  f  3  .  2 gray + 1 c l e a r gas model - Eq. (2.3) 0.70  0.46  0.49  0.22  0.16  0.16  0.08  0.38  0.35  1.41  1.35  0.89  30.90  27.85  30.28  0.0  0.0  0.0  Appendix A2  DERIVATION OF EQUATION (2.9)  162 As previously stated i n Chapter 2, Eq. (2.9) represents the projected area  of intersection between a hemisphere of radius r and  cylinder o f diameter D, divided by the total projected area of the hemisphere.  For the configuration shown i n Fig. A2-1, the equations of  interest f o r a sphere and cylinder are given below. sphere:  r = x +y + z  cylinder:  (|) = x + ( y - f )  2  2  2  2  (A2.1)  2  2  (A2.2)  2  The projected area of intersection onto the x - z plane f o r t h i s configuration i s given by AREA  =J  x(z)dz  (A2.3)  z where, by eliminating y from Eqs. (A2.1) and (A2.2) x(z)  [ r - z - (Xi^li)--] *  =  2  5  2  .  (A2.4)  By use of symmetry about the o r i g i n , Eqs.(A2.3) and (A2.4) may be combined such that  2  r  AREA = 4J  lr  -z - (  2  2  0  r  ~  Z  2 2~1 ^ ) J *dz  (A4.5)  for r < D  Vr - D 2  2  for r > D  where the l i m i t s of integration are shown schematically i n Fig. A2.2. The above i n t e g r a l , Eq. (A2.5), may now be rewritten i n terms of the dimensionless lengths z/D and r/D as follows  (o)  z (c)  F i g u r e A2-1  Orthogonal views o f c y l i n d e r and  hemisphere;  (a) e l e v a t i o n ; (b) end; and; ( c ) p l a n e .  164  For  r<D  lower limit = 0 upper limit = r  T lower limit = +Jr -D upper limit = r 2  For  r>D  2  T D  dA  Figure  A2-2  Schematic diagram of c y l i n d e r of diameter D and hemisphere of radius r showing the l i m i t s of i n t e g r a t i o n used in Eq. (A4.5).  r/D AREA = 4D'  7  2 2  for  r/D < 1  (£) - 1 f o r  r/D > 1  V U s i n g Eq.  -as-)  <#>  r  (A2.6) t h e p r o j e c t e d a r e a s f o r s e v e r a l  ted i n F i g . A2-3.  F i n a l l y , Eq.  area o f the e n t i r e hemisphere, (2.9).  9(r)  165 (A2.6)  d(§)  v a l u e s o f r/D a r e  (A2.6) must be d i v i d e d by t h e  plot-  projected  irr , t o a r r i v e a t t h e f i n a l form o f  Eq.  r/D d(§)  -  irr  irr  V  (£) -1  for  r/D<  for  r/D  1 > 1  (A2.7)  y/D  -1  -2  (\ / \ / M  [IA / \ 11  /D-I.5V /  V  X I /  1  >C  z/D  - - I  Figure A3-3  Projected area of intersection for a cylinder of diameter D and hemispheres with radii of r/D = 0.5, 1,1.5 and 2.  cn  CTl  Appendix A3  FINITE DIFFERENCE EQUATIONS FOR ROTARY KILN WALL  168 For t h e a c t i v e l a y e r  E q . ( 3 . 2 ) was s o l v e d  numerically using  the  25 explicit  f i n i t e - d i f f e r e n c e method  s c h e m a t i c a l l y i n F i g . 3-1. for 1)  each  t  _ . / L  x  A  h  •"-OR  difference equations  where M  =  Ar /k 2  w  V 1,1,  A  r  shown  below,  balances:  +,2 I- +_ A T ) T M R M Wl i >^ m  +  T  2  'rr-l-Rp'S,  \v  shown  g->w  w  I  ( A 3 J )  i >'  At  inner  wall:  cv  h  k  T  3  Nodes a t t h e c o v e r e d  2  r  )  V  +  g,s->w  configuration  wall:  u  C  i,l  inner  •'• + h• ^ ^2 ^ AT_ T R M k g a^w " "w w " ga.s-^w ,s+w ~'g+w +  2)  The f i n i t e  node t y p e , were o b t a i n e d by n o d a l  Nodes a t t h e exposed  A  and t h e nodal  w  M.  s  M  R  T  I  M  'w,  J  1,2 0  2Ar h C V  +  3)  {  Interior  k  1  wall  w  w^s  M "  nodes:  "  _ 2 _ _Ar_ "  f  "  R7H  l  i  (A3.2)  } T  'w- , 1,1  169. By examination o f Eqs. (A3.1 - A3.3) the o v e r a l l c r i t e r i o n f o r s t a b i l i t y i s given by «>2 (h +  +h  R  g,s->w  c v  +  g-»w  h  c  (A3.4)  v  w-*s  w  I  Appendix A 4  DETERMINATION OF RADIATIVE HEAT FLOWS USING THE NETWORK METHOD  171 . Solutions f o r the r e s i s t i v e networks used i n the present study are given below. parts.  For c l a r i t y , the presentation has been divided into two  The f i r s t section summarizes the view factors needed to evaluate  the branch resistances shown i n Figs. 2-16', 3-3, 3-4, 3-18, 4-2, 4-3 and 4-8.  Based on these values, the remainder of the appendix b r i e f l y out-  lines the method used to calculate heat flows within each of the systems. For brevity, only material deemed essential to the solution of these analogs i s discussed.  For a more rigorous treatment of radiative analogs 29  the interested reader i s referred to the work of Oppenheim. View factors for the resistive networks 1-zone networks To evaluate the 1-zone branch resistances shown i n Figs. 2-16, 3-3 and 3-18, the three view factors F  , F  , and F  must be determined.  By inspection of Fig. A4-l(a) F sg  =  F wg  -  F sw  =1  (A4.1)  where w  A  =  l 2  A  +  A  +  A  3  +  A  4  =  D(ir  " *L*  ( A 4 , 2 )  and A  s  =  Dsin<J>  L  (A4.3)  172  Figure  A4-1  S c h e m a t i c diagram o f t h e c r o s s - s e c t i o n o f a. r o t a r y k i l n used t o e v a l u a t e v i e w f a c t o r s f o r (a) and (b) 4-zone  analogs.  1-zone  173 4-zone network To evaluate the 4-zone branch resistances shown i n Fig. 3-4, the f i f t e e n view factors summarized in Table A4-1 must be determined. By inspection of Fig. A4-l(b) the number of unique view factors needed to characterize the 4-zone model may be reduced.  Since a l l the surfaces,  1, 2, 3, 4 and s, see the entire freeboard gas volume, by convention i t follows that F  Vg  " 2g - 3g ' % F  F  "  <">  1  M  4  By use of symmetry i t can also be shown that F  s4  F  s3  = s 2 = °- - sl  F„  = F,  2  F  23  F  s l  <-> M  F  5  F  (  -  6  )  (A4.7)  3  " 34 • 12 F  M  5  F  ( A 4  - ' 8  F i n a l l y , since the k i l n wall and solids surface form an enclosure, by use of reciprocity it' can be shown that F  14 ' " l l " 12 " 13 " i f s l 1  F  F  F  F  < - ' M  9  Using Eqs. (A4.4 - A4.9) the number of view factors needed to solve the 4-zone network may be reduced to four, these are, F^, F ^ F ^ and F -j. s  A summary of these view factors i s presented below.  174  TABLE A4-1.  Summary o f kiln  F  S9  F  'g  F  2g  F  3g  F  4g  wall  view  f a c t o r s needed to e v a l u a t e  sl  F  12  F  23  F  s2  F  13  F  24  F  14  F  S3  S4  4-zone  model  F  F  the  F  34  •  175  F1 1  '•  4 sin( -  F„  1.0-  TT  -  <f>  —)  4  (A4.10)  ir - <j> L  IT  sin ,  +  (  s i n  "  Tl  1  s l  3  IT +  !k) - s i n ( 4_ 2 sin <f  <f>,  4  (A4.ll)  4  L  F r  13-  l  2  TT  2 sin (  -  T  .  TT  —  TT  <f>,  — - ) - sin ( — — )  3  -  - sin (  5  Cf>^  |  )j (A4.12)  F  F  12  l  3  (* -  "  * ) L  :  -  TT  2 sin ( F  =  1 2  1-0 - F  (j>,  -)  - 1 —  in  TT  -  (f>  (A4.13)  L  where A  =  1  A  2  = A  3  = A  4  = D/4 (TT -  <^) L  (A4.14)  1 - dimensional and 16-zone networks To evaluate the branch resistances of the 1-dimensional and 16-zone networks of Figs. 4-2 and 4-3, respectively, the 256 view factors i=i  m u s t  b e  determined.  Using Fig. A4-2, i n a similar manner  to that previously described, with the aid of both symmetry and reciproc i t y the number of unique view factors needed to characterize the  Figure A4-2  Schematic diagram of concentric cylinders showing zonal configuration used in 16-zone analog.  179 16-zone analog may be reduced to the 29 view factors given i n Table A4-2 . A summary of these view factors i s presented below.  i s determined  F  i s determined  ] g  using Fig. A4-3 where D' = r y r ^ and L' = Ax/r-,  33  33  using Fig. A4-4 where D' = r^r^  F  18  F  18 " (l+2)(9+8) " 19  F  (l+2)(9+8)  and L' = A X / ^  :  F  F  i s  d e t e r m i n e d  u s i n  ( A 4  -  (  4  1 5 )  9 Fig. A4-4 where D' = ^ / r ^ and  L' = 2Ax/r-j F  H2F  12  F  l,19  =  F  l,21  =  =  1  -  l l  F  h  - 19- 18- l,19- l,21 F  {  2F  F  F  - l l " 19  }  F  F  F  }  (1 2),21 " l , 1 9 F  A  ( A 4  '  (  4  A  J  6  )  1 7 )  J  8  )  +  F  (l+2),21  h  n  F  (l+2)(l+2)  i s  =  " (l+2)(9+8) " ( l + 2 ) ( l + 2 ) F  F  d e t e r i T n n e d  }  ( A 4 , 1 9 )  using Fig. A4-3 where D' = r,,/^ and  L' = 2 Ax/r.,  F  17  F  1 7  :  - ^  ^  3 F  (T  + 2 + 3  )  (y+s+g) ' 1 8 " 1 9 4F  3F  (A4.20)  180 TABLE A4- 2 .  Summary of view factors needed to evaluate the 16-zone resistive analog of Fig.4-3.  45  F  56  4,12  F  57  4,13  F  58  4,14  F  59  35  F  3,12  F  3,13  F  3,14  F  F  ll  F  F  12  F  F  13  F  F  14  F  F  15  F  F  16  F  17  F  F  18  F  F  19  F  25  F  2,12  F  2,13  F  2,14  F  5,10 5,ll 5,12 5,13  181 (1+2+3)(7+8+9)  F  determined using Fig. A4-4 where D' = r ^ / r ^ and  i s  L' = 3Ax/r-,  !n  :  F  13  F  l,22  F  (l+2+3),22  F  =  ' l l " 12 " 17 ' 18 ~ 19 " l , 1 9 " l , 2 2  1  F  =  3F  F  F  !]5  F  (A4.21)  F  (A4.22)  2F  3Ax/r  " ( 1+2+3)(7+8+9) " (l+2+3)(1+2+3)  =  F  F  (A4.23)  }  determined using Fig. A4-3 where D' = t^/r-j and  i s  }  :  F  15  F  (l+2+3+4)5  F  F  (l+2+3),22 " (1+2),21  (l+2+3)(1+2+3)  L' =  F  =  4F  (1+2+  (l+2+3+4)5 " (l+2+3),22  (A4.24)  3F  3 4)( +  =  1+2  h  { 1  "(l+2+3+4)(6+7+8+9) "(1+2+3+4)(1+2+3+4) F  +3+4)  F  a n d  (l+2+3+4)(6+7+8+9)  F  a r e  d e t e r m i n e d  u s i n  Figs. A4-3 and A4-4, respectively, where D' = r ^ / r ^  a n d  !l6 F  =  9 4Ax/r.j  :  16  !i4  (A4.25)  }  ^^ (l+2+3+4)(6+7+8+9)- 19- 18- i7  =  4F  4F  6F  4F  }  ( A 4  '  2 6 )  :  F  14  F  25  F  25  =  " l l " 12 " 13 " 15 " 16 " 17 " 18 " 19 " l , 1 9  1  F  F  F  F  F  F  F  F  F  (A4.27)  :  "  F  l,22  (A4.28)  182 F  35  F  35  F  45  F  45  :  =F  (A4.29)  l,21  ;  F  (A4.30)  l,19  F  F  2,13 " (2+1+16+15+14),13 " (l+2+3+4)5 5F  4F  (2+l+16+15+14),13  F  =  h  { 1  " (2+1+16+15+14)(8+9+10+11+12) F  " (2+1 +16+15+14)(2+1 +16+15+14) F  ( 2 + 1 +16+15+14)(2+1+16+15+14)  F  (A4.32)  }  a n d  F  ( 2 +1+16+15+14)(8+9+10+11+12)  are determined using Figs. A4-3 and A4-4, respectively, where D  = r /r  1  2  1  and L' = 5Ax/r  1  F 3,13' 3,13 (3+2+1+16+15+14),13 " (2+1+16+15+14),13 r  F  =  6F  (3+2+l+16+15+14),13  F  (A4.33)  5F  =  h  { 1  " (3+2+1+16+15+14)(7+8+9+10+11+12) F  . " (3+2+1 +16+15+14) (3+2+1 +16+15+14). F  (3+2+1+16+15+14)(3+2+1+16+15+14)  F  }  a n d  (3+2+1+16+15+14)(7+8+9+10+11+12)  F  are determined using Figs. A4-3 and A4-4, respectively, where D' = r / r ^ and L' = 6Ax/r^ 2  ( 4 • 34) A  183 • F  4,13  :  F  4,13  =  (4+3+2+1+16+15+14),13 " (3+2+1+16+15+14),13  7F  (A4.35)  6F  (4+3+2+l+16+15+14),13  F  =  { 1  " (4+3+2+1+16+15+14)(6+7+8+9+10+11+12) F  " (4+3+2+l+16+15+14)(4+3+2+l+16+15+14) ( * ) F  }  (4+3+2+l+l6+15+14)(4+3+2+1+16+15+14)  a n d  (4+3+2+1+16+15+14)(6+7+8+9+10+11+12)  are determined using Figs.  F  F  A4  36  A4-3 and A4-4, respectively, where D' = r^/r-j and L' = 7Ax/r^ F h  2,12'  F  2,12  =  h  {5F  (2+1+16+15+14) (8+9+10+11+12) " 1 9 " 1 8 5F  -6F  F  3,12  F  3,12 ~  17  8F  - 4F >  '(A4.37)  ]6  :  (3+2+l+16+15+14)(7+8+9+10+ll+12) ' 1 9 "  h  {6F  6F  1 0 F  18  - 1 7 " 1 6 " 2,12> 8F  F  4,12  :  F  4,12  =  h  6F  2,14  :  F  2,14  =  1  ( M  (4+3+2+l+l6+15+14)(6+7+8+9+10+11+12)" 1 9 "  {7F  ?F  " 10F  F  4F  1?  - 8F  1 6  - 6F  - 4F  2 J 2  3 J 2  1 2 F  -  3 8 )  18  }  (A4.39)  - l l ' 1 2 " 1 3 " 25 " 1 7 " 18 " 19 " 16 F  2F  2F  F  2F  " 2,12 " 2,13 - 14 F  F  F  F  F  F  ( A 4  -  4 0 )  184 3,14" 3,H  F  =  " l l * 1 2 ~ 35 " 1 8 " 19 " 17 " 16 " 2,12  1  F  2F  F  2F  F  F  F  F  (A4.41)  " 3,12 " 3,13 " 2,14 " 14 " 13 F  F  4,14  :  F  4,14  =  F  56  F  56  F  F  " l l " 45 " 19 " 18 " 17 " 16 " 2,12 " 3,12 " 4,12  1  F  F  F  F  F  F  F  F  F  F  F  F  4,13" 3,14" 2,14' 14" 13" 12 F  F  F  F  F  (A4.42)  :  A  (A4.43)  \  5  57' F = 57 f  A. A— _1 _ -1 F A 18 A 19  — A  hF  r  ?  c  ?  1  ,  2  1  .  1  (A4.44)  F  A 19 F  6  58'  58  ^8 i A c  4  *8  A.  17  ~ V*8  1  <^F  A  C  1  8  "V  1  1  _ -L; 9  2  (  — A  6  F  (A4.45)  19  59" F59 h  =-A.- i 9  1 -  A. A, 4c 3 A g ~ 1 6 ~ Ag~ 17 F  F  h  A, 1 F 2 ) A 19  1  —  1  r  6  A r Ag 18  A, Lp A 19 r  g  (A4.46)  185 5,10'  '5,10  *10  A  A,  A. 1  ^  F  A,«/19 10  1  c  A10  p  _ I |  A^ 2,12  ZY  4  A  F  " io  1 8  A  .  