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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 Un iver s i ty , 1975 M.S., Michigan Technological Un iver 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 Meta l lurg ica l Engineering We accept t h i s thesis 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of M e t a l l u r g i c a l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 October 27, 1982 ABSTRACT The overall heat-transfer mechanism within a direct-fired 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 interior surface of a rotary kiln 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 wall, 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 deter-mination of the gas mean beam!ength and the total heat flux to the wall and to the solids. These charts can be used to account for 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 is a very localized phenomenon. Radiation to a surface element from gas more than half a k i l n in diameter away is quite small and, as a result, even large axial gas temperature gradients have a negligible effect on total heat flux. Results are also presented which show that the radiant energy either reflected or emitted by a surface element is limited to regions less than 0.75 k i l n diameters away. The radiative exchange integrals have been used, together with a i i modified reflection method, to develop a model for the net heat flux to the solids and to the k i l n wall from a nongray gas. This model is compared to a simple resistive network/gray-gas model and i t i s shown that sub-stantial errors may be incurred by the use of the simple models. To examine the overall heat-transfer mechanism in the absence of free-board 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 heat-transfer coefficients in a k i l n , has been employed to examine the effect of different kil 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 solids, convective heat-transfer coefficients at the exposed and covered wall, and thermal d i f f u s i v i t y of the wall. 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 inde-pendent of these variables. Owing to this insensitivity i t has been pos-sible to simplify the model with the aid of a resistive analog. Calcula-tions are presented indicating that both the shell loss and total heat flow to the bed may be estimated within 5 per cent using this simplified model. Finally, to examine the overall heat-transfer mechanism in the pre-sence 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 is based on the one-dimensional furnace approximation, has been employed to examine the i i i effects of fuel type, f i r i n g rate, primary a i r , oxygen enrichment and secondary air temperature on the flame temperature, solids heat flux shell losses, and overall flame length. iv TABLE OF CONTENTS Abstract Table of Contents List of Tables ... List of Figures .. List of Symbols ., Acknowledgement ., Page i i v vi i i ix xv xix Chapter 1 INTRODUCTION .. . 2 RADIATIVE HEAT TRANSFER WITHIN THE KILN FREEBOARD 6 2.1 Introduction • •• 2.3.1 Total radiative heat flux from an isothermal gas 1 6 2.2 Representation of the emissive characteristics of the gas, solids and kiln wall 7 2.2.1 Emissive characteristics of the freeboard gas... 7 2.2.2 Emissive characteristics of the k i l n wall and solids 11 2.3 Radiative exchange between freeboard gas and k i l n wall. 11 12 2.3.2 Radiative heatflux from a non-isothermal free-board gas 16 2.3.3 Reflected gas radiation 22 2.4 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 38 2.6.2 Comparison between real and gray gas radiative models in rotary kilns 46 v Chapter Page OVERALL HEAT TRANSFER IN THE ABSENCE OF FREEBOARD FLAMES. 3.1 Introduction 3.2 Previous work < 3.3 Model development • 3.4 Heat-transfer coefficients in rotary kilns 3:4.1 Radiative heat-transfer coefficients 3.4.2 Convective heat-transfer coefficients 3.5 Model predictions 3.5.1 Convective heat-transfer coefficient at the covered wall • 3.5.2 Kiln speed 3.5.3 Convective heat-transfer coefficient at the exposed wall 3.5.4 Kiln loading 3.5.5 Emissivities of solids and wall 3.5.6 Thermophysical properties of wall 3.6 Simplified resistive network OVERALL HEAT TRANSFER IN THE PRESENCE OF A FREEBOARD FLAME. 4.1 Introduction 4.2 Previous work 4.3 Model development 4.3.1 Selection of modelling technique 4.3.2 Model assumptions 4.3.3 Model formulation and solution 50 50 51 53 60 60 68 72 79 82 84 86 88 91 94 100 100 101 103 103 113 118 vi Chapter Page 4.4 Model predictions.. 123 4.4.1 Fuel type 129 4.4.2 Firing rate 134 4.4.3 Secondary a i r temperature... 136 4.4.4 Use of primary a i r 141 4.4.5 Oxygen enrichment. 146 5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK.. 151 5.1 Conclusions 151 5.2 Suggestions for future work 154 REFERENCES 155 Appendix Al Radiative properties for equimolal C02-H20 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 A5 Solution for the resistive network used to predict heat flows and temperatures in the absence of a freeboard flame.. 189 A6 Solution for the resistive 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 A8 FORTRAN source l i s t i n g and sample output for k i l n flame model 246 v i i 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„-H90 gas mixtures at 830, 110 and 1390 K . . . t . . . . 159 Al - 2 Summary of absorptivity data for equimolal CO2-H2O. gas mixtures at 1110 K for blackbody radiation at 277, 555 and 833 K 160 A4-1 Summary of view factors needed to evaluate the 4-zone k i l n wall model 174 A4-2 Summary of view factors needed to evaluate the 16-zone resistive analog of Fig. 4-3 180 v i i i LIST OF FIGURES Figure 1- 1 Schematic diagram of rotary kil n showing; (a) axial cross section; (b) major heat flow paths in non-flame zone I I ; and; (c) major heat flow paths in flame zone I 2 2- 1 Emissivity, e q, and emissive character, de g (r)/dr, for an equimolal C0 2 - H20 gas mixture at 1390, 1710 and 830 K... 9 2-2 Absorptivity of an equimolal CO2 - H2O gas mixture at 1110 K for blackbody radiation at 277, 555 and 833 K 10 2-3 Exchange integral for gas-to-kiln wall radiation versus k.pD 15 2-4 Top: Schematic cross-section of an empty rotary k i l n showing slices used in evaluating the non-isothermal heat transfer from the freeboard gas to a differential element, dA, on the k i l n wall. Bottom: Assumed temperature pro-f i l e in freeboard gas 17 2-5 Normalized cumulative gas-to-wall radiative heat flux versus axial position, z/D, for both an isothermal and non-isothermal equimolal C02 - H20 freeboard gas mixture 2-6 Heat-flux distribution densities for gas-to-wall radia-tive heat flux versus axial position, z/D, for both an isothermal and a non-isothermal equimolal C0 2 - H^ O freeboard gas mixture 21 2-7 Schematic diagram of the cross-section of an empty rotary k i l n showing differential elements, dA-j and dA2» used to evaluate the nature of reflected gas radiation from the ki l n wall 23 2-8 Exchange integral for reflected gas radiation-to-kiln wall versus k.pD 26 i 2-9 Flux surface of reflected gas radiation incident on kiln wall for a small pilot 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 2-10 Schematic diagram of the cross-section of a rotary kil n showing differential elements, dA, and dA2, used to evaluate either the radiative exchange between the free-board gas and the solids or the radiative exchange be-tween the solids and k i l n wall 29 ix Figure Page 2-11 Exchange integral for gas-to-sol ids radiation versus k.pD as a function of f i l l ratio, F/D. 3 3 2-12 Heat flux distribution 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 3 4 2-13 Modified solids surface-to-wall view factor versus axial position, z/D, for a pilot 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.... 3 7 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.. 4 0 2-15 Schematic diagram of the cross-section of a rotary k i l n showing the path of radiant energy emitted from the free-board gas 4' 2-16 Radiative resistance network for a rotary k i l n where the freeboard gas, kiln wall and solids surface are assumed to be gray 47 2- 17 Radiant exchange, for a rotary-kiln, 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 3- 1 Schematic diagram of rotary kiln showing nodal configura-tion used to determine temperature distributions in the kil n wall 5 6 3-2 Computer flow-diagram used to determine temperature distributions in the rotary-kiln wall 68 3-3 1-zone radiation analog of rotary-kiln wall 62 3-4 4-zone radiation analog of rotary-kiln wall 53 3-5 Circumferential kil n wall temperature profiles used in the 1- and 4-zone radiation analogs 55 3-6 Comparison of 1- and 4-zone radiant heat flows incident on the inner kil n wall 56 x Figure Page 3-7 Axial temperature profiles for a dir e c t - f i r e d , lime k i l n with no preheater. Sankey diagrams show the relative contribution of freeboard and regenerative heating of the solids within the calcination, preheat and drying zones of the rotary kil n 73 3-8 (a) Circumferential inner and (b) radial wall temperature profiles for the calcination or high temperature zone of a rotary k i l n 76 3-9 (a) Circumferential inner and (b) radial wall temperature profiles for the drying or low temperature zone of a rotary k i l n 77 3-10 The influence of the convective heat transfer coefficient at the covered wall on the inner wall circumferential temperature profile within the low temperature region of the rotary k i l n 80 3-11 The influence of k i l n speed on the inner wall circum-ferential temperature profile within the low temperature region of the rotary kil n 83 3-12 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 . .. 85 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 . . . . 87 3-14 The influence of solids emissivity on the inner wall circumferential temperature profile within the low temperature region of the rotary k i l n 89 3-15 The influence of wall emissivity on the inner wall c i r -cumferential temperature profile within the high tempera-ture region of the k i l n 90 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 92 3-17 The influence of the specific heat of the wall on the radial temperature profile within the low termperature region of the k i l n . 93 xi Figure Page 3-18 Simplified resistive network used to predict heat flows within the rotary kil n 95 3-19 Exposed wall temperatures predicted by resistive analog versus the integrated exposed wall temperature within the calcination, preheat and drying zones 96 3-20 Heat received by the solids predicted using the resis-tive analog versus the heat received by the solids based on the integrated average wall temperatures within the calcination, preheat and drying zones.. 3-21 Predicted heat loss from the kiln shell using the resis-tive analog versus the integrated heat loss from the shell within the calcination, preheat and drying zones.., 98 99 4-1 Zonal configurations and temperature distributions 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 one-dimensional flame model 110 4-5 The influence of relative flame size on the one-dimensional flame model. H I 4-6 The influence of k i l n wall r e f l e c t i v i t y on one-dimensional flame model 112 4-7 Variation of flame emissivity with distance from burner... 114 4-8 Simplified resistive network used to predict heat flows within the flame zone of a rotary kil n 117 4-9 Schematic diagram of the cross-section of a rotary k i l n showing heat flow paths and resistive elements for (a) non-flame zone and (b) flame zone 119 4-10 Schematic diagrams of rotary kil n showing (a) zonal configuration for one-dimensional flame model and (b) the major heat flows within each s l i c e 120 4-11 Computer flow-diagram used to determine temperatures and heat flows within the flame zone of a rotary k i l n 124 x i i Figure Page 4-12 The influence of fuel type on flame temperatures within the rotary kil n 130 4-13 The influence of fuel type on the solids heat flux within the flame zone of a rotary kiln .... 131 4-14 Bar diagrams showing the influence of fuel type on the energy distribution within the flame zone of a rotary kiln 133 4-15 Schematic diagram of natural gas burner used in flame model calculations 135 4-16 The influence of f i r i n g rate on flame temperatures within the rotary kil n 137 4-17 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 138 4-18 The influence of secondary a i r temperature on flame temperatures within the rotary k i l n 139 4-19 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 4-20 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 4-21 The influence of primary air on flame temperatures within the rotary k i l n 143 4-22 The influence of primary air on the solids heat flux within the flame zone of the rotary kil n 144 4-23 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 . . . 145 4-24 The influence of oxygen enrichment on flame temperatures within the rotary k i l n 147 4-25 The influence of oxygen enrichment on the solids heat flux within the flame zone of a rotary k i l n 148 4-26 Bar diagrams showing the influence of oxygen enrichment on the energy distribution within the flame zone of a rotary k i l n 149 x i i i Figure Page A2-1 Orthogonal views of cylinder and hemisphere; (a) elevation; (b) end; and; (c) plane 163 A2-2 Schematic diagram of cylinder of diameter D and hemisphere of radius r showing the limits of integration used in Eq. (A4.5) 164 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 I 6 6 A4-1 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 A4-2 Schematic diagram of concentric cylinders showing zonal configuration used in 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 ) Axs 2 cross-sectional area of solids (m ) A x t 2 cross-sectional area of empty kiln (m ) AF stoichiometric air-to-fuel ratio (kg/kg) b weighting factor for gray-plus-clear gas model, absorptivity C P specific heat (J/kg) d distance between two elements (m) do equivalent burner diameter (m) D kiln diameter (m) E 2 emissive power (W/m ) E 2 emissive power, locally defined (W/m ) f absorption coefficient, absorptivity (nf 1 atm 1) F view factor FL flame length (m) 9 function, locally defined G 2 momentum flowrate (kg m/s ) G9 2 gas mass flux (kg/m hr) Gr Grashof number hcv 2 convective heat-transfer coefficient (W/m K) hR 2 radiative heat-transfer coefficient (W/m K) HF gross heating value of fuel (J/kg) Hloss hydrogen loss XV h exchange integral for gas-to-kiln wall radiation exchange integral for gas-to-kiln wall radiation, T 1 l2 exchange integral for gas-to-kiln wall slice radiation l3 exchange integral for reflected gas radiation-to-kiln wall *4 exchange integral for gas-to-sol ids radiation J radiosity (W/m ) J o radiosity, locally defined (W/m ) k absorption coefficient, emissivity (nf 1 atm"1) - chapter 2 k thermal conductivity (W/m K) - chapters 3 and 4 K extinction coefficient (nf 1) Lm average mean beamlength (m) m mass flowrate (kg/s) N unit vector normal to element, locally defined P partial pressure (atm) PA per cent stoichiometric a i r as primary PO per cent primary air as oxygen Pr Prandtl number q heat flow (W/m) Q' heat flow (W) r radial position (m) R I inner radius (m) R f radius separating steady and unsteady state regions (m) Ro outer radius (m) Re transverse flow Reynold's number Re rotational Reynold's number R vector between two elements, locally defined xv i s width of s o l i d s surface (m) t time (s) T temperature (K) ^cov in tegra ted average covered wal l temperature T e x in tegrated average exposed wal l temperature V volume (m ) w d i s tance from center o f s o l i d s surface to element on s o l i d s wa l l (m) W ha l fwidth of s o l i d s sur face (m) z d i s tance along k i l n ax i s (m) a a b s o r p t i v i t y a g a b s o r p t i v i t y o f rea l gas f o r i t s own r a d i a t i o n B dimensionless d i s tance along k i l n a x i s , z / r e e m i s s i v i t y 0 ang le , def ined l o c a l l y (rad) dimensionless d i s tance along k i l n a x i s , z/D P r e f l e c t i v i t y or dimensionless d i s t ance , r/D, def ined l o c a l l y p e equ iva lent f ue l gas dens i ty (kg/m ) • P dens i ty (kg/m 3) k i l n s lope K 2 thermal d i f f u s i v i t y (m /s ) a -8 2 4 Stefan-Boltzman constant (5.67 x 10 W/m K ) T t r a n s m i s s i v i t y t r a n s m i s s i v i t y of rea l gas f o r i t s own r a d i a t i o n * angle (rad) h a l f angle subtended by so l i d s (rad) n s o l i d angle ( sr ) x v i i 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 air sh outer shell T total w wall A wave length (um) x v i i i 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 dis-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 li k e to thank Stelco Inc. and NSERC for their assistance in the form of financial support of this project. Special thanks must also be given to Mrs. M. Curtis for her assistance in helping me to obtain U.S. student loans during my stay in Canada. Finally, I would li 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 their help i t would have been impossible for me to complete my education. In response to the love they have shown, I would li k e to dedicate my thesis to both of them. 1 Chapter 1 INTRODUCTION Rotary kilns are primarily heat exchangers in which solids 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 in Fig. 1-1(a), a rotary kiln is 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 is 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 is 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 is referred to as being direct-fired in that a l l the fuel is burnt within the confines of the vessel. This i s in contrast to an indirect-fired kiln in which the fuel i s burnt externally to the vessel. In this study, only direct-fired 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 kil n showing; (a) axial cross section; (b) major heat flow paths in non-flame zone I I ; and; (c) major heat flow paths in flame zone I. provided incentives to more clearly define the fundamental principles 2-11 that govern kiln operation. This has led some investigators to mathematically model rotary kilns. In these studies attempts have been made to predict or examine operating conditions which w i l l enable the kiln industry to better understand their operating practices while simultaneously improving both heat u t i l i z a t i o n and product quality. While a l l of these investigators have clearly 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 kil n performance for more than a very limited range of operating conditions. Inasmuch as rotary kilns are primarily heat exchangers, the ultimate success of any kil n model in predicting either temperature profiles or product quality is clearly dependent on i t s a b i l i t y to characterize accurately the various heat flows within the k i l n . For this reason, the emphasis of this thesis i s placed on how to best characterize the mechanisms of heat transfer within a direct-fired rotary k i l n . The heat transfer within the rotary k i l n is 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 is believed to be the dominant heat-transfer mechanism.1 Examination of Fig. 1-1 (a) reveals that two distinct regions of heat transfer within the kiln may be easily identified: the flame zone I and the non-flame zone II. In the latter region, shown schematically in Fig. 1—1(b), heat is 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 directly 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 la t t e r heat-transfer path is part of the regenerative cycle.of the k i l n wall which, as i t rotates through the freeboard, also receives thermal energy via radiation and convection from the hot combustion gas. Of course a portion of this heat is lost to the surroundings as depicted in 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 is 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 vi s i b l e flame, shown schematically in Fig. 1-1(c). The primary objective of this study is 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 is 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 in 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, kiln wall and sol ids. 2) The determination of the radiative exchange between free-board gas and kiln wall, gas and solids,and kil n wall and solids. 3) The prediction of total radiative exchange amongst the freeboard gas, kiln 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 is demonstrated. 2.2 Representation of the emissive characteristics of the gas, solids  and kiln wall 2.2.1 Emissive characteristics of the freeboard gas The freeboard gases common to rotary kiln operation are composed mainly of CC^ - h^ O mixtures in nitrogen generated by the com-bustion of hydrocarbon fuels with a i r . Because these gases emit and absorb radiation in distinct bands, the use of gray-gas approximation, i.e. the emissivity and absorptivity are equal and constant at a given gas temperature, is not valid. 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, is 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 sufficient number of gray and clear gas components to approximate the banded characteristics of a real gas. Thus according to their approach the gas emissivity may be represented by e = Ea (1 - e " k i p r ) (2.1) y i subject to the restrictions that the a^ are a l l positive and za i = 1 (2.2) since emissivity approaches 1 with increasing pr. 8 In a similar manner, the absorptivity of a real gas at temperature Tg for blackbody radiation from a surface at T"s can be represented by a = Zb ( l - e _ f i P r ) (2.3) 9 i 1 where again the b. are a l l positive and Eb. = 1 (2.4) In the present study, equimolal C0 2 - H20 gas mixtures have been used to represent the freeboard gas in 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 in 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. It may be noted that a simpler method outlined by Hottel and Sarofim could be employed to f i t the emis-si v i t y data i f computer time is unavailable. Fig. 2-2 shows the absorptivity of an equimolal C0 2 - HgO gas mixture at 111 OK for black-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 for the equimolal COg * H^ O gas mixtures used in this study is presented in Appendix A l . Owing to the diversity of ki l n operations, the gas temperatures arbi-t r a r i l y chosen here are typical of those found in 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 C02 to H20 is one to one. Although the use of an equimolal gas at the selected temperatures w i l l affect the numeri-cal values reported below, the results of this work are general in P { C O 2 + H ^ ) D ( f t - a t m ) 0 2 4 6 . 8 1 0 0 0.5 I 1.5 2 2 .5 3 P ( C O 2 + H 2 O ) D ( m - a t m ) Figure 2-1 Emissivity, £ g, and emissive character, d e g(r)/dr, for an equimolal C09 - H„0 gas mixture at 1390, 1110 and 830 K. p ( c o 2 + H 2 o ) D ( f t - a t m ) _0-LO a 0.6 Tg = 1110 K 0.75 1.00 p ( c o 2+H 2 o )D ( m - a t m ) Figure 2-2 Absorptivity of an equimolal C02 - HgO gas mixture at 1110 K 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 kiln wall and solids In the current work the kil 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 is 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 simplification, and is not necessary to the final solution. 2.3 Radiative exchange between freeboard gas and ki l n wall The characterization of radiative exchange between the freeboard gas and kiln wall has been studied by considering three cases. 1) A black cylinder f i l l e d with an isothermal gas for which the total radiative flux incident on the k i l n wall has been estimated using a differential 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 in the freeboard gas on the total heat flux. 3) A gray cylinder f i l l e d with an isothermal gas to 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 later, this has a minor effect on the final solution. 12 2.3.1 Total radiative heat flux from an isothermal gas For the evaluation of the radiative exchange between an isothermal freeboard gas and k i ln wall consider an empty k i l n where the ki ln wall is assumed black to eliminate surface ref lect ions . Then the radiative flux from the freeboard gas to a d i f ferent ia l element, dA, 14* located on the k i ln wall is given by cosedA -K r K,E„AdV j— e A dx (2.5) irr o A V where dV = 2Trr2sin<J>d4dr = r 2 dttdr (2.6) Integrating Eq. (2.5) over a l l wavelengths and rearranging yields -v H A r r d £ n ( r ) c o s e = y y ^aui n i ^ d n d r ( 2 . 7 ) fi dA J J dr TT 9 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 ferent ia l hemispherical shells of radius r and thickness dr centered above the d i f ferent ia l element dA. The contribution to the total emissivity that each of these shells would make i f they were f u l l hemispheres is (de g (r ) /dr)dr. However, as only a fraction of these shells l i e ent ire ly within the k i ln the emissive contributions from each must be multiplied by a correction factor, g (r ) , equal to the area projected onto the base of the hemisphere by the section of the hemisphere within the ki ln wal l , divided by the projection of the entire hemisphere. For volume-surface radiative exchange Hottel *see l i s t of symbols for def init ion of terms used and Sarofim have shown that g(r) be rewritten as 13 = /cosedft/V, therefore Eq. (2.7) may de (r) dr %+M_f g ( r ) 9 E ^ J Av 9 dA J dr r | . I [ geometric emissive emissive correction character power factor of the gas For the present configuration g(r) is 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 this case r D 2 /*' 2 2 2 2 2 % g ( r ) = ^ 2 J ' [ ( J ) •-(§)• - (r-^-L") ] d ( f ) (2.9) 0 for £ <_ 1 V 9 " - 1 for £ > 1 where for hemispheres with radii larger than the k i l n diameter the lower l i m i t is 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 differentiating Eq. (2.1) as follows de (r) -k.pr - r 9 = za.k.pe 1 (2.10) dr . l v 14 Substituting Eqs. (2.9) and (2.10) into Eq. (2.8) and lettering p = r/D and 5 = z/D CO P VvdA . 4 /* Ea ik.pDe"S p D p/ [ P 2 - S 2 - ( P 2 - C 2 ) 2 J % d? dp (2:11) E dA TTP 2 •/ 1 ^ g 0 0 for p < 1 VP 2 - ] for p > 1 Final ly, lett ing 6 = 5/P the total radiative flux from an isothermal freeboard gas to a d i f ferent ia l element located on a black k i l n wall may be evaluated as follows }g dA _ Ea.I,. (2.12) • . i • 11 : ( g where E dA 1 -1 1 I = i = 2 M i x 2 i | -kiPDy* e ' k i p D p y [ l - B 2 - p 2 ( l - B 2 ) 2 f 2 dg d P (2.13) for p < 1 Vl - (1/P 2) for P > 1 Due to symmetry along the kiln axis Eq. (2.13) represents half the total 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 kiln diameters since contributions to the total flux beyond this point were found to be insignificant. Results of these integrations are presented in Fig. 2.3 where 1^ is plotted as a 1—I—I I I f T T T Figure 2-3 Exchange integral for gas-to-kiln wall radiation versus k.pD. 16 function of k^pD. Using this graph and Eq. (2.12), the total radiative flux from the freeboard gas to wall may be evaluated for any gas composi-tion whose emissivity can be described by Eq. (2.1). To verify this ap-proach, using the method outlined by Hottel and Sarofim, 1^ the mean beam-length for 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 in determining mean beamlengths, compares favorably with the value of 0.94D reported by Hottel and Sarofim thereby confirm-ing the val 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 consider-ing a black cylinder divided into slices as shown in Fig. 2-4. For this configuration the contribution to the radiant flux from each sl 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, de g (r)/dr, for a real gas has been shown in Fig. 2-1 where the derivative is 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 is negligible in 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 slice may be evaluated using Eq. (2.13) where the limits of integration, shown in Fig. 2-4, are now Figure 2-4 \ \ Z / D Top: Schematic cross-section of an empty rotary k i l n showing slices used in evaluating the non-isothermal heat transfer from the freeboard gas to a dif -ferential element, dA, on the ki l n wall. Bottom: Assumed temperature pro-f i l e in freeboard gas. 18 chosen in such a way as to eliminate radiant contributions from freeboard gas lying outside the slice of interest. Rewriting Eq. (2.13) the radiant contribution from the j sl i c e is given by w Ea.Ii. i i 2i a i ! 2 i ( M ) (2.14) where 1 for p < 1 ! 2 i j for p > 1 [l-g 2-p 2(l-3) 2] 2d3dp 0 for p i l Vl - ( V P 2 ) for p > 1 (2.15) and E, ..th j is the emissive power of the gas for the j 1 " ' s l i c e . For a non-a isothermal gas with a temperature gradient shown in Fig. 2-4 the emissive power of each individual slic e is evaluated at the midpoint temperature. As an example, the radiant exchange between the freeboard gas and a differential element on a black kiln wall was evaluated for a small pilot k i l n , (D = 0.4m), using Eq. (2.14) for both an isothermal (Tg=1110K) and non-isothermal (250K/m) equimolal gas ( P ^ Q = Q = 0-12 atm). In the non-isothermal calculation, the temperature of the gas directly 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 in Fig. 2 - 5 which shows the normalized cumulative heat flux from the freeboard gas to a differential 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 - H90 freeboard gas mixture. 20 located on the k i l n wall plotted as a function of position along the kiln axis. Clearly the influence of temperature gradients i s small. Comparison, for the small pilot k i l n , between the total heat flux received by a differential 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 kiln axis are present. The negligible influence of temperature gradients on the total heat flux received by a differential spot located on the k i l n wall may be ex-plained by considering the distance along the kil n axis over which radiant exchange between the freeboard gas and wall occurs. Fig. 2-6 shows the heat flux distribution densitites for both the isothermal and non-isother-mal cumulative heat flux curves plotted in Fig. 2-5. As can be seen in 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 diam-eters long. Because there is no long range radiant exchange between the freeboard gas and kil n wall, the gas seen by a spot on the wall appears to be roughly isothermal even though an axial temperature gradient exists. 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 typically less than 0.2 while the transmissivity for gray or blackbody radiation may be as high as 0.6, Fig. 2-2. -1,5 -0.9 ~0.3 0.3 0.9 1.5 A x i a l p o s i t i o n , Z / D Figure 2-6 Heat-flux distribution densities for gas-to-wall radiative heat flux versus axial position, z/D, for both an isothermal and a non-isothermal ro equimolal C09 - H90 freeboard gas mixture.. ~* 22 2.3.3 Reflected gas radiation To examine the nature of reflected gas radiation from the kiln wall consider a gray cylinder f i l l e d with a real gas. For this configuration the total flux of gas radiation incident on a d i f f e r -ential element, dA^, located on the ki l n wall may be evaluated using Eq. (2.12). Since the kiln 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 s the total flux of reflected gas radiation leaving dA^, is given by ^reflected _ y a T c o iz\ — d A " p s - V i i ^ g ( 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 differential ele-ments shown in Fig. 2-7. For this configuration the amount of reflected energy leaving dA-j, which is transmitted to the surrounding wall i s given by the following equation ^reflected (dA1 -> w) /_ Y cr «-k^ pd cose, cose 0 ,„ p s T V l i E g e 1 1 2 dA 2 ( 2 J 7 ) A, u d 2 2 reflected transmissivity view factor radiant energy dA-j •*• wall leaving dA^  23 Figure 2-7 Schematic diagram of the cross-section of an empty rotary k i l n showing differential elements, dA-j and dA2, used to evaluate the nature of reflected gas radiation from the kiln wall. 24 Eq. (2.17) may be rewr i t ten i n terms of c y l i n d r i c a l coordinates <j> and z using the fo l lowing transformations dA 2 = | d<j,dz (2.18) d 2 = D 2 ( l y o s ^ + Z (2/|9) R | ( l - C O S * ) cose = _ x c _'- = — (2.20) 1 I R 1 2|| N , | d cos9 = y "*2 - f ( 1 - C O S ^ (2.21) 2 I R2111 \ \ d Subst i tut ing Eqs. (2.18-2.21) into Eq. (2.17) and l e t t i n g £ = z/D the radiant energy re f l ec ted by dA^, which s t r i kes the k i l n wall may be evaluated as fol lows R e f l e c t e d (dA •=>- w) —wrr-^ - " s r V i i ' s i ( 2 - 2 2 ) i g where oo Tf j . I f f n - co s^ ) 2 e - k l P D [ ( ! f 5 i ) + C 2 ] \ D , (2.23) 3 1 2*J J H -cos * + ? 2 "12 0 0 L 2 J Note that the l i m i t s of integrat ion used in Eq. (2.23) take into account symmetry both about and along the k i l n ax i s . Eq. (2.23) was integrated numerically as a funct ion of k^ .pD where the upper l i m i t of £ was set equal to 5 since contr ibut ions beyond t h i s point were found to be insignificant. Results of these integrations are presented in Fig. 2-8 where I_. is plotted as a function of k.pD. This graph may be used together with Fig. 2-3 and Eq. (2.22) to calcu-late the total reflected flux received by the surrounding k i l n wall from a differential 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 differential spot is directed toward and i s received by the surrounding wall. In other words, for a small pi l o t k i l n 80 per cent of reflected gas radiation from the wall is reabsorbed by the freeboard gas. Note that for larger kilns, where the absorption paths are much greater than the 0.4 m of the pilot k i l n nearly a l l the re-flected gas radiation to the overall heat transfer in rotary kilns i s expected to be negligible. The distribution of reflected freeboard gas radiation from a di f -ferential spot located on the kiln wall to the surrounding wall may be evaluated using Eq. (2.23). For the small pilot k i l n f i l l e d with an isothermal gas (T = 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 kiln wall has been unfolded into the plane of the paper. 1^, the reflected energy, is 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 pre-sented. The flux surface may be taken to represent the distribution of ro =5 I.Ofc" O c o T3 O o cn -o o 0} CTl <U O c a JC o X UJ ^ reflected gas- -wall J >s2 ajlli^i Eg dA kjpD I3i 0 1.0 0.5 0 . 6 2 3 1.0 0 . 4 0 4 ZD 0 . 1 8 6 3.0 0 . 0 9 5 4.0 0 . 0 5 4 6J0 0 . 0 2 3 8.0 0 . 0 1 2 1 0 . 0 0 . 0 0 8 8 k|PD Figure 2-8 Exchange integral for reflected gas radiation-to-kiln wall versus k..pD. TT/2 T3 O C o to o Q. O 5 3 — C < -TT/2r— -25 -2.0 -1.5 -1.0 -0.5 0 Q5 Axial position,Z/D 1.0 1.5 2.0 25 Figure 2-9 Flux surface of reflected gas radiation incident on kiln wall for a small pilot 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 kiln wall where the volume bounded by the flux surface represents the total flux of re-flected energy received by the wall. For the flux surface shown here the iso-flux lines, as a function of both angular and axial wall posi-tion, are drawn on the l e f t side of Fig. 2-9 and may be used to map the regions on the k i l n wall where the flux of reflected energy is greatest. Thus for the small pilot k i l n , the reflected radiant gas energy incident on the wall is limited to an area within one k i l n diameter of dA^, and as expected, the areas receiving the greatest flux are directly adjacent to dA^. The localized nature of reflected gas radiation again results from the low transmissivity of a real gas for i t s own radiation. 2.4 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 iso-thermal 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 differential element, dA^, located on the solid surface, may be calculated using Eq. (2.24) (2.24) a r where by definition (2.25) w h 3 1 "Figure 2 - 1 0 Schematic diagram of the cross-section of a rotary k i l n showing differential elements, dA-j and dA2» used to evaluate either the radiative exchange between the free-board gas and the solids or the radiative exchange be-tween the solids and kiln 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 same b a s i c e q u a -t i o n c a n b e a g a i n u s e d t o c a l c u l a t e t h e r a d i a n t e n e r g y r e c e i v e d b y t h e s o l i d s . As r e p r e s e n t 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 g a s may be c o n s i d e r e 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 o n a d i f f e r e n t i a l e l e m e n t l o c a t e d o n t h e s o l i d s s u r f a c e . A t t h i s p o i n t , i t m u s t be e m p h a s i z e d t h a t b o t h E q s . (2.7) and (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 as g i v e n by E q . (2.5). H o w e v e r , 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 t o e v a l u a t e t h e r a d i a n t f l u x a s a f u n c t i o n o f p o s i t i o n o n t h e s o l i d s s u r -f a c e . 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 ; g ( r ) / d r ) d r = e g ( d ) , t h e r a d i a n t e n e r g y f r o m t h e f r e e b o a r d g a s i n c i d e n t o n t h e 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 a s f o l l o w s 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 -c a l c o o r d i n a t e s <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 t r a n s f o r m a t i o n s . (2.26) dA, d<}>,dz ( z , = a x i a l p o s i t i o n s o l i d s ) (2.27) cos2<f> ( z 9 = a x i a l p o s i t i o n w a l l ) (2.28) [§cos<j> 2 - ' ( § - F ) ] 2 + [ | s i n * 2 - ( § - F H a n ^ + f Z g z j 2 (2.29) COS0 N = (| - F - \ cos<i> 2) (2.30) d 31 c o s e 2 = cos^L"!" cos<(.2 - - F)].+ s i n $ 2 [ | - s i n ^ - (| - F j t a n ^ ) (2.31) d S u b s t i t u t i n g Eqs. (2.27-2.31) i n t o Eq. (2.26) and l e t t i n g 5 = z^/D, Eq. (2.32) i s ob ta ined which may be employed t o eva l ua te the 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 s l i c e across the s o l i d s s u r f a c e ^ "* 5 = z a . I , . (2.32) S d z l E g i 1 4 1 where cf> « (2TT-<J>L) L . DAE(Bsiny+ Ccos*2) _ ^ p D ^ t f ^ ( 2 . 3 3 ) 41 J J J W l T ( B 2 + C2 + ^2)2 2 1 o o * L A = \ ' U " \ C 0 S h (2.34) B = z\ sin<j)2 - (]r - ]j) tan<j>1 (2.35) C = \ c o s ^ - (1 - £ ) (2.36) E = (1 - § ) / c o s 2 ^ (2.37) Note t h a t the 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 e lement on the s o l i d s s u r f a ce w i l l change on l y as a f u n c t i o n of angu la r p o s i t i o n , <J>, s i n c e the gas temperature has been assumed to be i so thermal a long the k i l n a x i s . Therefore i n d e r i v i n g Eq. ( 2 . 32 ) , the value o f z ^ , the p o s i t i o n of dA^ measured a long the s o l i d s s u r f a c e , has been s e t equal t o 0. A l s o , the l i m i t s o f i n t e g r a t i o n i n Eq. (2.32) take i n t o account symmetry both about and a long the k i l n a x i s . 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 both the f i l l 32 ratio, F/D, and k.pD where the upper lim i t of z, was again set equal to 5 since contributions beyond this point were found to be insignificant. Results of these integrations are presented in Fig. 2-11 where for a given f i l l r a t i o , 1^. is plotted as a function of k^D. Using this graph to-gether with Eq. (2.32) the total radial flux received by the ki l n solids may be evaluated for any gas composition described by Eq. (2.1). To verify the results plotted in 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 free-board gas to a differential element, dA^, on the solids surface may be evaluated as a function of angular position ^ by rewriting Eq. (2.31) as fol1ows % + d A l d A l E g • (27r-<f>L) y C f a-A(Bsin<j)? + Ccos<j>J _k n n / R 2 + r 2 + r 2 \ J s . "J J T r(B 2+C 2H 2) 2 0 <frL Eq. (2.38) was numerically integrated both with respect to f i l l r a t i o , 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 for dA-j is plotted against position across the solid surface for three f i l l ratios. Thus at 1.0 - 0.8 c o o -5 o tt •a 2 0.6 o (A O Cn O c o» e: o <J X UJ 0 0 . 1 0 0.2 0 0 . 3 0 K i P D / I 4 i 0 0 . 0 0 . 0 0 . 0 0 . 0 0 2 5 Q 2 5 0 0 2 0 0 Q I 8 4 0 1 6 5 0 . 7 5 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 . 4 9 1 2 J 0 Q 8 I 4 0 . 7 8 7 0 . 7 6 0 0 . 7 1 3 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 . 9 2 9 0 . 9 1 8 0 . 8 8 3 5.0 0 . 9 6 2 0 - 9 5 2 0 9 4 7 0 . 9 1 6 I Q O 0 . 9 9 2 0 - 9 9 5 0 9 9 0 Q 9 6 7 0 0 . 0 5 0 . 1 5 0 . 2 5 AxS / A X T 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 ra t io , F/D. o o n JO CO •o X 3 O X 0.08 1 1 1 F/D -0.05 1 1 1 -0.8 -0.6 -0.4 -0.2 0 Q2 0.4 Radial position, w/W 0.6 0.8 Figure 2-12 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 solids 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 is seen to decrease with increasing f i l l ratio. In either case the decrease in radiant energy results from there being less radiating gas directly 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 kiln wall, gray solids and a real gas. To simplify the mathematics the wall and solids are assumed to be isothermal. Referring to Fig. 2-10, the radiant exchange between the solids surface and k i l n wall i s given by q . [ f e D T (d) C 0 S 6 1 C O S 9 2 dA2A (2.39) Mg + s / / s s 9 c 1 r J J , I I I I TTOA I A, A2 energy transmissivity view factor emitted solids -+ wall by solids where from Eq. (2.3). x g(d) = 1 - 0 g ( d ) = 1 - Eb.(l - e - f i p d ) (2.40) As before Eq. (2.39) may be rewritten in terms of cylindrical coordinates. Substituting Eqs. (2.27-2.31) and Eq. (2.40) into Eq. (2.39) and letting £ = z/D, Eq. (2.41) i s obtained which gives the radiant flux received by the kiln wall from a slice on the solids surface. 