1  1  A  F  1 7  io  A  1 6  (A4.47)  q  A 19 F  5  5,IT A,  A, 5,11  A  1 - A ^ 3,12  A ^ '2,12  r  t  15  16  %  A  A  n  18  A  n  19  2  ll  A  1 5  ,.1-F  1  A,  1  ll  A  1 7  (A4.48)  1  'l9i  5,12' A. 4 p AT^ 4,12  '5,12  A_ _3_ p A,« 3,12 *12  h  M2  2 . ! h _ A 12 ^2,12 A,o 12 16 A  F  I J T A F 2j A 19  16 15 14 A^ 17-A^ 18-A^ 19  A  A  F  A  F  F  F  F  A  F F  1  F  (A4.49)  g  F r  5,13" A  A, F  5,13 -  l  1  A _o.p h  9  2F  A^ 15"A7 25 cF  -  r  5  ^14 M5 A F, ' 4,13 " A 5  A  F  A 35 A 45  F  5  g  A 16 '3,13 " A 5  ' 5 8 " 59 " 5,10 " 5 , l l " 5,13 F  F  F  F  F  57  2,13 -  F  56  (A4.50)  186 where f o r Eqs. (A4.15 - A4.50) A =A =A =A =A 1  A  6  A  5  2  =  =  A  7  A  =  A  13  3  8  =  A  = A  9  17  4  A  =  =  A  10  18  =  =  A  A  ll  =  19  =A  ] 4  =  A  A  12  20  1 5  =A  = 2AXlTr  =  A  21  ] 6  = 2AXirr  (A4.51)  2  T =  (A4.52) A  22  =  *  ( r  2  " A  ]  (  A  4  '  5  3  )  flame zone analog To evaluate the branch resistances shown i n Fig. 4-8, the four view factors F , F ,., F,-, and F must be determined. sw s f fw fs x  By inspection of  Fig. A4-5 i t follows that  F  fs  F  fw  F  sf  (A4.54)  -  - fs  1  F  A =  (A4.55)  f fs A  (A4.56)  F  s  F sw  =  1 - Fsf  (A4.57)  A  =  d ,  (A4.58)  D sin  ( 4.59)  where  A  f  s  f  A  Analog solutions Having determined the branch resistances the heat flow or current through each branch may be determined by applying Kirchoff's current law at each of the unknown nodes. The result i s a set of simultaneous  Figure A4-5  Schematic diagram of the cross-section of a rotary k i l n used to evaluate view factors within the flame zone.  188 equations which may be solved to y i e l d the unknown potentials.  To  i l l u s t r a t e this technique, solutions for the combined heat-flow networks of Figs. 3-18 and 4-8 are presented i n Appendices A5 and A6, respectively. As outlined in references 25 and 29, solutions for the remaining analogs are developed in a similar manner.  Appendix A5  SOLUTION FOR THE RESISTIVE NETWORK USED TO PREDICT HEAT FLOWS AND TEMPERATURES IN THE ABSENCE OF A FREEBOARD FLAME  190 The modified resistance network shown i n Fig. 3-18 has been used to estimate the inner wall temperature, the overall heat flow to the s o l i d s , and the heat loss through the k i l n w a l l , i n the absence of a freeboard flame.  For a given set of operating conditions  (T , T , T , Ii , _ sh+a h ,h » h „ ) the unknown variables of interest ( T ) Vw V s V s qs' 1oss may be determined by application of Kirchoff's current law at the 4 9  s  a  cv  o v  c  c  c  ex,  nodes F^, E , J $ h  w  q  and J shown i n Fig. 3-18. The result i s a set g  simultaneous nonlinear equation as follows:  J  w  A  :  ~  (fvA * p  + p A t + e F A )J - ( ^ ) a T sw s g g wg w' w P W r  w  n  l  -  - ( wVg> s F  J  S  - < sw sV w F  A  J  1  W w  < swVg  +  F  V  h  +  s  E  g  )  E  g sg s F  +  A  ( A 5  1 )  >s  +  J  S  ( h  -  (A5.2)  cv^ cov> s A  E  ( A 5  -  3 )  W-*-S  g->W sh  g  r  P,  g sg s' g  <K'M*h = < cv  E  w  W  :  (K')^Mb;  u t  A  s h  *^)«4 (h; A )E V  ut  sh  a  (A5.4)  191 where 2 T r k  w  K' = l  n  W < W s h H  T  w  (A5.5)  P  ?  +  1h  ) o  This set of equations was then solved numerically for the unknowns J , J , 32  T  w  and T ^ using a generalized secant method.  Based on these values  the desired solution i s then obtained using the following set of equations. q,  = h'  T  =T  H  loss e x  A , v( T.  out sh  W  4  sh  - E y)  a  v  (A5.6)  '  (A5.7)  Appendix A6  SOLUTION FOR THE RESISTIVE NETWORK USED TO PREDICT HEAT FLOWS AND TEMPERATURE IN THE PRESENCE OF A FREEBOARD FLAME  193  The modified resistance network shown i n Fig. 4-8 has been used to estimate the overall heat flow to the solids and total heat loss from the freeboard flame. T , ti (q > $  ,h  For a given set of operating conditions  ,h  , h  (T , T , f  s  ) the unknown variables of interest  g ) may be determined by application of Kirchoff's current law at f  the 4 nodes E , E J and J shown i n Fig. 4-8. The result i s a set w sh' w s of simultaneous nonlinear equations as follows:  V  e A  ( . + F J\ T, + F A T * ef J™f>w Af f s f <V; ff "sw s i /sV ga  _  :f  r  M  r  'e  o.. P  ri  c f  J w  + -^-) W  A  (A6.1)  ~ J  s  e  f fw f f F  A  E  :  < sfVf F  +  F  swVg w )J  £  ^ sf s f  +  F  F  A  T  A  E  +  +  f fs f f —  '  (  ^7  }  j  w  J  F  swVg  f A  E  P  + ( h(h'  S  A  cv ^ w A  +  (  h  f fs f F  A  +  s s, A  "V  }  J  s (A6.2)  s + ^ + h '  P  f  -C'>'1h -  e  cv  A  cv ^ cov  :  i  V f' ^W .' E  f  •TK )OT , 4  w  W  >  w  +  E  (A6  '  3)  E -  -(K')aTjMh  where  i u t  A  s h  +  K')aTj  h  =  (h^A^jE,  (A6.4)  194  This set of equations was then solved numerically for the unknowns J , w  32  J  s >  T  and T  w  using a generalized secant method.  sh  Based on these  values the desired heat flows were then obtained using the following set of equations. q M  S  = - - - ( J - E ) + h' p s s' cv^ 5  §  ,  x  S  A s e  (E---E) f s'  v  T-^S  (A6.6)  q- = h' A (E, - E ) + h' A (E, - E j f cv ^ w f w' cv ^ s f s' f  +  e  f  f f w f < f - s> F  A  E  J  +  E  f fs f F  A  ( E  f - s J  )  ( A 6  '  7 )  Appendix A7  FORTRAN SOURCE LISTING AND SAMPLE OUTPUT FOR KILN WALL MODEL  FORTRAN SOURCE LISTING KILN WALL MODEL  1  2 3  4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51  c c c F I N I T E D I F F E R E N C E MODEL TO C A L C U L A T E c D I S T R I B U T I O N I N ROTARY K I L N WALL • c c J . P . GOROG c OCT. 2 3 , 1 9 8 0 c c  TEMPERATURE  I M P L I C I T R E A L * 8 ( A - H,0 - 2 ) DIMENSION R ( 5 0 0 ) , T ( 4 0 0 . 5 0 ) . T S S ( 5 0 0 ) INTEGER ! S T E P S . C S T E P S . S T E S T , C Y C L E INTEGER ! T E S T 2 ( 3 ) , T E S T 3 ( 3 ) . T E S T 4 ( 3 ) INTEGER DUM1. DUM2, UUM3. FNODE L O G I C A L P L O T . REFRAC R E A L » 8 K, M. K 1 , KP COMMON / B L K 1 / TA. T S . TG. AW, AS. EMW, EMG. EMS COMMON / B L K 2 / R I . RO. DT. DR. RPM COMMON / B L K 3 / KP. K. THETA, FD. ALPHA, E S H . K l . COMMON / B L K 4 / HEX. HCOV. HOUT, U S E . H0UT2 COMMON / B L K 5 / T, T S S , HRAD. R COMMON / B L K 6 / OWECV. QWER. OWCCV. OTHRU, QTHRU1 COMMON / B L K 7 / QG, OW, OS. OWC. OWE. OWT. QSE COMMON / B L K 8 / O I E T . 0 1 E T . Q1ST COMMON / B L K 9 / TAVE. TAVC COMMON / B L K 1 0 / S T E P S . C S T E P S . NODES. FNODE COMMON / B L K 1 1 / QW4. QWE4. 0 4 E T . 0 S 4 . 0 4 S T . 0WT4 COMMON /BLK 12/ T 2 . T 3 . T 4 . T 5 COMMON /BLK 14/ HRADS COMMON / B L K 1 5 / TWAB, TIME CALL S E T L I O ( ' 6 ' , '-A ' ) CALL S E T L I 0 C 8 '. '*SINK* ') C A L L CMD('$EMPTY -A OK 13) 1  c c c c c c c c c c c c c c c  READ  INPUT DATA  10 = TS = TG* TA = EMS" EMW = EMGRI« R0 = RPM= T HE TA = Y.=  c;>»  RUN I D E N T I F I C A T I O N NO. S O L I D S TEMPERATURE ( K ) GAS TEMPERATURE ( K ) AMBIENT TEMPERATURE ( K ) SOLIDS EMISSIVITY WALL E M I S S I V I T Y GAS E M I S S I V I T Y INNER RADIUS (M) OUTER RADIUS (M) K I L N SPEED ( R E V . PER MIN.) HALF ANGLE SUBTENDED BY SOLIDS THERMAL C O N D U C T I V I T Y (W/M K ) S P E C I F I C HEAT (J/KGR K )  (PAD.)  52 53 54 55 56 57 58 59 GO 61 62 63 64 65 66 67 68 60 70 " 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103  C C C C C C C C C C  RHO= Kt= RF = HEX= HCOV= HOUT= HSE= REFRAC= PLOT=  '  READ READ READ READ READ READ READ  DENSITY (KGR/M**3) THERMAL C O N D U C T I V I T Y (W/M K ) INNER T H I C K N E S S (M) HTC GAS TO WALL ( W / M « * 2 K ) H T C WALL TO S O L I D S (W/M**2 K ) HTC OUTER WALL TO AMBIENT (W/M**2 K ) HTC SOLIDS SURFACE (W/M**2 K ) ONE SHELL PLOT DRAW (5.310) (5,320) (5.320) (5,320) (5,320) (5,320) (5.300)  ID T S . T G . TA EMS, EMW. EMG R I , RO, RPM, THETA K, C P , RHO, K l , RF HEX. HCOV. HOUT. HSE REFRAC. PLOT  C C I N I T I A L I Z E ACCUMULATORS C E S H = 1.DO S T E P S = 161 . FNODE = 25 MAX I = 4 0 0 MAX 12 = 5 0 DO 10 I * 1, 3 TEST2(I ) ' 0 TEST3(I ) = 0 TEST4(I ) = 0 10 CONTINUE C CYCLE = 1 STEST = 1 C C S E T NODES C DR = .1D-02 T8AR = ( T G + T S ) / 2.DO NODES = I D I N T ( ( R O - R I ) / D R ) + 1 C DD 2 0 I = 1, NODES 20 R ( I ) » RI + F L O A T O - 1 ) * DR C C S E T TIME S T E P S C RPS = RPM • (1.DO/60.DO) TIME « 1.00 / RPS FRAC = THETA / D P I ( O . D O ) C T I M E = FRAC * TIME DT = TIME / F L O A T ( S T E P S - 1) C S T E P S «= S T E P S - IDI NT (CT IME/DT ) C  KD  CO  104  C DETERMINE  105 106 107 108  C  EFFECTIVE  I F ( .NOT. REFRAC) T10 = DLOG(RO/RI) T l 1 = DLOG(RF/RI)  109  T12  110  KP  111 112 113 1 14 115 116 117  GO TO 4 0 KP = K CONTINUE  30 40 C C SET C  123 124 125 126 127 128 129 130 13 1 132 133 134 135 136 137 138 139 140 14 1 142 '143  144 145 146 147 148 MO 150 151 152 153 154 155  = »  GO  CONDUCTIVITY  TO  OF  WALL  30  DLOG(RO/RF) TIO  /  INITIAL TBAR  118 1 19 120 121 122  THERMAL  (T11/K  +  T12/K1)  TEMPERATURE  = (TG + TS) /  DISTRIBUTION 2.DO  T 4 0 0 = ( D L O G ( R O / R I ) ) / KP  +  1.D0  /  (HOUT»RO)  C DO * 50 C  5 0 I = 1, NODES T401 = ( D L O G ( R O / R ( I ) ) ) / KP + 1.DO / (HOUT*RO) T S S ( I ) = ( ( ( T B A R - T A ) * T 4 0 1 ) / T 4 0 0 ) + TA  CONTINUE TFIX  C C SET C  =  TSS(FNODE)  RADIATION  PARAMETERS  AND  BED  DEPTH  EMS 1 = EMS EMW1 = EMW I F (EMS .GT. .999) EMS => . 9 9 9 9 D 0 I F (EMW .GT. .999) EMW = .9999D0 AW = ( D P I ( 0 . D 0 ) » 2 . D 0 * R I ) * ( F L 0 A T ( C S T E P S - 1 ) / F L 0 A T ( S T E P S DTHETA = ( D P I ( 0 . D O ) • 2 . 0 0 ) / F L O A T ( S T E P S - 1) AC = ( 2 . D O * R I * D P I ( O . D O ) ) - AW THETA = AC / ( 2 . D 0 * R I ) FD «= ( 1 .DO-DCOS(THETA) ) / 2 . DO AS - 2.DO * RI * D S I N ( T H E T A ) FSW • 1.DO FWS = D S I N ( T H E T A ) / ( D P I ( O . D O ) - T H E T A ) FWW » 1.D0-FWS SIGMA ' 5.67D-08 x  C C SET C  ROW  •  ROS  - 1.DO-EMS  1.DO-GMW  INITIAL  RADIATIVE  CALL HCOEFF(TBAR,  HTC HRAD,  1.  HRADS)  C C S E T HTC A T COVERED WALL C C C SET F I N I T E D I F F E R E N C E PARAMETERS C ALPHA = ( K / ( R H 0 C P ) ) +  - 1))  15G 157 150 159 1G0 161 1G2 163 164 165 166 1G7 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 1B3 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 20G 207  N2 = FNODE - 1 N = NODES - 1 N4 = FNODE - 2 N5 = FNODE / 2 M « DR •* 2 / (ALPHA *DT) C C BEGIN F I N I T E DIFFERENCE CALCULATIONS C DO 6 0 I » I, N2 GO T( 1 ,I) = T S S ( I ) C C 70 DO 100 d = 2. S T E P S C C I N T E R I O R NODES C DO 8 0 I = 2. N4 80 T ( J . I ) = (1.D0/M) * ( T ( 0 - 1.1 - 1) + T ( J - 1,1 + 1 ) ) + l 1 . D 0 - ( 1 2.DO/M)) • T ( d - 1.1) + ( D R / ( M * 2 . D 0 * R ( I ) ) ) • ( T ( d - 1.1 + 1) 2 T ( J - 1,1 - 1) ) C C I N T E R I O R F I X E D NODE C T ( d . N 2 ) = (1.D0/M) * ( T ( d - 1.N4) + T F I X ) + ( 1 . D O - ( 2 . D 0 / M ) ) * T ( 1 d - 1.N2) + ( D R / ( 2 D 0 * M * R ( N 2 ) ) ) • ( T F I X - T ( d - 1 .N4) ) C IF ( d .GE. C S T E P S  AND.  d  .LT. S T E P S ) GO  TO  90  C C EXPOSED INNER SURFACE NODE C T 1 0 0 » ( 2 . D 0 * H R A D » R I * D R ) / (M*K*(RI + DR/4.D0)) T102 = (2.D0*HEX*RI*DR) / (M*K*(RI + DR/4.D0)) T 1 0 4 = ( 2 . D 0 » R I + DR) / ( M * ( R I + D R / 4 . D 0 ) ) T ( d . l ) = ( T 1 0 0 + T 1 0 2 ) • TG + T104 * T ( d - 1.2) + ( 1 . 0 0 - T 1 0 0 1 T 1 0 2 - T 1 0 4 ) * T ( d - 1.1) GO TO 100 C C COVERED INNER SURFACE NODE C 90 T 1 5 0 •> ( 2 . D 0 * H C 0 V * R I * D R ) / ( M * K * ( R I + D R / 4 . D 0 ) ) T 1 5 2 = ( 2 . 0 0 * R I + DR) / ( M * ( R I + OR/4.DO)) T ( d . 1 ) ='T152 * T ( d - 1.2) + T 1 5 0 * TS + (1.DO-T152 - T 1 5 0 ) * T ( 1 d - 1 . 1 ) lOO CONTINUE C C ADdUST ACCUMULATORS C 00 H O 1 = 1.2 T E S T 2 ( I ) = TEST2(I + 1 ) T E S T 3 ( I ) = T E S T 3 ( I + 1) TEST4(I) = TEST4(I +1) 1IO CONTINUE  O O  200 203 2 10 21 1 2 12 2 13 214 215 216 217 218 2 19 220 22 1 222 223 224 225 226 227 228 229 230 23 1 232 233 234 235 236 237 238 239 240 24 1 242 243 244 245 246 247 248 249 250 25 1 252 253 254 255 256 257 258 259  C  120 130  140 150  160 170  T200 = DABS(T(1.1) - T ( S T E P S . D ) T201 - D A B S ( T ( 1 . N 2 ) - T ( S T E P S . N 2 ) ) T202 = DABS(T(1.N5) - T ( S T E P S . N 5 ) ) IF ( T 2 0 0 . L T . .1D-04) GO TO 120 TEST2(3) = 2 GO TO 130 TEST2(3) = 1 IF ( T 2 0 1 . L T . .1D-04) GO TO 140 TEST3(3) = 2 GO TO 150 TEST3(3) = 1 IF ( T 2 0 2 . L T . .1D-04) GO TO 160 TEST4(3) = 2 GO TO 170 TEST4(3) = 1 CONTINUE IF ( C Y C L E . L E . 4 ) GO TO 190 DUM1 = 0 DUM2 = 0 DUM3 = 0  C DO  180 I DUM1 = DUM2 = DUM3 = CONTINUE  = 1. DUM1 DUM2 DUM3  3 + TEST2(I) + TEST3(I) + TEST4(I)  180 C C CHECK FOR CONVERGENCE C IF (DUM1 .EQ. 3 .AND. DUM2  .EQ. 3  .AND. DUM3 .EQ  1 GO TO 2 3 0 CONTINUE  190 C C CHECK C Y C L E C CYCLE = CYCLE + 1 IF ( C Y C L E .GT. MAX I ) GO TO 2 0 0 GO TO 2 1 0 IF ( S T E S T .EQ. 1) H0UT2 = HOUT 200 C A L L PRINT 1 ( 1 0 . MAXI, REFRAC. PLOT. EMS 1, EMW1) GO TO 2 9 0 C C R E S E T TEMPERATURE D I S T R I B U T I O N AND RETURN TO C F I N I T E DIFFERENCE CALCULATIONS C DO 2 2 0 I = 1. N2 2 10 T(1.I) » T(STEPS.I) 220 C C DETERMINE AVERAGE EXPOSED WALL TEMPERATURE AND S E T C R A D I A T I V E HTC C  ro o  260 261 262 263 264 265 266 267 268 269 270 27 1 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 29 1 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 3 IO 3 1 1  C A L L T A V ( T A V E , TAVC. T, C S T E P S . S T E P S ) C A L L H C O E F F ( T A V E . HRAD. 1. HRADS) GO TO 7 0 C C CHECK FOR F I N A L SOLUTION - COMPARE HEAT FLOWS INTO C NODE, A D J U S T TEMPERATURES AND RESTART C A L C U L A T I O N S C I F NEEDED  c  230  •  23 1  FIXEO  CALL FLOWS(RI) OC = OTHRU QSS = ( ( T F I X - T S S ( F N O D E + 1 ) )/DLOG(R(FNODE + 1 ) / R ( F N O D E ) ) ) * KP 1 2.DO * D P I ( O . D O ) QTEST * DABS((OC - OSS)/OSS) I F ( S T E S T .EQ. 1)GOTO 231 IF ( O T E S T . L E . .5D-02 .AND. OC .GT. 0.DO .AND. OSS .GT. O.DO) 1 GO TO 2 8 0 CONTINUE TBAR = ( ( Q C / ( 2 . D O * D P I ( O . D O ) * K P ) ) * D L 0 G ( R ( F N 0 D E ) / R I ) ) + T F I X H0UT2 = ( E S H * S I G M A * ( T S S ( N 0 D E S ) * * 4 - T A * * 4 ) ) / ( T S S ( N O O E S ) - T A ) H0UT1 = HOUT + H0UT2 T 4 0 0 » ( D L 0 G ( R 0 / R I ) ) / KP + 1.D0 / (H0UT1*R0)  C DO  240 1 = 1 . NODES T 4 0 1 = ( D L O G ( R O / R ( I ) ) ) / KP + 1.D0 / (H0UT1*RO) T S S ( I ) = ( ( ( T B A R - T A ) * T 4 0 1 ) / T 4 0 0 ) + TA CONTINUE  240 C C C CHECK C Y C L E C IF ( S T E S T .GT. MAX 12) GO TO 250 GO TO 2 6 0 250 C A L L P R I N T 2 ( I D , MAX 12. REFRAC. PLOT. EMS 1, GO TO 2 9 0 260 STEST = STEST + 1 TFIX = TSS(FNODE) CYCLE = 1 C DO 2 7 0 I = 1, N2 270 T(1.1) = TSS(I) C GO TO 7 0 280 C A L L T A V ( T A V E . TAVC, T, C S T E P S . S T E P S ) C C C INTEGRATE HEAT FLOWS C CALL FLOWS(RI) C C CALCULATE 1-ZONE HEAT FLOWS C C A L L Z O N E ( T A V E , TAVC. R I , THETA. AC. 2)  EMW1)  312 313 3 14 3 IS 3 16 317 318 319 320 32 1 322 323 324 .325 326 327 328 329 330 331 . 332 333 334 335 336 337  C C C A L C U L A T E 4-ZONE HEAT FLOWS C C A L L ZONE4 C C PRINT F I N A L SOLUTION C C A L L P R I N T ( I O . REFRAC. EMS 1, EMW1) C C C A L C U L A T E HEAT FLOWS U S I N G ANALOG APPROXIMATIONS C C A L L ANAL 1 >. C A L L ANAL2 C C 9 7 0 0 FORMAT C 290 CALL P 9 7 0 0 U D ) C C PLOT R E S U L T S C IF (PLOT) CALL P L O T I T ( I D ) STOP 300 FORMAT ( 2 L 1 ) 310 FORMAT ( 4 1 3 ) 320 FORMAT ( 8 G 1 5 . 5 ) END  ro  o  CO  338  £»+••+•»•**  339 340 34 1  C C C SUBROUTINE  342 343  C C  344  345 346 347 348 349 350 351 352 353 354 355 3'JG 357 358 359 360 36 1 362 363 364 365 366 367 368 369 370 371 372 373 374  TO OETERMINE  AVERAGE  WALL  TEMPERATURES  c****«*«*»*  SUBROUTINE T A V ( T A V E , TAVC, T, C S T E P S . I M P L I C I T R E A L * 8 ( A - H.O - Z ) DIMENSION T ( 4 0 0 , 5 O ) , A C ( 6 4 ) INTEGER C S T E P S , S T E P S  '  STEPS)  C C EXPOSED WALL C ' DUMMY = D A C S U M ( A C . T , 0 ) N = CSTEPS - 1 C DO 10 I = 1. N IO DUMMY - DACSUM( AC . T ( I . 1 ) ) C TAV1 = DACSUM(AC) TAVE = TAV1 / F L O A T ( N ) C C COVERED WALL C DUMMY = D A C S U M ( A C . T , 0 ) C N1 = S T E P S - 1 C 0 0 2 0 I = C S T E P S , N1 20 DUMMY = D A C S U M ( A C . T ( I , 1 ) ) C TAV1 = OACSUM(AC) TAVC = TAV1 / F L O A T ( S T E P S - C S T E P S ) DUMMY = DACSUM(AC.T,O) RETURN END  ro o  375 37G 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 ' 4 10 4 11 4 12 413 4 14 4 15 4 16 417 4 18 4 19 420 421 422 423 424 425  Q* ********  *  Q  c c c SUBROUTINE c c Q**********  TO C A L C U L A T E  INTEGRATED HEAT  FLOWS  SUBROUTINE F L O W S ( R I ) I M P L I C I T R E A L * 8 ( A - H.O - Z ) DIMENSION T ( 4 0 0 . 5 0 ) . T S S ( 5 0 0 ) . R ( 5 0 0 ) . A C ( 6 4 ) R E A L * 8 K. K 1 , KP INTEGER FNODE. S T E P S . C S T E P S COMMON /BLK1 / TA, T S , TG. AW, AS. EMW, EMG, EMS COMMON / B L K 3 / KP. K. THETA. FD. ALPHA, E S H . K I , COMMON / B L K 4 / HEX, HCOV. HOUT. H S E . H0UT2 COMMON / B L K 5 / T, T S S . HRAD. R COMMON / B L K 6 / QWECV, QWER, QWCCV, QTHRU. QTHRU1 COMMON / B L K 8 / Q l E T , Q 1 E T , Q1 ST COMMON / B L K 1 0 / S T E P S . C S T E P S . NODES. FNODE DTHETA = ( 2 . D 0 * D P I ( 0 . D 0 ) ) / F L O A T ( S T E P S - . 1 ) N = CSTEPS - 1  c c c c 10 C  C O N V E C T I O N GAS TO EXPOSED QTEMP DUMMY  c c 20 C  = O.DO = DACSUM(AC.QTEMP,0)  DO  10 I = 1. N QTEMP = HEX • RI * DTHETA DUMMY = DACSUM(AC,QTEMP) QWECV  C C  WALL  RADIATION DUMMY DO  * (TG - T ( I . D )  = DACSUM(AC) TO EXPOSED  WALL  = DACSUM(AC,QTEMP.0)  2 0 I = 1.  N  QTEMP = HRAD • RI • DTHETA * ( T G DUMMY = DACSUM(AC.QTEMP) QWER  T(I.D)  = DACSUM(AC)  C  c c c 30  C O N V E C T I O N COVERED WALL  TO S O L I D S  DUMMY = DACSUM(AC.QTEMP.0) N2 = S T E P S - 1 DO  ro o tn  3 0 I = C S T E P S . N2  QTEMP = HCOV • RI * DTHETA DUMMY « = DACSUM( AC , QTEMP )  * (TS -  T(I.D)  42G 427 428 429 430 43 1 432 433  C OWCCV = DACSUM(AC) OTHRU = QWER + OWECV + OWCCV OTHRU t = ( 2 . D O * D P 1 ( O . D O ) * K P * ( T S S ( F N O D E 1DLOG(R(NODES - 1)/R(FNODE 0 1 E T = QWER + OWECV RETURN END  + 1) - TSS(NODES  - 1))) /  • 1))  ro o cn  434  £•*•*••••••  435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 45 1 452 453 454 455 456 457 458 459 460 4G1 462 463  C C C SUBROUTINE TO C A L C U L A T E R A D I A T I V E HTC USING 1-ZONE GRAY C GAS ANALOG C C C********** SUBROUTINE HCOEFF(TW. HRAD, N, HRADS) I M P L I C I T R E A L * 8 ( A - H,0 - Z ) DIMENSION A ( 2 , 2 ) , X ( 2 ) , IPERM(4), T ( 2 . 2 ) , R ( 5 ) , B ( 2 ) COMMON /BLK 1/ TA. T S . TG, AW. AS. EPW. E P G . EPS COMMON / B L K 4 / HEX. HCOV, HOUT, H S E . H0UT2 COMMON / B L K 8 / O I E T , Q 1 E T . Q1ST COMMON / B L K 7 / OG. OW, OS. QWC. OWE. OWT, OSE SIGMA = 5.67D-08 C C ENTRY FOR 1-ZONE GRAY GAS HEAT FLOWS C ENTRY ZONE(TW,TWC.RI,THETA.AC,N) ES = SIGMA * TS •* 4 EW = SIGMA * TW * * 4 . EG = SIGMA * TG *• 4 TRG = 1.DO-EPG FWG » 1 .DO F S G = 1 .DO FSW = 1.DO R ( 1 ) = (EPW*AW) / (1.D0-EPW) R ( 2 ) » AW » FWG * EPG R ( 3 ) = AS • FSG * EPG  464  R(4)  465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 48 1 482 483 484  R ( 5 ) = AS * FSW • TRG A(1.1) =R(5) A(1.2) = -(R(3) * R(4) + R(5)) A(2.1) = -(R(2) + R(1) + R(5)) A(2,2) «R(5) B(1) = -(R(3)*EG + R(4)*ES) B ( 2 ) = - ( R ( 2 ) * E G + R(1)*EW) C A L L S L E ( 2 , 2. A, 1. 2. B. X. I PERM, 2, T. DEP. J E X P ) I F ( D E P ) 10, 2 0 . 10 OW = ( X ( 1 ) - EW) * R ( 1 ) OS = ( X ( 2 ) - E S ) * R ( 4 ) HRADS = OS / ( A S * ( T G - T S ) ) HRAD = OW / (AW*(TG - TW)) I F ( N .EQ. 1) RETURN QG = ( X ( 2 ) - EG) * R ( 3 ) + ( X ( 1 ) - E G ) • R ( 2 ) QWC = HCOV * AC * ( T S - TWC) OWE = HEX * AW * ( T G - TW) QWT = QW + QWC + OWE QSE = H S E * AS • ( T G - T S ) Q 1 E T = QW * QWE  10  •  (EPS'AS)  /  (1.D0-EPS)  ro o  485 486 487 488 489 490 491  20  30  0 1 S T = OS + OSE RETURN WRITE ( 6 . 3 0 ) CALL EXIT STOP FORMAT ( ' ' . '1-ZONE  SOLUTION  FAILED')  END  ro O CO  492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 0 10 511 512 513 514 515 5 16 517 518 519 520 52 1 522 523 524 525 526 527 528 529 530 531 532 533 534 535 53G 537 538 539 540 54 1 542  c c c c c  SUBROUTINE  TO PRINT  RESULTS  c**********  SUBROUTINE P R I N T ( I D . REFRAC. EMS 1. EMW1) I M P L I C I T R E A L * 8 ( A - H.O - Z ) DIMENSION T ( 4 0 0 . 5 0 ) . T S S ( 5 0 0 ) , R ( 5 0 0 ) LOGICALM STARS(6) /6*'*'/ INTEGER FNODE. S T E P S . C S T E P S L O G I C A L REFRAC, PLOT R E A L * 8 K, K l . KP COMMON /BLK 1/ TA, T S . TG. AW. AS. EMW. EMG. EMS COMMON / B L K 2 / R I . RO. DT, DR, RPM COMMON / B L K 3 / KP, K. THETA, FD. ALPHA, ESH, K 1 . COMMON / B L K 4 / HEX, HCOV, HOUT. HSE , H0UT2 COMMON / B L K 5 / T. T S S , HRAD, R COMMON / B L K 6 / OWECV, OWER, OWCCV. OTHRU. OTHRU1 COMMON / B L K 7 / OG. OW. OS. QWC. OWE. QWT. OSE COMMON / B L K 8 / O I E T , 0 1 E T . 0 1 S T COMMON / B L K 9 / TAVE. TAVC COMMON /BLK 10/ S T E P S . C S T E P S . NODES. FNODE COMMON /BLK 11/ QW4. QWE4, 0 4 E T . 0 S 4 . Q4ST, QWT4 COMMON /BLK 12/ T 2 . T 3 . T 4 . T 5 COMMON /BLK 14/ HRADS C A L L HEADER WRITE ( 6 . 1 1 0 ) ID N = 1 GO TO 10  c c c  ENTRY  c c c  ENTRY  10  WHEN CONVERGENCE  NOT REACHED  IN MAXI  ENTRY P R I N T 1(ID.MAXI,REFRAC,PLOT,EMS PLOT = . F A L S E . C A L L HEADER WRITE ( 6 . 1 2 0 ) ID, MAXI N = 2 GO TO 10 WHEN F I N A L SOLUTION  NOT REACHED  CYCLES  1,EMW1)  IN MAX 12 CYCI.I  ENTRY PR INT2(ID,MAX I 2,REFRAC,PLOT.EMS 1.EMW1) PLOT = . F A L S E . C A L L HEADER . WRITE ( 6 , 1 3 0 ) I D . MAXI2 N » 2 N1 = FNODE - 1 RT1 = RF - RI RT2 = RO - RF  ro o io  543 544 545 546 547 54S 549 550 551 552 553 554 555 556 557 558 559 5GO 56 1 562 563 564 565 566 567 568 569 570 571 572 573 574 .575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594  20 30-  '  40 50  60  70  80 90  100  110 120 130 140  WRITE ( 6 . 1 5 0 ) RI . RO, D T . OR. RPM WRITE ( 6 . 1 4 0 ) T S . EMS 1, TG. EMG. TA. ESH, EMW1 WRITE ( 6 . 1 6 0 ) THETA. FD I F ( R E F R A C ) GO TO 20 GO TO 30 WRITE ( 6 , 5 0 ) RT1, K. ALPHA, RT2, K I WRITE ( 6 . 1 7 0 ) K. ALPHA WRITE ( 6 . 1 8 0 ) HEX. HCOV, HOUT, HSE. HRAD. H0UT2. HRADS WRITE ( 6 , 1 9 0 ) C A L L DPRMAT(T, 4 0 0 . 5 0 . S T E P S . N I . 1, 1, 4 0 0 , 1) WRITE ( 6 . 2 0 0 ) C A L L DPRMAT(TSS, 1, 2 5 0 . 1. NODES. 1. 1. 1. 1) IF (N .NE. 1) GO TO 40 WRITE ( 6 . 6 0 ) TAVE. T 2 . T 3 . T4, T 5 . TAVC WRITE ( 6 . 7 0 ) QWER. QW, QW4, QWECV, OWE, QWE4, Q l E T , Q1ET, Q4ET WRITE ( 6 , 8 0 ) QWCCV. QWC. QWC WRITE ( 6 . 9 0 ) STARS. QS. QS4. STARS, OSE. QSE, STARS, Q1ST, Q4ST WRITE ( 6 . 1 0 0 ) QTHRU. QWT. QWT4. OTHRU1. STARS. STARS RETURN FORMAT (' '. T 5 , 'COMPOSITE REFRACTORY WALL:'/TIO, 1 'THICKNESS OF INNER REFRACTORY=', IX. G 1 2 . 5 . IX, '(M)'/T10, 2 'INNER THERMAL C O N D U C T I V I T Y ' ' , IX, G 1 2 . 5 . 1X, 3 '(W/M S ) ' / T 1 0 , 'INNER THERMAL D I F F U S I V I T Y =' , IX, G 1 2 . 5 . IX, 4 ' ( M * * 2 / S ) ' / T 1 0 , 'OUTER THICKNESS=', 1X, G 1 2 . 5 . IX. 5 ' ( M ) ' / T 1 0 , 'OUTER THERMAL CONDUCTIVITY'', IX, G12.5, 1X, 6 ' (W/M K) '/) FORMAT ('1'. 'HEAT FLOW MODLES:'///T5, 'AVERAGE EXPOSED WALL TEMPE 1RATURES ( K ) : ' / T 1 0 . '1-Z0NE=', G12.5, T 3 5 , '4-ZONE'. T 4 1 , '#1=', 2 G 1 2 . 5 / T 4 1 , '/f2='. G 1 2 . 5 / T 4 1 . ' *3=' . G 1 2 . 5 / T 4 1 . '04=', 3 G 1 2 . 5 / / T 5 . 'AVERAGE COVERED WALL TEMPERATURE=', G12.5//) FORMAT (' '. T5, 'CALCULATED HEAT FLOWS (W/M):'//T25. 1 'INTEGRATED', T 5 0 . '1-ZONE', T 7 5 , ' 4 - Z 0 N E ' / T 5 . 2 'EXPOSED WALL: ' / T I O . 'RADIATION'. T 2 5 , G12.5, T 5 0 . G 1 2 . 5 . 3 T 7 5 . G 1 2 . 5 / T 1 0 . 'CONVECTION', T 2 5 , G 1 2 . 5 . T 5 0 . G12.5. T 7 5 , 4 G 1 2 . 5 / T 1 0 , 'TOTAL'. T 2 5 , G 1 2 . 5 . T 5 0 . G12.5, T 7 5 . G 1 2 . 5 / / ) FORMAT (' '. T 5 . 'COVERED WALL:'/TIO, 'CONVECTION', T 2 5 . G 1 2 . 5 . 1 T 5 0 , G12.5, T 7 5 . G 1 2 . 5 / / ) FORMAT (' ', T 5 . 'SOL I D S : ' / T 1 0 . 'RADIATION', T 2 5 , 6A1, T 5 0 , G 1 2 . 5 , 1 T 7 5 , G 1 2 . 5 / T 1 0 , 'CONVECTION', T 2 5 , 6 A 1 . T 5 0 , G 1 2 . 5 . T 7 5 . 2 G 1 2 . 5 / T 1 0 . 'TOTAL'. T 2 5 . 6 A 1 . T 5 0 . G 1 2 . 5 . T 7 5 . G 1 2 . 5 / / ) FORMAT (' '. T 5 , 'THROUGH W A L L : ' / T 1 0 . ' D I F F E R E N C E ' , T 2 5 . G12.5, 1 T 5 0 . G 1 2 . 5 . . T 7 5 . G 1 2 . 5 / T 1 0 . ' P R O F I L E ' . T 2 5 . G12.5. T 5 0 , 2 6A1, T75. G A I / ' I ' , 'ANALOG APPROXIMAT'ONS:'/) FORMAT (' '. 'FINAL SOLUTION FOR RUN NO.'. IX. 14//) FORMAT (' ', 'FOR RUN NO.'. 1X. 14. 1X, 'NO CONVERGENCE I N ' , IX. 1 14, IX, ' C Y C L E S ' / / ) FORMAT (' ', 'FOR RUN NO.'. IX. 14. IX. 'FINAL SOLUTION NOT REACHE ID I N ' , IX. 14, 1X. ' C Y C L E S ' / / ) FORMAT (' ', T 5 . 'SOLIDS TEMPERATURE''. IX, G 1 2 . 5 . IX. '(K)'/T5, 1 'SOLIDS E M I S S I V I T Y = ' . IX. G 1 2 . 5 / T 5 . 'GAS T E M P E R A T U R E ' , 1X, 2 G 1 2 . 5 . IX, ' ( K ) ' / T 5 , 'GAS E M I S S I V I T Y ' ' . 1X. G 1 2 . 5 / T 5 . 3 'AMBIENT TEMPERATURE''. 1X. G 1 2 . 5 . IX, '(K)'/T5. 3  ,  595 59G 597 590 599 600 601 602 603 604 605 606 607 608 609 610 611 612 6 13 614 6 15  150  160 170  180  190 200  4 'SHELL E M I S S I V I T Y ' ' , IX, G 1 2 . 5 / T 5 . 'WALL E M I S S I V I T Y = '. 1X, 5 G12.5/) FORMAT (' '. T 5 . ' K I L N INNER RADIUS''. G12.G. IX, ' ( M ) ' / T 5 , 1 ' K I L N OUTER RADIUS''. G12.5, 1X, ' ( M ) ' / T 5 , 'TIME S T E P ' ' , 2 G 1 2 . 