36 » (2TT-<p. ) q r r r • DAE(Bsin<j>, • + Ccos<j>0) Sdz l £ E J J J Wir L 1 1 s s 0 0 * i _ • ' <J>^  =° (2ir - <j>^) -Z /* f f b.DAECBsin*, + Ccos<j>J _ f n n , R 2 + ri + i J J J ~ — r M l - e f i p D ( B + C + ^ J d ^ d ^ , (2.41) Again the limits of integration in Eq. (2.41) reflect symmetry both about and along the axis. By inspection the f i r s t term of Eq. (2.41) is re-cognized to be, in the absence of a freeboard gas, the view factor between a slice on the solid surface and the kiln 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 for 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 - ^ - = e(l - Eb.I,.) (2.42) SdzE s . 1 4 1 s 1 Using Eq. (2.42) together with Fig. 2-11 the total radiant flux received by the wall from the solids surface may be evaluated for any gas composi-tion whose absorptivity is described by Eq. (2.3). Eq. (2.42) has been employed to calculate the solids-to-wall radiant flux for the small p i l o t kiln containing an isothermal gas at 1110K and solids with an emissivity of 0.8 and a temperature of 833K. Fig. 2-13 shows the modified view Figure 2-13 Modified solids surface-to-wall view factor versus axial position, z/D, for a pilot 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 ratio, F/D. CO ^1 38 view factor of the so l id s surface f o r the wall of t h i s k i l n , as determined by Eq. (2.42), p lotted as a function of ax ia l po s i t i on . Thus the rad iant f l ux received by the wal l i s seen to increase with increas ing f i l l r a t i o re su l t i ng from a decrease in the opt i ca l thickness of the gas. A l s o , 89 per cent of the rad ia t i ve exchange between the s o l i d s l i c e and k i l n wa l l i s l im i t ed 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 that predicted f o r the rad iat i ve 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 resu l t s from the increased t ransmis s i v i ty of a real gas for gray-body rad ia t ion as compared to that for i t s own r ad i a t i on . The l o c a l i z a t i o n of r ad i a t i ve t r an s fe r again would have the e f fec t of minimizing the inf luence of temperature gradients i n both the so l id s and k i l n w a l l . 2.6 Mathematical model of the to ta l rad ia t i ve exchange i n rotary k i l n s 2.6.1 Real gas model development In t h i s sect ion , a mathematical model i s presented which, based on previous d i scuss ion, may be used to describe the t o t a l r ad i a t i v e interchange amongst the freeboard gas, k i l n wal l and s o l i d s . The f o l l o w -ing assumptions have been made in the model: 1) The freeboard gas i s r ad i a t i v e l y a rea l gas. 2) The freeboard gas, k i l n wal l and so l i d s are taken to be isothermal since the inf luence of temperature gradients along the k i l n axis 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 so l id s are r ad i a t i v e l y gray surfaces. Based on these assumptions a modified r e f l e c t i o n method 1 7 was used to describe the net radiant loss for the freeboard gas, k i l n wa 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, kiln wall and solids, the net radiant loss may then be calculated for each by taking the difference between the total amount of radiant energy emitted and the total amount absorbed. For the kiln solids, the path of emitted energy is shown in Fig. 2-14. As seen here radiant energy leaves the solids surface, travels through the gas where a portion is absorbed, and strikes the kiln wall. At the wall a portion of the energy is absorbed with the remainder being reflected back through the gas to either the solids or the remaining wall. This process is repeated for each reflected ray until a l l the energy leaving the solids has been absorbed. Two reflections have been considered, shown by solid lines in Fig. 2-14, where the energy remaining after the second reflection is distributed between the solids and wall as i f they were black, as described below. The amount of radiant energy attenuated by the gas for 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 is decreased since for each reflection there i s less radiant energy lying within the wavelength regions where absorption • occurs. For this 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 S c h e m a t i c d i a g r a m 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 s h o w i n g t h e p a t h o f r a d i a n t e n e r g y e m i t t e d f r o m t h e f r e e -b o a r d g a s . 42 second reflection since a l l the radiant energy within the banded regions has previously been absorbed, i.e. the gas becomes transparent after only two reflections. 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 solid as i f they were black as indicated on Fig. 2-14 by dashed lines. The error introduced by this approximation is small, especially i f the solid and wall emissivities are high. In a similar manner the path of radiant energy from the kiln wall may also be traced. For the freeboard gas the path of emitted energy i s shown in Fig. 2-15. As seen here radiant energy emitted by the gas is received by both the k i l n wall and solids. At the wall or solids a portion of this incident radia-tion i s absorbed with the remainder being reflected. Since the absorp-; t i v i t y of a real gas for i t s own radiation i s high only one reflection need be considered. The energy remaining after the second reflection i s com-pletely absorbed by the freeboard gas. Using this development the net radiant loss for the freeboard gas, kiln wall and solids was determined as follows - FwsT<VF™Fswpwes¥s energy emitted by solids and reabsorbed by solids -FwsT<VFWsFsw<'s^sAsEs "(2.4,3) 43 V(Lm)FwsewAwEw V(VFwsFwwpwEwAwEw V(VFwsFLpwewAwEw V (V Fws Fsw ps pwS Aw Ew FwsT(3Lm)FwwpwewAwEw 2 FwsT(3Lm)FwsFwwFswpspwewAwEw energy emitted by wall and absorbed by solids V g < L r A E g energy emitted by gas and absorbed by solids E A E www energy emitted by wall V ( Lm ) Fsw es As Es V ( 2 Lm ) Fww Fsw pw es As Es V' 3 Lm ) F 2ww Fsw pw es As Es V ( 3 Lm ) Fws Fsw ps pw £s As Es FwwT(3Lm)F2wwFswpwesAsEs Fsw T ( 3 Lm ) Fws Fww ps pw es As Es Fww T ( 3 Lm ) Fws Fsw ps pw es As Es energy emitted by solids and absorbed by wall 44 V(Lm)FwwewAwEw V(2Lm ) FLewAwEw V ( 2 Lm ) Fws F Sw ps ew Aw Ew V(VFwwpwewAwEw 2V(VFwsFwwFswpspw£wAwEw 3 3 Fwwx(3Lm)FwwpwEwAwEw 2FwwT(3Lm)FwsFwwFswpspwewAwEw FswT(3Lm)FLFwspspwewAwEw FswT(3Lm)FwsFwwFswpspwewAwEw Vg ( Lm ) Aw Eg % FwwVg^ £g ( Lm ) Aw Eg Vsw ps Tg< Lm> £g ( Lm ) As Eg e g ( L m ) A g E g H " ^ Lm^ Fsw es As Es [ x ( g - T ( 2 L M)]F s wP ws SA SE S • [ t ( 2 L m ) - T O L J I F ^ F ^ ^ E m'J ww sw w s s s " - ^ 3 Lm ) ] Fws Fsw ps pw £s As Es energy emitted by wall and reabsorbed by wall energy emitted by gas and absorbed by wall (2 energy emitted by gas energy emitted by solids and absorbed by gas 45 - [T(2L m) - T ( 3 L m ) ] F w s F s w P 5 P w E w A w E w - " T < 3 L m » F w s F w w F s , A V s A w E w " <"gVg<Ln,>AwEg - tVww+ ' syvi | i.i £ , 1 L . v , " ["5 + V g ' L n , ' ] F w s ' V g < L , A E g energy emitted by wall and absorbed by gas (2.45) energy emitted by gas and reabsorbed by gas where, referring to Fig. 10 sw ws WW = 1 sin<f). TT - (f>, = 1 WS (2.46) (2.47) (2.48) by The transmissivity, t g , of a real gas for i t s own radiation is given a . k . e " k i p r (2.49) __ k. J J J The average mean beamlength for a rotary-kiln was evaluated, as out-lined by Hottel and Sarofim, 1 6 using the charts developed in the previous sections. Based on these calculations the average mean beamlength for a rotary-kiln may be determined as follows ^ = 0.95(1 - (2.50) where F/D represents the f i l l ratio as defined by Fig. 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 solids. 4 g 2.6.2 Comparison between real and gray gas radiative models in  rotary kilns 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 in Fig. 2-16 and compared to that obtained using the real gas model developed here, Eqs. (2.44-2.5D). The freeboard gas used in these calculations was again taken to be an equimolal ( P C Q = P^ Q = 0.12 atm) isothermal gas mixture at 111 OK where for simplicity the solids were assumed to be black and the kil n wall to be gray at temperatures of 833 and 944K, respectively. Results showing the radiative flux received by the solids from the gas for a kil n 6m in diameter with a f i l l ratio F/D, of 0.30 are presented in Fig. 2-17. As can be seen the radiant energy received by the solids 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 kiln wall because a larger fraction of the gas radiation incident on the wall is reflected back to the solids. 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 re-sults 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 is 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 solids. However, for a gray gas, the trans-missivity is equal to 1 minus the emissivity, shown in Fig. 2-1 to be as AsFsv/Tg A w € w Figure 2-16 Radiative resistance network for a rotary kil n where the freeboard gas, kiln wall and solids surface are assumed to be gray. 48 220 E \ JSC to T3 200 O O </» o C7> 180 cr> c o J= o X 0) «_ c D 160 o or T~T K i l n d i a m e t e r = 6 m F/D =0.30 A x s / A x t = 0.25 / / / / / / R e a l g a s — G r a y g a s / / / / / 0 0.2 0.4 0.6 R e f l e c t i v i t y o f k i l n w a l l , y 0 w Figure 2-17 Radiant exchange, for a rotary-kiln, 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 in predicting the radiative exchange in rotary-kilns may lead to significant 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 is suggested the real gas model be used to evaluate the total radiative exchange in rotary kilns where i f the emissivities of the ki 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 th i s chapter i s to develop a fundamental understanding of the overa l l heat-transfer mechanism fo r that region of the k i l n in which there are no flames present w i th in the f r e e -board area. Toward th i s goa l , a deta i led mathematical model has been developed which takes into account a l l of the heat-transfer steps pre-v ious ly described and shown schematically in F ig. 1-1(b). Inasmuch as rad iat ion was the central top ic of the previous chapter and that the convective heat t rans fer at the upper surface of the bed i s described 1 1 8 elsewhere, ' in analyzing the overa l l heat flow in t h i s reg ion, emphasis i s placed here on character iz ing the regenerative heat t r an s fe r . Therefore, the model developed in th i s chapter, has l a rge l y been used to explore the regenerative act ion of the k i l n wall and i t s importance r e l a t i v e to the other heat-transfer steps, the e f fec t of d i f f e r en t k i l n var iables on regenerative as well as overa l l heat flow to the bed, and the p o s s i b i l i t y of employing a s imp l i f i ed model to predict the ins ide wal l temperature, heat loss through the ref ractory wall and ove ra l l heat t r an s -fe r to the s o l i d s . The approach taken in 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 div ided into four major sect ions: 1) The development of a mathematical model to pred ict the temperature f i e l d in a k i l n w a l l . 51 2) The determination of heat-transfer coefficients for 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 kiln variables. 4) Development of a simplified model to predict average inside wall temperature, heat transfer to the solids and heat loss through the wall. 3.2 Previous work The earliest attempts to predict the regenerative action of the kil n 19 wall were relatively 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 wall. 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 investiga-22 tions has been presented by Vaillant who went on to consider the more refined case of a rotating wall of f i n i t e thermal conductivity, which in-cluded shell losses and wall radiation to the solids surface. A set of dimensionless equations for a slab being alternately heated and cooled, with constant heat-transfer coefficients at the slab surfaces, was derived, and solved using an analog simulator. Results of his study indicate that the amount of regenerative heating of the solids increases with both 52 rotational speed and l in ing thermal conductivity, whereas the f i l l ra t io and l in ing thickness have no ef fect . 3 More recently, Cross and Young have developed a one-dimensional heat-flow model to predict the temperature variation in the wall of an induration k i l n . Their findings are in agreement with Vai l lant in that higher rotational speeds result in more e f f i c ient heat transfer to the underside of the burden. The heat transfer in a rotary heat exchanger 23 has been studied by Kern for the case of a non-radiating freeboard gas. Results of th is study also indicate improved regenerative e f f i c iency 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 direct-reduction ki ln reveal the variat ion of the inner-wall temperature to be typ ica l ly 50-100 K. In a l l of these theoretical studies, 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 radiative heat-transfer coeff ic ients which must be speci f ied. Un-fortunately, although c r i t i c a l to model success, previous investigators offer l i t t l e ju s t i f i ca t i on for the heat-transfer coeff ic ients used in their work. In these studies, convective heat-transfer coef f ic ients have either been calculated from untested equations or chosen on a ru le -o f -thumb basis, while the equations needed to determine the radiative heat-transfer coeff ic ients are either unspecified, incomplete or in a form which is not easi ly understood. Therefore, i t becomes d i f f i c u l t to apply, or assess the accuracy of these ear l ier models. For this reason, a de-ta i led evaluation of a l l heat-transfer coeff icients needed for the ca lcu la -tions in the present work has been included in this chapter. 53 3.3 Model development Heat conduction in a r o t a t i n g k i l n wal l i s governed by the f o l l o w i n g equation 8T k 3T ' 3T <L ( k __w^ + J«L _ w 1 3_ f k _Wv ar K w a r . . ' r ' ar r 2 3$ V l w a<j> ' - »; c P i r ( 3- l ) w The problem can be s i m p l i f i e d cons iderab ly by not ing that the k i l n wa l l may be d i v ided in to two reg ions: (I) a th in a c t i ve l aye r a t the i nner surface which undergoes a regu lar c y c l i c temperature change as the wa l l rotates through the freeboard and then passes beneath the burden; and (II) a s teady- s ta te l a y e r extending from the a c t i ve region to the outer surface which does not experience any temperature v a r i a t i o n as a f u n c t i o n of k i l n r o t a t i o n , i . e . 3T w /3 t = 0. These two regions are shown schemat ic -a l l y in F i g . 3-1 where the c y l i n d r i c a l i n t e r f a c e ( t h i r d dimension i n f i n i t e ) between them i s located at R .^ The problem can be f u r t h e r s i m p l i f i e d by making the fo l l ow ing assumptions: 1) The thermophysical p roper t ie s of the k i l n wal l are cons tant . 2) Conduction of heat in the l ong i tud ina l d i r e c t i o n i s n e g l i g i b l e , i . e . aT w /3z = 0. 3) Conduction o f heat in the c i r cumferen t i a l d i r e c t i o n i s n e g l i g i b l e , i . e . aTw/3(}> = 0. 22 V a i l l a n t has shown the e r ro r introduced by neg lect ing conduct ion i n both the l ong i tud ina l and c i r cumfe ren t i a l d i r e c t i o n s to be smal l ( < 2 p e t . ) . Apply ing the appropr iate s i m p l i f i c a t i o n s f i r s t to the a c t i v e reg ion 54 of the wall, the governing partial differential equation, Eq. (3.1), is reduced as follows: (I) For Rj <_ r <_ Rf: 32T , 3T , 3T, w . 1 w _ J _ w , f-i o\ ar 2 r 8 r Kw 3 t This equation can be solved to yield the radial temperature distribution in a sl 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 wall, for 0 <_ 4> <_ 2 (IT - <j>L) and t > 0: - * » F l „ • <hcv + H R NW r=Rj g-*w g,s->w 3 2) At the covered inner wall, for 2(TT - < < f> L < 2TT and t > 0: 3T - k - r - ^ [ = h (T - T ) (3.2b) w 3 r 'r=RI c V s w s 3) At the interface separating the two regions, r = R^., for 0 < . <j> < 2TT and t > 0: T w = T f (3.2c) An i n i t i a l condition is defined as follows for t = 0: T = T (r) (3.2d) w w ' where T (r) is given later in the text. 55 For the steady-state region the governing partial differential equation, Eq. (3.1), is reduced as follows: (II) For R. < r < R : v / f — — o d 2T , dT w + 1 ' w = o (3.3) dr 2 r 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 Q = ' ^ / V a ' ^ - ^ (3"3a) 2) At the interface separating the two regions, r = R^ , for 0 <_(()<_ 2 rr T w = T f (3.3b) For the active layer Eq. (3.2) was solved numerically using the 25 exp l i c i t finite-difference method and the nodal configuration shown schematically in Fig. 3-1. The finite-difference equations for each node type are summarized in Appendix A3. Having evaluated the temperature profile within the active region, the heat loss per meter of k i l n length, P i o s s > is taken to be the di 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 kiln wall. cn 5 7 q ' ° s s = < < h V _ + V » V - ^ v _ ° » L W (3.4) where T and T represent the integrated average temperatures of the GX CO V exposed and covered wall, respectively. For the steady-state region Eq. (3.3) can be solved analytically to yield the radial temperature profile in the wall as shown below T f - T a T(r) = T f l n ( ~ ) . ( 3 . 5 ) T R k R-,„/ 0\ , w f R f hout Ro where h . = h D ^ + h o u t Rsh->a c vsh+a The heat loss per meter of kiln length in the steady-state region, '"''loss' c a n b e 0 D , t a i n e d by differentiating Eq. ( 3 . 5 ) with respect to r and substituting into Fourier's law of conduction as follows 2TT k (T - T ) ^'loss ' R* ' k , ^ ln<ir>+ < h - V o out o y The flow diagram for the computer algorithm is shown in 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 their 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 distribution within the kiln wall is 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 Eq 3 . 7 Solve for cycl ic temperature distribution L Print results No Adjust temperature at R f using Eq 3 . 8 and reset steady state temperatures 7 C s,°p ) Figure 3-2 Computer flow-diagram used to determine temperature distributions in the rotary-kiln wall. 59 Eq. (3.5) as f o l l o w s T - T r T ( r ) = T R a k 1 n ( ^ - ) (3.7) •,„/ 0\ i w o R I " o u t K o where T = (T + T $ ) / 2 . Commencing w i t h t h i s temperature d i s t r i b u t i o n t he f i n i t e - d i f f e r e n c e techn ique then i s employed t o c a l c u l a t e t h e temperature f i e l d i n the a c t i v e l a y e r . Heat f lows i n t o and out o f the i n t e r f a c e sepa ra t i ng the two reg ions are then compared us ing Eqs. (3.4) and (3.6) . I f the heat f l ows are not e q u a l , the i n t e r f a c e tempera tu re , T^, i s c o r -r ec ted as shown below fln(^)+i^V,q,i°=s T f - ° ° + T a <3-8> w and the temperature d i s t r i b u t i o n i n the w a l l i s r e c a l c u l a t e d . Th i s p r o -cess cont inues u n t i l the f i n a l s o l u t i o n i s reached. The main advantages o f t h i s a l g o r i t hm are t h a t both t h e number o f nodes and t ime steps needed t o c a l c u l a t e the f i n a l temperature d i s t r i -bu t i on are g r e a t l y reduced compared to t h a t r e q u i r e d i f t he e n t i r e k i l n w a l l i s assumed to be t r a n s i e n t . Th is r e s u l t s i n a t e n - f o l d r e d u c t i o n i n computing t ime r e l a t i v e to the complete t r a n s i e n t s o l u t i o n ; and because a k i l n t y p i c a l l y may take up to 20 hours to reach s t e a d y - s t a t e o p e r a t i o n , the sav ings i n computer t ime are c o n s i d e r a b l e . The main d i sadvantage o f t h i s approach i s t h a t i t cannot be used to examine s t a r t - u p p rocedure s . A complete FORTRAN source l i s t i n g o f the 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 the programs output are presented i n Appendix A 7 . 60 3.4 H e a t - t r a n s f e r c o e f f i c i e n t s i n r o t a r y k i l n s In o rder t o p r e d i c t temperature 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 va lues 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 the 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 the c o e f f i c i e n t s have been d i v i d e d i n t o two major groups: 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 i n n e r w a l l , burden s u r f a ce and ou te r s h e l l , and the co r re spond ing c o n v e c t i v e c o e f f i c i e n t s . 