5 . IX. ' ( S E C ) ' / T 5 , 'RADIAL STEP=', G 1 2 . 5 . 1X, ' ( M ) ' / T 5 . 3 ' K I L N SPEED'', G 1 2 . 5 . IX, ' ( R P M ) ' / ) FORMAT (' '. T 5 , 'HALF ANGLE SUBTENDED BY S O L I D S = ' . G 1 2 . 5 / T 5 . 1 'RATIO BED D E P T H : K I L N DIAMETER''. G12.5/) FORMAT (' '. T 5 . 'WALL THERMAL C O N D U C T I V I T Y ' ' . G12.5, 1X, 1 '(W/M K ) ' / T 5 , 'WALL THERMAL D I F F U S I V I T Y = ' , G12.5, IX. 2 '(M**2/S)'/) FORMAT (' '. T 5 . 'CONVECTIVE HTC (W/M**2 K ) : ' / T 8 , 1 , 'GAS TO EXPOSED WALL''. G 1 2 . 5 / T 8 . 'SOLIDS TO COVERED WALL=' 2 . G 1 2 . 5 / T 8 . 'OUTER S H E L L TO ATMOSPHERE''. G 1 2 . 5 / T 8 . 3 'GAS TO S O L I D S ' ' . G 1 2 . 5 / / T 5 . 'RADIATIVE HTC (W/M**2 K ) : ' / 4 T 8 . 'EXPOSED WALL''. G 1 2 . 5 / T 8 . 'OUTER SURFACE''. G 1 2 . 5 / T 8 . 5 'EXPOSED S O L I D S ' ' . G 1 2 . 5 / ) FORMAT ('1'. T 5 . 'THE UNSTEADY S T A T E TEMPERATURE F I E L D FOLLOWS:'// 1 ) FORMAT ('1'. T 5 . 'THE STEADY STATE TEMPERATURE F I E L D FOLLOWS:'//) ENO  ro  6 16 617 6 18 6 19 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 64 1 64 2 64 3 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666  c cC  SUBROUTINE TO CALCULATE 4- ZONE RADIATIVE HEAT FLOWS  C C Q**********  SUBROUTINE Z0NE4 IMPLICIT REAL«8(A - H,0 - Z) REAL*8 K, K l . KP INTEGER CSTEPS. STEPS, FNODE DIMENSION R(500). T(400.50). TSS(500), AC(64) DIMENSION A ( 5 , 5 ) . B ( 5 ) , X ( 5 ) . IPERM(10), TT(5.5) COMMON /BI.K1/ TA, TS,TG. AW. AS. EMW. EMG, EMS• COMMON /BLK3/ KP. K, THETA, FD. ALPHA, ESH. K l . COMMON /BLK4/ HEX, HCOV, HOUT. USE. H0UT2 HRAD. R COMMON /BLK5/ T. TSS. COMMON /BLK 11/ 0W4. 0WE4. Q4ET, 0S4. 04ST, 0WT4 COMMON /BLK 12/ T1, T2,T3. T4 COMMON /BLK10/ STEPS. CSTEPS. NODES. FNODE COMMON /BLK7/ QG. QW, OS. OWC. OWE. OWT, OSE J4 REAL*8 US. J l . J2. J 3 . N - CSTEPS - 1 N1 - N / 4 N2 = N l • 2 N3 = Nl * 3 11 = 1 + N1 12 = I 1 + N1 13 = 12 + N l C 10  c  c 20 C  c 30 C  c  DO 10 I = 1. Nl DUMMY = DACSUM(AC,T(I , T l = DACSUM(AC) T1 - T1 / FLOAT(N1) DUMMY = DACSUM(AC.T.O) DO 20 I = I 1. N2 DUMMY = DACSUM(AC.T(I,, 1)) T2 = DACSUM(AC) T2 - T2 / FLDAT(Nl) DUMMY » DACSUM(AC.T.O) DO 30 I = 12. N3 DUMMY » DACSUM(AC.T(I . 1 )) T3 = DACSUM(AC) T3 = T3 / FLOAT(N1) DUMMY » DACSUM(AC.T,0)  ro  G67 668 G69 670  40 C  DO 40 I = 13, N DUMMY = DACSUMfAC.T ( I . 1 ) ) T4 = DACSUM(AC)  671  N7 » C S T E P S  672 673 674 675 676 677 678 679 680 68 1 682 683 684 685 686 687 G88 689 690 69 1 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 7 13 714 715 716 717 718  T4 = T4 / FL0AT(N7) DUMMY = DACSUM(AC,T.0J SIGMA = 5.67D-08 ES = SIGMA * TS •* 4 EG = SIGMA * TG ** 4 E1 « SIGMA • T l 4 E2 » SIGMA * T2 •* 4 E3 = SIGMA * T3 ** 4 E4 = SIGMA * T4 ** 4 TRG «• I.DO-EMG A1 = AW / 4.DO A2 = A1 A3 = A1 A4 = A l EP1 = EMW EP2 = EP 1 EP3 = EP 1 EP4 = EP1 F1G = 1 .DO F2G = 1 . 0 0 F3G = 1.DO F4G = 1 .DO FSG » 1.DO TP 1 = DSIN((PI(O.OO) - THETA)/4.DO) F11 = 1.D0-(4.D0+TP1) / (PI(O.DO) - THETA) TP2 = DSIN((PI(O.DO) - THETA)/2.DO) F12 = 1.D0-F11 - (2DO*TP2) / (PI(O.DO) - THETA) TP3 = DSIN((PI(O.DO) + 3.DO*THETA)/4.DO) F13 = (2.DO*(2.DO*TP2 - TP 1 - TP3) ) / (PI(O.DO) - THETA) TP4 « DSIN(THETA) FS1 = (TP4 • TP 1 - TP3) / (2.DO*TP4) FS4 = FS1 FS2 = .5D0-FS1 FS3 = FS2 F24 = F13 F23 = F12 F34 = F23 F1S = (AS/A 1 ) * FS1 F14 = 1.D0-F11 - F12 - F13 - F1S R1( 1) = A2 * F2G * EMG R1(2) = A l * FIG * EMG R1(3) = AS * FSG * EMG R1(4) = A4 * F4G * EMG R1(5) = A2 * F23 * TRG R1(6) = A3 • F3G * EMG R1(7) = EP2 * A2 / (1.D0-EP2) R1(8) = EP1 * A l / (1.D0-EP1)  - 13  —• w  719 720 721 722 723 724 725 72G 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770  _  50  R 1(9) «> EPS * A3 / (1.D0-EP3) R1(10) = A3 * F34 » TRG R1(11) = EP4 * A4 / (1.D0-EP4) R1( 12) « AS • FS4 * TRG R l ( 13) = AS * FS1 » TRG R K 1 4 ) = EMS * AS / (1.DO-EMS) R1 (15) " A2 • F24 • TRG R1( 16) * A l * F13 * TRG R1(17) = AS « FS3 • TRG R1 ( 18) = AS • FS2 * TRG R1( 19) = A l • F14 * TRG R1(20) = A1 • F12 * TRG A(1,1) = R1(20) A( 1,2) = R1( 13) A(1.3) » R1( 19) A( 1 ,4) = R1( 16) A(1.5) « -(R1(20) + R1(13) + R1(19) • R1(16) + R1(2) + R1(8)) A(2, 1 ) • R 1 ( 18) A(2,2) = -(R1(18) + R1(12) + R1(17) + R1(13) + R1(14) + R1(3)) A(2.3) « R K 12) A(2.4) = R1(17) A(2,5) = R K 13) A(3. 1) = R1( 15) A(3.2) = R1 ( 12) A(3,3) = -(R1(15) + R K 1 2 ) + R K l O ) + R1{19) + R1 ( 1 1 ) + R K 4 ) ) A(3.4) = R1(10) A(3.5) = R1( 19) A(4, 1) = R K 5) A(4.2) = R1( 17) A(4,3) = R K 10) A(4,4) = -(R1(5) + R1(17) + R1(10) + R1(16) + R1(9) + R1(6)) A(4.5) = R1(16) A(5.1) = -(R1(15) + R1(18) + R1(5) + R1(20) + R1(7) + R1 ( 1 ) ) A(5.2) = R1( 18) A(5,3) = R1(15) A(5,4) «= R1(5) A(5,5) « R1(20) B ( 1 ) = -( R 1 (8) *E 1 + R1.(2)*EG) B(2) " -(R1(14)*ES + R1(3)*EG) B ( 3 ) = -(R1(11)*E4 + R1(4)*EG) B(4) * -(R1(9)*E3 + R1(6)*EG) B ( 5 ) " -(R1(7)*E2 + R1(1)*EG) CALL SLE(5. 5, A, 1. 1, B, X. IPERM. 5, TT, DET, JEXP) IF (DET) 50, 60. 50 J2 = X( 1 ) JS = X(2) J4 = X(3) J3 = X(4) J1 = X(5) 0S4 = (OS - ES) * R K 14) 01 = ( J l - E l ) * R1(8) 02 = (J2 - E2) * R1(7)  ,  -  — i  771 772 773 774 775 77G 777 778 779 780 781 782 783  60 70  03 = ( J 3 - E3) • R1(9) 04 = (J4 - E4) * R1( 1 1 ) QW4 =01+02+03+04 0WE4 » A l * HEX * (TG - T1) + A2 • HEX * (TG - T2) + A3 * HEX * ( 1TG - T3) + A4 • HEX * (TG - T4) Q4ET = QW4 + QWE4 Q4ST • 0S4 + 05E QWT4 = Q4ET + QWC RETURN WRITE (6.70) STOP FORMAT (' '. '4-ZONE SOLUTION FAILED.') END  i cn  784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 8 11 8 12 8 13 8 14 815 8 16 817 818 819 820 82 1 822 823 824 825 826 827 828 829 830 831 832 833 834  C** » • * • * * * * C C C SUBROUTINE TO PLOT MODEL OUTPUT C C SUBROUTINE PLOTIT(ID) IMPLICIT REAL*8(A - H.O - Z) REAL*8 K, K 1 . KP LOGICAL PLOT DIMENSION T(400.50), R(500) DIMENSION 0W1(400), 0W2(400). AC(64), Y(30) DIMENSION TSS(500) INTEGER CSTEPS. STEPS, FNODE REAL*4 T0(400), Y1(400). Y2(400). Y5(400), Y10(400) R E A L M X(400). YPLOT1(400), YPL0T2(4OO), YPL0T3(400) R E A L M YPLOT4(400), YPL0T5(400), XPL0T(400). YMIN R E A L M ID1. DY, YTHRU1(100). YTHRU2(100). DX, XMIN. ' WSH(400) R E A L M Y0(400). YSC, YMX, YMN. RID COMMON /BLK1/ TA. TS. TG. AW. AS. EMW. EMG. EMS COMMON /BLK2/ RI, RO. DT, DR. RPM COMMON /BLK3/ KP. K. THETA. FD, ALPHA. ESH. KI. RF COMMON /BLK4/ HEX. HCOV. HOUT. HSE. H0UT2 COMMON /BLK5/ T, TSS, HRAO, R COMMON /BLK10/ STEPS, CSTEPS, NODES. FNODE COMMON /BLK15/ TWAB. TIME  c  10  c c  20 C  DO  10 I = 1. STEPS YO(I) = SNGL((T(I.1) - TS)/(TG - T S ) ) Y 1 ( I ) = SNGL((T(I,2) - TS)/(TG - T S ) ) Y 2 ( I ) = SNGL((T(I,3) - TS)/(TG - T S ) ) Y 5 ( I ) = SNGL((T(I,6) - TS)/(TG - T S ) ) Y 1 0 ( I ) * SNGL((T(I.11) - TS)/(TG - T S ) ) YWSH(I) = SNGL((TSS(NODES) - TS)/(TG - T S ) ) CONTINUE X(1) » 0.0 TP 1 = TIME / FLOAT(STEPS - 1) DO 20 I = 2 . STEPS X ( I ) = SNGL(TP1*FL0AT(I - 1))  DO 30 I = 1, STEPS X(I ) = X ( I ) / SNGL(TI ME) 30 C C PLOT / f i C CALL AXIS(2., 2.. '(T-TS)/(TG-TS)'. CALL PL0T(2. .. 7 . . 3 )  14. 5.. 90.. 0.. .2) .. .125)  835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 86 1 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886  CALL PLOT(10., 7. . 2) CALL PL0T(10., 2. . 1) CALL PL0T(2., 2.. 3) RID - FLOAT(ID) CALL SYMB0L(2.. 7.1. .15. 'PLOT 1-'. 0.. 7) CALL NUMBER(2.92, 7.1. .15. RID. 0 . , -1) C  40 C  DO 40 I ' 1. STEPS XPLOT(I) = ( X ( I ) / . 1 2 5 ) + 2. YPLOTI(I) = ( Y 0 ( I ) / . 2 ) + 2 . Y P L 0 T 2 U ) = (YWSH(I)/.2) + 2. CONTINUE CALL LINE(XPLOT. YPL0T1. STEPS. 1) CALL PL0T(XPL0T( 1 ) , Y P L 0 T 2 U ) . 3) CALL DASHLN(.125. .125. .125, .125)  C 50  DO 50 I = 1. STEPS CALL PLOT(XPLOT(I), YPL0T2(I). 4) YPLOT1(55) = YPL0T1(55) + .IEO YPL0T2(55) » YPL0T2(55) + .IEO CALL SYMBOL(XPLOT(55), YPLOT1(55). .15, 'TW. 0.2) CALL SYMB0L(XPL0T(55), YPL0T2(55), .15, 'TSH', O.. 3)  C CALL PL0T(16., 0.0, -3) C C PLOT H2 C YMX - -1.E30 YMN = 1.E30 DO 60 I = 1. STEPS IF (YO(I) .GT. YMX) YMX = Y 0 ( I ) IF ( Y 1 ( I ) .GT. YMX) YMX = Y 1 ( I ) IF ( Y 2 ( I ) .GT. YMX) YMX = Y2(I) IF ( Y 5 ( I ) .GT. YMX) YMX = Y5( I ) IF ( Y 1 0 ( I ) GT. YMX) YMX = Y10(I) IF (YO(I) .LT. YMN) YMN = Y 0 ( I ) IF ( Y 1 ( I ) .LT. YMN) YMN = Y 1 ( I ) IF ( Y 2 ( I ) .LT. YMN) YMN = Y 2 ( I ) IF ( Y 5 ( I ) .LT. YMN) YMN = Y5( I ) IF ( Y 1 0 ( I ) .LT. YMN) YMN = Y10(I) 60 CONTINUE YPLOT 1 (1) = YMN YPLOT1(2) = YMX CALL SCALE(YPL0T1. 2. 6., YMIN. DY. 1) CALL AXIS(2.. 2.. '(T-TS)/(TG-TS)'. 14. 6., 90.. YMIN. DY) CALL AXIS(2., 2.. ' FRACTION OF CYCLE'. -17. 8.. 0.. 0... .125) CALL PL0T(2., 8.. 3) CALL PL0T(10.. 8.. 2) CALL PLOT(10. . 2 . . 1 ) CALL PL0T(2., 2.. 3) CALL SYMB0L(2.. 8.1. .15. 'PLOT 2-'. 0.. 7)  8B7 888 889 890 891 892 893 894 895 89G 897 898 899 900 901 902 903 904 905 906 907 908 909 910 9 1 1 912 913 9 14 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938  CALL NUMBER(2.92. 8.1.  .15. RIO. 0.. - I )  C  70 C  '  00 70 I = 1. STEPS YPLOT 1(1) = ( ( Y O ( I ) - YMIN)/OY) + 2. Y P L 0 T 2 U ) = ( ( Y 1 ( I ) - YMINJ/DY) +• 2. Y P L 0 T 3 U ) = ( ( Y 2 ( I ) - YMIN)/DY) + 2. Y P L 0 T 4 U ) = ( ( Y 5 ( I ) - YMINI/DY) + 2. Y P L 0 T 5 O ) = ( ( Y 1 0 ( I ) - YMIN)/DY) .+ 2. CONTINUE CALL CALL CALL CALL CALL  LINEfXPLOT. LINE(XPLOT. LINE(XPL0T. LINE(XPLOT. LINE(XPLOT.  YPLOT1. YPL0T2. YPL0T3. YPL0T4. YPL0T5.  STEPS. STEPS. STEPS, STEPS. STEPS.  1) 1) 1) 1) 1)  C 80 C 90 C 100 C 110 C 120  C  DO 80 I = 5. STEPS, 25 CALL SYMBOL(XPLOT(I), YPLOT1(I).  .08, 30. O..  DO 90 I = 5, STEPS, 25 CALL SYMBOL (XPLOT ( I ) , Y P L 0 T 2 U ) .  .08. 3, 0..  DO 100 1 = 5 . STEPS, 25 CALL SYMBOL(XPLOT(I), Y P L 0 T 3 ( I ) .  .08. 2. 0..  DO 110 I = 5. STEPS. 25 CALL SYMBOL(XPLOT(I), YPL0T4(I),  .08. 11. 0..  • DO 120 I = 5. STEPS, 25 CALL SYMBOL(XPLOT(I). Y P L 0 T 5 ( I ) . .08, 0, YPLOT1(55) = YPLOT1(55) + .3E0 YPL0T2(55) = YPL0T2(55) + .1E0 YPL0T3(55) = YPL0T3(55) + . 1E0 YPL0T4(55) = YPLOT4(55) + .1E0 YPL0T5(55) = YPL0T5(55) + . 1EO CALL SYMB0L(XPL0T(55), YPL0T1(55). .15, CALL NUMBER(XPL0T(55), YPL0T2(55), .15. CALL NUMBER(XPL0T(55), YPL0T3(55), .15. CALL NUMBER(XPL0T(55). YPL0T4(55). .15. CALL NUMBER(XPL0T(55), YPL0T5(55). .15,  CALL C C PL0T/C3 C CALL CALL CALL CALL CALL CALL CALL  PL0T(16., 0.,  -3)  PCIRC(7.. 6.5, .63, 0) PCIRC(7.. 6.5. .75. 0) PL0T(7..6.5.3) PL0T(6.38. 5.88. 2) PL0T(7.. 6.5. 3) PL0T(7.88. 6.5. 2) PL0T(7.63, 6.5. 3)  0..  'O-MM 1.. 0 2.. 0 5., 0 10..  939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 ' 958 959 960 961 962 9G3 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978  CALL CALL CALL Nl =  PL0T(6.55. G.05, 2) SYMBOL(7.94, 6.5. .19, 'B'. 0.. SYMB0L(6.3, 5.63. .19. 'A'. 0.. FNODE - 1  1) 1)  C DO 130 C 140 C C  150 C  130 I = 1. Nl YTHRUI(I) - SNGL((T(CSTEPS - 1,1) - TA)/(TG - TA)) YTHRU2(I) = SNGL((T(STEPS - 1.1) - TA)/(TG - TA)) CONTINUE DO  140 I = 1. Nl XPLOT(I) = SNGL(R(I) - R ( 1 ) ) CONTINUE CALL SCALE(YTHRU1. N l , 6., YMIN, DY. 1) CALL SCALE(XPLOT. N l . 8., XMIN. DX, 1) DO  150 I = 1, N1 YTHRUI(I) = YTHRU1(1) + 2. YTHRU2U) = ((YTHRU2(I) - YMIN)/DY) + 2. XPLOT(I) = XPLOT(I) + 2 . CONTINUE  CALL AXIS(2.. 2.. '(T-TA )/(TG-TA)', 14. 6.. 90.. YMIN. DY) CALL AXIS(2.. 2.. 'DISTANCE FROM WALL SURFACE (MM)'. -31. 8., 1 XMIN, DX) CALL PL0T(2.. 8.. 3) CALL PLOT( 10. . 8. . 2) CALL P L O T ( 1 0 , 2 . 1 ) CALL SYMB0L(2., 8.1. .15. 'PLOT 3-'. 0.. 7) CALL NUMBER(2.92. 8.1, .15. RID, O.. -1) CALL LINE(XPL0T. YTHRU 1, N l . 1) CALL LINE(XPLOT, YTHRU2. N1, 1) YTHRU1(4) = YTHRU1(4) + .IEO CALL SYMB0L(XPL0T(4). YTHRU1(4). .19. 'A'. O.. 1) YTHRU2(4) = YTHRU2(4) - .25EO CALL SYMBOL(XPL0T(4). YTHRU2(4), .19. 'B'. 0., 1) CALL PLOTND RETURN END  *  O..  ro  979 980 981 982 983 984 985 986 987 988 989 990 99 1  992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016  £*»••*•**•* C C SUBROUTINE TO PRINT HEADER C £*»**•**»** SUBROUTINE HEADER L0GICAL*1 STARS(34) /34* ' * ' / WRITE (6. 10) STARS. STARS 10 FORMAT ('1'. T48. 34A1/T48. UNIVERSITY OF BRITISH COLUMBIA 1 T48. '* METALLURGICAL ENGINEERING *'/T48. 2 '• WALL.PROFILES VERSION 1 •'/T48. 34A1//) RETURN ENO  C*.*«*««*** C C SUBROUTINE C  10 20  *'  TO SET 9700 OUTPUT  SUBROUTINE P9700UD) LOGICAL* 1 A(132) INTEGER*4 CNT. PG INTEGER*2 LEN REWIND 6 CALL G E T L S T C 6 '. CNT) CNT = CNT / 1000 PG • 1 NUMB = 1 DO 20 I " 1. CNT CALL READ(A. LEN, O. LNUMB, 6) L " 0 CALL FINDST(A, 2. '1'. NUM8. L + 1. L, 810) CALL PAGE(LNCK, ID. PG. A) LNCK » LNCK + 1 IF ((LNCK - 60) .EQ. 0) CALL PAGE(LNCK. ID. PG, A) CALL WRITE(A. LEN, 0. LNUMB, 8) CONTINUE RETURN END ro ro o  1017  £*•••****•*  1018 1019 1020 1021  C C SUBROUTINE TO PRODUCE PAGE NUMBERS C C**********  1022 1023  1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038  SUBROUTINE P A G E ( L N C K , ID, L O G I C A L * 1 A ( 1 3 2 ) . IBLANK.  •  P G , A) FMT(37).  FI(3)  INTEGERM PG DATA IBLANK /' '/. FI /ZF 1 , ZF2, ZF3/ DATA FMT /Z4D. Z/D. ZF1. Z7D. Z6B. ZE3. ZF1. ZF2. ZF1, Z6B, Z7D, 1 ZD7, ZC1, ZC7, ZC5, Z40. Z7D, Z6B. ZC9. ZF3, Z6B, Z70, Z60, 2 Z7D, Z6B, ZC9. ZF2. Z61, Z61. Z61. Z61. Z61, Z61, Z61. Z61. 3 Z61. Z5D/ IF (PG .LE. 9) FMT(27) = F I ( 1 ) IF (PG .GE. 10 .AND. PG .LE. 99) FMT(27) = F I ( 2 ) IF (PG .GE. 100) FMT(27) = F I ( 3 ) WRITE (8,FMT) ID. PG A ( 1 ) = IBLANK PG » PG + 1 LNCK » 10 RETURN END  ro  1039 1040 104 1 1042 1043 1044 1045 1046 1047 1048 1049 1050 105 1 1052 1053 1054 1055 105G 1O07 1058 1059 1060 1061 1062 1063 1064 1065 10G6 1067 1068 1069 1070 107 1 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090  c»**».**«** C C SUBROUTINE TO PREDICT HEAT FLOWS USING MODIFIED ANALOGC WITH CONVECTIVE BRANCHES C (;«**••«*»•*• SUBROUTINE ANAL 1 IMPLICIT REAL*8(A - H,0 - Z) REAL*8 K, K l , KP EXTERNAL FN5 DIMENSION X ( 4 ) . F ( 4 ) , ACCEST(4) COMMON /BLK1/ TA, TS. TG, AW. AS. EMW. EMG, EMS . COMMON /BLK2/ RI, RO, DT. DR. RPM COMMON /BLK3/ KP, K. THETA, FD. ALPHA, ESH, K l , RF COMMON /BLK4/ HEX, HCOV. HOUT, HSE. HOUT2 TR " 1.D0-EMG RS = 1 .DO-EMS SIGMA - 5.67D-08 ES • SIGMA • TS * * 4 EG = SIGMA * TG ** 4 EA = SIGMA * TA ** 4 T l = AW * EMG .T2 = AS * TR T3 = AS * EMG T4 = (EMS*AS) / RS G = T3 + T2 + T4 EAD = (T1 * EG + T2*((T3*EG + T4*ES)/G)) / ( T l + T2 - (T2**2)/G) X(4) = ES X(3) = EAD X(2) = ( ( X ( 3 ) - .25D0*(X(3) - EA))/SIGMA) ** . 25D0 X(1) = ( ( X ( 3 ) - .75DO*(X(3) - EA))/SIGMA) ** .2500 CALL NDINVT(4. X. F. ACCEST; 5000. 5.D-04. FN5. &20) C 1 » TA / X(1) C2 = 1.D0+C1 + C1 ** 2 + C1 ** 3 HO = HOUT + (C2*SIGMA*X(1)**3) OL » HO » 2.DO • RO • PI(O.DO) * ( X ( 1 ) - TA) HSEP = HSE / (((TG + TS)*(TG**2 + TS**2))*SIGMA) HCOVP = HCOV / ( ( ( X ( 2 ) + T S ) * ( X ( 2 ) * * 2 + TS•*2 ) )*SIGMA) ACOV = (2.D0*PI(0.D0)*RI) - AW OS * ( X ( 4 ) - ES) * AS * EMS / RS + (SIGMA*X(2)**4 - ES) * HCOVP * 1ACOV + (EG - ES) * HSEP * AS WRITE (6.10) X ( 2 ) . X ( l ) , OS. OL 10 FORMAT (' '. 'MODIFIED ANALOG:(INCLUDES CONVECTIVE BRANCHES)'/T8. 1 'EXPOSED WALL TEMPERATURE''. G12.5. ' (K)'/T8. 2 'OUTER SHELL TEMPERATURE'', G12.5//T8. 3 'HEAT RECEIVED BY SOLIDS''. G12.5. ' (W/M)'/T8, 4 'HEAT LOSS TO SURROUNDINGS'', G12.5///) GO TO 40 20 WRITE (6.30) 30 FORMAT ('1'. 'SOLUTION TO MODIFIED ANALOG FAILED (INCL CONV)') 40 RETURN ' END  1091 1092 1093 1094  1095  1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1 1 10 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1 132 1 133 1134 1 135 1136 1137 1138 1139 1140  C* * * « * • * * • * C C EXTERNAL SUBROUTINE FOR ANAL 1 C  c**********  SUBROUTINE FN5(X, F) IMPLICIT REAL*8(A - H.O - Z) REAL*8 K, K l . KP DIMENSION X ( 1 ) . F ( 1 ) COMMON /BLK1/ TA. TS. TG. AW. AS, EMW. EMG. EMS COMMON /BLK2/ RI. RO. DT. DR, RPM COMMON /BLK3/ KP. K. THETA. FD, ALPHA, ESH. K l , RF COMMON /BLK4/ HEX, HCOV. HOUT, HSE. H0UT2 TR = 1.D0-EMG RS = 1.DO-EMS RW = 1.D0-EMW SIGMA = 5.67D-08 ES = SIGMA * TS *• 4 EG « SIGMA * TG •• 4 EA = SIGMA • TA *• 4 T l = (1.D0-EMW) / (EMW*AW) T2 = (DL0G(R0/RI)*(X(1) + X(2 ) ) *(X( 1 )* *2 + X ( 2 ) * * 2 ) ) / (2.DO*PI(0. 1DO)*KP) * SIGMA T3 = 1.DO / (EMG*AW) T4 = 1.DO / (AS'TR) T5 * 1.DO / (EMG*AS) T6 = (1.DO-EMS) / (EMS*AS) C1 = TA / X(1) C2 = 1.D0+C1 + C1 ** 2 + C1 •• 3 HO = HOUT + (C2*SIGMA*X(1)**3) HOP = HO / ( ( ( X ( 1 ) * T A ) * ( X ( 1 ) * * 2 + TA* *2)) *SIGMA) HCOVP = HCOV / ( ( ( X ( 2 ) + T S ) * ( X ( 2 ) * * 2 + TS**2))*SIGMA ) HSEP » HSE / ( ( ( T G + TS)*(TG**2 + TS**2))*SIGMA) HEXP = HEX / ( ( ( T G * X(2))*(TG**2 + X(2 ) **2 ) ) *SIGMA) T15 = 1.D0 / (HEXP*AW) ASH = 2.DO * PI(O.DO) • RO T8 = 1.00 / (HOP*ASH) ACOV = (2.DO*PI(O.DO)*RI) - AW T9 = 1.D0 / (HCOVP*ACOV) TIO • 1.00 / (HSEP*AS) F ( 1 ) =• (EG - X(3)) / T3 + (SI GMA * X (2 ) * *4 - X(3)) / T l + (X(4) - X( 13)) / T4 T i l = SIGMA * X(2) ** 4 F ( 2 ) " ( X ( 3 ) - T i l ) / T1 + (ES - T1 1) / T9 + (SIGMA*X( 1)* *4 - T11) 1 / T2 + (EG - T11) / T15 F ( 3 ) = (ES - X ( 4 ) ) / T6 + (EG - X ( 4 ) ) / T5 + (X(3) - X ( 4 ) ) / T4 T 12 - T2 + T8 F ( 4 ) = (SIGMA'X(2)**4 - EA) / T12 - (SIGMA * X( 1 )* + 4 - EA) / T8 RETURN END  oo  *********  1 142 1143 • 1144 1145 1146 1147 1148 1 149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159* 1160 1 IG1 1 162 1 163 1 164 1165 1 166 1167 1160 1169 1170 1171 1 172 1173 1174 • 1175 1176 1177 1178 1179 1180 1181 1182 1183 1104 1185 1186 1187 1 188 1189 1190 1191  C C SUBROUTINE TO PREDICT MEAT FLOWS USING MODIFIED ANALOGC RADIATION ONLY C £•••***••*• SUBROUTINE ANAL2 IMPLICIT REAL*8(A - H,0 - Z) REAL*8 K. K1, KP DIMENSION X ( 4 ) , F ( 4 ) , ACCEST(4) EXTERNAL FN 1, FN2, FN3, FN4 COMMON /BLK1/ TA, TS, TG. AW. AS, EMW. EMG. EMS COMMON /BLK2/ RI. RO. DT, DR, RPM ' COMMON /BLK3/ KP. K, THETA. FD, ALPHA, ESH, KI, RF COMMON /BLK4/ HEX, HCOV. HOUT. HSE. H0UT2 COMMON /BLK16/ EG. ES. EA. ASH1. HOUTP. H0UTP1 OLOSS(TA.TSH.HOUTPL.ASH) = ASH * HOUTPL • (TSH - TA) SIGMA * 5.67D-08 ASH » 2.DO • Pl(O.OO) • RO 10 CONTINUE TR = 1.OO-EMG RS = 1.DO-EMS . RW = 1.DO-EMW EG = SIGMA • TG ** 4 ES = SIGMA * TS * * 4 EA = SIGMA * TA * * 4 T1 = AW * EMG T2 • AS • TR T3 = AS * EMG IF (EMS .LT. .9999) GO TO 20 EAD = (T1*EG + T2*ES) / (T1 + T2) GO TO 30 20 T4 = (EMG*AS) / RS G = T3 + T2 + T4 EAD = (T1*EG + T2*((T3*EG + T4*ES)/G)) / (T1 + T2 - (T2**2)/G) 30 IF (EMS .GE. .999DO .AND. EMW .GE. .99900) GO TO 40 IF (EMS .GE. .99900 AND. EMW .LT. .999D0) GO TO 50 IF (EMS .LT. .999D0 .AND. EMW .GE. .999D0) GO TO 60 X ( 4 ) * SIGMA • TS ** 4 X(3) » EAO X(2) = ( ( X ( 3 ) -. ,25D0*(X(3) - EA))/SIGMA) ** . 25DO X ( 1 ) = ( ( X ( 3 ) - .7500*(X(3) - EA))/SIGMA) *• .25DO CALL NDINVT(4, X, F. ACCEST. 5000. 5.0-04. FN1, &80) HOUTPL = HOUTP1 OL = 0LOSS(TA.X(1).HOUTPL.ASH) QS = ( X ( 4 ) - ES) * ((AS*EMS)/RS) WRITE (6.70) X ( 2 ) . X ( i ) , OS, OL GO TO 100 40 X(2) = (E AD/SI GMA) . 25DO X(1) = X(2) - .7500 * ( X ( 2 ) - TA) CALL NDINVT(2. X. F. ACCEST. 5000. 5.D-04. FN2, &80)  ro  -P*  1192 1193 1194 1195 1196 1197 1198  1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 122 1 1222 1223  50  HOUTPL = HOUTP1 OL = OLOSS(TA.X(I),HOUTPL.ASH) OS » ((SIGMA*X(2)*»4 - ES)*AS*TR) + ((EG - ES)»EMG*AS) WRITE (6,70) X ( 2 ) . X ( 1 ) . OS. OL GO TO 100 X ( 3 ) = EAD X(2)  = ((X(3)  -  .25D0*(X(3)  -  EA))/SIGMA)  **  25D0  ' X(1) = ( ( X ( 3 ) - .75D0'(X(3) - EA))/SIGMA) ** .2500 CALL NDINVT(3, X. F. ACCEST, 5000, 5.D-04. FN3. &80) HOUTPL = HOUTP1 OL = OLOSS(TA,X(1),HOUTPL,ASH) OS = ( ( X ( 3 ) - ES)*AS*TR) + ((EG - ES)'EMG*AS) WRITE (6.70) X ( 2 ) . X(1). OS. OL GO TO 100 60 X(3) = SIGMA * TS 4 X(2) = (EAD/SIGMA) ** .25D0 X ( 1 ) = X(2) - .75D0 + (X(2) - TA) CALL NDINVTO. X. F. ACCEST. 5000. 5.D-04. FN4, 8.80) HOUTPL = HOUTP1 OL = OLOSS(TA,X(1).HOUTPL,ASH) OS = ( X ( 3 ) - ES) * ((AS*EMS)/RS) WRITE (6,70) X ( 2 ) , X ( 1 ) . OS. OL 70 FORMAT (' '. 'MODIFIED ANALOG:(RADIAT ION 0NLY)'/T8, 1 'EXPOSED WALL TEMPERATURE''. G12.5. ' (K)'/T8, 2 'OUTER SHELL TEMPERATURE'', G12.5//T8, 3 'HEAT RECEIVED BY SOLIDS''. G12.5. ' (W/M)'/T8. 4 'HEAT LOSS TO SURROUNDINGS''. G12.5/'1') GO TO 100 80 WRITE (6.90) 90 FORMAT ('1', 'SLOUTION TO MODIFIED ANALOG FAILED (NO CONV)') . 100 RETURN END  ro ro cn  1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248  1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270  c***«***«*« C C EXTERNAL SUBROUTINE FOR ANAL2 C £**•*•***•» SUBROUTINE FN1(X, F) IMPLICIT REAL*8(A - H,0 - Z) REAL *8 K. KI. KP DIMENSION X(1). F ( 1 ) . T ( 9 ) COMMON /BLK 1 / TA, TS. TG. AW. AS, EMW, EMG. EMS COMMON /BLK2/ RI. RO. DT, DR, RPM COMMON /BLK3/ KP, K. THETA, FD, ALPHA. ESH, KI, RF . COMMON /BLK4/ HEX. HCOV. HOUT, HSE, H0UT2 ' COMMON /BLK16/ EG. ES. EA. ASH. HOUTP, HOUTPL SIGMA = 5.67D-08 CALL TEMP(X. T) F ( 1 ) = (SIGMA*X(2)**4 - X ( 3 ) ) / T(1) + (EG - X ( 3 ) ) / T(3) + ( X ( 4 ) 1- X ( 3 ) ) / T(4) F ( 2 ) = (EG - X(4)) / T ( 5 ) + (ES - X(4)) / T ( 6 ) + (X(3) - X ( 4 ) ) / 1T(4) F ( 3 ) = (HOUTP*ASH*SIGMA*(X(1)**4 - TA**4)) - ( T ( 7 ) * ( X ( 3 ) - EA)) F ( 4 ) = (2.*PI(O.)*KP*SIGMA*(X(2)**4 - X ( 1 ) * * 4 ) ) / (DLOG(RO/RI)*T( 19)) - ( T ( 7 ) * ( X ( 3 ) - EA)) RETURN END  C EXTERNAL SUBROUTINE FOR ANAL2 C SUBROUTINE FN2(X. F) IMPLICIT REAL*8(A - H.O - Z) REAL*8 K. K1. KP DIMENSION X(1). F ( 1 ) . T ( 9 ) COMMON /BLK1/ TA. TS. TG. AW. AS. EMW, EMG. EMS COMMON /BLK2/ RI. RO, DT, DR, RPM COMMON /BLK3/ KP, K. THETA. FD. ALPHA, ESH KI , RF COMMON /BLK4/ HEX; HCOV, HOUT, HSE, H0UT2 COMMON /BLK 16/ EG, ES, EA. ASH, HOUTP. HOUTPL SIGMA = 5.67D-08 CALL TEMP(X. T) F ( 1 ) = (HOUTP*ASH*SIGMA*(X( 1 )**4 - TA**4)) - (T(7)*SIGMA*(X(2)**4 1- TA**4)) SIGMA*X(2)**4) / T ( 4 ) + F ( 2 ) = (EG - SIGMA»X(2)**4) / T ( 3 ) + (ES 1 (SIGMA*(X( 1)**4 - X ( 2 ) * * 4 ) ) / T ( 2 ) RETURN END  ro ro cn  1271  Qtt*t******  1272 1273 1274 1270 1276 1277 1278 1279 1280 128 1 1202 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293  C C EXTERNAL SUBROUTINE FOR ANAL2 C £****•*»••* SUBROUTINE FN3(X, F) IMPLICIT REAL '8(A - H.O - Z) REAL *8 K, K1 , KP DIMENSION X ( 1 ) , F ( 1 ) . T ( 9 ) COMMON /BLK 1 / TA, TS, TG, AW. AS. EMW. EMG. EMS COMMON /BLK2/ RI. RO. DT. DR. RPM COMMON /BLK3/ KP. K, THETA. FD, ALPHA. ESH. K l . RF COMMON /BLK4/ HEX. HCOV. HOUT. HSE, HOUT2 COMMON /BLK 16/ EG. ES. EA, ASH. HOUTP, HOUTPL SIGMA = 5.67D-08 CALL TEMP(X, T) F ( 1 ) = ( (SIGMA*X( 1 ) * M - X ( 3 ) ) / ( T ( 1 ) + T ( 2 ) ) ) + (EG - X ( 3 ) ) / T ( 3 ) 1 + (ES - X ( 3 ) ) / T(4) F ( 2 ) = (HOUTP*ASH«SIGMA«(X(1)*»4 - T A ' M ) ) - ( T ( 7 ) ' ( X ( 3 ) - E A ) ) F ( 3 ) = ((2.00*PI(O.DO)*KP*SIGMA*(X(2)**4 - X(1)**4))/(OLOG(RO/RI)* 1 T ( 9 ) ) ) - ( T ( 7 ) * ( X ( 3 ) - EA)) RETURN END  1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 13 10 1311 13 12 1313 1314 1315 1316 1317  C EXTERNAL  SUBROUTINE FOR ANAL2  C* * * * * * * * * * * SUBROUTINE FN4(X, F) Z) IMPLICIT REAL*8(A - H.O REAL*8 K, K1, KP DIMENSION X ( 1 ) , F( 1) T(9) COMMON /BLK1/ TA. TS TG. AW, AS, EMW, EMG, EMS COMMON /BLK2/ RI. RO DT. DR. RPM COMMON /BLK3/ KP, K. THETA. FD. ALPHA, ESH, K l RF COMMON /BLK4/ HEX, HCOV. HOUT, HSE, HOUT2 COMMON /BLK16/ EG, ES, EA. ASH. HOUTP, HOUTPL SIGMA » 5.67D-08 • CALL TEMP(X. T) F( 1) » (EG - X ( 3 ) ) / T ( 5 ) • (ES - X ( 3 ) ) / T(6) + (SIGMA*X(2)**4 1X(3)) / T ( 4 ) F ( 2 ) = (HOUTP*ASH*SIGMA*(X( 1)**4 - TA**4)) - T(7) • (SIGMA*X(2)< 14 - EA) X( 1 )**4)) / (DLOG(RO/RI ) F(3) » (2.DO*PI(O.DO)*KP*SIGMA*(X(2)* 1*T(9)) - (T(7)*(SIGMA*X(2)**4 - EA)) RETURN END  ro ro  1318 1319 1320 132 1 1322 1323 1324 1325 1326 1327 1328 1329 1330 133 1 1332 1333 1334 1335 1336 1337 1338 1339 1340 134 1 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352  C• • » • * * * + * * C C SUBROUTINE TO CALCULATE NODAL TERMS FOR ANAL2 C C SUBROUTINE TEMP(X, T) IMPLICIT REAL*8(A - H,0 - Z) REAL*B K, K1, KP DIMENSION T( 1), X( 1) COMMON /BLK1/ TA. TS. TG, AW. AS. EMW. EMG. EMS COMMON /BLK2/ RI. RO. DT. DR, RPM COMMON /BLK3/ KP, K. THETA, FD. ALPHA, ESH, K l , RF COMMON /BLK4/ HEX. HCOV, HOUT, HSE, HOUT2 COMMON /BLK 16/ EG, ES, EA, ASH. HOUTP, HOUTPL TR = 1.D0-EMG SIGMA = 5.67D-08 EG = SIGMA * TG ** 4 ES = SIGMA • TS •* 4 EA = SIGMA • TA 4 ASH = 2.DO • PI(O.DO) * RO T ( 1 ) = (1.D0-EMW) / (EMW*AW) T(9) = (X(1)**2 + X(2)»*2> * (X(1) + X ( 2 ) )* SIGMA T ( 2 ) = (DLOG(RO/RI)*T(9)) / (2.DO*PI(0.DO)* KP) T ( 3 ) = 1.DO / (EMG*AW) T ( 4 ) = 1.DO / (TR * AS) T ( 5 ) = 1.DO / (EMG*AS ) T ( 6 ) = (1.DO-EMS) / (EMS*AS) CI - TA / X(1) C2 = 1.DO+C1 + C 1 * * 2 * C 1 * * 3 HOUTPL = HOUT + (C2*SIGMA *X(1)* *3) HOUTP = HOUTPL / ( ( ( X ( 1 ) * * 2 • TA**2)*(X(1) + T A ) ) * T ( 8 ) = 1.D0 / (HOUTP*ASH) T ( 7 ) = 1.00 / ( T ( 1 ) + T ( 2 ) + T ( 8 ) ) RETURN END  ro ro  CO  420 (K) 1060 298 0.8 0.8  "w  0. 