3.4.1 Rad i a t i ve , h e a t - t r a n s f e r 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 he f r e e b o a r d area were s i m p l i f i e d by making the f o l l o w i n g assumptions: 1) Both the k i l n s o l i d s and w a l l a re taken t o be r a d i a t i v e l y g ray because the 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 w a l l r e f r a c t o r i e s are not w e l l known. Th i s assumption i s thought t o i n t r oduce on l y a smal l e r r o r p rov ided t h e e m i s s i v i t y o f both t he s o l i d s and w a l l are h i gh . 2) The f reeboard gas i s taken to be r a d i a t i v e l y gray even though i t c on ta i n s CO^ and H^O which emit 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 should 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 roduced by the gray-gas assumption may be g r e a t e r than 20 per c e n t , f o r the reasons de sc r i bed i n Chapter 2 , but i n t h i s work where the o b j e c t i v e i s to examine the nature o f the o v e r a l l h e a t - t r a n s f e r mechanism, gray-gas behav ior i s adequate t o p r o p e r l y c h a r a c t e r i z e the r a d i a t i v e i n t e r a c t i o n o f the gas, w a l l and s o l i d s . I f the o b j e c t i v e was e i t h e r the p r e d i c t i o n o f des i gn o r p rocess c o n t r o l then the r e a l - g a s t reatment i s p r e f e r a b l e . 61 3) The presence of flames within the freehoard has been ignored; and hence the model is valid only for regions remote from flames. 4) There are no radial temperature gradients in the solids bed or free-board gas so that either phase can be characterized by a unique temperature at a given point in the k i l n . 5) The influence of axial temperature gradients in the solids, wall and freeboard gas is 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 pre-sence of circumferential temperature gradient, the wall temperature i s taken to be an integrated average over the entire exposed wall. Alter-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 wall. In the multi-zone model the wall is divided up into zones or sections of equal area each at an isothermal temperature so as to approximate the circumferential tempera-ture gradient. In this study, the 1-zone model has been used to determine the radiant heat flows, and hence the radiative heat-transfer coefficients, within the freeboard area. To estimate the error associated with this 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 E X € F A € F A <J w g w g g w " g e„Fc A c 6 F A g sg s g gs € A W W A F T A F T H s sw L g ' ws c g £ A s Figure 3-3 1-zone radiation analog of rotary-kiln wall. Figure 3-4 4-zone radiation analog of rotary-kiln wall. 64 are given in Appendix A4. To make the 1- and 4-zone predictions di r e c t l y comparable, as shown in Fig. 3-5, the integrated wall temperature, T , employed in 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. Using these temperatures the temperature gradients used 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 for each calculation. Results of the calculations are presented in Fig. 3-6 where the percentage d i f -ference between the 4- and 1-zone heat flows, { Q 4 _ z o n e - °^-zone^4-zone}10°' are plotted as a function of the circumferential 4-zone wall temperature gradient, AT where w A Tw = Tw,max " Tw,min = T4 " T l (3-9> As can be seen the error introduced in using the 1-zone approximation in estimating the exposed wall heat flow is less than 30 per cent for 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 in predict-ing radiative heat transfer at the exposed wall should be accurate to within 5 per cent. For circumferential gradients of TOO K, Fig. 3-6 also shows the error introduced in using the 1-zone approximation in pre-dicting the solids heat flow to be less than 3 per cent. Thus the 1-zone model shown in Fig. 3-3, has been adopted to characterize the 4 - Zone model I-Zone model OJ k_ O OJ CL e OJ OJ c c 0 - Tn V , T . + T 2 + T 3 + T 4 ) / 4 05 10 0 Fraction of exposed wall 0 5 -T„ T EX = TW 10 Figure 3-5 Circumferential k i l n wall temperature p ro f i l e s used in the 1- and 4-zone rad iat ion analogs. Figure 3-6 Comparison of 1- and 4-zone radiant heat flows incident on the inner kiln wall. 67 r a d i a t i v e h e a t - t r a n s f e r c o e f f i c i e n t s as d e s c r i b e d b e l o w . Exposed i n n e r w a l l The r a d i a n t h e a t - t r a n s f e r c o e f f i c i e n t a t t h e e x p o s e d w a l l , h R , g ,s->w i s g i v e n by E ( J - E ) : _ e w v ex e x ' . . Rg,s-w pw(VW where t h e t e m p e r a t u r e o f t h e w a l l , T . i s t a k e n t o be t h e i n t e g r a t e d a v e r a g e o v e r t h e e n t i r e e x p o s e d w a l l and t h e r a d i o s i t y , J g x , may be d e t e r m i n e d u s i n g s t a n d a r d m a t r i x methods as d e s c r i b e d i n A p p e n d i x A4. Note t h e e m i s s i v e power o f t h e w a l l , E y , i s e v a l u a t e d f o r t h e i n t e g r a t e d a v e r a g e t e m p e r a t u r e o f t h e e x p o s e d w a l l . Exposed s o l i d s s u r f a c e A l t h o u g h n o t emp loyed d i r e c t l y i n t h e w a l l m o d e l , t h e r a d i a t i v e h e a t - t r a n s f e r c o e f f i c i e n t a t t h e s o l i d s s u r f a c e , h R , i s g i v e n by g,w->s e s ( J s " E s ) n R = n tT - T ) (3.11) R g , w - s V ' g V where t h e r a d i o s i t y , J s , may a l s o be d e t e r m i n e d u s i n g s t a n d a r d m a t r i x me thod s . O u t e r S h e l l 22 V a i l ! a n t has shown t h a t t h e r a d i a t i v e h e a t - t r a n s f e r c o e f f i c i e n t s a t t h e o u t e r s h e l l , h , may be e v a l u a t e d as f o l l o w s R s h + a <3.,2, 68 where C = {1 t ' J 3 — + ( ^ - ) 2 + (^-) 3> (3.13) 'sh 'sh 'sh 3.4.2 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 wall,* h , is given by w*s 2 11.6 k co.R, n H « • H I T (-30V1) <3-™> w-*s L s for ID Rj <}i/30 K $ < 10 . Eq. (3.14) was employed to evaluate the convec-tive heat-transfer coefficient for a lime k i l n where the solids thermal conductivity, k s > and thermal d i f f u s i v i t y , < s > were taken to be 0.692 -7 2 18 W/m K and 5.1 x 10" m /s, respectively. Results of these calculations for a 3.5 m I.D. kiln indicate that the heat-transfer coefficient at the 2 covered wall typically 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 coefficient at the exposed wall can be determined from the 18 following correlation. „ c v . 0 . 0 3 6 ^ R e0.8 p i,0.33 ,0,0.055 ( 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 apply. 5 5 in a rotary k i l n where Reynolds numbers are typically 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 kiln with Re = 2.5 (10 5), Eq. 2 (3.15) yields a heat-transfer coefficient of about 12 W/m K. In the 2 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 wall, convective heat transfer from the freeboard gas to the solids is 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 = 0.4 G' 0 , 6 2 (3.16) cv ^. g v ' f i t s measurements obtained with kilns of 0.19^ 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 in scale-up is uncertain. Nonetheless i f 2 a 3.5 m ID k i l n is again considered, with a gas mass flux of 8275 kg/m hr, 2 Eq. (3.16) gives a value of 107 W/m K for h . This is considerably larger than the value that was calculated for the exposed inner wall, T O: oc and conforms broadly with experimental measurements reported previously. ' The high value is believed to result from the large surface area pre-sented 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 relative to the exposed inner wall. In this work h has been allowed values in the range of o g+s 50 to 100 W/rri K. Outer shell The convective heat-transfer coefficient at the outer s h e l l , h , 27 sh-*a is given by0.36 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 is less than 0.2 the influence of kiln rotation can be ignored 31 and the convective coefficient may be determined from k P r 0 ' 3 hrv = ~Si C (Re> (3*18> cvsh->a u where the values of the coefficient C and the exponent N are presented in Table 3-1. For most kiln operations the influence of kiln speed on h may be ignored. c sh-^ a TABLE 3-1. Constants for 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 profile shown in Fig. 3-7, which is thought to be typical 28 of that for a direct-fired lime k i l n with no preheater. For these calculations, pairs of gas and solids temperatures were taken along the kiln 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 in the k i l n wall over the range of operating conditions summarized in Table 3-2. Then using these temperature distributions, both the regenerative and free-board heat transfer to the solids together with heat losses at the outer surface were calculated for each simulation. In this way, the [ I Radiation to burden surface Convection to burden underside |jg3 Convection to burden surface Figure 3-7 Ax ia 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 ime k i l n with no preheater. Sankey diagrams show the r e l a t i v e cont r ibu t ion of f reeboard and regenerat ive heating o f the so l i d s wi th in the c a l c i n a t i o n , preheat and dry ing zones of the ro ta ry k i l n . CO 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 a 298 K % (0.5 + 0.9) e s (0.5 + 0.9) h r v (10 + 30) W/m2 K h (50 100) W/m2K c V s h c v (5 + 15) W/m2K cvsh+a h... (50 100) W/m2 K (1 + 3) rpm F/D (0.1 + 0.3) k w (1 + 3) W/m K Rj 1.75 m R. 1.98 m 75 role each process variable plays in determining the amount of heat trans-ferred to the solids by the regenerative action of the wall may be esta-blished. Furthermore, by comparing the magnitudes of both the regenera-tive and freeboard heat-transfer rates, the relative importance of re-generative 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 is to examine general trends in kiln operation. For th i s reason, no attempt has been made to compare model predictions to specific operating data. A typical pair of circumferential and radial wall temperature pro-f i l e s , calculated using the model, are plotted in Figs. 3-8 and 3-9 for both the calcination and drying zones, respectively. As seen in Fig. 3-8, for the calcination zone the circumferential inner wall temperature variation is 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 thick-ness remains nearly constant at 9 mm. Examination of the 100 simula-tions performed in this study reveals that the circumferential tempera-ture variations at any point along the kiln axis are typically 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 different zones reveals that the amplitude of temperature cycling at the inner wall i s directly proportional to both the gas temperature, T , and the 76 2 0 0 0 1900 eoo 1700 - 1600 1500 W O O B O O 1200 Colcinotion zone — high temperolura W 2 R P M € , • € ^ • 0 ^ 7 5 O,«2-6*l0"'tnVi F/0«0-IS ^ • 5 0 V * ' m * K 0 2 5 0 5 0 0 7 5 Fraction of told cyete ( a ) IO 0 f 7 5 160 HJ5 » 0 r95 2 0 Radio l wal l position (m) ( b ) F igure 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 high temperature zone o f a r o t a r y k i l n . 77 1000 Drying zone — low temperature * 800 | 600 * 400 F 200 OJ-ZRPM €,=€.= 0-75 \-\WmK ru, =50 IOW/rn"K "cv, 0—* a w =2-8X10 mVs n cv 5 -w =20 F/D = 015 "cv s = 50 0-25 050 075 Fraction of total cycle ( O 1-0 ~ 800 a> 5 o 600 a> a. J 400 200 0 1-75 1-80 1-85 Radial wall 1-90 1-95 position (m) (b) 20 Figure 3-9 (a) C i rcumferent ia l inner and (b) r a d i a l wal l temperature p r o f i l e s f o r the drying or low temperature zone of a ro tary k i l n . 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 significantly influences the depth of the active wall region. The above is illustrated by comparing the temperature profiles 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 in the wall i s expected to be greatest in the hotter regions of the kiln or in areas where there exists a large difference between the gas and solids temperatures. In order to determine the role of regenerative heating in the overall heat flow to the solids, temperature profiles similar to those shown in Figs. 3-8 and 3-9 were used to calculate the total heat trans-ferred 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 Fig. 3-7 where an "average" Sankey Diagram has been constructed for each zone. Thus the relative amounts of energy transferred to the solids directly from the freeboard and indirectly by the regenerative action of the wall is a function of k i l n position. In the calcination zone, where high gas and solid temperatures exist, the dominant path of heat transfer to the solids is freeboard radiation. In this region roughly 84 per cent of the total energy received by the solids results from their radiative interaction with the freeboard gas and exposed wall. Moving along the kiln axis toward the feed end this fraction i s seen to decrease such that, in the drying zone the radiative contribution is only 34 per cent. It follows that changes in either the regenerative or convective heat transfer to the solids w i l l influcence kiln 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 in cv both the exposed and covered wall temperatures, and hence also a de-crease 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 is 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 in Table 3-3 because regenerative heating accounts for only 34 per cent of the t o t a l . At the exposed wall, 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 is only a 13.5 per cent increase in the total heat received by the solids as summarized in Table 3-3. On this 1100 T 'g Drying zone — low temperature 900 . A v = 50 W/m2K —s o T Tw,i h c v = 100 emperatui 700 OJ= 2 RPM €w=V0-75 kw=IW/mK aw=2-8xl0"7 mVs F/D = 015 h c v = 50W/m*K c vg—s h-500 -T 's 300 i 1 » 0 0 2 5 0-50 075 10 Fraction of total cycle Figure 3-10 The influence of the convective heat-transfer coefficient at the covered Wall „ on the inner wall circumferential temperature profile within the low tempera-ture region of the rotary k i l n . TABLE 3-3. Influence of process var iables on regenerative act ion of the wal l and to ta l heat flow to so l ids near feed end of a rotary k i l n Change (per cent) Variable Range (rpm) F/D h (W/nf K) L w+s h (W/nT K) cv g+W W k(W/m K) a(m2/s) x IO 7 1 -* 2 0.08 •* 0.18 50 -* 100 10 •> 30 0.75 -> 0.9 0.75 0.9 1 + 3 5.3 -> 8.04 Difference between Heat flow average exposed exposed wall and covered-wall to so l id s temperature Regenerative heat flow to so l id s •30 +36 +10 n i l n i l -65 +19 -18 +18 -13 +6.0 +7.5 n i l -4.1 +1.0 +38 +4.0 +53 +15 n i l n i l -6.7 -1.0 Total heat flow to so l ids + 10 + 10 + 14 + 10 + 5 n i l - 5 < +2 oo 82 basis, changes in h are not expected to strongly influence heat c v w * s transfer within rotary kilns. In other words, significant changes in regenerative heating w i l l not always lead to markedly improved kiln performance. As w i l l be seen in the remaining discussion there i s a recurrent "trade-off" between the freeboard and regenerative heating in that an increase in one i s normally accompanied by a decrease in the other. 3.5.2 Kiln speed The influence of ki l n speed, co, on the inner wall tempera-ture is shown in Fig. 3-11. It is 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 total heat flow received by the solids by less than 10 percent, Table 3-3; and thus, changes in ki l n speed are not expected to significantly influence the solids heat flow at any point along the kiln axis. The effect of ki l n speed can be seen in terms of i t s influence on the heat-transfer coefficient at the covered wall, Eq. (3.14), and on the heating cycle during rotation. With increases in k i l n speed, the heating cycle changes because the time available for the wall to heat and cool, as i t moves through the freeboard and beneath the solids respectively, is decreased. Therefore, at higher kiln speeds, the difference between the integrated exposed and covered wall temperatures is reduced. For the case shown in Fig. 3-11, increasing co from 1 to 1100 r 900 Drying zone — low temperature 0) k_ D O w. a> QL 6 <u I— 700 5 0 0 £ s =e w =075 kw=IW/mK a = 2 8xiO~V/s F/D =015 h r v =50 W/m*K h c v =20 r-0J=IRPM h_ =IOOW/m*K h =122 «•* h CV fc S ' ° c v s h — a 3 0 0 025 050 0-75 Fraction of total cycle 10 Figure 3-11 The influence of k i ln speed on the inner wall circumferential temperature profi le 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 same time, according to Eq. (3.14), h increases with co raised to c V s the 0.3 power which, as discussed in the previous section, decreases both the exposed and covered wall temperatures. The two effects -reduced cycle time and increased h - thus account for the observed cv w+s influcence of kiln speed seen in Fig. 3-11. The small influence of kiln 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 with rotation speed more than offsets cv w+s the decrease in the integrated covered-wall temperature. However the net radiant exchange between the exposed wall and solids surface is 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 significantly, i t must be mentioned that co does strongly affect the residence time of the solids and hence overall k i l n per-formance. 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 wall, h , is shown in Fig. 3-12. In-c g-*w creasing h increases both the exposed and covered wall temperatures; CVg->w and thus the temperature driving force beneath the solids and that 0 025 050 07f 10 Fraction of total cyclr Figure 3 - 1 2 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 ef fect , 2 however, is re lat ively small since increasing h :f rom 10 Jto 30 W/m K c g^w 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 ln performance a problem may arise i f the increased freeboard gas velocit ies 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 rat ios , 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 re lat ive ly minor effect. Like k i ln speed, the effect of solids f i l l rat io can be seen in terms of changes to the convective heat-transfer coeff ic ient 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 de-creased 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 coeff ic ient is also reduced during rotation. The covered-wall heat transfer coeff ic ient is also reduced by increasing F/D since according to Eq. (3.14), 1100 r 900 OJ k_ Z3 o w. 0> Q. E CU r -700 500 J00 Drying zone—low temperolure 0J=2RPM € s =€w=0 75 kw=IW/mK aw=28xiO"V/s = 50W/m*K h c v =50 cvw—s =65W/m*K F/D =008 T C Ysh—o h c v =20 c v g —-w cv, j _ 025 0-50 075 Fraction of total cycle 10 Figure 3-13 The influence of solids f i l l rat io on the inner wall circumferential temperature prof i le within the low temperature region of the rotary k i l n . 00 •^1 .4 88 h depends on the half angle subtended by the solids raised to the cV>s -0.7 power. Thus the heat flux to the exposed and covered surfaces of the bed is reduced. However this effect is overcome by the fact that the area available for heat transfer at both the covered wall and exposed solids surface is increased with larger f i l l ratios. The increase in heat transfer rate to the solids is most pronounced on the freeboard side of the bed where, for the case shown in 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 solids). Regenera-tive heating, on the other hand, remains effectively unchanged because the combination of a reduced temperature driving force and covered-wall heat-transfer coefficient is sufficient 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 solids, i t does significantly alter 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 is 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 900 Drying zone —low temperature 700 OJ Z3 D k. OJ CL E CO 500 OJ =2RPM kw = IW/mK aw=2-8x|0"Tm*/^ F/D =015 h c v =50 W/m* K h c v =20 h,.. =100 cv, w-c v sh—a 300 J L JL 0-25 050 . 075 F r a c t i o n o f t o t a l c y c l e Figure 3-14 The influence of solids emissivity on the inner wall circumferential temperature profi le within the low temperature region of the rotary k i l n . CO U3 2000 r 1800 £ 1600 Z3 Q L 6 OJ H 1400 1200 1000 Calcinalion zone—high temperature 0J=2RPM kw = 20W/mK aw=6-7x|0"7m^s ?€w =0-75 T w , x 50 W/m* K w—s h c v = 50 = 10 •cv. 'sh—o cy. = 10 g — w X 025 050 Fraction of 075 total cycle Figure 3-15 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 in the fraction of radiant energy absorbed at the solids sur-face. The same can be said for increasing the emissivity of the wall. 3.5.6 Thermophysical properties of wall The influence of the thermal conductivity, k w, and specific heat, C , of the wall on the wall temperatures are shown in Figs. 3-16 pw and 3-17, respectively. In Fig. 3-17 the radial temperature profiles within the kiln wall are plotted for two refractories having thermal conductivities of 1 and 3 W/m K respectively, and a constant thermal -7 2 dif f u s i v i t y of 6.674(10 ) m /s. As can be seen, an increase in 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 is that the total heat received by the solids is reduced by 5 per cent, Table 3-3. This is clearly an unwanted result but even more serious is the fact that increasing k w increases both the shell tempera-tures and heat losses. 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 pw 30 per cent with k held constant, i.e. a 30 per cent reduction in w 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 de-creased by 8 K or 23 per cent. However, the total heat flow received by the solids is lowered by less than 2 per cent, Table 3-3. Based on these calculations, the kiln lining should be constructed using a refractory with low thermal conductivity and high thermal inertia (low thermal d i f f u s i v i t y ) . The f i r s t property results in 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 0 175 180 185 190 195 2 0 Radial wall position (m) Figure 3-17 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 rise to a small amplitude of the temperature cycle and shallow depth of the active layer thickness. This could be important i f thermal spalling is significantly influenced by wall-temperature cycling. 3.6 Simplified resistive network As shown in Table 3-3, the overall heat flow received by the solids is not significantly altered by any of the process variables examined in this study. This insensitivity suggests that the mathemati-cal model used to predict kiln heat flows and temperature profiles may be greatly simplified. In pursuit of this goal, the network shown in Fig. 3-18 was devel-oped by extension of the standard radiative analog to include both con-vective and conductive heat flows. Using the method outlined in 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 earlier in this chapter. In Fig. 3-19 the exposed wall temperature predicted using the modified network is plotted against the integrated exposed wall temperature, T . Thus i t is seen that along the entire k i l n axis the 6X modified analog of Fig. 3-18 accurately predicts the exposed wall tempera-ture. For any of the cases considered in this investigation the error introduced by the use of this approximation is 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 Figure 3-19 _ 1800 o o c o CJ _> "«) 0> JO •a o O . o. 6 O CO w & X UJ 1400 1000 600 200 1 1 1 1 l 1 1 - m Calcination zone / — © Preheat / A Drying / — / — / / — / — / 4 — f — / / / — / _ / / / 1 l l f 1 1 1 1 200 600 1000 1400 Integrated exposed wall temperature (K ) 1800 Exposed wall temperatures predicted by resistive analog versus the integrated exposed wall temperature within the calcination, preheat and drying zones. v o c n 97 For the same operating condit ions F i g . 3-20 shows the t o t a l heat r e -ceived by the so l i d s predicted using the modified network.plotted against the integrated heat flows ca lcu lated using the 1-zone model. Again the so l i d s heat flow i s seen to be accurately predicted using the analog so lu t i on . F i n a l l y , as shown in F ig . 3-21, predict ions o f the heat losses through the k i l n wal l based on the analog c i r c u i t are w i th in 5 per cent of the more complex f i n i t e - d i f f e r e n c e ca l cu l a t i on s . There-fo re , in the absence of a freeboard flame, the modified r ad i a t i ve net -work of F ig. 3-18 may be used to accurately pred ict both wal l tempera-tures and heat flows at any point along the k i l n ax i s . !200r Figure 3-20 0 400 800 1200 Heat received by the solids based on integrated average wall temperatures (kW/m) Heat received by the solids predicted using the resist ive analog versus the heat received by the solids based on the integrated average wall temperatures within the calcination, preheat and drying zones. vo CO 4> JZ V) E cn o o c O Q) > -O 3 0) 120 80 - ""E. 40 XL 1 i Calcination zone Preheat Drying 1 i i i 0 40 80 Integrated hoat loss from kiln Figure 3-21 1210 (kW/m) Predicted heat loss from the kiln shell using the resistive 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 in this chapter is to develop a fundamental understanding of the overall heat-transfer mechanism for that region of the kiln in which flames are present in 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 schemati-cally in 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 in the freeboard are largely found within the confines of the visible flame as opposed to the entire free-board 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 in the overall mechanism. The regenerative heating is 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 is very similar to that developed in the preceeding chapter for zone II-type heat transfer. Thus, the study of flames is really only an extension of the concepts previously developed in this text. 101 The approach taken in developing a flame model is 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 kiln 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 in furnaces are too numerous to be considered within the context of this study. Therefore, in writing this review only that work relating directly to flames within rotary kilns has been included. The interested reader is referred to a variety of other 39-42 sources for a more complete review of flames and furnace systems. As is the case with other aspects of rotary k i l n s , few studies dealing solely with flame characteristics in 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 plexi-glass vessel was designed in which the mixing and combustion processes of the kiln 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 in 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 kiln as a function of burner and ki 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 ki l n diameters. In his work, average values for both flame temperature and emissivity were based on direct measure-ments of kiln-type flames. The success of this approach has never been f u l l y tested in that the author was unable to collect a comprehensive set of temperature or heat flux distributions from a production kiln against which to test his model. However, the model was par 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 air and water models to examine the flow patterns at the hot end of rotary kilns. In an attempt to verify their work the authors collected and analyzed a considerable amount of industrial ki l n data. The results of their study indicate that the flame characteristics are significantly influenced by the secondary air path which is largely determined by the shape or design of the kiln f i r i n g hood through which the secondary air is 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 con-ditions yield 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 for a large enclosed flame in a rotary 103 k i l n . T h e mo d e l d e v e l o p e d i n t h i s s t u d y i s b a s e d o n t h e z o n e m e t h o d o f 38 H o t t e l m o d i f i e d t o t a k e i n t o a c c o u n t t h e 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 . U s i n g t h i s m o d e l , b o t h t e m p e r a t u r e a n d 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 w e r e p r e d i c t e d a n d t h e n c o m p a r e d t o t h o s e m e a s u r e d 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 -a b l e s u c c e s s . T h e m a j o r d r a w b a c k t o t h i s a p p r o a c h r e s u l t s f r o m t h e a p p r o x i m a t i o n s u s e d i n d e f i n i n g t h e z o n e s t r u c t u r e . To a v o i d c o m p l e x f l u x g e o m e t r y c a l c u l a t i o n s J e n k i n s a p p r o x i m a t e d 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 b u r d e n . H e n c e , t h e m o d e l may n o t be 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 o f t h e s o l i d s . 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 m o d e l f o r an i n d u r a t i o n k i l n , Y o u n g 3 a n d C r o s s u s e d a o n e - d i m e n s i o n a l f l a m e m o d e l t o e s t i m a t e t h e s o l i d s h e a t f l o w . 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 h e a t r e l e a s e p a t t e r n o f t h e f l a m e b y 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 o f a s i m i l a r k i l n t y p e f l a m e . U n f o r t u n a t e l y , t h e a u t h o r s g a v e v e r y l i t t l e i n f o r m a t i o n c o n c e r n i n g t h e e x a c t n a t u r e o f t h e i r h e a t r e l e a s e d i s t r i b u t i o n s a n d 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 t h e o v e r a l l m o d e l . 4.3 M o d e l d e v e l o p m e n t 4.3.1 S e l e c t i o n o f m o d e l l i n g t e c h n i q u e T h e 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 m o d e l 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 t h e p a r a m e t e r s 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 is then written for each s l i c e . Using an assumed heat release pattern for the flame the heat balances are then solved consecutively for each s l i c e to yield both flame temperatures and solids heat flows. 3) Hottel zone method: Using this technique the gases and bound-ing surfaces of the flame zone are divided into an arbitrary number of gas and solids zones such that each is isothermal. Heat and energy balances are then written for each zone. The resulting set of simultaneous nonlinear equations are then solved to yield 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, is more robust in i t s a b i l i t y to radiatively account for both axial and radial temperature gradients but, the flux geometry calculations needed to include the solids burden are both complex and tedious thereby diminishing the overall appeal of this approach. Further, i t should be noted that in the absence of any temperature gradients, the two methods are equivalent in their a b i l i t y to estimate heat flows in 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 in zone II-type heat transfer, the one-dimensional approach is preferred because i t provides the desired level of accuracy in a more straight-forward manner. To check the accuracy of the one-dimensional model relative to the zone method, a number of calculations were performed in which the flame and surrounding wall were approximated by two concentric cylinders where for simplicity, the solids burden was ignored. The configurations used for 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 directly 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, relative to the zone-9 temperature, using a linear temperature gradient, AT/Ax, I - Zone 1 © (o) 16—Zone o. E T 9 « I773 K T, -1173 K Distance ® ! @ 1 © 1 © 1 ®i® 1 ; 1 © : © i • i 0) w O k. 4) a. E - — I ^ T P T T T V K A Z ^ T , - 1173 K (b) Distance 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 in 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 resistive networks shown in Figs. 4-2 and 4-3, respectively, where the view factors needed to solve these analogs are summarized in Appendix A4. Aside from con-sidering 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 in Figs. 4-4 to 4-6 in 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 Figs« 4-1 (a) and (b), are plotted as a function of axial temperature gradient, AT/AX, relative flame size, r , / r 9 , and wall r e f l e c t i v i t y , p , respec-tively. As shown in 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 for a relatively long thin flame, r-j / r 2 = °-25» the error introduced by the one-dimensional model does not exceed 25 per cent. Finally, Fig. 4-6 shows that for a highly reflective wall, P w = 0.75, the error in using the one-dimensional approach i s again less than 25 per cent. Based on these calculations, for an operating kiln where typically the maximum temperature gradients rarely exceed 300 K/m, the flame to kiln 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 one-dimensional approach in modelling the flame should be accurate to til- — A W €, A , A W I F | 9A, Figure 4-2 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 0 2 0 0 4 0 0 Axial temperature gradient, A T / A Z (K/m) Figure 4-4 The influence of axial temperature gradients on the one-dimensional flame model. Figure 4-5 The influence of relative flame size on the one-dimensional flame model. 112 Wall reflectivity, p w Figure 4-6 The in f luence of k i l n wal l r e f l e c t i v i t y on one-dimensional flame model. 113 within 20 per cent. Thus, in 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 kiln solids and wall are taken to be radiatively gray because the spectral emissivities of the solid materials and wall refractories are not well known. This assumption is thought to introduce only a small error. 2) The flame is taken to be radiatively gray and of constant emissivity at any position along the kiln 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 in Fig. 4-7 which represents a typical relationship between distance from the 39 burner and flame emissivity. Trinks has suggested this relation-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 ratio of the fuel. This is shown by a dashed line in Fig. 4-7. Aside from fuel composition, the emissivity of a flame i s also a func-tion of other variables, the most important of which are: fuel-to-a i r r a t i o , temperature of fuel and a i r , rate of mixing of the fuel 114 Figure 4-7 Var iat ion of flame emiss iv i ty with distance from burner. 115 and air and the thickness or shape of the flame. Thus, for a given set of operating conditions, the final shape of the emis-s 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 is chosen solely on the basis of composition using the method outlined by Trinks. The error associated with an approach of this type is thought to be small and may be easily changed as more information becomes available or to suit a speci-f i c set of operating conditions. The flame is 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 is composed primarily of and ^ from the supply of secondary a i r . The presence of gases which are given off by the reacting solids into the freeboard volume has been ignored. The overall flame length, F L > is calculated using the equation 45 of Bee*r as follows P P . h P p . * ! e • v / e F. = 6d (1 + AF )(-^-) (-^) L 0 P Cp psa (4.1) where for a double coaxial-type burner 0 = _ f PA . (4.2) pF ppa 116 and AF* = ( " F + %a> (4.3) ( A F ) m F - m p a (4.4). V In developing Eq. (4-1) the term (p /p i s taken from the e sa 43 work of Ricou and Spaulding and takes into account the i n -fluence of secondary a i r temperature on the overa l l flame length. The.flame lengths predicted by Eq. (4.1) are in broad agreement "57 with those measured by previous invest igators ' ' and a l l thought to be accurate to with in 20 per cent of the actual length. A i r entrainment by the fuel gas j e t i s instantaneously mixed and burns a s to ich iometr ic amount of f u e l . By extension of the work described in Chapter 3 the heat flows wi th in the flame zone may be approximated by the r e s i s t i v e net-work shown in F i g . 4-8. A deta i led comparison between th i s net-work and that previously described and seen i n F i g . 3-18, shows them to be very s im i l a r where the gas node of F i g . 3-18, Eg, has been replaced by a s im i l a r node E^, which i s used to represent the flame. The remaining d i f ference between these networks, re su l t s from the rad iat i ve in te ract ion of the exposed wal l and so l id s surface. For zone I I-type heat t r an s f e r , when no flame i s present in the freeboard, r ad i a t i ve exchange n' c vf_w A w h c v f , $ A s F igure 4-8 S i m p l i f i e d r e s i s t i v e network used to p r e d i c t heat f l ows w i t h i n the f lame zone o f a r o t a r y k i l n . 118 between the wall and solids occurs via the single path shown in Fig. 4-9(a); and hence a single resistor is used to j o i n the radiosities. However for zone I, where there i s now a flame in the freeboard, radiative exchange between the solids and wall occurs via the two paths shown schematically in Fig. 4-9(b). As a result, for the flame zone, the resistance between the wall and solids radiosities must be altered as shown in Fig. 4-9(b). These same changes may also be seen in Figs. 4-8 and 3-18. 8) The specific heat of the fuel j e t mixture is taken to be a constant at any axial position. The value used in the present study was that of the combustion products at the adiabatic flame temperature. The error associated with an approximation of this type is 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 in Fig. 4-10(a), the flame is divided into n slices of equal size. In this study n was chosen such that each sl i c e was approximately 0.5 m thick; dividing the flame into thinner slices offered no significant improvement in the f i n a l solution. Having divided the flame into s l i c e s , a heat balance may then be con-structed around each sli 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 F igure 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 heat f l ow paths and r e s i s t i v e elements f o r (a) non-f lame zone and (b) f lame zone. Secondary air d.ecA>"*(*»"8 •a (a) Flame ^ Qq«n r — * — — **-1 Solids Figure 4-10 (b) Schematic diagram of rotary k i ln showing (a) zonal configuration for one-dimensional flame model and (b) the major heat flows within each s l i ce. ro 121 b a l a n c e a s f o l l o w s : Q ' + Q ' = Q ' + Q ' L . + Q ' . (4.5) y z v g e n v s y s h V z + A z V H , o ; i i i i I : _L h e a t i n h e a t g e n e r a t e d h e a t o u t w i t h i n s l i c e where T z Q' = m T f C n dT (4.6) w z T , z J p c p 298 • \ H F ( 1 " H l o s s ) /* 7 A Qgen = ( m e n , z + A z " m e n , z > ( 4 ' 7 ) %n - A s h W ( T s h " T a> (4.9) and T z+Az ^ + A z = ™ T , z + A z / C P c p d T ( 4 ' 1 0 > 298 C o m b i n i n g Eqs . ( 4 . 5 - 4 . 10 ) and r e a r r a n g i n g y i e l d s t h e f o l l o w i n g e q u a t i o n w h i c h may be u sed t o c a l c u l a t e T 2 + A z g i v e n t h e v a l u e o f T 7 f o r any s l i c e . W K , z C P c p ( V 2 9 8 » + ( S e n , z + i Z - % , z > - £ f l F » " Ash hout <Tsh " V " X 1 (°s " Es' M l - H l o s s ) • 8 , . . _ C . 298} { m T > z + z C p } (4.11) "T,z+ Z cp I v ' r c p 122 To solve Eq. (4.11) both the rate of entrainment, men, and the total mass flowrate of the fuel gas j e t , m-p must be determined. Based on Eq. (4.1), the entrainment of air by the fuel gas jet i s taken to be a linear function of axial distance as follows m en (4.12) where (4.13) nu + m pa and z> 6 (4.14) 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 specific kiln/burner configuration as the need dictates. Finally, by extension of Eq. (4.12), the total mass flow of the fuel gas jet i s given by Based on the i n i t i a l fuel and air temperatures at the k i l n discharge together with Eqs. (4.12) and (4.15), Eq. (4.11) may be used to deter-mine the flame temperature consecutively for each s l i c e . The radiosity of the solids, J s > needed to solve Eq. (4.11) together with the solids and outer shell heat flows may be determined using the resistive network (4.15) 123 of Fig. 4-8 as outlined in 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. The flow diagram for the computer algorithm is shown in Fig. 4-11. 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 kil 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 air 4) the amount of primary air 5) oxygen enrichment. The solids temperature, adopted in this study, was again taken from the temperature profile shown in 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 in the preceding Read input data 1 2 4 Calculate flame length using 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 for heat Eqs ( A 6.5 -flows using A 6.7) Print results c Stop Figure 4-11 Computer f low-diagram used to determine temperatures and heat flows wi th in the flame zone of a r o t a ry k i l n . 125 TABLE 4-1 Combustion properties of natural gas Composition (% by volume): CH 4 97.38 C 2 H g 2.17 C 3 H 8 0.15 N 2 0.30 Gross heating value = 5.521 x 10 7 J/kg = 992.9 BTU/ft 3 Stoichiometric a i r to fuel rat io = 16.97 kg/kg Hydrogen loss = 9.82% Average flame emissivity = 0.25 Average specif ic heat •= 1550 J/kg K 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 7 J/kg = 18145 BTU/lb Stoichiometric a i r to fuel ratio = 13.26 kg/kg Hydrogen loss = 5.06% Average flame emissivity = 0.85 Average specific heat = 1400 J/kg K 127 TABLE 4-3 Combustion properties producer gas  (Lurgi-Air Blown) Composition (% by volume): CH4 5.0 CO 16.0 H 2 25.0 C0 2 14.0 N 2 40.0 Gross heating value = 6.974 x 10 6 J/kg = 177 BTU/ft3 Stoichiometric a i r to fuel ratio =1.81 kg/kg Hydrogen loss = 6.61% Average flame emissivity = 0.25 Average specific heat = 1490 J/kg K 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 = 1200 K s T = 298 K a h = 50 W/m2K c vf+s h = 20 W/m2 K c vf+w h = 1 0 W/m2 K c vsh+a Rj = 1.52 m T p = 298K = 0.8 w 0.8 T = 298 to 773 K sa PA = 20 to 40 per cent of stoichiometric FR = 9.2 to 16 MW = 36 to 55 (x 106) BTU/hr Oxygen enrichment*— 21 to 39 volume per cent of Or, in primary a i r *In this study, a i r is taken to be 21 per cent 0 2 and 79 per cent N 2 on a volume basis. 