24 1.75 (m) 1.98 2 (RPM)  03  .64  w  1. (W/m K) 800 ( J / k g r K)  w 1800 (kgr/m ) 3  w  20 (W/m K) 2  cv  cv,w->s out  cv  30 10 50  230  SAMPLE OUTPUT  UNIVERSITY OF BRITISH COLUMBIA METALLURGICAL ENGINEERING WALL.PROFILES VERSION 1  FINAL SOLUTION FOR RUN NO.  1  KILN INNER RADIUS' * 1.7500 KILN OUTER RADIUS' 1.9800 1 IMF STEP' 0.18750 (SEC) RADIAL STEP' O.IOOOOE-02 (M) KILN SPEED' 2.0000 (RPM)  (M) ( M)  (K ) SOLIDS TEMPERATURE' 420.00 SOLIDS EMISSIVITY' 0.80000 GAS TEMPERATURE = 1060.0 (K) CAS EMISSIVITY' 0.24000 AMBIENT TEMPERATURE = 298.00 (K) SHELL EMISSIVITY' 1.OOOO WALL EMISSIVITY' 0.80000 NAIF ANGLE SUP.TFNDF.D BY SOLIDS' 0.62832 RAIIO BED DEPTH-.KILN DIAMF.TF.R" 0.95492E-01 WAIL Tl IEPMAI. CONDUCTIVITY' 1.0000 (W/M K) WALL THERMAL DIFFUSIVITY' 0.69444F-0G (M**2/S) CONVECTIVE HTC (W/M**2 K ) : GAS TO EXPOSED WALL' 20.OOO SOLIDS TO COVERED WALL' 50.000 OUTER SHELL TO ATMOSPHERE = 10.000 GAS TO SOLIDS' 50.000 RADIATIVE HTC (W/M'*2 K ) : EXPOSED WALL' 24.457 OUTER SURFACE' 9.7664 EXPOSED SOLIDS' 52.294  THF. UNSTEADY STATE TEMPERATURE FIELD FOLLOWS:  1 035 .06 19 840. 8499 0 14 .5 100 4 847 .4 156 04 9 .7555 5 7259 6 85 1 . 7 853'. 4 297 054 .9331 0 9 856 .2005 857 .5026 10 1 1 850 .62 19 12 , 051. 6549 13 060. 6 144 14 OG 1 . 5 103 15 862 .3508 16 863 . 1423 17 863. 8904 864 5995 10 19 865 .2735 065 .9 158 ?0 2 1 8G6 .5292 007 . 1 162 22 23 867 . .6790 24 868 .2194 868 . 7393 25 26 869 . 2400 27 869 .7230 28 870 '. 1895 29 870 . 6405 07 1.0772 30 87 1. 5002 31 87 1. 9 106 32 33 872 .3090 31 872 . 696 1 35 873 .0725 87 3 .4389 36 37 873 . 7957 074 . 1434 30 39 871 . 4025 074 .8 135 10 4 1 875 . 1367 875 . 1524 •12 43 075 .76 11 44 87G .0G30 t 2 3  2 845 .05 15 845 .2780 846 .004 7 846 .9513 847 .9766 849 .0111 850. 02 12 850. 9922 9188 85 1 . 852 .8003 853 . 6300 854 .4346 855 . 1928 855 .9156 856 . .6055 857 . 2653 857 . 897 1 858 .5031 859 .0851 859 .64 50 860 . 1843 86D . 704 4 86 1.2065 861 .6918 862 . 1615 862 .6163 863 .0574 863 . 4853 863 . 9010 8G4 . 3050 864 .698 1 865 .0807 865 .4535 865 .8168 866 . 1713 866 .5172 866 .8551 867 . 1853 867 .5081 067 .8239 868 . 1329 868 .4355 868 . 7319 869 .0224  3 055 .9742 855. 2836 854 .7361 854 .3597 854 . 14 08 854 .0533 854 .0702 854 . 1678 854 .3271 854 .5326 854 .7729 .0380 855 , 855 . 3234 855. 62 14 855 . 9288 856 . 2425 856 , 5599 856 .8792 857 . 1990 857 .5182 057 . 8359 858 . 1513 858 . 464 1 858 . 7737 859 .0800 859 . 3826 859 . 68 14 859 .9764 860 . 2674 860 . 55-15 860 . 8375 86 1. 1 166 86 1.3917 86 1.6629 86 1 .9303 062 . 1938 862 . 4536 862 . 7096 862 . 962 1 863 . 2 1 10 863 . 4564 863 . 6985 063 . 9372 864 . 1726  4 5882 86 1 . 86 1 . 0801 860. 5726 860. 0834 859 .6296 859. 2222 858. 8667 858 .5640 858 .3124 858. 1086 857 .9486 857 .8279 857 . 7422 857..6876 857 .6605 857 . .6573 857 . .6753 857 . .7119 857 .7646 . 857 .8315 857 . 9 108 858 .0010 858 . 1006 858 . 2085 858 . 3237 858 . 4452 858 . 5722 858 . 7039 858 . 8398 858 . 9793 859 .12 19 859 . 2671 859 .4 146 859 .5639 859 . 7 149 859 .8672 860 .02Q6 860 . 1748 860 . 3298 860 .4852 860 . 6409 860 . 7969 860 .9530 86 1. 1090  5 863 .298 1 862 .9764 862 .6500 862 .3 195 86 1 9875 . 86 1 . 6583 861 .3364 861 .0261 860. 73 10 860. 4536 860,. 1956 .9578 859 . .7406 859 . 859,.5439 859 , .3672 . 2097 859 , 859,.0705 858 .9488 858 .8434 858 .7535 858 .6779 858 .6 158 858 .566 1 858 .5280 858 . 5007 858 .4833 858 . 4752 858 .4756 858 .4840 858 .4997 858 .5221 858 .5508 858 .5854 858 .6253 858 .6701 858 .7 196 858 . 7733 858 .8309 858 .8922 858 .9568 859 .0245 859 .0951 859 . 1683 859 . 2440  6 862 .5368 862 .3660 862 .1888 862 .0054 861 .8161 62 15 86 1 . 86 1 . 4228 861 .22 12 861 .0184 860. 8160 860. 6 158 860., 4 192 860., 2275 860.,0420 859., 8635 859 , 6928 859 . .5302 . 3762 859 , 859 .2311 859 .0948 858 .9674 858 .8488 858 . 7389 858 .6374 858 . 5442 858 . 4590 858 .3816 858 .3116 858 .2487 858 . 1927 858 . 1433 858 . 1002 858 .0630 858 .0316 858 .0057 857 .9850 857 . 9692 857 .9581 857 .9516 857 .9492 857 .9510 857 .9566 857 . 9658 857 . 9786  7 860. 4649 860. 3951 860. 3195 860. 2381 860. 1512 860. 0588 859. 9613 859. 8588 859. 7518 859.,6407 859 .5263 859. 4091 859., 2899 859. 1694 .0484 859 . ,9275 858 . 858 . .8075 858 , ,6888 858 , .5721 858 .4578 858 . 3462 858 . 2377 858 . 1327 858 .0313 857 .9337 857 .8400 857 .7504 857 .6650 857 . 5836 857 . 5065 857 .4335 857 .3647 857 .2999 857 .2302 857 . 1825 857 . 1296 857 .0806 857 .0353 856 .9937 B56 .9556 856 .9210 856 . 8897 856 . 86 18 856 .8369  8 857 .8585 857 .8446 857 .8266 857 .8045 857 .7783 857 .7481 857. 7138 857. 6755 857. 6333 857. 5873 857 .5376 857 .4843 857 .4278 857. 3682 857. 3058 857 .241 1 857. 1742 857 . , 1056 857 . ,0356 856.,9645 856..8927 856 . .8205 856..7482 856,.6761 856 .6044 856 .5335 856 .4634 856 .3944 856 .3266 856 .2603 856 . 1955 856 . 1324 856 .071 1 856 .01 16 855 .9540 855 . 8985 855 .8449 855 . 7934 855 . 744 1 855 .6968 855 .6516 855 . 6086 855 . 5677 855 . 5289  9 855. 1468 855. 1573 855. 1655 855. 1713 855 .1746 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OB28 830. 0828 830. 0828 830. 0829 830. 0830 830. 0830 830. 0832 830. 0833 830..0834 830..0836 830 0838 830..0840 030,.084 2 8 30 .0844 830 .0847 830 .0850 830 .0853 830 .0856 830 .0859 830 .0863 830 .0866 830 .0870 830 .0874 830 .0878 830 .0882 830 .0886 830 .0891 830 .0895 830 .0900 830 .0905 830 .0909 830 .0914 830 .09 19 830 .0924 830 .0929 830 .0934 830 .0939  827 .9 109 827. 9 107 827 .9 t04 827. 9102 827 .9 100 827 .9097 827 .9095 827 .9093 827. 9092 827 .9090 827 .9088 827 .9087 827 .9085 827 .9084 827 .9083 827 .9082 827 .908 1 827. 9080 827 .9079 .9079 827 . 827 . 9078 827 . 9078 827 .9078 827 .9070 827 .9078 827 . 9070 027 .9079 827 . 9079 827 .9080 827 .908 1 827 .9082 827 .9083 827 .9084 827 . 9086 827 . 9087 827 .9089 827 .9091 827 .9093 827 .9095 • 827 .9097 827 .9099 827 .9102 827 . 9 104 827 .9107 827 .9109 827 .9112 827 .9115 827 .9118 827 .912 1 827 .9124  825. 7367 825. 7365 825. 7363 825. 7361 825. 7309 825 .7357 825 .7355 825. 7353 825. 7351 825. 7349 825. 7347 825 .7346 825. 7344 825 .7343 825. 734 1 825. 7340 825 .7339 825 .7337 825. 7336 825. 7335 825 .7334 825. 7333 825. 7332 825 .7332 825..7331 825 . 7330 825 .7330 825 . 7330 825 .7329 825 . 7329 825 . 7329 825 .7329 825 .7329 825 .7329 825 . 7330 825 .7330 825 .7331 825 .7331 825 .7332 825 .7333 825 .7334 825 .7335 825 . 7336 825 . 7337 825 . 7338 825 . 7339 825 . 734 1 825 .7342 825 .7344 825 .7345  823 .5620 823. 56 19 823. 5617 023. 5616 823. 5615 823 .5613 823 .5612 823. 5610 823. 5609 823. 5608 823. 5607 823. 5605 823 .5604 823 .5603 823. 5602 823. 5600 823. 5599 823 .5598 823 .5597 823. 5596 823 .5595 823..5594 823 . 5594 .5593 823 . 823..5592 823 .5591 823 .5591 823 . 5590 823 .5590 823 . 5589 823 .5589 823 . 5588 823 . 5588 823 . 5588 823 .5588 823 .5587 823 .5587 823 .5587 823 .5587 823 .5588 823 .5588 823 . 5588 823 .5588 823 .5589 823 .5589 823 .5590 823 . 5590 823 .5591 823 .5591 823 . 5592  821 .3875 821 .3874 821 ,3874 821 .38/3 821 .3872 82 1 . 3872 3871 82 1 . 821 .3870 82 1 . 3869 3869 82 1 . 821 .3868 821 .3867 3867 82 1 . 3866 82 1 . 821 .3865 821 .3865 82 1 . 3864 3864 82 1 . 82 1 .. 3863 82 1 ..3862 82 1. 3862 821 .3861 82 1. 386 1 82 i . 3860 821 . 3860 82 1. 3859 821 . 3859 821 . 3859 821 . 3858 821 . 3858 821 . 3858 821 .3857 82 1. 3857 82 1. 3857 821 . 3857 821 .3856 821 .3856 82 1.3856 821 .3856 821 .3856 821 . 3856 821 . 3856 821 . 3856 82 1. 3856 821 . 3856 821 .3856 82 1. 3856 82 1. 3857 821 . 3857 821 . 3857  83? .28 12 69 832 .28 19 70 7 1 832 .2826 72 832 .2033 73 832 .2839 74 832 .2846 75 832 .2852 76 832 .2858 77 832 .2864 832 .2869 78 79 832 .2875 832 !2880 80 8 1 832 .2885 832 .2889 82 832 .2094 83 R-l 032 .2098 2902 05 " 03 2 . , 2906 86 832 . 07 032 , .29 10 832 .29 13 88 832 . 29 16 89 832 . 29 19 90 91 832 . 292 1 92 832 . 2923 832 . 2925 93 94 832 . 2927 95 832 . 2928 832 . 2929 96 97 832 . 2930 98 832 . 293 1 99 832 . 293 1 832 . 293 1 100 832 . 293 1 101 102 832 . 2930 103 032 . 2929 832 . 2920 104 832 . 2927 105 106 832 . 2925 107 832 . 2923 103 832 . 292 1 832 . 29 19 109 832 . 29 16 1 to 1 t 1 832 . 29 13 1 12 832 .2910 1 13 832 . 2907 1 14 832 . 2904 832 . 2900 1 15 1 16 832 . 2096 1 17 832 . 2092 1 18 832 . 2008  830. 094 4 830. 0949 830. 0954 830. 0959 830. 0964 830. 0969 830. 0974 830. 0979 830. 0984 830. 0989 830. 0994 830. 0998 830. 1003 830. 1007 830. 1012 03O. 1016 030. 1020 830., 1024 830.. 1020 830.. 1032 830 . 1036 830 . 1039 830 .1043 830 . 1046 830.. 1049 830 . 1052 830 . 1054 830 . 1057 830 . 1059 830 . 1062 830 . 1064 830 . 1066 830 . 1067 830 . 1069 030 . 1070 830 . 107 1 830 . 1072 830 . 1073 830 . 1074 830 . 1074 830 . 1074 830 . 1074 830 . 1074 830 . 1074 830 . 1073 830 . 1073 830 . 1072 830 . 1071 830 . 1070 830 . 1068  027 .9127 827 .9130 827 .9134 827. 9137 827 .9140 827 .9 144 827 .9 147 827 .9150 827 .9 154 827 .9157 827 .9161 027 .9 164 827 .9 168 827 .9171 827 .9 175 027 .9 t78 027 .9 182 827 .9 . 185 827 9 100 027 . .9192 827 .9195 827 .9 198 827 .9201 827 .9204 827 .9207 827 .92 10 827 .92 13 827 . 92 16 827 .9218 827 .922 1 827 .9223 827 .9226 827 .9228 827 .9230 827 .9232 827 .9234 827 .9236 827 .9238 827 .9240 827 .924 1 827 .924 2 827 . 9244 827 .9245 027 .9246 827 .9247 827 .9248 827 .9248 827 .9249 827 . 9249 827 .9249  825 .7347 825 .7349 825. 7351 825. 7353 825. 7354 825. 7356 825 .7359 825 .7361 825 .7363 825 .7365 825 .7367 825. 7370 825 .7372 825 .7374 825 .7377 825 .7379 825 ,730 1 825 .7384 825.. 7 306 825 . 7388 825 . 7391 825,.7393 825 . 7396 825 .7398 825 .7400 825 .7403 825 . 7405 825 . 7407 825 . 7409 825 . 74 12 825 .7414 825 . 74 16 825 . 74 18 825 . 7420 825 .7422 825 . 7424 825 .7426 825 . 7428 825 .7429 825 .7431 825 . 7433 825 . 7434 825 . 7436 825 . 7437 825 .7438 825 . 7440 825 . 744 1 825 . 7442 825 . 7443 825 . 7444  823. 5593 823 .5594 823. 5595 823. 5596 823. 5597 823. 5598 823. 5599 823. 5600 823 .5601 823. 5602 823. 5603 823. 5605 823 .5606 823. 5607 823 .5609 023 .5610 023 .56 12 823 . ,5613 823. 5615 823..5616 823 ,5618 823..5619 823 .5621 823 . 5622 823 .5624 823 .5625 823 .5627 823 .5628 823 .5630 823 .5631 823 .5633 823 .5634 823 . 5636 823 .5637 823 . 5639 823 .5640 82.3 . 5642 823 .5643 823 . 5644 823 .5646 823 .5647 823 .5648 823 . 5649 823 .5651 823 .5652 823 .5653 823 .5654 823 .5655 823 . 5656 823 .5657  821 .3857 821 .3858 821 .3858 821 .3858 821 .3859 821 .3859 821 .3860 821 .3860 82 1 . 386 1 821 .3861 821 . , 3862 821 . ,3862 82 1 ,.3863 821 . 3863 82 1 .3864 , 82 1 . 3065 82 1. 3065 82 1. 3866 821 . 3867 821 . 3867 821 .3868 821 .3869 821 .3870 82 1.3870 821 .3871 821 .3872 821 . 3873 82 1. 3873 821 .3874 821 .3875 821 .3876 82 1. 3876 821 .3877 821 .3878 82 1. 3879 821 . 3879 82 1. 3800 821 .3881 821 . 3882 821 .3882 821 .3883 821 . 3804 82 1. 3884 821 .3885 821 .3886 821 . 3886 821 . 3887 821 .3887 821 . 3888 82 1. 3888  11^ 832 .2383 832 20/8 i?o 1? 1 832 2874 12? R32 .2R69 832 .2863 123 832 .2050 12 1 832 .2053 125 0<2 .20 17 1 26 127 832 .284 1 832 .2036 128 120 832 .2830 832 .2824 130 832 .28 17 13 1 832 .28 1 1 132 832 .2805 133 13 1 032 .2798 1 35 ' 832 .2792 1 30 032 .2786 137 032 .2779 032 2772 138 832 . 2766 139 832 . 2759 1 10 1-1 1 832 . 2753 832 . 2746 1'I2 832 .2739 143 032 . 2733 14 4 145 832 . 2726 146 832 .2720 147 832 .2713 832 . 2707 148 149 832 . 2700 832 . 2694 150 832 . 2688 15 1 832 . 2682 152 832 . 2676 153 154 832 . 2670 832 . 2664 155 832 .2658 156 157 832 . 2652 . 158 832 . 2647 832 . 2642 159 832 . 2636 160 032 .2631 16 1  830. 1067 830. 1065 830. 1063 830. 1061 830. 1059 830. 1057 830. 1054 830. 1051 830. 10-19 830. 1046 830. 104 3 830. 1040 830. 1036 830. 1033 830. 1029 8 30. 1026 830. 1022 830. 1018 830. 1014 0 30. 1010 830. 1006 830. 1002 830. 0998 830. 0993 830. 0989 830. 0985 830. 0980 830. 0976 830..0971 830. 0967 830..0962 830..0958 830 .0953 830 .0948 830 .0944 830 .0939 830 .0935 830 .0930 830 .0926 830 .0921 830 .09 17 830 .0912 830 .0908  827 .9249 827 .9249 827 .9249 827. 9249 827 .9249 827 .9240 827 .9248 827 .9247 827 •9246 027 .9240 827 .9244 827 .9243 827 .924 1 827 .9240 827 .9238 827 .9237 827 .92 . 35 827 9233 827 .9231 827 . 9229 827 . 9227 827 .9225 827 . 9222 827 . 9220 827 . 92 17 827 . 92 15 827 . 92 12 827 .9210 827 .9207 827 . 9204 827 . 9201 827 . 9 198 827 .9195 827 .9192 827 . 9 189 827 . 9 186 827 . 9 1B3 827 .9180 827 .9176 827 9173 827 .9170 827 . 9 167 827 .9164  825 .7444 825 .7445 825. 7-146 825. 7446 825. 7447 825 .7447 825. 7448 825 .7448 825, 7448 825 .7448 825. 7448 825. 7.448 825 .7448 825. 7447 825 .7447 825 .7446 825. 7446 825 .7445 825 .7444 825 .7444 825. 7443 825 .7442 .7441 825 , 825 . 7440 825.. 7438 825 .7437 825 . 7436 825 .7434 825 . 7433 825 .7431 825 . 7430 825 .7428 825 . 7426 825 .7425 825 . 7423 825 .7421 825 . 74 19 825 .7417 825 .7415 825 .7413 825 .74 1 1 825 .7409 825 . 7407  823 .5657 823. 5658 823 .5659 823 .5660 823. 5660 823. 5661 823 .5662 823. 5662 823. 5662 823. 5663 823 .5663 823. 5663 823 .5664 823 .5664 823. 5664 823 .5664 823 .5664 823 .5664 823 .5664 823 , 5664 .5663 823 . .5663 823 , 823 . 5663 823 . 5662 823 .5662 823 .5661 823 . 566 1 823 .5660 823 .5659 823 .5659 823 . 5658 823 .5657 823 .5656 823 .5655 823 .5654 823 .5653 823 .5652 . 823 .5651 823 .5650 823 .5649 823 .5648 823 .5647 823 . 5646  821 .3089 82 1 . 3989 3890 82 1 . 321 .3890 821 .3891 821 .389 1 821 .3892 82 1 . 3892 3892 82 1 . 821 .3893 82 1 .. 3893 821 .3893 . 82 1 ..3893 82 1 .. 3893 82 1. 3894 82 1. 3894 821 . 3894 821 . 3894 82 1. 3894 82 1. 3894 82 1.3894 821 .3894 821 . 3894 821 .3894 82 1. 3894 821 . 3893 82 1. 3893 821 . 3893 821 .3893 821 . 3893 821 . 3892 821 .3892 82 1. 3892 821 .3891 821 .3891 821 .3891 82 1. 3890 821 .3890 82 1. 3889 821 .3889 82 1. 3888 821 .3888 821 . 3887 ro  THF  r> T F At) / S T A T E  TEMPERATURE  FIELD  FOLLOWS:  1 87 1 .6430  869.444 1  3 867.2465  865.0501  5 862 ,8550  860.66 12  858.4686  8 856.2772  854.0871  10 85 1.8982  1 1 849.7 106  12 847.5242  13 845.3391  14 843 1552  15 840.9725  16 838.7911  17 836.6109  18 834.4319  19 832 2542  20 830.0777  2 1 827 .9024  22 825.7284  23 823.5556  24  25  26  27  82 1 .3840  819.2137  817.0445  814.8766  28 8 12 7099  29 8 10.5441  30 808 3802  31 806.2 172  32 804 .0553  37 793 2644  38 79 1 . 1098  39 788 9565  40 786.8043  4 1 784 .6534  33 801.8947 42 782.5036  34 799.7353 43 780.3551  35 797.5771 44 778.2077  36 795.4202 45 776.0616  773 .9166  77 1 .7728  48 769 .6303  49 767.4889  50 765 . 3487  51 763.2097  52 761.0719  53 758.9353  54 756.7998  55 754 . 6656  56 752.5325  57 750 4006  58 748.2699  59 746 1404  60 744.0120  61 74 I .8848  62 739.7588  63 737.6340  64 735 . 5 103  65 733.3879  66 731 . 2665  67 729.1464  68 727 .0274  69 724.9096  70 722.7930  71 720.6775  72 718.5631  73 7 16 4500  74 7 14. 3380  75 7 12 .227 1  76 710.1174  77 708 0089  78 705.9015  79 703.7953  80 701.6902  81 699.5863  82 697 . 4835  83 695.3819  84 693 28 14  85 691 . 1821  86 689 .0839  87 686.9869  88 684.8910  89 682.7962  90 680.7026  91 678 .6101  92 676.5 187  93 674 4285  94 672.3394  95 670 2515  96 668.1647  97 666.0790  98 663.9944  99 661 .9110  100 659 8287  101 657.7476  102 655 .6675  103 653.5886  104 651 .5108  105 649.4341  106 647 .3585  107 645.2e41  108 643.2108  109 . 1386 6 1 1  1 10 639.0675  1 1 1 636 9975  112 634.9286  1 13 632 .8609  1 15 628.7287  1 16 626.6643  1 18 622 5387  1 19 620.4776  120 618 4176  12 1 616.3587  124 610.1886  125 608.1341  4G  127 604.0283  47  128 601.9771  129 599.9270  130 597.8779  614  122 .3009  131 595 .8300  1 14 630.7942 123 612.2442 132 593.7831  133 591.7373  134 589.6926  1 17 624.6009 126 606.0806 135 587.6490  ro ro  143 571.3391  144 569.3053  152 553.0727  551.0485  536 .9084  161 534.8926  532.8779  169 518 ,8043  170 516.7980  171 514.7927  177  178  502 7 8 2 9  5 0 0 .7849  179 498.7880  180 496.792 1  186 484 .8303  187 482 . 8 4 9 6  188 480.8619  189 478.8752  194 468.9574  195 466 .9768  196 464 .9973  197 463.0189  198 461 .0414  202 453.14 17  203 45 1 . 1693  204 449 . 1979  205 447 .2276  206 445.2582  207 443.2898  210 437 . 390R  2 1 1 435 . 42G4  212 4 33.463 1  2 13 431 . 5008  2 14 429 .5394  215 427.5791  216 425.6197  218 42 1.7040  2 19 4 19 7477  220 417.7923  221 4 15.8380  222 413 .8846  223 41 1 .9322  224 409.9808  225 408.0304  227 4 0 4 . 1326  228 402.1851  229 400.2386  230 398.2932  14 1  142  136 585.6065  137 583.5651  581 , 5247  139 579.4854  140 577.4473  145 567 . 2724  146 565.2407  563  147 2 100  148 561.1804  559.1519  557  154 549.0253  155 547.0032  156 544 . 9821  157 542.9621  158 540.9431  159  160  538 .9252  163 530.8642  164 5 2 8 .8516  165 526 . O'lO 1  166 524.8295  167 522.8201  168 5 2 0 8 117  172 512 .7885  173 510.7853  508  174 7831  175 506.7820  176 504.7820  18 1 •in4 . 797 2  182 492.8033  183 4 90 8 105  184 488.0 187  185 48G . 820O  190 476.0096  191 474.9050  192 472 . 92 14  193 470.9389  199 459.0650  200 457.0895  455  201 1 151  208 4 1 .3225  209 4 39.3561  2 17 4 2 3 . 66 14 226 4 0 6 . 0 8 IO  138  5 7 5 . 4101  149  150 1245  573 ,3741 151 55E .098 1  153 162  ro co  AT FI.OW MOULFS AVERAGE EXPOSED WALL TEMPERATURES ( K I : 1-ZONE= 877.70 4-ZONE/M* *2 = #3 = 04= AVLRARE  CUVF.REO WALL  J E MP ERA I URE  ;  860.95 877.11 884.04 888.70  047.84  CALCUIATFO HFAT FLOWS (W/M): INTEGRA T ED EXPOSED WALL : HAD I A I 1 ON CONVECTION TOTAL COVERED WALL: CONVICTION ",01. IDS: RADIATION CONVECTION TOTAL THROUGH WALL: DIFFERENCE PROF ILE  1-ZONE  -ZONE  39219. 32072. 7 1290.  39219. 3207 2. 71290.  39093. 32072 . 71165.  •4 704 3.  -4 704 3.  -47043 .  68853. 65832 . O.13468E+06  6892 1 . 65832 . 0. 13475E+06  *•»*.• ......  24247. 24 185.  24247 .  24122.  AfJAI 00 APPROXIMATIONS: MODIFIED ANALOG:(INCLUDES CONVECTIVE BRANCHES) EXPOSED WALL TEMPERATURE* 873,40 (K) OUTER SHELL TEMPERATURE- 396.90 HFAT RECEIVED BY SOLIDS' 0.18379E+06 HEAT LOSS TO SURROUNDINGS' 24246.  MODIFIED ANALOG:(RADIATION ONLY) EXPOSED WALL TEMPERATURE' 904.76 OUTER SHELL TEMPERATURE' 401.40 HEAT RECEIVED BY SOLIDS' 73805. HfcAF LOSS TO SURROUNDINGS' 25613  (W/M)  (K) (W/M)  Appendix  A8  FORTRAN SOURCE L I S T I N G AND SAMPLE OUTPUT FOR K I L N  FLAME MODEL  FORTRAN SOURCE  LISTING  K I L N FLAME MODEL  1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 2.2 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51  c c c c c c c c  FLAME MODEL FOR ROTARY KILN J.P. GOROG NOV. 27. 1981  IMPLICIT REAL*8(A - H.O - Z) DIMENSION X(500), T ( 5 0 0 ) . TW(500). 0L(500) DIMENSION X1(4). F ( 4 ) , ACCE5T(4) DIMENSION TSH(500). 0FS(50O). QFW(500). 0FSH(500) REAL'8 L, MWF, MF. MAT. MAP. MO. MS. MEN, KP REAL *8 M(500), MOP. MWCP. MST. MAEP EXTERNAL FN2. FN3 . COMMON /BLK1/ ASAT. D COMMON /BLK2/ R COMMON /BLK3/ EMG, EMS. EMW COMMON /BLK4/ HOUT, HCOV. HSE. HEX COMMON /BLK5/ KP, TS. TA. RO COMMON /BLK6/ TF. OLOSS. TW1, TSA COMMON /BLK7/ FFS. DR. AF1. AS. AW. FSF. FSW, FFW COMMON /BLK8/ X, M. GHV. AF, 11 COMMON /BLK9/ SPHEAT, H2L0 COMMON /BLK10/ OSOLID. QWALL, QSHELL COMMON /BLK 11/ MF. MO LOGICAL* 1 NAME(20) LOGICAL LZ1 CALL SETLIO('6 '. '-A ') CALL SETLIO('8 '. '*SINK+ ') CALL CMD('$EMPTY -A OK '. 13)  c c c c c c c c c c c c c c c c c c  READ INPUT AS FOLOWS: NAME  = Fuel  type  ID  = Run I d e n t i f i c a t i o n  0 L BTH RES OS  = = = = =  K i l n diameter (m) K i l n l e n g t h (m) T h i c k n e s s o f k i l n l i n i n g (m) Residence time o f s o l i d s ( h r ) Mass f l o w r a t e o f s o l i d s ( k g r / s )  FR TF TSA TS  = = * "  F i r i n g r a t e of f u e l (d/kgr s o l i d ) I n l e t f u e l temperature (K) Secondary a i r temperature (K) S o l i d s temperature (K)  number  ro 00  52 53 54 55 5G 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103  C C C C C C C c c c c c c c c c c c c c c c  TA  = Ambient  EMW EMS EMG  = Wall e m i s s i v i t y = sol Ids e m l s s l v ! t y = Flame e m i s s i v i t y  HOUT HCOV HSE HEX KP  o = = = =  PPA ; PPO PTA  = Percent o f s t o i c h i o m e t r i c a i r as primary = Percent o f p r i m a r y a i r as oxygen = Percent o f s t o i c h i o m e t r i c a i r t o t a l  AF DO H2L0 GHV ROJET  = = = = =  10 20 30  temperature  Combined HTC at o u t e r s h e l l (W/m**2 K) C o n v e c t i v e HTC a t covered wall (W/m**2 K) C o n v e c t i v e HTC a t s o l i d s s u r f a c e (W/m»*2 K) C o n v e c t i v e HTC at exposed wall (W/m**2 K) Thermal c o n d u c t i v i t y of l i n i n g (W/m K)  A i r to f u e l r a t i o f o r I n i t i a l f u e l (kgr/kgr E q u i v a l e n t diameter of burner (m) Hydrogen l o s s Gross h e a t i n g v a l u e o f f u e l ( J / k g r ) J e t d e n s i t y (kgr/m**3)  READ (5.20) (NAME(I).1=1.20) READ (5.30) ID READ (5.10) D, L, BTH. RES. OS READ (5.10) FR. TF. TSA. TS. TA READ (5,10) EMW, EMS. EMG READ (5.10) HOUT. HCOV. HSE. HEX. KP READ (5.10) PPA, PPO. PTA READ (5,10) AF, DO, H2L0, GHV. ROJET SPHEAT = 1500.00 RHOS = 600.DOO FORMAT (5G12.5) FORMAT (20A1) FORMAT (13) TIF = 1850.DO AFO = AF TFO = TF 0S1 = (OS + 100.DO/56.DO'QS) / 2.DO RES = RES * 3600.DO ASF » (0S1*RES) / (L*RH0S) ASAT = ASF / (PI(O.DO)*(D/2.DO)**2) RO = D / 2.D0+BTH MF = OS * FR * (1.DO/GHV) MAT = MF * AF MAP = (1.DO-PP0) * PPA * MAT MOP = PPO * PPA • MAT MAEP = 4.31DO * MOP MEN = MAT - MAP - MAEP MO = MF + MOP + MAP ROAIR = 28.82DO / (0.08205D0*TSA) ROCP = 0.20D0  ro vo  104 105 106 107 108 109 1 10 111 112 113 114 115 116 117 1 18 1 19 120 121 122 123 124 125 126 127 128 129 130 13 1 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155  40  50  60  70  TERM = (R0JET/R0CP) ** 0.500 * (ROJET/ROAIR) *• 0.5D0 FL = (1.DO+MEN/MO) * DO • 6.DO * TERM BL = FL - 6.00 • DO * TERM DDXX = .25D0 N = IDINT(BL/DDXX) + 1 NI * N * 1 X( 1 ) = 6 . DO * 00 • TERM DO 40 I = 2, N X ( I ) = X ( I - 1 ) + DDXX X(N1) = FL M(1) = MO DO 50 I • 2. N1 M ( I ) = ((1.D0/6.D0)*X(I)«M0) / DO * (1.DO/TERM) AF = MEN / MO T ( 1 ) - TF CALL VIEW QTOTS = O.DO QTOTW = O.DO 0T0T5H = O.DO QTOT = 0 . 0 0 DO 80 I = 1. N TR « 1.DO-EMG RS = 1.DO-EMS SIGMA = 5.67D-08 ES = SIGMA * TS ** 4' EF = SIGMA * T ( I ) * * 4 EA = SIGMA * TA * * 4 T1 = AW * EMG T2 = AS * TR T3 = AS * EMG T4 = (EMS*AS) / RS G = T3 + T2 + T4 EAD = (T1 *EF + T2*((T3»EF + T4*ES)/G)) / ( T l + T2 - (T2**2)/G) IF ( I .GT. 1) GO TO 60 X1(4) = ES X1(3) • EAD X1(2) = ((X1(3) - .25D0*(X(3) - EA))/SIGMA) •* .25DO X1(1) = ((X1(3) - .75DO*(X(3) - EA))/SIGMA) ** .2500 CONTINUE TF * T ( I ) CALL NDINVT(4, X1. F, ACCEST. 800, .1, FN2. 6210) TFL = O.DO TFU = 1.D05 E l l = 5.0-05 11=1 CALL 2ER01(TFL. TFU, FN3, E l l . LZ1) IF ( .NOT. LZ1) GO TO 230 T ( I + 1 ) = TFL IF (I .GT. 1) GO TO 70 TW(1) ' X1(2) TSH(1) • X1(1) T ( I + 1 ) » TFL  ^ en  °  156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 17 1 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 19 1 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207  TW(I + 1) = X1(2) TSH(I + 1) = X1(1) O F S ( I ) = OSOLID / AS QFW(I) = QWALL / AW OFSH(I) = OSHELL / (PI(O.DO)'(D + 2.D0»BTH)) Q U I ) = OLOSS 0L( 1 ) = 0.00 QTOT = QTOT + ( x ( I * 1) - X ( D ) • OLOSS OTOTS = OTOTS + ( X ( I + 1) - X ( I ) ) * OSOLID 1) - X ( D ) * OWALL QTOTW = OTOTW + (X( I OSHELL QTOTSH + ( X ( I + 1) - X ( I ) ) OTOTSH CONTINUE 80 OCALC * QS * 100.DO / 56.DO * 1.637D06 FRN = FR * (1 .DO-H2L0) CALL HEADER WRITE (6.90) (NAMEM ) . 1 = 1, 20) , GHV. H2L0, AFO, FR. FRN T5, 'Fuel t y p e . ' . 1X IX, . 20A1/T5. 'Gross h e a t i n g v a l u e FORMAT ( 90 1f f u e l =', 1X, E12.5. 1X, ' ( d / k g r ) ' / T 5 . 'Hydrogen l o s s ='. 1X. E12.5/T5. ' A i r to f u e l r a t i o of f u e l ='. IX, E12.5, IX, 2 ' ( k g r / k g r ) ' / T 5 , 'Gross f i r i n g r a t e of f u e l ='. 1X, E12.5, 3 IX. '(J/kgr s o l 1 d ) ' / T 5 . 'Net f i r i n g r a t e of f u e l =' IX , 4 E12.5. 1X. ' ( J / k g r s o l i d ) ' / / ) 5 WRITE (6.100) D. L, DO, OS, BTH, ASAT FORMAT (' '. T5. ' K i l n diameter ='. 1X. E12.5. IX, '(m)'/T5, 100 'K1ln l e n g t h ='. IX, E12.5. IX. '(m)'/T5. 1 ' E q u i v a l e n t burner diameter ='. 1X, E12.5, 1X, '(m)'/T5, 2 'Mass f l o w r a t e of s o l i d s =', 1X, E12.5, 1X, ' ( k g r / s ) ' / T 5 , 3 • L i n i n g t h i c k n e s s ='. IX. E12.5, 1X. '(m)'/T5. 4 'Percent s o l i d s l o a d i n g ='. IX, E12.5, //) 5 WRITE (6, 110) EMS, EMW, EMG ', IX, E12.5/T5, FORMAT (' ' . T5, 'Wal 1 em I s s 1 v I t y 1 10 ' Wa11 em 1ss1v1ty =' 1X, E12 5/T5, 'Flame e m i s s i v i t y =' IX, E12.5//) WRITE (6.120) TFO TSA, TA, PPA, PPO. FL FORMAT ( ' ' . T5. I n i t i a l f u e l temperature =', 1X, E12.5. IX, 120 Secondary a i r tmeperature =', IX, E12.5, IX, '(K)'/T5. Ambient a i r temperature =', 1X, E12.5, IX, '(K)'/T5. Percent of s t o i c h i o m e t r i c a i r as primary =', 1X, '(K)'/T5. E12.5/T5. 