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 K, respectively. 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 equi-valent diameter, d Q, and the gross f i r i n g rate of the fuel were fixed 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 results 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 com-40 pare favorably with those previously reported in the literature. 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 lime-44 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 in the 130 ?l I I I I i I L 0 2 4 6 8 10 12 14 Axial distance from kiln discharge (m) Figure 4-12 The influence of fuel type on flame temperatures within the rotary k i l n . 131 300 CM E JSC X 3 o JC •o o cn 150 i r Natural gas / "#G Fuel oil Firing rate = 14.5 MW =(49.5 X 10 B.T.U./hr) T f = 298 K T P A =298 K T S A = 298 K % PA = 20 deq =0.160 m -150' 1 1 ^ 0 2 4 6 8 10 12 14 Axial distance from kiln discharge (m) Figure 4-13 The influence of fuel type on the so l ids heat f l ux wi th in the flame zone of a rotary k i l n . 132 effective fuel gas density, p g . In other words, the denser fuel o i l jet is less able to entrain the secondary air needed for combustion. It is this 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 significantly reduced stoichiometric air requirements. Because no kilns are currently using producer gas as a fuel i t is impossible to compare these results to any real operating data. As shown in Fig. 4-13, the solids heat flux is 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 kiln 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 is significantly higher using fuel o i l , care must be taken to avoid overburning of the k i l n product. Since vi r t u a l l y no heat is transferred to the solids within the flame zone using producer gas, i t is 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 this fuel on k i l n 70.82 5.89-# 6 Fuel oil 9 . 9 6 ^ 9 . 8 2 ^ Natural gas 9 2 . 2 6 0.75y 0.38-^6£( Lurgi producer g I 1 Sensible heat of combustion products Heat transferred to solids £ 3 ^ ^ ^ Heat loss from shel l H 2 I O S S 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 character-i 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 kiln axis. Primary a i r is introduced through the outer annulus. For this configuration using natural gas as the fuel, Eqs. (4.2) to (4.4) take the following form. 2{1 + PA(AF*)} do = P < 4 J 6 ) <(V + 6 PaK } 2 where (PA(AF*)} 2 G P + G n a = 1414 + (4.17) P a ( l _ I_P0 + ^ 0 _ ) o.065 1.224 1.308 1 + PA(AF*) P (4.18) 6 _ L _ + (1 - P0)PA(AF*) + P0(PA)(AF*) 0.67 1.224 1.308 and {AF - (1 - PO)PA(AF) - 4.31(P0)PA(AF)} AP* = (4.19) • c 1 ( H - PO)PA(AF) + (PO)PA(AF) } 0.67 1.224 1.308 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 for 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 significantly 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 air 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 air temperature significantly increases the flame length while having only a minor effect on both the maximum flame temperature and solids heat flux. The increase in flame length results from a decrease in the secondary a i r density since the fuel gas je t is 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 is transferred to the Axial distance from discharge end (m) Figure 4-16 The influence of f i r i n g rate on flame temperature within the rotary k i l n . Axial distance from discharge end (m) Figure 4-17 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 air temperature on flame temperatures within the rotary k i l n . Figure 4-19 The influence of secondary air temperature on the solids heat flux within the flame zone of a rotary k i l n . 141 heat sinks. However, because the secondary a i r adds additional energy in the form of sensible heat, this effect is 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 relative energy distribution within the flame zone. As seen here, by increasing the secondary air temperature from 298 to 773K the total heat received by the solids is increased by nearly a factor of two. Since the maximum heat flux at any position is not significantly in-creased the use of preheated secondary air appears to be a reasonably safe method for improving kiln 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 character-i 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 in 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 is shown in Fig. 4-23. Thus, by increasing the primary air from 20 to 40 per cent of stoichiometric the total amount of heat received by the solids is decreased from 9.8 to 5.0 per cent of the total energy input. This results from the cor-responding decrease in flame length because the distance available for Natural gas f ir ing rate = I4.5MW(49.5 X I 0 6 b t u / h r ) % PA = 2 0 T p = 2 S G K T s a = 2 9 8 K 142 76.31 T. f l=778K 8.69' Sensible heat of combustion products Heat transferred to solids Heat lost by outer shell H 2 loss Figure 4-20 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 air on flame temperatu within the rotary k i l n . Axial distance from discharge end (m) Figure 4-22 The influence of primary air on the solids heat flux within the flame zone of the rotary k i l n . Natural gas F i r i ng rate = 14.5 MW (49.5 X 10 b t u / h r ) T p = 2 9 8 K T P A * 2 9 8 K T S A = 2 9 8 K 76.31 79.69 82.7 ! 9.83 ' 9.82 7.3r 9.82 2.39 5 . 0 8 ^ 9.8 2 / % P A =20 %PA=30 %PA=40 C J w//m Sensible heat of combustion products Heat transferred to solids Heat lost from outer shell H 2 lost Figure 4-23 Bar diagrams showing the influence of primary air on the energy distribution within the flame zone of a rotary k i l n . heat transfer is reduced. Even though the maximum flame temperature and solids heat flux are increased with additional primary a i r i t is not enough to offset the decrease in heat transfer length. Hence the total amount of energy received by the solids is decreased. On this basis, the primary air 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 air 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 enrich-ment would correspond to 21 per cent oxygen in the primary a i r while 20 per cent enrichment corresponds to 39 per cent oxygen in the primary stream. The results of these calculations are presented in Fig. 4-24 and 4-25 where the flame temperatures and solids heat flux are plotted as a function of axial distance from the kiln discharge. It is seen that, by increasing the level of oxygen in the primary stream both the maximum flame temperature and solids heat flux are significantly increased while the flame length is 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 distance from discharge end (m) Figure 4-24 The influence of oxygen enrichment on flame temperatures within the rotary k i l n . Axial distance from discharge end (m) Figure 4-25 The influence of oxygen enrichment on the solids heat flux within the flame zone of a rotary k i l n . Natural gas Fi r ing rate = 14.5 MW (49.5 X l O btu/hr) % PA = 2 0 T F = 2 9 8 K T P A = 2 9 8 K T S A =298K 76.3 75.31 % PA as 0 2= 0 9.8 5 ^ 9.8 2 ^ % P A as 0 2 = 10 11.0' 74.2 5 ^ 3.( 2 ^ 9 5 5 ^ . 8 2 ^ k 1 %PA as 0 2 = 2 0 12.28 9 5 2 / Sensible heat of combustion products Heat transferred to solids Heat lost by outer shell H 2 loss Figure 4-26 Bar diagrams showing the influence of oxygen enrichment on the energy distribution within the flame zone of a rotary k i l n . 150 e f f e c t , t h i s behav ior r e s u l t s from an increase i n the ad i aba t i c flame temperature. By us ing oxygen in the primary a i r stream the increase in the ac tua l f lame temperature i s great enough to overcome the reduced h e a t - t r a n s f e r length of the f lame. This i s in contras t to the use o f increased primary a i r where because the ad iaba t i c temperature remained constant t h i s inc rease i n flame temperature was not l a rge enough to overcome the reduced h e a t - t r a n s f e r length o f the f lame. Therefore oxygen may be used to increase s o l i d s output; however because the maximum va lue of the s o l i d s heat f l u x i s i nc reased , care must be used to avoid overburning of the product. Chapter 5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 5.1 Conclusions The following conclusions concerning heat transfer in a direct-fired rotary k i l n may be drawn from this study. 1) Because the freeboard gases common to rotary-kiln operation contain CO^ and r^O which emit and absorb radiation in dist i n c t bands, the gray-gas approximation is not valid; 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 slic e +0.3 kiln 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 kiln wall or the solids surface is negligible. This is again due to 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 re-flected from either the k i l n wall or the solids surface is re-absorbed by the gas on the f i r s t reflection. Hence, the i n -fluence of reflected gas radiation on the total radiant heat transfer in rotary kilns is minimal and is a local phenomenon. 5) The radiant exchange between the solids surface and the k i l n wall occurs over an axial slice + 0.75 kil n diameters along the k i l n axis. Hence, the influence of temperature gradients along the k i l n wall or the solids surface has a negligible effect on the radiative exchange between the k i l n wall and solids surface. 6) 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 kiln diameter, gas composi-t i o n , temperature and solids f i l l ratio. 7) Using the results shown here, a modified reflection method has been used to develop a model which describes the total radia-tive exchange in rotary kilns 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 this 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 ki l n wall and solids exceed 0.2. 9) During a single k i l n rotation, the unsteady state or cyclic temperature region is limited to a thin layer at the inner wall that rarely exceeds a depth of 15 mm. 10) At any point along the ki l n axis, the cyclic temperature variation at the inner surface of the kiln wall l i e s in 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 w a l l . 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 w a l l . 13) The temperature distribution within the wall at any point, along the ki l n axis and the total heat flows received by the solids and lost to the surroundings, are largely i n -dependent of the process variables considered in this investigation. 154 14) Since the temperature distribution of the kiln 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 simplified by using the resistive network of Fig. 3-18 which incorporates radiative, convective and conductive heat flows. Using this 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 this approach is less than 20 per cent. 16) The resistive 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 for future work Having analyzed the various heat flows within the direct-fired 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 resis-tive networks developed in this text. 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Read, Ed.: North American Combustion Handbook, North American Mfg. Co., Cleveland, 1978. 41. M. W. Thring: The Science of Flames and Furnaces, Chapman and . Hall, 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: J. Fluid Mech., 1961, Vol. 11, pp. 21 - 32. 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 C02-H20 GAS MIXTURES 159 TABLE Al-T.. Summary of em i s s i v i t y data f o r equimolal C0 ? -H o 0_gas mixtu re s at 830, 1110 and 1390 K P ( C 0 2 + H2O) L T = 830 (K) g T = 1110 (K) g T = 1390 (K) g (atm - m) e ( C 0 2 + H 20) e ( C 0 2 + H 20) e ( C 0 2 + H 20) 0.0 0.06 0.09 0,12 0.18 0.24 0.30 0.37 0.49 0.61 0.91 1.22 1.83 3.05 0.0 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.28 0.32 0.36 0.42 0.49 0.0 0.13 0.16 0.18 0.20 0.23 0.25 0.27 0.30 0.33 0.38 0.43 0.50 0.57 0.0 0.16 0.18 0.21 0.25 0.28 0.30 0.32 0.36 0.38 0.44 0.43 0.54 0.64 C o e f f i c i e n t s : 4 gray + 1 c l e a r gas model - Eq. (2.1) a l a2 a3 a4 a5 k l k3 k4 k5 0.40 0.04 0.04 0.06 0.46 0.66 3.67 13.71 59.38 0.0 0.43 0.09 0.04 0.08 0.36 0.59 3.58 14.57 43.18 0.0 0.48 0.06 0.O5 0.10 0.31 0.69 10.17 6.73 26.90 0.0 160 TABLE A1-2:. Summary of absorptivity data for eguimolal C02-H2P_gas mixtures at 1110 K for blackbody radiation at 277, 555 and 833 K p ( C 0 2 + H20) L (at m-m) T s = 277 (K) T g = 1110 (K) a ( C 0 2 + H20) T = 555 (K) s T = 1110 (K) g a ( C 0 2 + H20) T s = 833 (K) T = 1110 (K) g a ( C 0 2 + H20) 0.0 0.03 0.09 0.18 0.30 0.46 0.61 0.91 1.52 0.0 0.16 0.30 0.39 0.45 0.57 0.63 0.70 0.85 0.0 0.11 0.21 . 0.26 0.32 0.37 0.41 0.48 0.57 0.0 0.10 0.18 0.23 0.28 0.32 0.36 0.42 0.52 Coeff ic ients : 2 gray + 1 clear gas model - Eq. (2.3) b l b 2 b 3 f l f 2 f 3 0.70 0.22 0.08 1.41 30.90 0.0 0.46 0.16 0.38 1.35 27.85 0.0 0.49 0.16 0.35 0.89 30.28 0.0 Appendix A2 DERIVATION OF EQUATION (2.9) 162 As previously stated in Chapter 2, Eq. (2.9) represents the pro-jected area of intersection between a hemisphere of radius r and cylinder of diameter D, divided by the total projected area of the hemi-sphere. For the configuration shown in Fig. A2-1, the equations of interest for a sphere and cylinder are given below. sphere: r 2 = x 2 + y 2 + z 2 (A2.1) cylinder: (|) 2 = x 2 + ( y - f ) 2 (A2.2) The projected area of intersection onto the x - z plane for this configura-tion 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 2 - z 2 - ( X i ^ l i ) - - ] 5 * . (A2.4) By use of symmetry about the origin, Eqs.(A2.3) and (A2.4) may be combined such that r 2 2 2~1 ^  AREA = 4J lr2 - z 2 - ( r ~ Z ) J *dz (A4.5) 0 for r < D Vr2 - D2 for r > D where the limits of integration are shown schematically in Fig. A2.2. The above integral, Eq. (A2.5), may now be rewritten in terms of the dimensionless lengths z/D and r/D as follows ( o ) z (c) F i g u r e A2-1 O r t h o g o n a l v i e w s o f c y l i n d e r a n d h e m i s p h e r e ; ( a ) e l e v a t i o n ; ( b ) e n d ; a n d ; ( c ) p l a n e . 164 For r<D lower limit = 0 upper limit = r T For r>D lower limit = +Jr2-D2 upper limit = r T D dA Figure A2-2 Schematic diagram of cy l inder of diameter D and hemisphere of radius r showing the l i m i t s of integrat ion used in Eq. (A4.5). r/D AREA = 4D' 7 <#> -as-) 2 2 r d(§) 165 (A2.6) f o r r/D < 1 V (£) - 1 f o r r/D > 1 Us ing Eq. (A2.6) the p r o j e c t e d areas f o r s eve ra l va lues of r/D are p l o t -t ed i n F i g . A2-3. F i n a l l y , Eq. (A2.6) must be d i v i d e d by the p r o j e c t e d a rea o f the e n t i r e hemisphere, irr , to a r r i v e at the f i n a l form of Eq. ( 2 . 9 ) . r/D 9 ( r ) -irr irr d ( § ) (A2.7) f o r r /D< 1 V (£) -1 f o r r/D > 1 y/D -1 (\ / \ / X 1 1 1 - 2 [I M I / A / \ >C /D-I.5V / V - - I z/D 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 Eq . ( 3 . 2 ) was s o l v e d n u m e r i c a l l y u s i n g t h e 25 e x p l i c i t f i n i t e - d i f f e r e n c e method and t h e noda l c o n f i g u r a t i o n shown s c h e m a t i c a l l y i n F i g . 3-1. The f i n i t e d i f f e r e n c e e q u a t i o n s shown b e l o w , f o r e a c h node t y p e , were o b t a i n e d by n o d a l b a l a n c e s : 1) Nodes a t t h e e x p o s e d i n n e r w a l l : A t _ / L . •'• x u • ^ 2 A T _ T + ,2 _ A T + h ^ ^ A r T + I- + m ) T R + h C V ) M k g M R T M W - 2 a.s-^w a^w w 3 l i >^  i , l " g , +  ~'g+w " "  • " - O R +\v ' r r - l - R p ' S , ( A 3 J ) g,s->w g->w w I i >' w h e r e M = A r 2 / k A t w 2) Nodes a t t h e c o v e r e d i n n e r w a l l : 2 A r h c v V , k M . s M R T M J ' w , 0 1,1 w I 1,2 2Ar h + { 1 k M " " " f " R7Hi 'w - , w l 1,1 C V w ^ s _ 2 _ _ A r _ } T ( A3 . 2 ) 3) I n t e r i o r w a l l n o d e s : 169. By examination of Eqs. (A3.1 - A3.3) the overa l l c r i t e r i o n fo r s t a b i l i t y i s given by « > 2 + ( h R + h c v + h c v (A3.4) g,s->w g-»w w-*s w I Appendix A4 DETERMINATION OF RADIATIVE HEAT FLOWS USING THE NETWORK METHOD 171 . Solutions for the resistive networks used in the present study are given below. For c l a r i t y , the presentation has been divided into two parts. The f i r s t section summarizes the view factors needed to evaluate the branch resistances shown in 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 briefly 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 is discussed. For a more rigorous treatment of radiative analogs 29 the interested reader is referred to the work of Oppenheim. View factors for the resistive networks 1-zone networks To evaluate the 1-zone branch resistances shown in 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 = F - F = 1 (A4.1) sg wg sw where Aw = A l + A2 + A3 + A4 = D ( i r " *L* ( A 4 , 2 ) and A s = Dsin<J>L (A4.3) 172 F i gu re A4-1 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 used to eva lua te view f a c t o r s f o r (a) 1-zone and (b) 4-zone ana logs . 173 4-zone network To evaluate the 4-zone branch resistances shown in Fig. 3-4, the fifteen 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 FVg " F2g - F3g ' % " 1 <M"4> By use of symmetry i t can also be shown that Fs4 F s l <M-5> Fs3 = Fs2 = °- 5- Fsl ( M - 6 ) F2„ = F,3 (A4.7) F23 " F34 • F12 ( A 4- 8' Finally, since the k i l n wall and solids surface form an enclosure, by use of reciprocity i t ' can be shown that F14 ' 1 " F l l " F12 " F13 " i f F s l < 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 s-j. A summary of these view factors is presented below. TABLE A4-1. 174 Summary o f v i e w f a c t o r s n e e d e d t o e v a l u a t e t h e 4 - z o n e k i l n w a l l m o d e l FS9 F s l F12 F23 F34 F'g Fs2 F13 F24 F2g F S 3 F14 • F3g F S4 F4g 175 F 1 1 '• TT - <f> F„ - 1.0-4 sin( 4 — ) (A4.10) ir - <j>L sin IT " Tl IT + 3 <f>, , + s i n ( !k) - sin ( 4 (A4.ll) 1 4_ 4 s l 2 sin <fL F r13- l TT - T . TT — <f>, TT - 3 Cf>^  | 2 2 sin ( — - ) - sin ( — — ) - sin ( 5 ) j F l 3 " (* - * L ) (A4.12) F12 : TT - (j>, 2 sin ( -) F 1 2 = 1-0 - F i n - 1 — (A4.13) TT - (f>L (A4.14) where A 1 = A 2 = A 3 = A 4 = D/4 (TT - <^ L) 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, in a similar manner to that previously described, with the aid of both symmetry and recipro-c 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 in Table A4-2 . A summary of these view factors is presented below. is determined 3 3 using Fig. A4-3 where D' = r y r ^ and L' = Ax/r-, F ] g i s determined 3 3 using Fig. A4-4 where D' = r^r^ and L' = A X / ^ F18 : F18 " F(l+2)(9+8) " F19 ( A 4 - 1 5 ) F(l+2)(9+8) i s d e t e r m i n e d u s i n 9 Fig. A4-4 where D' = ^ / r ^ and L' = 2Ax/r-j F H2-F12 = 1 - F l l - F 1 9 - F 1 8 - F l , 1 9 - F l , 2 1 ( A 4 J 6 ) Fl,19 = h { } - F l l " F19 } ( A 4 ' 1 7 ) Fl,21 = 2 F(1 +2),21 " Fl,19 ( A 4 J 8 ) F(l+2),21 = h n " F(l+2)(9+8) " F(l+2)(l+2) } ( A 4 , 1 9 ) F(l+2)(l+2) i s d e t e r i T n n e d using Fig. A4-3 where D' = r,,/^ and L' = 2 Ax/r., F17 : F 1 7 - ^ ^ 3 F ( T + 2 + 3 ) (y+s+g) ' 4 F18 " 3 F19 (A4.20) 180 TABLE A4- 2 . Summary of view factors needed to evaluate the 16-zone  resistive analog of Fig.4-3. F l l F25 F35 F45 F56 F12 F2,12 F3,12 F4,12 F57 F13 F2,13 F3,13 F4,13 F58 F14 F2,14 F3,14 F4,14 F59 F15 F5,10 F16 F 5 , l l F17 F5,12 F18 F5,13 F19 181 F(1+2+3)(7+8+9) i s determined using Fig. A4-4 where D' = r^/r^ and L' = 3Ax/r-, ! n : F13 = 1 ' F l l " F12 " F17 ' F18 ~ F19 " Fl,19 " Fl,22 (A4.21) Fl,22 = 3F(l+2+3),22 " 2F(1+2),21 (A4.22) F(l+2+3),22 = " F ( 1+2+3)(7+8+9) " F(l+2+3)(1+2+3)} (A4.23) F(l+2+3)(1+2+3) i s determined using Fig. A4-3 where D' = t^/r-j and L' = 3Ax/r} ! ] 5 : F15 = 4F(l+2+3+4)5 " 3F(l+2+3),22 (A4.24) F(l+2+3+4)5 = h { 1 " F(l+2+3+4)(6+7+8+9) " F(1+2+3+4)(1+2+3+4)} (A4.25) F ( 1 + 2 +3 +4)( 1 + 2+3+4) a n d F(l+2+3+4)(6+7+8+9) a r e d e t e r m i n e d u s i n 9 Figs. A4-3 and A4-4, respectively, where D' = r^/r^ a n d = 4Ax/r.j ! l 6 : F16 = ^^ 4 F(l+2+3+4)(6+7+8+9)- 4 F19- 6 F18- 4 Fi7 } ( A 4 ' 2 6 ) ! i 4 : F14 = 1 " F l l " F12 " F13 " F15 " F16 " F17 " F18 " F19 " Fl,19 (A4.27) F25 : F25 " Fl,22 (A4.28) F35 : F35 = Fl,21 F45 ; F45 Fl,19 F 182 (A4.29) (A4.30) F2,13 " 5F(2+1+16+15+14),13 " 4F(l+2+3+4)5 F(2+l+16+15+14),13 = h { 1 " F(2+1+16+15+14)(8+9+10+11+12) " F(2+1 +16+15+14)(2+1 +16+15+14)} (A4.32) F ( 2 + 1 +16+15+14)(2+1+16+15+14) 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 D1 = r 2 / r 1 and L' = 5Ax/r 1 F r3,13' F3,13 = 6F(3+2+1+16+15+14),13 " 5F(2+1+16+15+14),13 (A4.33) F(3+2+l+16+15+14),13 = h { 1 " F(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).