'Percent of prImary a i r as oxygen «=' , 1X, E12.5// T5. ' C a l c u l a t e d f 1 ame length « ', IX, E12.5. 1X. '(m)'/) WRITE (6.130) ' D i s t a n c e (m)', T22, FORMAT ('1', T47, ^Temperature (K) /T5. 130 'Sol Ids', T38, 'Wal1'. T52, 'Flame', T68. 'Shel1', T82, 1 'Ambient'/) 2 DO 140 I = 1. NI 140 WRITE (6, 150) X ( I ) , TS. TW(I). T ( I ), TSH(I), TA 150 FORMAT (' ', T4, E 12.5. T19. E12.5, T34. E12. 5. T49. E12.5. T64. 1 E12.5, T79, E12.5) WRITE (6,160) 160 FORMAT ('1'. T31, Heat f l u x (W/m **2)'/T5, ' D i s t a n c e (m)'. T22, 'Sol Ids'. T39, 'Wal 1 , T54 'Shel I '/) 1 DO 170 I = 1 . N  ro on  208 209 2 IO 2I 1 212 213 214 215 2 is 2 17 2 18 2 19 220 22 1 222 223 224 225 226 227 228 229 2 30 23 1 232 233 234 235 236 237 238 239 240 24 1 242 243 244 245 246 247 248 249 250 25 1 252 253 254 255 256  170 180  190  200  2 10 220 230 240 250  WRITE ( 6 . 1 8 0 ) X ( I ) , O F S ( I ) . QFW(I), O F S H ( I ) FORMAT (' '. T4. E12.5. T19. E12.5. T34. £12 5. T50. E12.5) QTOTW - OTOTSH OREGN HINSF MO * SPHEAT • (TFO - 298.DO) HINSA MEN •* SPHEAT * (TSA - 298. DO) HINFL FR • OS TOTIN HINSF + HINSA + HINFL PHINSF = (HINSF/TOTIN) 100.DO PHINSA = (HINSA/TOT IN) 100.DO PHINFL = (HINFL/TOTIN) 100.DO ASHELL = PI(O.DO) * (D 2.DO'BTH) HOTS = 0 .00 HOTSH = 0.00 HOTCP = M(N1) SPHEAT • (T(N1) - 298.DO) DO 190 1 = 1 . N HOTS » HOTS + (<X(I + 1 ) - X ( I ) ) * A S * Q F S ( I ) ) HOTSH = HOTSH + <<X(I + 1) - X(I))*ASHELL*OFSH(I)) CONTINUE H0TH2 = HINFL * H2L0 TOTOUT = HOTCP + HOTS + HOTSH + H0TH2 PHOTCP = (HOTCP/TOTOUT) * 100.DO PHOTS = (HOTS/TOTOUT) * 100.D0 PHOTSH = (HOTSH/TOTOUT) * 100.00 PH0TH2 = (H0TH2/T0T0UT) * 100.DO * 100.DO PTHEO = (HOTS/OCALC) • PHINSF. HINSA. PHINSA, HINFL , PHI NFL TOTIN. WRITE (6.200) HINSF 1H0TCP. PHOTCP. HOTS PHOTS. HOTSH, PHOTSH. H0TH2, PH0TH2. TOTOUT, 2PTHE0 FORMAT ( ' 1 ', T5. 'Heat b a l a n c e on f 1 ame'///T5 , 'Heat In (W):', T59, 'Percent of t o t a l : ' / T 5 , ' S e n s i b l e heat of f u e l m i x t u r e 1 T40. E12.5, T63, F6.2/T5, ' S e n s i b l e haet of secondary a i r =', 2 = T40. E12.5. T63, F6.2/T5, 'Chemical energy of f u e l ='. T40, 3 E12.5, T63, F6.2//T15. 'Total =', T40. E12.5///T5, 4 'Heat out (W):'/T5. ' S e n s i b l e heat of combustion p r o d =', 5 T40. E12.5. T63. F6.2/T5. 'Heat r e c e i v e d by s o l i d s ='. T40, 6 E12.5, T63. F6.2/T5. 'Heat l o s s from s h e l l =', T40. E12.5. 7 8 T63, F6.2/T5, 'Hydrogen l o s s ='. T40. E12.5. T63. 9 F6.2//T15. ' T o t a l ='. T40. E12.5////T5. * '% of t h e o r e t i c a l c a l c i n a t i o n energy t o s o l i d s =', 1X, F6.2/'<'/'4'/) 1 CALL P9700(ID) STOP WRITE (6.220) FORMAT ('1'. ERROR RETURN FORM NDINVT') GO TO 250 WRITE (6.240) ERROR RETURN FROM FLAME TEMP SOLUTION') FORMAT ('1', STOP END  ro ro  257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280  28 1 282 283 284 285 286 287 288 289 290 291 292  ********** C C SUBROUTINE TO CALCULATE SOLIDS FILLRATE C £*+*••***** SUBROUTINE FIIL(ASAT, FD. PHI. 0) IMPLICIT REAL*8(A - H.O - Z) EXTERNAL FN 1 LOGICAL LZ COMMON /BLK2/ R X = O.DO Y = 1.002 E1 * 5.D-04 CALL ZEROKX. Y. FN 1 . E l . LZ) IF ( .NOT. LZ) GO TO 10 PHI = X / 2.DO R » D / 2.DO FD = (R - R*DCOS(PHI)) / D RETURN 10 WRITE (6,20) 20 FORMAT ('1', 'SOLUTION CALL FROM FILL FAILED') CALL EXIT STOP END  c  £*•**«*+*** C C EXTERNAL ROUTINE FOR FILL C £«***•***»* FUNCTION FN1(X) IMPLICIT REAL*8(A - H.O - Z) COMMON /BLK2/ R COMMON /BLK 1/ ASAT. D FN1 * ASAT * PI(O.DO) * R * * 2 - R * * 2 / RETURN END  2.DO * (X - DSIN(X))  ro cn co  293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 3 11 312 313 314  315 316 317 318 319 320 321 322 323 324 325 326 ' 327 328 • 329 330 331 332 333 334  £•***•*»•*** C C SUBROUTINE TO CALCULATE VIEW FACTORS C £•*****•»•** SUBROUTINE VIEW IMPLICIT REAL+8(A - H.O - Z ) COMMON /BLK1/ ASAT. D COMMON /BLK2/ R COMMON /BLK7/ FFS, DR, AF. AS. AW, FSF. FSW, FFW R = D / 2.DO CALL FILL( ASAT, FD. PHI. D) FFS = (2.D0*PHI) / Pl(O.DO) DR = D / 3.DO AF = PI(O.DO) * DR AS = D * DSIN(PHI) AW = D * (PI(O.DO) - PHI) FSF = (AF * FFS) / AS FSW " 1.DO-FSF FFW = 1 .DO-FFS RETURN END  C*»*»****•» C C EXTENAL ROUTINE TO CALCULATE FLAME TEMPERATURE C C********** SUBROUTINE FN3(X1) IMPLICIT REAL*8(A - H.O - Z) REAL*8 X(500), M(500). MF. MO COMMON /BLK6/ TF, OLOSS. TW, TSA COMMON /BLK8/ X, M, GHV, AF, 11 COMMON /BLK9/ SPHEAT, H2L0 COMMON /BLK11/ MF, MO H2L01 = 1.00-H2L0 TERM2 » MF / MO - Mill)) FN 1 = M(I1) * SPHEAT * (TF - 298.DO) + TERM2 • (M(I1 + 1) * (TSA 1 * (1.D0/AF) * GHV * H2L01 + (M(I1 + 1) - M(I1)) * SPHEAT X(I1)) 2 298.) - M(I1 + 1) * SPHEAT * (XI - 298.00) - ( X ( I 1 * 1) 1  3* OLOSS RETURN END  ro cn  335  336 337 338 339 340 341 342 343 344 345 346. 347 348 349 350 35 1 352 353 354 355 356 357 358 359 360 36 1 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 301 382 383 384 385  (;******•**•  C C EXTERNAL FUNCTION TO EVALUATE HEAT FLOW CIRCUIT C £*»*•***•*« SUBROUTINE FN2(X. F) IMPLICIT REAL*8(A - H.O - Z) REAL*8 KP DIMENSION X( 1 ). F ( 1 ) COMMON /BLK1/ ASAT. D COMMON /BLK2/ R COMMON /BLK3/ EMG. EMS, EMW COMMON /BLK4/ HOUT, HCOV, HSE. HEX COMMON /BLK5/ KP. TS, TA. RO COMMON /BLK6/ TF, OLOSS. TW. TSA COMMON /BLK7/ FFS. DR. AF. AS, AW. FSF, FSW. FFW COMMON /BLK10/ OSOLID, OWALL. QSHELL TR = 1.D0-EMG RS = 1.DO-EMS o RW = 1.DO-EMW SIGMA = 5.67D-08 ES = SIGMA * TS *» 4 EF = SIGMA * TF ** 4 EA = SIGMA • TA ** 4 T1 = (1.DO-EMW) / (EMW*AW) T2 = (DL0G(R0/R)*(X( 1) + X( 2 ) ) * (X ( 1)•*2 + X(2 ) **2)/(2.D0*PI(O.DO)* 1KP) ) • SIGMA T3 = 1.D0 / (FFW*AF*EMG) T4 = 1.DO / (AS*FSW) T5 = 1.D0 / ( F S F * A S * E MG) T6 = (1.DO-EMS) / (EMS*AS) CI = TA / X ( 1 ) C2 • 1.D0+C1 + C1 ** 2 + C1 •* 3 HO » HOUT + (C2•SIGMA*X( 1 ) •*3) HOP = HO / ( ( ( X ( 1 ) + TA)*(X(1)**2 + TA**2))*SIGMA) HCOVP = HCOV / ( ( ( X ( 2 ) + T S ) * ( X ( 2 ) * * 2 + TS**2))*SIGMA) HSEP • = HSE / ( ( ( T F + TS)*(TF**2 • TS**2) )*SIGMA) HEXP * HEX / ( ( ( T F • X ( 2 ) ) * ( T F * * 2 + X(2 ) **2 )) *SIGMA) ACOV = (2.DO*PI(O.DO)*R) - AW ASH = 2.DO * PI(O.DO) * RO T8 = 1.DO / (HOP*ASH) T9 = 1.D0 / (HCOVP'ACOV) T10 = 1.D0 / (HSEP*AS) T i l - 1.D0 / (HEXP*AW) T12 = 1.00 / (FSF *AS*TR) RE = (T4*T12) / (T4 + T12) T13 = SIGMA * X(2) ** 4 F ( 1 ) « (EF - X ( 3 ) ) / T3 + (T13 - X(3)) / T l + (X(4) - X(3)) / R E T14 = SIGMA * X(1) ** 4 F ( 2 ) = (X(3) - T13) / T l + (T14 - T13) / T2 * (EF - T13) / T11 + ( 1ES - T13) / T9  .  v '  tn ^  386 387 388 389 390 391 392 393 394 395 396 397 398  399  400 401 402 403 404 405 406 407 408 409 4 10 411  10  .  F{3) = (EF - X ( 4 ) ) / T5 + (ES - X ( 4 ) ) / TG + (X(3) - X(4)) / RE T20 = T2 + T8 F ( 4 ) = ( T I 3 - EA) / T20 - (T14 - EA) / T8 FORMAT (12(G12.5.2X)) OLOSS = (EF - X ( 3 ) ) / T3 + (EF - T13) / T i l + (EF - ES) / TIO + ( 1 EF - X ( 4 ) ) / T5 OSOLID = ( X ( 4 ) - ES) / T6 + (EF - ES) / T10 + (T13 - ES) / T9 T30 = SIGMA * X(2) ** 4 T31 = SIGMA * X(1) ** 4 QWALL = ( X ( 3 ) - T30) / T l + (EF - T30) / T i l OSHELL = (T31 - EA) / T8 RETURN END  c******»»»* C C SUBROUTINE TO PRINT HEADER C  10  SUBROUTINE HEADER LOGICAL*1 STARS(34) /34*'*'/ WRITE (6.10) STARS. STARS FORMAT (' ', T48. 34A1/T48, '* UNIVERSITY OF BRITISH COLUMBIA *'/ 1 T48. '* METALLURGICAL ENGINEERING *'/T48. 2 '* FLAME.MODEL VERSION 1 *'/T48, 34A1//) RETURN END  ro cn cn  412 4 13 4 14 4 15 4 16 4 17 418 4 19 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444  C* ****•*»•* C  C SUBROUTINE TO SET 9700 OUTPUT C (;*•***••*•* SUBROUTINE P97OO(I0) LOGICAL * 1 A(132) INTEGER*4 CNT. PG INTEGER*2 LEN REWIND 6 CALL GETLSTC'6 '. CNT) CNT = CNT / 1000 PG = 0 NUMB = 1 DO 50 I = 1. CNT CALL READ(A, LEN. 0. LNUMB. 6) IF (PG .EQ. O) GO TO IO L = O CALL FINOST(A. 2, '1'. NUMB. L + 1, L. »40) IF (PG .GE. 1) GO TO 30 10 WRITE (8.20) 20 FORMAT ('1'//////////) PG = PG • 1 LNCK = 10 GO TO 40 30 CONTINUE CALL PAGE(LNCK, ID. PG, A) 40 LNCK = LNCK + 1 IF ((LNCK - 60) EQ. O) CALL PAGE(LNCK, ID, PG. A) CALL WRITE(A, LEN, O, LNUMB, 8) 50 CONTINUE RETURN END  ro  4«15 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466  £**•*****•• C C SUBROUTINE TO PRODUCE PAGE NUMBERS C  .  SUBROUTINE PAGE(LNCK, 10, PG, A) LOGICAL*! A(132). I BLANK, FMT(41), F l ( 3 ) INTEGERM PG DATA I BLANK /' '/. FI /ZF 1 , ZF2, ZF3/ DATA FMT /Z4D, Z7D, Z4C, Z7D, Z6B, ZE3. ZF1, ZF2, ZF1, Z6B. Z7D, 1 ZD7. ZC1, ZC7, ZC5, Z40, Z7D, Z6B, ZC9. ZF3, Z6B, Z7D, Z60, 2 Z7D, Z6B, ZC9. ZF2. Z61. Z7D, ZF1. Z7D. Z61, Z61. Z61 , Z61, 3 Z61, Z6 1, Z61, Z6 1. Z61, Z5D/ IF (PG .LE. 9) FMT(27) = F I ( 1 ) IF (PG .GE. 10 .AND. PG .LE. 99) FMT(27) • F I ( 2 ) IF (PG .GE. 100) FMT(27) = F I ( 3 ) WRITE (8.FMT) ID. PG A ( 1 ) = IBLANK PG * PG • 1 LNCK = 10 RETURN END  ro cn  oo  SAMPLE INPUT DATA  fuel  type  Rj  = 1.52  L  =  R  : natural (m) 80  =  Q  gas  1.60 Q  gross f i r i n g rate inlet  fuel  temperature  secondary a i r T  T  =  a  w  £  e  = 1100  s  e  temperature  =  0  =  (K) 298  ,  8  8  0.25  per cent o f  stoichiometric  air  per c e n t oxygen enrichment  = 0  Air/fuel  r a t i o = 16.97  hydrogen l o s s equivalent gross  solid)  (K)  °' =  g  = 298  (J/kgr  298  =  s  = 1 . 1 0 . x 10  burner  heating  equivalent  = 9.82  (kgr  per  =  20  fuel)  cent  of fuel  j e t density  primary  air/kgr  diameter =  value  as  .17 = 5.52  = 1.03  (m) x 10^  (kgr/m )  (J/kgr)  260  SAMPLE OUTPUT  +  * *  UNIVERSITY OF BRITISH COLUMBIA * METALLURGICAL ENGINEERING * FLAME.MODEL VERSION 1 *  Fuel t y p e : NATURAL GAS G r o s s h e a t i n g v a l u e of f u e l = 0.55200E+08 ( J / k g r ) Hydrogen l o s s = 0.98200E-01 A i r to f u e l r a t i o of fuel = 0.16970E+02 ( k g r / k g r ) G r o s s f i r i n g r a t e of f u e l = O.11050E+O8 (.J/kgr s o l i d ) Net f i r i n g r a t e of f u e l = 0.99649E+07 ( J / k g r s o l i d ) K i l n d i a m e t e r = 0.30480E+01 (m) K i l n l e n g t h = 0.80000E+02 (m) E q u i v a l e n t b u r n e r diameter = 0.16930E+00 (m) Mass f l o w r a t e o f s o l i d s = 0.13130E+01 ( k g r / s ) L i n i n g t h i c k n e s s = 0.15200E+00 (m) P e r c e n t s o l i d s l o a d i n g = 0.56394E-01 Wall  e m i s s i v i t y = 0.80000E+00 e m i s s i v i t y = 0.80000E<00 f l a m e e m i s s i v i t y = 0.25000E+00  W-Tll  I n i t i a l f u e l temperature = 0.29800E+03 (K) Secondary a i r tmeperature = 0.29800E+03 (K) Ambient a i r temperature = 0.29800E+O3 (K) P e r c e n t of s t o i c h i o m e t r i c a i r as p r i m a r y = 0.20000E+00 P e r c e n t of p r i m a r y a i r as oxygen = 0.0 Calculated  flame  length =  0.88129E+01  (m)  ro  o o o o o o o o o o o o o o o o o o o o o o o o o o o o  o  co cn ^ — c o c n x i - i i a c n x i - i co cn i — a o 6 ^ i f l o i t - i f l i n t > - ' «• - u i O u i O u i O u O O O U O ' J ' C i a o u i O t n o u i O u i O O O u i t» |OXiXiXiXiXiXiXiXiXiXiXiXiX-i-X.XiXiXiXiXiXiXiXiXiXiXiXi 2 CQIDCOlDOCDtOCOCOCOCDCOCDiDIJCDCOCOCDUJCOCOCDCDtOCDCOlD O m m m m m m m m m m nvrn r n r n r r r n r n m r n r n r r i r n r n r n r n r n r n r n ID + + ^ + + + + + + + + - t - + + -r + T + <- + + + + + + + + +  o o o o o o o o o o o o c o c o o o o o o o o o o o o o —• o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1.-. — L.L — — — — — —  co 0  o o o o o o o o o o o o o o o o o o o o o o o o o o o o — o o o o o o o o o o o o o o o o o o o o o o o o o o o o — o o o o o o o o o o o o o o o o o o o o o o o o o o o o a mmmrnmmmmrnmmmmmr~immmmmrnrnrnrnrnrnrnrti w + ++++ + ++++++ + ++ + ++ ++ + +++ + o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O o o o o Xi & co IO CO co O m m + + O O x. u  O O O O O o  O OO O O O o O O O O O O O O  & Xi ii Xi Xi CJ CO CO co CJ CO CJ CO ro ro o o CO OO ^1 cn cn Xi ro o CO CJ cn cn cn CJ O *. c,cn CJ to u ^1 - j o ~ j 00 CO CD ~ j ii CO CO ro -o cs CJ m m m m m m m m m m rr? m m m — + + + +• 4- + + +• + + i- + O O O o o O O O O O O O X i X i X i X i ii X . ii x. x-  -  9  o O O O O O O O O O O o O o 03 cn O CO m + O Xi  09 cn ~j CO m + O X*  CO co CO cn x> CO Xi cn . ro cn m m m + + + O O o Xi Xi i i  CD ro 00 ro m + O Xi  00 CO O 00 co ~1 IO m m + + O O Xi Xi  -J oo ro oo m + O O Xi Xi -o CD Ul CJ m  -j cn -j cn m + O ii  ^1 Ul O CO m  o 00 O m •*- + O O O Xi Xk x> CO O cn m  Ol Ol Ol Ul Ul Ul O o o CO CD CO CO o CO ~ 4cn cn Ol O CO Ol o ro ~ l CO Ul m m m m m m + + + + + O o O O O O CO CO CO CO CO CO  cn Ul Ul CD CO co CO . 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