} (A4 • 34) F(3+2+1+16+15+14)(3+2+1+16+15+14) a n d F(3+2+1+16+15+14)(7+8+9+10+11+12) are determined using Figs. A4-3 and A4-4, respectively, where D' = r 2 / r ^ and L' = 6Ax/r^ 183 • F4,13 : F4,13 = 7F(4+3+2+1+16+15+14),13 " 6F(3+2+1+16+15+14),13 (A4.35) F(4+3+2+l+16+15+14),13 = { 1 " F(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)} ( A 4* 3 6) F(4+3+2+l+l6+15+14)(4+3+2+1+16+15+14) a n d F(4+3+2+1+16+15+14)(6+7+8+9+10+11+12) are determined using Figs. A4-3 and A4-4, respectively, where D' = r^/r-j and L' = 7Ax/r^ F h2,12' F2,12 = h {5F(2+1+16+15+14) (8+9+10+11+12) " 5 F19 " 8 F18 -6F 1 7 - 4F ] 6> '(A4.37) F3,12 : F3,12 ~ h {6F(3+2+l+16+15+14)(7+8+9+10+ll+12) ' 6 F19 " 1 0 F18 - 8 F17 " 6 F16 " 4F2,12> ( M - 3 8 ) F4,12 : F4,12 = h {7F(4+3+2+l+l6+15+14)(6+7+8+9+10+11+12)" ? F19 " 1 2 F18 " 10F 1 ? - 8F 1 6 - 6 F 2 J 2 - 4 F 3 J 2 } (A4.39) F2,14 : F2,14 = 1 - F l l ' 2 F12 " 2 F13 " F25 " 2 F17 " F18 " F19 " F16 " F2,12 " F2,13 - F14 ( A 4 - 4 0 ) 184 3,14" F3,H = 1 " F l l * 2 F12 ~ F35 " 2 F18 " F19 " F17 " F16 " F2,12 " F3,12 " F3,13 " F2,14 " F14 " F13 (A4.41) F4,14 : F4,14 = 1 " F l l " F45 " F19 " F18 " F17 " F16 " F2,12 " F3,12 " F4,12 F4,13" F3,14" F2,14' F14" F13" F12 (A4.42) F56 : F56 A 5 \ (A4.43) 57' F = — f57 A c A. A— _1 F _ -1 F A ? h18 A ? r19 1 , . 1 F 2 1 A 6 F 1 9 (A4.44) 58' 8^ 58 Ac i 4 C <^F _ - L ; 1 — F *8 1 7 ~ V 1 8 " V 1 9 2 ( A6 1 9 A *8 A. 1 (A4.45) 59" F =-9-h59 A. A. A, A A, i 4 c 3 F r Lp 1 -Ag~ F16 ~ Ag~h17 Ag 18 A g r19 A, 1 1 - — F 2 ) 1 A 6 r19 (A4.46) 185 5,10' *10 '5,10 Ac 1 ^ F 1 A,«/19 A A. A, 10 10 1 8 " A i o 1 7 A i o 1 6 A4 p _ I | 1 . A 1 F q A^ F2,12 ZY A 5 F 1 9 (A4.47) 5,IT 5,11 At A, A, 1 - A^ r3,12 A^ '2,12 A l l 1 5 A l l 1 7 A16 %15 An 1 8 An 1 9 2 1 , . 1 - F 1 1 A, ' l 9 i (A4.48) 5,12' '5,12 A. A-4 p _ _3_ p AT^ h4,12 A ,« 3,12 A M2 *12 A2 F . ! h _ F ^2,12 A,o 16 12 A12 A 1 6 F A 1 5 F A 1 4 F I J T A F A ^ F 1 7 - A ^ F 1 8 - A ^ F 1 9 2 j 1 A g F19 (A4.49) F r5,13" F5,13 - 1 A, A 9 A A l c 2 F _o.p - F A^ F15"A7 F25 A 5 h35 A g r45 1^4 F, M5 A 16 A 5 ' 4,13 " A 5 '3,13 " A 5 2,13 ' F58" F59 " F5,10 " F 5 , l l " F5,13 57 - F 56 (A4.50) 186 where for Eqs. (A4.15 - A4.50) A1 = A 2 = A 3 = A 4 = A ] 4 = A 1 5 = A ] 6 = 2AXirr2 (A4.51) A6 = A7 = A8 = A9 = A10 = A l l = A12 = 2 A X l T rT (A4.52) A5 = A13 = A17 = A18 = A19 = A20 = A21 = A22 = * ( r 2 " A ] ( A 4 ' 5 3 ) flame zone analog To evaluate the branch resistances shown in Fig. 4-8, the four view factors F , F ,., F,-, and F x must be determined. By inspection of sw sf fw fs Fig. A4-5 i t follows that F f s Ffw - 1 - F f s (A4.54) (A4.55) F s f = A s F = 1 - F sw A f F f s (A4.56) sf (A4.57) where A f = d f , (A4.58) A s D sin (A4.59) 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 is a set of simultaneous Figure A4-5 Schematic diagram of the cross-section of a rotary kil n used to evaluate view factors within the flame zone. 188 equations which may be solved to yield 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 in 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 in Fig. 3-18 has been used to estimate the inner wall temperature, the overall heat flow to the solids, and the heat loss through the kiln wall, in the absence of a freeboard flame. For a given set of operating conditions (T , T , T , Ii , 9 s a_ cvsh+a h , h » h „ ) the unknown variables of interest (T o v ) c V w c V s c V s e x ,qs' q1oss may be determined by application of Kirchoff's current law at the 4 nodes F^, E $ h , J w and J g shown in Fig. 3-18. The result is a set simultaneous nonlinear equation as follows: J w : A ~ (fvA + p A t + e F A ) J - ( ^ ) a T w * p w rsw ns lg g wg w' w P W W - (FSwVg>Js - W w ) E g ( A 5 - 1 ) - <FswAsVJw + <FswVg + sgFsgAs + >Js r (A5.2) Esh : 1 g sg s' g P, S <K 'M*h = <hcv V E g + ( h c v ^ A c o v > E s ( A 5 - 3 ) g->W W-*-S ( K ' ) ^ M b ; u t A s h *^)«4 V(h; u tA s h)E a (A5.4) 191 where 2 T r kw K' = ? P (A5.5) l n W < W s h H T w + 1 h ) 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 is then obtained using the following set of equations. q, = h' A , ( T4. - E ) (A5.6) H l o s s out sh v sh a y v ' T e x =T W (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 in Fig. 4-8 has been used to estimate the overall heat flow to the solids and total heat loss from the freeboard flame. For a given set of operating conditions (T f, Ts, T , ti , h , h , h ) the unknown variables of interest (q$> g f) may be determined by application of Kirchoff's current law at the 4 nodes E , E J and J shown in Fig. 4-8. The result is a set w sh' w s of simultaneous nonlinear equations as follows: V e A _ (eJ> A . + F J \ T , + Fri A T + - ^ - ) J * f ™ f c f <; f sw s a o.. w : f r f w M f r s f V f " s i / V g P W 'e A ~ e f F f w A f E f J s : < FsfVf + F swVg ) Jw £ s A s , E (A6.1) + ^FsfAsTf + FswVg + ef Ffs Af + " V } Js F A E + f A E (A6.2) f fs f f P S s — j + (h' A + ^ + h ' A • T K i ) O T 4 , ' ( ^ 7 } Jw ( h c v f ^ Aw P W cvw^ cov -C ' > ' 1h - ( h c v f : > w V E f ' + ^ W E . ' ( A 6 ' 3 ) - - ( K ' ) a T j M h i u t A s h + K')aTjh = ( h ^ A ^ j E , (A6.4) where 194 This set of equations was then solved numerically for the unknowns J w , 32 J s > T w and T s h using a generalized secant method. Based on these values the desired heat flows were then obtained using the following set of equations. q =- 5- §- ,(J - E ) + h' Ae (E---E) (A6.6) M S p x s s' cv^ s v f s' S T-^ S q- = h' A (E, - E ) + h' A (E, - E j f c v f ^ w f w' c v f ^ s f s' + e f F f w A f <Ef - Js> + E f F f s A f ( E f - J s ) ( A 6 ' 7 ) Appendix A7 FORTRAN SOURCE LISTING AND SAMPLE OUTPUT FOR KILN WALL MODEL FORTRAN SOURCE LISTING KILN WALL MODEL 1 2 c 3 c 4 c FINITE DIFFERENCE MODEL TO CALCULATE TEMPERATURE 5 c DISTRIBUTION IN ROTARY KILN WALL 6 • c 7 c J.P. GOROG 8 c OCT. 23, 1980 9 c 10 c 1 1 12 IMPLICIT REAL*8(A - H,0 - 2) 13 DIMENSION R(500), T ( 4 0 0 . 5 0 ) . TSS(500) 14 INTEGER ! STEPS. CSTEPS. STEST, CYCLE 15 INTEGER ! T E S T 2 ( 3 ) , T E S T 3 ( 3 ) . TEST4(3) 16 INTEGER 1 DUM1. DUM2, UUM3. FNODE 17 LOGICAL PLOT. REFRAC 18 REAL»8 K, M. K1, KP 19 COMMON /BLK1/ TA. TS. TG. AW, AS. EMW, EMG. EMS 20 COMMON /BLK2/ RI. RO. DT. DR. RPM 2 1 COMMON /BLK3/ KP. K. THETA, FD. ALPHA, ESH. K l . 22 COMMON /BLK4/ HEX. HCOV. HOUT, USE. H0UT2 23 COMMON /BLK5/ T, TSS, HRAD. R 24 COMMON /BLK6/ OWECV. QWER. OWCCV. OTHRU, QTHRU1 25 COMMON /BLK7/ QG, OW, OS. OWC. OWE. OWT. QSE 26 COMMON /BLK8/ OIET. 01ET. Q1ST 27 COMMON /BLK9/ TAVE. TAVC 28 COMMON /BLK10/ STEPS. CSTEPS. NODES. FNODE 29 COMMON /BLK11/ QW4. QWE4. 04ET. 0S4. 04ST. 0WT4 30 COMMON /BLK 12/ T2. T3. T4. T5 31 COMMON /BLK 14/ HRADS 32 COMMON /BLK15/ TWAB, TIME 33 CALL SETLIO('6 ' , '-A ' ) 34 CALL S E T L I 0 C 8 '. '*SINK* ') 35 CALL CMD('$EMPTY -A OK 13) 36 c 37 c READ INPUT DATA 38 c 39 c 10 = RUN IDENTIFICATION NO. 40 c TS = SOLIDS TEMPERATURE (K) 41 c TG* GAS TEMPERATURE (K) 42 c TA = AMBIENT TEMPERATURE (K) 43 c EMS" SOLIDS EMISSIVITY 44 c EMW = WALL EMISSIVITY 45 c EMG- GAS EMISSIVITY 46 c RI« INNER RADIUS (M) 47 c R0 = OUTER RADIUS (M) 48 c RPM= KILN SPEED (REV. PER MIN.) 49 c T HE TA = HALF ANGLE SUBTENDED BY SOLIDS (PAD.) 50 c Y.= THERMAL CONDUCTIVITY (W/M K) 51 c;>» SPECIFIC HEAT (J/KGR K) 52 C RHO= DENSITY (KGR/M**3) 53 C Kt= THERMAL CONDUCTIVITY (W/M K) 54 C RF = INNER THICKNESS (M) 55 C HEX= HTC GAS TO WALL (W/M«*2 K) 56 C HCOV= HTC WALL TO SOLIDS (W/M**2 K) 57 C HOUT= HTC OUTER WALL TO AMBIENT (W/M**2 K) 58 C HSE= HTC SOLIDS SURFACE (W/M**2 K) 59 C REFRAC= ONE SHELL GO C PLOT= PLOT DRAW 61 C 62 READ (5.310) ID 63 READ (5,320) TS. TG. TA 64 READ (5.320) EMS, EMW. EMG 65 ' READ (5,320) RI, RO, RPM, THETA 66 READ (5,320) K, CP, RHO, K l , RF 67 READ (5,320) HEX. HCOV. HOUT. HSE 68 READ (5.300) REFRAC. PLOT 60 C 70 " C INITIA L I Z E ACCUMULATORS 71 C 72 ESH = 1.DO 73 STEPS = 161 74 . FNODE = 25 75 MAX I = 400 76 MAX 12 = 50 77 DO 10 I * 1, 3 78 T E S T 2 ( I ) ' 0 79 T E S T 3 ( I ) = 0 80 TEST4(I ) = 0 8 1 10 CONTINUE 82 C 83 CYCLE = 1 84 STEST = 1 85 C 86 C SET NODES 87 C 88 DR = .1D-02 89 T8AR = (TG + TS) / 2.DO 90 NODES = IDINT((RO - RI)/DR) + 1 91 C 92 DD 20 I = 1, NODES 93 20 R ( I ) » RI + FLOATO - 1 ) * DR 94 C 95 C SET TIME STEPS 96 C 97 RPS = RPM • (1.DO/60.DO) 98 TIME « 1.00 / RPS 99 FRAC = THETA / DPI(O.DO) 100 CTIME = FRAC * TIME 101 DT = TIME / FLOAT(STEPS - 1) 102 CSTEPS «= STEPS - IDI NT (CT IME/DT ) 103 C KD CO 104 C DETERMINE EFFECTIVE THERMAL CONDUCTIVITY OF WALL 105 C 106 IF ( .NOT. REFRAC) GO TO 30 107 T10 = DLOG(RO/RI) 108 T l 1 = DLOG(RF/RI) 1 0 9 T 1 2 = D L O G ( R O / R F ) 1 1 0 K P » T I O / ( T 1 1 / K + T 1 2 / K 1 ) 111 GO TO 40 112 30 KP = K 113 40 CONTINUE 1 14 C 115 C SET INITIAL TEMPERATURE DISTRIBUTION 116 C 117 TBAR = (TG + TS) / 2.DO 118 T400 = (DLOG(RO/RI)) / KP + 1.D0 / (HOUT»RO) 1 19 C 120 DO 50 I = 1, NODES 121 T401 = (DLOG(RO/R(I))) / KP + 1.DO / (HOUT*RO) 122 * T S S ( I ) = (((TBAR - TA)*T401)/T400) + TA 123 50 CONTINUE 124 C 125 TFIX = TSS(FNODE) 126 C 127 C SET RADIATION PARAMETERS AND BED DEPTH 128 C 129 EMS 1 = EMS 130 EMW1 = EMW 13 1 IF (EMS .GT. .999) EMS => . 9999D0 132 IF (EMW .GT. .999) EMW = .9999D0 133 AW = (DPI(0.D0)»2.D0*RI) * (FL0AT(CSTEPS - 1 )/FL0AT(STEPS - 1)) 134 DTHETA = (DPI(0.DO)•2.00) / FLOAT(STEPS - 1) 135 AC = (2.DO*RI*DPI(O.DO)) - AW 136 THETA = AC / (2.D0*RI) 137 FD «= ( 1 .DO-DCOS(THETA) ) / 2 . DO 138 AS - 2.DO * RI * DSIN(THETA) 139 FSW • 1.DO 140 FWS = DSIN(THETA) / (DPI(O.DO) - THETA) 14 1 FWW » 1.D0-FWS 142 SIGMA ' 5.67D-08 ' 1 4 3 x ROW • 1.DO-GMW 144 ROS - 1.DO-EMS 145 C 146 C SET INITIAL RADIATIVE HTC 147 C 148 CALL HCOEFF(TBAR, HRAD, 1. HRADS) MO C 150 C SET HTC AT COVERED WALL 151 C 152 C 153 C SET FINITE DIFFERENCE PARAMETERS 154 C 155 ALPHA = (K/(RH0 +CP)) 15G N2 = FNODE - 1 157 N = NODES - 1 150 N4 = FNODE - 2 159 N5 = FNODE / 2 1G0 M « DR •* 2 / (ALPHA *DT) 161 C 1G2 C BEGIN FINITE DIFFERENCE CALCULATIONS 163 C 164 DO 60 I » I, N2 165 GO T( 1 ,I) = T S S ( I ) 166 C 1G7 C 168 70 DO 100 d = 2. STEPS 169 C 170 C INTERIOR NODES 171 C 172 DO 80 I = 2. N4 173 80 T ( J . I ) = (1.D0/M) * ( T ( 0 - 1.1 - 1) + T ( J - 1,1 + 1)) + l1.D0-( 174 1 2.DO/M)) • T(d - 1.1) + (DR/(M*2.D0*R(I))) • ( T ( d - 1.1 + 1) -175 2 T ( J - 1,1 - 1) ) 176 C 177 C INTERIOR FIXED NODE 178 C 179 T(d.N2) = (1.D0/M) * ( T ( d - 1.N4) + TFIX) + (1.DO-(2.D0/M)) * T( 180 1 d - 1.N2) + (DR/(2 D0*M*R(N2))) • (TFIX - T ( d - 1 .N4) ) 181 C 182 IF ( d .GE. CSTEPS AND. d .LT. STEPS) GO TO 90 1B3 C 184 C EXPOSED INNER SURFACE NODE 185 C 186 T100 » (2.D0*HRAD»RI*DR) / (M*K*(RI + DR/4.D0)) 187 T102 = (2.D0*HEX*RI*DR) / (M*K*(RI + DR/4.D0)) 188 T104 = (2.D0»RI + DR) / (M*(RI + DR/4.D0)) 189 T ( d . l ) = (T100 + T102) • TG + T104 * T ( d - 1.2) + (1.00-T100 -190 1 T102 - T104) * T ( d - 1.1) 191 GO TO 100 192 C 193 C COVERED INNER SURFACE NODE 194 C 195 90 T150 •> (2.D0*HC0V*RI*DR) / (M*K*(RI + DR/4.D0)) 196 T152 = (2.00*RI + DR) / (M*(RI + OR/4.DO)) 197 T ( d . 1 ) ='T152 * T ( d - 1.2) + T150 * TS + (1.DO-T152 - T150) * T( 198 1 d - 1 . 1 ) 199 lOO CONTINUE 200 C 201 C ADdUST ACCUMULATORS 202 C 203 0 0 H O 1 = 1.2 O 204 T E S T 2 ( I ) = TEST2(I + 1 ) O 205 T E S T 3 ( I ) = TEST3(I + 1) 20G T E S T 4 ( I ) = TEST4(I + 1 ) 207 1IO CONTINUE 200 C T200 = DABS(T(1.1) - T ( S T E P S . D ) 203 2 10 T201 - DABS(T(1.N2) - T(STEPS.N2)) 2 1 1 T202 = DABS(T(1.N5) - T(STEPS.N5)) 2 12 IF (T200 .LT. .1D-04) GO TO 120 2 13 TEST2(3) = 2 214 GO TO 130 215 120 TEST2(3) = 1 216 130 IF (T201 .LT. .1D-04) GO TO 140 217 TES T 3 ( 3 ) = 2 218 GO TO 150 2 19 140 TEST3(3) = 1 220 150 IF (T202 .LT. .1D-04) GO TO 160 22 1 TEST4(3) = 2 222 GO TO 170 223 160 TES T 4 ( 3 ) = 1 224 170 CONTINUE 225 IF (CYCLE .LE. 4) GO TO 190 226 DUM1 = 0 227 DUM2 = 0 228 DUM3 = 0 229 C 230 DO 180 I = 1. 3 23 1 DUM1 = DUM1 + TE S T 2 ( I ) 232 DUM2 = DUM2 + TE S T 3 ( I ) 233 DUM3 = DUM3 + T E S T 4 ( I ) 234 180 CONTINUE 235 C 236 C CHECK FOR CONVERGENCE 237 C IF (DUM1 .EQ. 3 .AND. DUM2 .EQ. 3 .AND. DUM3 .EQ 238 239 1 GO TO 230 240 190 CONTINUE 24 1 C 242 C CHECK CYCLE 243 C 244 CYCLE = CYCLE + 1 245 IF (CYCLE .GT. MAX I) GO TO 200 246 GO TO 210 247 200 IF (STEST .EQ. 1) H0UT2 = HOUT 248 CALL PRINT 1(10. MAXI, REFRAC. PLOT. EMS 1, EMW1) 249 GO TO 290 250 C 25 1 C RESET TEMPERATURE DISTRIBUTION AND RETURN TO 252 C FINITE DIFFERENCE CALCULATIONS 253 C 254 2 10 DO 220 I = 1. N2 255 220 T ( 1 . I ) » T(STEPS.I) 256 C 257 C DETERMINE AVERAGE EXPOSED WALL TEMPERATURE AND SET 258 C RADIATIVE HTC 259 C r o o 260 CALL TAV(TAVE, TAVC. T, CSTEPS. STEPS) 261 CALL HCOEFF(TAVE. HRAD. 1. HRADS) 262 GO TO 70 263 C 264 C CHECK FOR FINAL SOLUTION - COMPARE HEAT FLOWS INTO FIXEO 265 C NODE, ADJUST TEMPERATURES AND RESTART CALCULATIONS 266 C IF NEEDED 267 c 268 230 CALL FLOWS(RI) 269 OC = OTHRU 270 QSS = ( ( T F I X - TSS(FNODE + 1 ) )/DLOG(R(FNODE + 1 )/R(FNODE))) * KP 27 1 1 2.DO * DPI(O.DO) 272 QTEST * DABS((OC - OSS)/OSS) 273 • IF(STEST .EQ. 1)GOTO 231 274 IF (OTEST .LE. .5D-02 .AND. OC .GT. 0.DO .AND. OSS .GT. O.DO) 275 1 GO TO 280 276 23 1 CONTINUE 277 TBAR = ((QC/(2.DO*DPI(O.DO)*KP))*DL0G(R(FN0DE)/RI)) + TFIX 278 H0UT2 = (ESH*SIGMA*(TSS(N0DES)**4 - TA**4)) / (TSS(NOOES) - TA) 279 H0UT1 = HOUT + H0UT2 280 T400 » (DL0G(R0/RI)) / KP + 1.D0 / (H0UT1*R0) 281 C 282 DO 240 1 = 1 . NODES 283 T401 = (DLOG(RO/R(I))) / KP + 1.D0 / (H0UT1*RO) 284 T S S ( I ) = (((TBAR - TA)*T401)/T400) + TA 285 240 CONTINUE 286 C 287 C 288 C CHECK CYCLE 289 C 290 IF (STEST .GT. MAX 12) GO TO 250 29 1 GO TO 260 292 250 CALL PRINT2(ID, MAX 12. REFRAC. PLOT. EMS 1, EMW1) 293 GO TO 290 294 260 STEST = STEST + 1 295 TFIX = TSS(FNODE) 296 CYCLE = 1 297 C 298 DO 270 I = 1, N2 299 270 T ( 1 . 1 ) = T S S ( I ) 300 C 301 GO TO 70 302 280 CALL TAV(TAVE. TAVC, T, CSTEPS. STEPS) 303 C 304 C 305 C INTEGRATE HEAT FLOWS 306 C 307 CALL FLOWS(RI) 308 C 309 C CALCULATE 1-ZONE HEAT FLOWS 3 IO C 3 1 1 CALL ZONE(TAVE, TAVC. RI, THETA. AC. 2) 312 C 313 C CALCULATE 4-ZONE HEAT FLOWS 3 14 C 3 IS CALL ZONE4 3 16 C 317 C PRINT FINAL SOLUTION 318 C 319 CALL PRINT(IO. REFRAC. EMS 1, EMW1) 320 C 32 1 C CALCULATE HEAT FLOWS USING ANALOG APPROXIMATIONS 322 C 323 CALL ANAL 1 >. 324 CALL ANAL2 .325 C 326 C 9700 FORMAT 327 C 328 290 CALL P 9 7 0 0 U D ) 329 C 330 C PLOT RESULTS 331 . C 332 IF (PLOT) CALL PLOTIT(ID) 333 STOP 334 300 FORMAT (2L1) 335 310 FORMAT (413) 336 320 FORMAT (8G15.5) 337 END ro o C O 338 £»+••+•»•** 339 C 340 C 34 1 C SUBROUTINE TO OETERMINE AVERAGE WALL TEMPERATURES 342 C 343 C 344 c****«*«*»* 345 SUBROUTINE TAV(TAVE, TAVC, T, CSTEPS. STEPS) 346 IMPLICIT REAL*8(A - H.O - Z) 347 DIMENSION T(400,5O), AC(64) 348 INTEGER CSTEPS, STEPS 349 C 350 C EXPOSED WALL 351 C ' 352 DUMMY = DACSUM(AC.T,0) 353 N = CSTEPS - 1 354 C 355 DO 10 I = 1. N 3'JG ' IO DUMMY - DACSUM( AC . T ( I . 1 ) ) 357 C 358 TAV1 = DACSUM(AC) 359 TAVE = TAV1 / FLOAT(N) 360 C 36 1 C COVERED WALL 362 C 363 DUMMY = DACSUM(AC.T,0) 364 C 365 N1 = STEPS - 1 366 C 367 00 20 I = CSTEPS, N1 368 20 DUMMY = DACSUM(AC.T(I,1)) 369 C 370 TAV1 = OACSUM(AC) 371 TAVC = TAV1 / FLOAT(STEPS - CSTEPS) 372 DUMMY = DACSUM(AC.T,O) 373 RETURN 374 END r o o 375 Q* ******** * Q 37G c 377 c 378 c SUBROUTINE TO CALCULATE INTEGRATED HEAT FLOWS 379 c 380 c 381 Q********** 382 SUBROUTINE FLOWS(RI) 383 IMPLICIT REAL*8(A - H.O - Z) 384 DIMENSION T(400.50). TSS(500). R(500). AC(64) 385 REAL*8 K. K 1 , KP 386 INTEGER FNODE. STEPS. CSTEPS 387 COMMON /BLK1 / TA, TS, TG. AW, AS. EMW, EMG, EMS 388 COMMON /BLK3/ KP. K. THETA. FD. ALPHA, ESH. KI, 389 COMMON /BLK4/ HEX, HCOV. HOUT. HSE. H0UT2 390 COMMON /BLK5/ T, TSS. HRAD. R 391 COMMON /BLK6/ QWECV, QWER, QWCCV, QTHRU. QTHRU1 392 COMMON /BLK8/ QlET , Q1ET, Q1 ST 393 COMMON /BLK10/ STEPS. CSTEPS. NODES. FNODE 394 DTHETA = (2.D0*DPI(0.D0)) / FLOAT(STEPS -.1) 395 N = CSTEPS - 1 396 c 397 c CONVECTION GAS TO EXPOSED WALL 398 c 399 QTEMP = O.DO 400 DUMMY = DACSUM(AC.QTEMP,0) 401 c 402 DO 10 I = 1. N 403 QTEMP = HEX • RI * DTHETA * (TG - T ( I . D ) 404 10 DUMMY = DACSUM(AC,QTEMP) 405 C 406 QWECV = DACSUM(AC) 407 C 408 C RADIATION TO EXPOSED WALL 409 ' c 4 10 DUMMY = DACSUM(AC,QTEMP.0) 4 11 c 4 12 DO 20 I = 1. N 413 QTEMP = HRAD • RI • DTHETA * (TG - T ( I . D ) 4 14 20 DUMMY = DACSUM(AC.QTEMP) 4 15 C 4 16 QWER = DACSUM(AC) 417 C 4 18 c CONVECTION COVERED WALL TO SOLIDS 4 19 c 420 DUMMY = DACSUM(AC.QTEMP.0) 421 N2 = STEPS - 1 422 c 423 DO 30 I = CSTEPS. N2 424 QTEMP = HCOV • RI * DTHETA * (TS - T ( I . D ) 425 30 DUMMY «= DACSUM( AC , QTEMP ) ro o tn 42G C 427 OWCCV = DACSUM(AC) 428 OTHRU = QWER + OWECV + OWCCV 429 OTHRU t = (2.DO*DP1(O.DO)*KP*(TSS(FNODE + 1) - TSS(NODES - 1 ) ) ) / 430 1DLOG(R(NODES - 1)/R(FNODE • 1)) 43 1 01ET = QWER + OWECV 432 RETURN 433 END ro o cn 434 £•*•*•••••• 435 C 436 C 437 C SUBROUTINE TO CALCULATE RADIATIVE HTC USING 1-ZONE GRAY 438 C GAS ANALOG 439 C 440 C 441 C********** 442 SUBROUTINE HCOEFF(TW. HRAD, N, HRADS) 443 IMPLICIT REAL*8(A - H,0 - Z) 444 DIMENSION A ( 2 , 2 ) , X ( 2 ) , IPERM(4), T ( 2 . 2 ) , R ( 5 ) , B(2) 445 COMMON /BLK 1/ TA. TS. TG, AW. AS. EPW. EPG. EPS 446 COMMON /BLK4/ HEX. HCOV, HOUT, HSE. H0UT2 447 COMMON /BLK8/ OIET, Q1ET. Q1ST 448 COMMON /BLK7/ OG. OW, OS. QWC. OWE. OWT, OSE 449 SIGMA = 5.67D-08 450 C 45 1 C ENTRY FOR 1-ZONE GRAY GAS HEAT FLOWS 452 C 453 ENTRY ZONE(TW,TWC.RI,THETA.AC,N) 454 ES = SIGMA * TS •* 4 455 EW = SIGMA * TW * * 4 456 . EG = SIGMA * TG *• 4 457 TRG = 1.DO-EPG 458 FWG » 1 .DO 459 FSG = 1 .DO 460 FSW = 1.DO 4G1 R(1) = (EPW*AW) / (1.D0-EPW) 462 R(2) » AW » FWG * EPG 463 R(3) = AS • FSG * EPG 464 R(4) • ( E P S ' A S ) / ( 1 . D 0 - E P S ) 465 R(5) = AS * FSW • TRG 466 A(1.1) = R(5) 467 A ( 1 . 2 ) = - ( R ( 3 ) * R(4) + R ( 5 ) ) 468 A(2.1) = -(R ( 2 ) + R(1) + R ( 5 ) ) 469 A(2,2) « R(5) 470 B ( 1 ) = - ( R ( 3 ) * E G + R( 4 ) * E S ) 471 B ( 2 ) = -( R ( 2 ) * E G + R(1)*EW) 472 CALL S L E ( 2 , 2. A, 1. 2. B. X. I PERM, 2, T. DEP. JEXP) 473 IF (DEP) 10, 20. 10 474 10 OW = ( X ( 1 ) - EW) * R(1) 475 OS = ( X ( 2 ) - ES) * R(4) 476 HRADS = OS / (AS*(TG - T S ) ) 477 HRAD = OW / (AW*(TG - TW)) 478 IF (N .EQ. 1) RETURN 479 QG = ( X ( 2 ) - EG) * R(3) + (X(1) - EG) • R(2) 480 QWC = HCOV * AC * (TS - TWC) 48 1 OWE = HEX * AW * (TG - TW) 482 QWT = QW + QWC + OWE 483 QSE = HSE * AS • (TG - TS) 484 Q1ET = QW * QWE ro o 485 01ST = OS + OSE 486 RETURN 487 20 WRITE (6.30) 488 CALL EXIT 489 STOP 490 30 FORMAT ( ' ' . '1 -ZONE SOLUTION F A I L E D ' ) 491 END r o O CO 492 493 c 494 c 495 c SUBROUTINE TO PRINT RESULTS 496 c 497 c 498 c********** 499 SUBROUTINE PRINT(ID. REFRAC. EMS 1. EMW1) 500 IMPLICIT REAL*8(A - H.O - Z) 501 DIMENSION T(400.50). T S S ( 5 0 0 ) , R(500) 502 LOGICALM STARS(6) /6*'*'/ 503 INTEGER FNODE. STEPS. CSTEPS 504 LOGICAL REFRAC, PLOT 505 REAL*8 K, K l . KP 506 COMMON /BLK 1/ TA, TS. TG. AW. AS. EMW. EMG. EMS 507 COMMON /BLK2/ RI. RO. DT, DR, RPM 508 COMMON /BLK3/ KP, K. THETA, FD. ALPHA, ESH, K1. 509 COMMON /BLK4/ HEX, HCOV, HOUT. HSE , H0UT2 0 10 COMMON /BLK5/ T. TSS, HRAD, R 511 COMMON /BLK6/ OWECV, OWER, OWCCV. OTHRU. OTHRU1 512 COMMON /BLK7/ OG. OW. OS. QWC. OWE. QWT. OSE 513 COMMON /BLK8/ OIET, 01ET. 01ST 514 COMMON /BLK9/ TAVE. TAVC 515 COMMON /BLK 10/ STEPS. CSTEPS. NODES. FNODE 5 16 COMMON /BLK 11/ QW4. QWE4, 04ET. 0S4. Q4ST, QWT4 517 COMMON /BLK 12/ T2. T3. T4. T5 518 COMMON /BLK 14/ HRADS 519 CALL HEADER 520 WRITE (6.110) ID 52 1 N = 1 522 GO TO 10 523 c 524 c ENTRY WHEN CONVERGENCE NOT REACHED IN MAXI CYCLES 525 c ENTRY PRINT 1(ID.MAXI,REFRAC,PLOT,EMS 1,EMW1) 526 527 PLOT = .FALSE. 528 CALL HEADER 529 WRITE (6.120) ID, MAXI 530 N = 2 531 GO TO 10 532 c ENTRY WHEN FINAL SOLUTION NOT REACHED IN MAX 12 CYCI.I 533 c 534 c ENTRY PR INT2(ID,MAX I 2,REFRAC,PLOT.EMS 1.EMW1) 535 53G PLOT = .FALSE. 537 CALL HEADER . 538 WRITE (6,130) ID. MAXI2 539 N » 2 540 10 N1 = FNODE - 1 54 1 RT1 = RF - RI 542 RT2 = RO - RF ro o io 543 WRITE (6.150) RI . RO, DT. OR. RPM 544 WRITE (6.140) TS. EMS 1, TG. EMG. TA. ESH, EMW1 545 WRITE (6.160) THETA. FD 546 IF (REFRAC) GO TO 20 547 GO TO 30 54S 20 WRITE (6,50) RT1, K. ALPHA, RT2, KI 549 30- WRITE (6.170) K. ALPHA 550 WRITE (6.180) HEX. HCOV, HOUT, HSE. HRAD. H0UT2. HRADS 551 WRITE (6,190) 552 CALL DPRMAT(T, 400. 50. STEPS. NI. 1, 1, 400, 1) 553 WRITE (6.200) 554 CALL DPRMAT(TSS, 1, 250. 1. NODES. 1. 1. 1. 1) 555 IF (N .NE. 1) GO TO 40 556 ' WRITE (6.60) TAVE. T2. T3. T4, T5. TAVC 557 WRITE (6.70) QWER. QW, QW4, QWECV, OWE, QWE4, QlET, Q1ET, Q4ET 558 WRITE (6,80) QWCCV. QWC. QWC 559 WRITE (6.90) STARS. QS. QS4. STARS, OSE. QSE, STARS, Q1ST, Q4ST 5GO WRITE (6.100) QTHRU. QWT. QWT4. OTHRU1. STARS. STARS 56 1 40 RETURN 562 50 FORMAT (' '. T5, 'COMPOSITE REFRACTORY WALL:'/TIO, 563 1 'THICKNESS OF INNER REFRACTORY=', IX. G12.5. IX, '(M)'/T10, 564 2 'INNER THERMAL CONDUCTIVITY'', IX, G12.5. 1X, 565 3 '(W/M S)'/T10, 'INNER THERMAL DIFFUSIVITY = ' , IX, G12.5. IX, 566 4 '(M**2/S) ' / T10, 'OUTER THICKNESS=', 1X, G12.5. IX. 567 5 '(M)'/T10, 'OUTER THERMAL CONDUCTIVITY'', IX, G12.5, 1X, 568 6 ' (W/M K) '/) 569 60 FORMAT ('1'. 'HEAT FLOW MODLES:'///T5, 'AVERAGE EXPOSED WALL TEMPE 570 1RATURES (K):'/T10. '1-Z0NE=', G12.5, T35, '4-ZONE'. T41, '#1=', 571 2 G12.5/T41, '/f2='. G12.5/T41. ' *3=' . G12.5/T41. '04=', 572 3 G12.5//T5. 'AVERAGE COVERED WALL TEMPERATURE=', G12.5//) 573 70 FORMAT (' '. T5, 'CALCULATED HEAT FLOWS (W/M):'//T25. 574 1 'INTEGRATED', T50. '1-ZONE', T75, '4-Z0NE'/T5. .575 2 'EXPOSED WALL: '/TIO. 'RADIATION'. T25, G12.5, T50. G12.5. 576 3 T75. G12.5/T10. 'CONVECTION', T25, G12.5. T50. G12.5. T75, 577 4 G12.5/T10, 'TOTAL'. T25, G12.5. T50. G12.5, T75. G12.5//) 578 80 FORMAT (' '. T5. 'COVERED WALL:'/TIO, 'CONVECTION', T25. G12.5. 579 1 T50, G12.5, T75. G12.5//) 580 90 FORMAT (' ', T5. 'SOL IDS:'/T10. 'RADIATION', T25, 6A1, T50, G12.5, 581 1 T75, G12.5/T10, 'CONVECTION', T25, 6A1. T50, G12.5. T75. 582 2 G12.5/T10. 'TOTAL'. T25. 6A1. T50. G12.5. T75. G12.5//) 583 100 FORMAT (' '. T5, 'THROUGH WALL:'/T10. 'DIFFERENCE', T25. G12.5, 584 1 T50. G12.5..T75. G12.5/T10. 'PROFILE'. T25. G12.5. T50, 585 2 6A1, T75. G A I / ' I ' , 'ANALOG APPROXIMAT'ONS:'/) 586 110 FORMAT (' '. 'FINAL SOLUTION FOR RUN NO.'. IX. 14//) 587 120 FORMAT (' ', 'FOR RUN NO.'. 1X. 14. 1X, 'NO CONVERGENCE IN', IX. 588 1 14, IX, 'CYCLES'//) 589 130 FORMAT (' ', 'FOR RUN NO.'. IX. 14. IX. 'FINAL SOLUTION NOT REACHE 590 ID IN', IX. 14, 1X. 'CYCLES'//) 591 140 FORMAT (' ', T5. 'SOLIDS TEMPERATURE''. IX, G12.5. IX. ' ( K ) ' / T 5 , 592 1 'SOLIDS EMISSIVITY='. IX. G12.5/T5. 'GAS TEMPERATURE 3', 1X, 593 2 G12.5. IX, ' ( K ) ' / T 5 , 'GAS EMISSIVITY''. 1X. G12.5/T5. 594 , 3 'AMBIENT TEMPERATURE''. 1X. G12.5. IX, ' ( K ) ' / T 5 . 595 4 'SHELL EMISSIVITY'', IX, G12.5/T5. 'WALL EMISSIVITY = '. 1X, 59G 5 G12.5/) 597 150 FORMAT (' '. T5. 'KILN INNER RADIUS''. G12.G. IX, '(M)'/T5, 590 1 'KILN OUTER RADIUS''. G12.5, 1X, '(M)'/T5, 'TIME STEP'', 599 2 G12.5. IX. '(SEC)'/T5, 'RADIAL STEP=', G12.5. 1X, '(M)'/T5. 600 3 'KILN SPEED'', G12.5. IX, '(RPM)'/) 601 160 FORMAT (' '. T5, 'HALF ANGLE SUBTENDED BY SOLIDS='. G12.5/T5. 602 1 'RATIO BED DEPTH:KILN DIAMETER''. G12.5/) 603 170 FORMAT (' '. T5. 'WALL THERMAL CONDUCTIVITY''. G12.5, 1X, 604 1 '(W/M K)'/T5, 'WALL THERMAL DIFFUSIVITY= ' , G12.5, IX. 605 2 '(M**2/S)'/) 606 180 FORMAT (' '. T5. 'CONVECTIVE HTC (W/M**2 K):'/T8, 607 1 , 'GAS TO EXPOSED WALL''. G12.5/T8. 'SOLIDS TO COVERED WALL=' 608 2 . G12.5/T8. 'OUTER SHELL TO ATMOSPHERE''. G12.5/T8. 609 3 'GAS TO SOLIDS''. G12.5//T5. 'RADIATIVE HTC (W/M**2 K ) : ' / 610 4 T8. 'EXPOSED WALL''. G12.5/T8. 'OUTER SURFACE''. G12.5/T8. 611 5 'EXPOSED SOLIDS''. G12.5/) 612 190 FORMAT ('1'. T5. 'THE UNSTEADY STATE TEMPERATURE FIELD FOLLOWS:'// 6 13 1 ) 614 200 FORMAT ('1'. T5. 'THE STEADY STATE TEMPERATURE FIELD FOLLOWS:'//) 6 15 ENO r o 6 16 617 c 6 18 c 6 19 C SUBROUTINE TO CALCULATE 4-ZONE RADIATIVE HEAT FLOWS 620 C 621 C 622 Q********** 623 SUBROUTINE Z0NE4 624 IMPLICIT REAL«8(A - H,0 - Z) 625 REAL*8 K, K l . KP 626 INTEGER CSTEPS. STEPS, FNODE 627 DIMENSION R(500). T(400.50). TSS(500), AC(64) 628 DIMENSION A ( 5 , 5 ) . B ( 5 ) , X ( 5 ) . IPERM(10), TT(5.5) 629 COMMON /BI.K1/ TA, TS, TG. AW. AS. EMW. EMG, EMS• 630 COMMON /BLK3/ KP. K, THETA, FD. ALPHA, ESH. K l . 631 COMMON /BLK4/ HEX, HCOV, HOUT. USE. H0UT2 632 COMMON /BLK5/ T. TSS. HRAD. R 633 COMMON /BLK 11/ 0W4. 0WE4. Q4ET, 0S4. 04ST, 0WT4 634 COMMON /BLK 12/ T1, T2, T3. T4 635 COMMON /BLK10/ STEPS. CSTEPS. NODES. FNODE 636 COMMON /BLK7/ QG. QW, OS. OWC. OWE. OWT, OSE 637 REAL*8 US. J l . J2. J3. J4 638 N - CSTEPS - 1 639 N1 - N / 4 640 N2 = Nl • 2 64 1 N3 = Nl * 3 64 2 11 = 1 + N1 64 3 12 = I 1 + N1 644 13 = 12 + Nl 645 C 646 DO 10 I = 1. Nl 647 10 DUMMY = DACSUM(AC,T(I , 648 c 649 Tl = DACSUM(AC) 650 T1 - T1 / FLOAT(N1) 651 DUMMY = DACSUM(AC.T.O) 652 c 653 DO 20 I = I 1. N2 654 20 DUMMY = DACSUM(AC.T(I, , 1)) 655 C 656 T2 = DACSUM(AC) 657 T2 - T2 / FLDAT(Nl) 658 DUMMY » DACSUM(AC.T.O) 659 c 660 DO 30 I = 12. N3 661 30 DUMMY » DACSUM(AC.T(I . 1 )) 662 C 663 T3 = DACSUM(AC) 664 T3 = T3 / FLOAT(N1) 665 DUMMY » DACSUM(AC.T,0) 666 c ro G67 DO 40 I = 13, N 668 40 DUMMY = DACSUMfAC.T ( I . 1 ) ) G69 C 670 T4 = DACSUM(AC) 671 N7 » CSTEPS - 13 672 T4 = T4 / FL0AT(N7) 673 DUMMY = DACSUM(AC,T.0J 674 SIGMA = 5.67D-08 675 ES = SIGMA * TS •* 4 676 EG = SIGMA * TG ** 4 677 E1 « SIGMA • Tl 4 678 E2 » SIGMA * T2 •* 4 679 E3 = SIGMA * T3 ** 4 680 E4 = SIGMA * T4 ** 4 68 1 TRG «• I.DO-EMG 682 A1 = AW / 4.DO 683 A2 = A1 684 A3 = A1 685 A4 = Al 686 EP1 = EMW 687 EP2 = EP 1 G88 EP3 = EP 1 689 EP4 = EP1 690 F1G = 1 .DO 69 1 F2G =1.00 692 F3G = 1.DO 693 F4G = 1 .DO 694 FSG » 1.DO 695 TP 1 = DSIN((PI(O.OO) - THETA)/4.DO) 696 F11 = 1.D0-(4.D0+TP1) / (PI(O.DO) - THETA) 697 TP2 = DSIN((PI(O.DO) - THETA)/2.DO) 698 F12 = 1.D0-F11 - (2DO*TP2) / (PI(O.DO) - THETA) 699 TP3 = DSIN((PI(O.DO) + 3.DO*THETA)/4.DO) 700 F13 = (2.DO*(2.DO*TP2 - TP 1 - TP3) ) / (PI(O.DO) - THETA) 701 TP4 « DSIN(THETA) 702 FS1 = (TP4 • TP 1 - TP3) / (2.DO*TP4) 703 FS4 = FS1 704 FS2 = .5D0-FS1 705 FS3 = FS2 706 F24 = F13 707 F23 = F12 708 F34 = F23 709 F1S = (AS/A 1 ) * FS1 710 F14 = 1.D0-F11 - F12 - F13 - F1S 711 R1( 1) = A2 * F2G * EMG 712 R1(2) = Al * FIG * EMG 7 13 R1(3) = AS * FSG * EMG 714 R1(4) = A4 * F4G * EMG —• 715 R1(5) = A2 * F23 * TRG w 716 R1(6) = A3 • F3G * EMG 717 R1(7) = EP2 * A2 / (1.D0-EP2) 718 R1(8) = EP1 * Al / (1.D0-EP1) 719 R 1(9) «> EPS * A3 / (1.D0-EP3) 720 R1(10) = A3 * F34 » TRG 721 R1(11) = EP4 * A4 / (1.D0-EP4) 722 R1( 12) « AS • FS4 * TRG 723 Rl( 13) = AS * FS1 » TRG 724 RK14) = EMS * AS / (1.DO-EMS) 725 R1 (15) " A2 • F24 • TRG 72G R1( 16) * Al * F13 * TRG 727 R1(17) = AS « FS3 • TRG , 728 R1 ( 18) = AS • FS2 * TRG 729 R1( 19) = Al • F14 * TRG 730 R1(20) = A1 • F12 * TRG 731 _ A(1,1) = R1(20) 732 A( 1,2) = R1( 13) 733 A(1.3) » R1( 19) 734 A( 1 ,4) = R1( 16) 735 A(1.5) « -(R1(20) + R1(13) + R1(19) • R1(16) + R1(2) + R1(8)) 736 A(2, 1 ) • R 1 ( 18) 737 A(2,2) = -(R1(18) + R1(12) + R1(17) + R1(13) + R1(14) + R1(3)) 738 A(2.3) « RK 12) -739 A(2.4) = R1(17) 740 A(2,5) = RK 13) 741 A(3. 1) = R1( 15) 742 A(3.2) = R1 ( 12) 743 A(3,3) = -(R1(15) + RK12) + RKlO) + R1{19) + R1 ( 1 1 ) + R K 4 ) ) 744 A(3.4) = R1(10) 745 A(3.5) = R1( 19) 746 A(4, 1) = R K 5) 747 A(4.2) = R1( 17) 748 A(4,3) = RK 10) 749 A(4,4) = -(R1(5) + R1(17) + R1(10) + R1(16) + R1(9) + R1(6)) 750 A(4.5) = R1(16) 751 A(5.1) = -(R1(15) + R1(18) + R1(5) + R1(20) + R1(7) + R1 ( 1 ) ) 752 A(5.2) = R1( 18) 753 A(5,3) = R1(15) 754 A(5,4) «= R1(5) 755 A(5,5) « R1(20) 756 B(1) = -( R 1 (8) *E 1 + R1.(2)*EG) 757 B(2) " -(R1(14)*ES + R1(3)*EG) 758 B(3) = -(R1(11)*E4 + R1(4)*EG) 759 B(4) * -(R1(9)*E3 + R1(6)*EG) 760 B(5) " -(R1(7)*E2 + R1(1)*EG) 761 CALL SLE(5. 5, A, 1. 1, B, X. IPERM. 5, TT, DET, JEXP) 762 IF (DET) 50, 60. 50 763 50 J2 = X( 1 ) 764 JS = X(2) 765 J4 = X(3) 766 J3 = X(4) — i 767 J1 = X(5) 768 0S4 = (OS - ES) * RK 14) 769 01 = ( J l - El) * R1(8) 770 02 = (J2 - E2) * R1(7) 771 03 = (J3 - E3) • R1(9) 772 04 = (J4 - E4) * R1( 1 1 ) 773 QW4 = 0 1 + 0 2 + 0 3 + 0 4 774 0WE4 » Al * HEX * (TG - T1) + A2 • HEX * (TG - T2) + A3 * HEX * ( 775 1TG - T3) + A4 • HEX * (TG - T4) 77G Q4ET = QW4 + QWE4 777 Q4ST • 0S4 + 05E 778 QWT4 = Q4ET + QWC 779 RETURN 780 60 WRITE (6.70) 781 STOP 782 70 FORMAT (' '. '4-ZONE SOLUTION FAILED.') 783 END i cn 784 C** »•*•**** 785 C 786 C 787 C SUBROUTINE TO PLOT MODEL OUTPUT 788 C 789 C 790 791 SUBROUTINE PLOTIT(ID) 792 IMPLICIT REAL*8(A - H.O - Z) 793 REAL*8 K, K 1 . KP 794 LOGICAL PLOT 795 DIMENSION T(400.50), R(500) 796 DIMENSION 0W1(400), 0W2(400). AC(64), Y(30) 797 DIMENSION TSS(500) 798 INTEGER CSTEPS. STEPS, FNODE 799 REAL*4 T0(400), Y1(400). Y2(400). Y5(400), Y10(400) 800 REALM X(400). YPLOT1(400), YPL0T2(4OO), YPL0T3(400) 801 REALM YPLOT4(400), YPL0T5(400), XPL0T(400). YMIN 802 REALM ID1. DY, YTHRU1(100). YTHRU2(100). DX, XMIN. ' 803 REALM Y0(400). YSC, YMX, YMN. RID 804 COMMON /BLK1/ TA. TS. TG. AW. AS. EMW. EMG. EMS 805 COMMON /BLK2/ RI, RO. DT, DR. RPM 806 COMMON /BLK3/ KP. K. THETA. FD, ALPHA. ESH. KI. RF 807 COMMON /BLK4/ HEX. HCOV. HOUT. HSE. H0UT2 808 COMMON /BLK5/ T, TSS, HRAO, R 809 COMMON /BLK10/ STEPS, CSTEPS, NODES. FNODE 810 COMMON /BLK15/ TWAB. TIME 8 11 c 8 12 DO 10 I = 1. STEPS 8 13 YO(I) = SNGL((T(I.1) - TS)/(TG - TS)) 8 14 Y1(I) = SNGL((T(I,2) - TS)/(TG - TS)) 815 Y2(I) = SNGL((T(I,3) - TS)/(TG - TS)) 8 16 Y5(I) = SNGL((T(I,6) - TS)/(TG - TS)) 817 Y10(I) * SNGL((T(I.11) - TS)/(TG - TS)) 818 YWSH(I) = SNGL((TSS(NODES) - TS)/(TG - TS)) 819 10 CONTINUE 820 c 82 1 X(1) » 0.0 822 TP 1 = TIME / FLOAT(STEPS - 1) 823 c 824 DO 20 I =2. STEPS 825 20 X(I) = SNGL(TP1*FL0AT(I - 1)) 826 C 827 DO 30 I = 1, STEPS 828 30 X(I ) = X(I) / SNGL(TI ME) 829 C 830 C PLOT / f i 831 C 832 CALL AXIS(2., 2.. '(T-TS)/(TG-TS)'. 14. 5.. 90.. 0.. 833 834 CALL PL0T(2. .. 7 . . 3 ) WSH(400) .2) .. .125) 835 CALL PLOT(10., 7. . 2) 836 CALL PL0T(10., 2. . 1) 837 CALL PL0T(2., 2.. 3) 838 RID - FLOAT(ID) 839 CALL SYMB0L(2.. 7.1. .15. 'PLOT 1-'. 0.. 7) 840 CALL NUMBER(2.92, 7.1. .15. RID. 0 . , -1) 841 C 842 DO 40 I ' 1. STEPS 843 XPLOT(I) = (X(I)/.125) + 2. 844 YPLOTI(I) = (Y0(I)/.2) +2. 845 YPL0T2U) = (YWSH(I)/.2) + 2. 846 40 CONTINUE 847 C 848 CALL LINE(XPLOT. YPL0T1. STEPS. 1) 849 CALL PL0T(XPL0T( 1 ) , YPL0T2U). 3) 850 CALL DASHLN(.125. .125. .125, .125) 851 C 852 DO 50 I = 1. STEPS 853 50 CALL PLOT(XPLOT(I), YPL0T2(I). 4) 854 YPLOT1(55) = YPL0T1(55) + .IEO 855 YPL0T2(55) » YPL0T2(55) + .IEO 856 CALL SYMBOL(XPLOT(55), YPLOT1(55). .15, 'TW. 0 . 2 ) 857 CALL SYMB0L(XPL0T(55), YPL0T2(55), .15, 'TSH', O.. 3) 858 C 859 CALL PL0T(16., 0.0, -3) 860 C 86 1 C PLOT H2 862 C 863 YMX - -1.E30 864 YMN = 1.E30 865 DO 60 I = 1. STEPS 866 IF (YO(I) .GT. YMX) YMX = Y0(I) 867 IF (Y1(I) .GT. YMX) YMX = Y1(I) 868 IF (Y2(I) .GT. YMX) YMX = Y2(I) 869 IF (Y5(I) .GT. YMX) YMX = Y5( I ) 870 IF (Y10(I) GT. YMX) YMX = Y10(I) 871 IF (YO(I) .LT. YMN) YMN = Y0(I) 872 IF (Y1( I ) .LT. YMN) YMN = Y1(I) 873 IF (Y2( I ) .LT. YMN) YMN = Y2(I) 874 IF (Y5(I) .LT. YMN) YMN = Y5( I ) 875 IF (Y10(I) .LT. YMN) YMN = Y10(I) 876 60 CONTINUE 877 YPLOT 1 (1) = YMN 878 YPLOT1(2) = YMX 879 CALL SCALE(YPL0T1. 2. 6., YMIN. DY. 1) 880 CALL AXIS(2.. 2.. '(T-TS)/(TG-TS)'. 14. 6., 90.. YMIN. DY) 881 CALL AXIS(2., 2.. ' FRACTION OF CYCLE'. -17. 8.. 0.. 0... .125) 882 CALL PL0T(2., 8.. 3) 883 CALL PL0T(10.. 8.. 2) 884 CALL PLOT(10. . 2 . . 1 ) 885 CALL PL0T(2., 2.. 3) 886 CALL SYMB0L(2.. 8.1. .15. 'PLOT 2-'. 0.. 7) 8B7 CALL NUMBER(2.92. 8.1. .15. RIO. 0.. -I) 888 C 889 00 70 I = 1. STEPS 890 YPLOT 1(1) = ((YO(I) - YMIN)/OY) + 2. 891 YPL0T2U) = ((Y1(I) - YMINJ/DY) +• 2. 892 YPL0T3U) = ((Y2(I) - YMIN)/DY) + 2. 893 YPL0T4U) = ((Y5(I) - YMINI/DY) + 2. 894 YPL0T5O) = ((Y10(I) - YMIN)/DY) .+ 2. 895 70 CONTINUE 89G C 897 CALL LINEfXPLOT. YPLOT1. STEPS. 1) 898 CALL LINE(XPLOT. YPL0T2. STEPS. 1) 899 CALL LINE(XPL0T. YPL0T3. STEPS, 1) 900 ' CALL LINE(XPLOT. YPL0T4. STEPS. 1) 901 CALL LINE(XPLOT. YPL0T5. STEPS. 1) 902 C 903 DO 80 I = 5. STEPS, 25 904 80 CALL SYMBOL(XPLOT(I), YPLOT1(I). .08, 30. O.. 905 C 906 DO 90 I = 5, STEPS, 25 907 90 CALL SYMBOL (XPLOT ( I ) , YPL0T2U). .08. 3, 0.. 908 C 909 DO 100 1 = 5 . STEPS, 25 910 100 CALL SYMBOL(XPLOT(I), YPL0T3(I). .08. 2. 0.. 9 1 1 C 912 DO 110 I = 5. STEPS. 25 913 110 CALL SYMBOL(XPLOT(I), YPL0T4(I), .08. 11. 0.. 9 14 C 915 • DO 120 I = 5. STEPS, 25 916 120 CALL SYMBOL(XPLOT(I). YPL0T5(I). .08, 0, 0.. 917 YPLOT1(55) = YPLOT1(55) + .3E0 918 YPL0T2(55) = YPL0T2(55) + .1E0 919 YPL0T3(55) = YPL0T3(55) + . 1E0 920 YPL0T4(55) = YPLOT4(55) + .1E0 921 YPL0T5(55) = YPL0T5(55) + . 1EO 922 CALL SYMB0L(XPL0T(55), YPL0T1(55). .15, 'O-MM 923 CALL NUMBER(XPL0T(55), YPL0T2(55), .15. 1.. 0 924 CALL NUMBER(XPL0T(55), YPL0T3(55), .15. 2.. 0 925 CALL NUMBER(XPL0T(55). YPL0T4(55). .15. 5., 0 926 CALL NUMBER(XPL0T(55), YPL0T5(55). .15, 10.. 927 C 928 CALL PL0T(16., 0., -3) 929 C 930 C PL0T/C3 931 C 932 CALL PCIRC(7.. 6.5, .63, 0) 933 CALL PCIRC(7.. 6.5. .75. 0) 934 CALL PL0T(7..6.5.3) 935 CALL PL0T(6.38. 5.88. 2) 936 CALL PL0T(7.. 6.5. 3) 937 CALL PL0T(7.88. 6.5. 2) 938 CALL PL0T(7.63, 6.5. 3) 939 CALL PL0T(6.55. G.05, 2) 940 CALL SYMBOL(7.94, 6.5. .19, 'B'. 0.. 1) 941 CALL SYMB0L(6.3, 5.63. .19. 'A'. 0.. 1) 942 Nl = FNODE - 1 943 C 944 DO 130 I = 1. Nl 945 YTHRUI(I) - SNGL((T(CSTEPS - 1,1) - TA)/(TG - TA)) 946 YTHRU2(I) = SNGL((T(STEPS - 1.1) - TA)/(TG - TA)) 947 130 CONTINUE 948 C 949 DO 140 I = 1. Nl 950 XPLOT(I) = SNGL(R(I) - R(1)) 951 140 CONTINUE 952 C 953 CALL SCALE(YTHRU1. Nl, 6., YMIN, DY. 1) 954 CALL SCALE(XPLOT. Nl. 8., XMIN. DX, 1) 955 C 956 DO 150 I = 1, N1 957 ' YTHRUI(I) = YTHRU1(1) + 2. 958 YTHRU2U) = ((YTHRU2(I) - YMIN)/DY) + 2. 959 XPLOT(I) = XPLOT(I) +2. * 960 150 CONTINUE 961 C 962 CALL AXIS(2.. 2.. '(T-TA )/(TG-TA)', 14. 6.. 90.. YMIN. DY) 9G3 CALL AXIS(2.. 2.. 'DISTANCE FROM WALL SURFACE (MM)'. -31. 8., O.. 964 1 XMIN, DX) 965 CALL PL0T(2.. 8.. 3) 966 CALL PLOT( 10. . 8. . 2) 967 CALL P L O T ( 1 0 , 2 . 1 ) 968 CALL SYMB0L(2., 8.1. .15. 'PLOT 3-'. 0.. 7) 969 CALL NUMBER(2.92. 8.1, .15. RID, O.. -1) 970 CALL LINE(XPL0T. YTHRU 1, Nl. 1) 971 CALL LINE(XPLOT, YTHRU2. N1, 1) 972 YTHRU1(4) = YTHRU1(4) + .IEO 973 CALL SYMB0L(XPL0T(4). YTHRU1(4). .19. 'A'. O.. 1) 974 YTHRU2(4) = YTHRU2(4) - .25EO 975 CALL SYMBOL(XPL0T(4). YTHRU2(4), .19. 'B'. 0., 1) 976 CALL PLOTND 977 RETURN 978 END ro 979 £*»••*•**•* 980 C 981 C SUBROUTINE TO PRINT HEADER 982 C 983 £*»**•**»** 984 SUBROUTINE HEADER 985 L0GICAL*1 STARS(34) /34* ' * ' / 986 WRITE (6. 10) STARS. STARS 987 10 FORMAT ('1'. T48. 34A1/T48. UNIVERSITY OF BRITISH COLUMBIA *' 988 1 T48. '* METALLURGICAL ENGINEERING *'/T48. 989 2 '• WALL.PROFILES VERSION 1 •'/T48. 34A1//) 990 RETURN 99 1 ENO 992 C*.*«*««*** 993 C 994 C SUBROUTINE TO SET 9700 OUTPUT 995 C 996 997 SUBROUTINE P9700UD) 998 LOGICAL* 1 A(132) 999 INTEGER*4 CNT. PG 1000 INTEGER*2 LEN 1001 REWIND 6 1002 CALL GETLSTC6 '. CNT) 1003 CNT = CNT / 1000 1004 PG • 1 1005 NUMB = 1 1006 DO 20 I " 1. CNT 1007 CALL READ(A. LEN, O. LNUMB, 6) 1008 L " 0 1009 CALL FINDST(A, 2. '1'. NUM8. L + 1. L, 810) 1010 CALL PAGE(LNCK, ID. PG. A) 1011 10 LNCK » LNCK + 1 1012 IF ((LNCK - 60) .EQ. 0) CALL PAGE(LNCK. ID. PG, A) 1013 CALL WRITE(A. LEN, 0. LNUMB, 8) 1014 20 CONTINUE 1015 RETURN 1016 END r o r o o 1017 £ * • • • * * * * • * 1018 C 1019 C SUBROUTINE TO PRODUCE PAGE NUMBERS 1020 C 1021 C********** 1022 SUBROUTINE PAGE(LNCK, ID, PG, A) 1023 LOGICAL* 1 A ( 1 3 2 ) . IBLANK. FMT(37). F I ( 3 ) 1024 INTEGERM PG 1025 • DATA IBLANK /' '/. FI /ZF 1 , ZF2, ZF3/ 1026 DATA FMT /Z4D. Z/D. ZF1. Z7D. Z6B. ZE3. ZF1. ZF2. ZF1, Z6B, Z7D, 1027 1 ZD7, ZC1, ZC7, ZC5, Z40. Z7D, Z6B. ZC9. ZF3, Z6B, Z70, Z60, 1028 2 Z7D, Z6B, ZC9. ZF2. Z61, Z61. Z61. Z61. Z61, Z61, Z61. Z61. 1029 3 Z61. Z5D/ 1030 IF (PG .LE. 9) FMT(27) = FI(1) 1031 IF (PG .GE. 10 .AND. PG .LE. 99) FMT(27) = FI(2) 1032 IF (PG .GE. 100) FMT(27) = FI(3) 1033 WRITE (8,FMT) ID. PG 1034 A(1) = IBLANK 1035 PG » PG + 1 1036 LNCK » 10 1037 RETURN 1038 END ro 1039 c » * * » . * * « * * 1040 C 104 1 C SUBROUTINE TO PREDICT HEAT FLOWS USING MODIFIED ANALOG-1042 C WITH CONVECTIVE BRANCHES 1043 C 1044 (;«**••«*»•*• 1045 SUBROUTINE ANAL 1 1046 IMPLICIT REAL*8(A - H,0 - Z) 1047 REAL*8 K, K l , KP 1048 EXTERNAL FN5 1049 DIMENSION X(4). F(4), ACCEST(4) 1050 COMMON /BLK1/ TA, TS. TG, AW. AS. EMW. EMG, EMS 105 1 . COMMON /BLK2/ RI, RO, DT. DR. RPM 1052 COMMON /BLK3/ KP, K. THETA, FD. ALPHA, ESH, K l , RF 1053 COMMON /BLK4/ HEX, HCOV. HOUT, HSE. HOUT2 1054 TR " 1.D0-EMG 1055 RS = 1 .DO-EMS 105G SIGMA - 5.67D-08 1O07 ES • SIGMA • TS * * 4 1058 EG = SIGMA * TG ** 4 1059 EA = SIGMA * TA ** 4 1060 Tl = AW * EMG 1061 .T2 = AS * TR 1062 T3 = AS * EMG 1063 T4 = (EMS*AS) / RS 1064 G = T3 + T2 + T4 1065 EAD = (T1 * EG + T2*((T3*EG + T4*ES)/G)) / (Tl + T2 - (T2**2)/G) 10G6 X(4) = ES 1067 X(3) = EAD 1068 X(2) = ((X(3) - .25D0*(X(3) - EA))/SIGMA) ** . 25D0 1069 X(1) = ((X(3) - .75DO*(X(3) - EA))/SIGMA) ** .2500 1070 CALL NDINVT(4. X. F. ACCEST; 5000. 5.D-04. FN5. &20) 107 1 C 1 » TA / X(1) 1072 C2 = 1.D0+C1 + C1 ** 2 + C1 ** 3 1073 HO = HOUT + (C2*SIGMA*X(1)**3) 1074 OL » HO » 2.DO • RO • PI(O.DO) * (X(1) - TA) 1075 HSEP = HSE / (((TG + TS)*(TG**2 + TS**2))*SIGMA) 1076 HCOVP = HCOV / (((X(2) + TS)*(X(2)**2 + TS•*2 ) )*SIGMA) 1077 ACOV = (2.D0*PI(0.D0)*RI) - AW 1078 OS * (X(4) - ES) * AS * EMS / RS + (SIGMA*X(2)**4 - ES) * HCOVP * 1079 1ACOV + (EG - ES) * HSEP * AS 1080 WRITE (6.10) X(2). X ( l ) , OS. OL 1081 10 FORMAT (' '. 'MODIFIED ANALOG:(INCLUDES CONVECTIVE BRANCHES)'/T8. 1082 1 'EXPOSED WALL TEMPERATURE''. G12.5. ' (K)'/T8. 1083 2 'OUTER SHELL TEMPERATURE'', G12.5//T8. 1084 3 'HEAT RECEIVED BY SOLIDS''. G12.5. ' (W/M)'/T8, 1085 4 'HEAT LOSS TO SURROUNDINGS'', G12.5///) 1086 GO TO 40 1087 20 WRITE (6.30) 1088 30 FORMAT ('1'. 'SOLUTION TO MODIFIED ANALOG FAILED (INCL CONV)') 1089 40 RETURN ' 1090 END 1091 C* * *«*•**•* 1092 C 1093 C EXTERNAL SUBROUTINE FOR ANAL 1 1094 C 1095 c********** 1096 SUBROUTINE FN5(X, F) 1097 IMPLICIT REAL*8(A - H.O - Z) 1098 REAL*8 K, K l . KP 1099 DIMENSION X(1). F(1) 1100 COMMON /BLK1/ TA. TS. TG. AW. AS, EMW. EMG. EMS 1101 COMMON /BLK2/ RI. RO. DT. DR, RPM 1102 COMMON /BLK3/ KP. K. THETA. FD, ALPHA, ESH. K l , RF 1103 COMMON /BLK4/ HEX, HCOV. HOUT, HSE. H0UT2 1104 TR = 1.D0-EMG 1105 RS = 1.DO-EMS 1106 RW = 1.D0-EMW 1107 SIGMA = 5.67D-08 1108 ES = SIGMA * TS *• 4 1109 EG « SIGMA * TG •• 4 1 1 10 EA = SIGMA • TA *• 4 1111 T l = (1.D0-EMW) / (EMW*AW) 1112 T2 = (DL0G(R0/RI)*(X(1) + X(2 ) ) *(X( 1 )* *2 + X(2)**2)) / (2.DO*PI(0. 1113 1DO)*KP) * SIGMA 1114 T3 = 1.DO / (EMG*AW) 1115 T4 = 1.DO / (AS'TR) 1116 T5 * 1.DO / (EMG*AS) 1117 T6 = (1.DO-EMS) / (EMS*AS) 1118 C1 = TA / X(1) 1119 C2 = 1.D0+C1 + C1 ** 2 + C1 •• 3 1120 HO = HOUT + (C2*SIGMA*X(1)**3) 1121 HOP = HO / (((X(1) * TA)*(X(1)**2 + TA* *2)) *SIGMA) 1122 HCOVP = HCOV / (((X(2) + TS)*(X(2)**2 + TS**2))*SIGMA ) 1123 HSEP » HSE / (((TG + TS)*(TG**2 + TS**2))*SIGMA) 1124 HEXP = HEX / (((TG * X(2))*(TG**2 + X(2 ) **2 ) ) *SIGMA) 1125 T15 = 1.D0 / (HEXP*AW) 1126 ASH = 2.DO * PI(O.DO) • RO 1127 T8 = 1.00 / (HOP*ASH) 1128 ACOV = (2.DO*PI(O.DO)*RI) - AW 1129 T9 = 1.D0 / (HCOVP*ACOV) 1130 TIO • 1.00 / (HSEP*AS) 1131 F(1) =• (EG - X(3)) / T3 + (SI GMA * X (2 ) * *4 - X(3)) / Tl + (X(4) - X( 1 132 13)) / T4 1 133 T i l = SIGMA * X(2) ** 4 1134 F(2) " (X(3) - T i l ) / T1 + (ES - T1 1) / T9 + (SIGMA*X( 1)* *4 - T11) 1 135 1 / T2 + (EG - T11) / T15 1136 F(3) = (ES - X(4)) / T6 + (EG - X(4)) / T5 + (X(3) - X(4)) / T4 1137 T 12 - T2 + T8 1138 F(4) = (SIGMA'X(2)**4 - EA) / T12 - (SIGMA * X( 1 )* + 4 - EA) / T8 1139 RETURN oo 1140 END ********* 1 142 C 1143 • C SUBROUTINE TO PREDICT MEAT FLOWS USING MODIFIED ANALOG-1144 C RADIATION ONLY 1145 C 1146 £•••***••*• 1147 SUBROUTINE ANAL2 1148 IMPLICIT REAL*8(A - H,0 - Z) 1 149 REAL*8 K. K1, KP 1150 DIMENSION X(4), F(4), ACCEST(4) 1151 EXTERNAL FN 1, FN2, FN3, FN4 1152 COMMON /BLK1/ TA, TS, TG. AW. AS, EMW. EMG. EMS 1153 COMMON /BLK2/ RI. RO. DT, DR, RPM 1154 ' COMMON /BLK3/ KP. K, THETA. FD, ALPHA, ESH, KI, RF 1155 COMMON /BLK4/ HEX, HCOV. HOUT. HSE. H0UT2 1156 COMMON /BLK16/ EG. ES. EA. ASH1. HOUTP. H0UTP1 1157 OLOSS(TA.TSH.HOUTPL.ASH) = ASH * HOUTPL • (TSH - TA) 1158 SIGMA * 5.67D-08 1159* ASH » 2.DO • Pl(O.OO) • RO 1160 10 CONTINUE 1 IG1 TR = 1.OO-EMG 1 162 RS = 1.DO-EMS 1 163 . RW = 1.DO-EMW 1 164 EG = SIGMA • TG ** 4 1165 ES = SIGMA * TS * * 4 1 166 EA = SIGMA * TA * * 4 1167 T1 = AW * EMG 1160 T2 • AS • TR 1169 T3 = AS * EMG 1170 IF (EMS .LT. .9999) GO TO 20 1171 EAD = (T1*EG + T2*ES) / (T1 + T2) 1 172 GO TO 30 1173 20 T4 = (EMG*AS) / RS 1174 G = T3 + T2 + T4 • 1175 EAD = (T1*EG + T2*((T3*EG + T4*ES)/G)) / (T1 + T2 - (T2**2)/G) 1176 30 IF (EMS .GE. .999DO .AND. EMW .GE. .99900) GO TO 40 1177 IF (EMS .GE. .99900 AND. EMW .LT. .999D0) GO TO 50 1178 IF (EMS .LT. .999D0 .AND. EMW .GE. .999D0) GO TO 60 1179 X(4) * SIGMA • TS ** 4 1180 X(3) » EAO 1181 X(2) = ((X(3) -. ,25D0*(X(3) - EA))/SIGMA) ** . 25DO 1182 X(1) = ((X(3) - .7500*(X(3) - EA))/SIGMA) *• .25DO 1183 CALL NDINVT(4, X, F. ACCEST. 5000. 5.0-04. FN1, &80) 1104 HOUTPL = HOUTP1 1185 OL = 0LOSS(TA.X(1).HOUTPL.ASH) 1186 QS = (X(4) - ES) * ((AS*EMS)/RS) 1187 WRITE (6.70) X(2). X ( i ) , OS, OL 1 188 GO TO 100 ro 1189 40 X(2) = (E AD/SI GMA) . 25DO -P* 1190 X(1) = X(2) - .7500 * (X(2) - TA) 1191 CALL NDINVT(2. X. F. ACCEST. 5000. 5.D-04. FN2, &80) 1192 HOUTPL = HOUTP1 1193 OL = OLOSS(TA.X(I),HOUTPL.ASH) 1194 OS » ((SIGMA*X(2)*»4 - ES)*AS*TR) + ((EG - ES)»EMG*AS) 1195 WRITE (6,70) X(2). X(1). OS. OL 1196 GO TO 100 1197 50 X(3) = EAD 1198 X ( 2 ) = ( ( X ( 3 ) - . 25D0* (X (3 ) - EA) ) /S IGMA) * * 25D0 1199 ' X(1) = ((X(3) - .75D0'(X(3) - EA))/SIGMA) ** .2500 1200 CALL NDINVT(3, X. F. ACCEST, 5000, 5.D-04. FN3. &80) 1201 HOUTPL = HOUTP1 1202 OL = OLOSS(TA,X(1),HOUTPL,ASH) 1203 OS = ((X(3) - ES)*AS*TR) + ((EG - ES)'EMG*AS) 1204 WRITE (6.70) X(2). X(1). OS. OL 1205 GO TO 100 1206 60 X(3) = SIGMA * TS 4 1207 X(2) = (EAD/SIGMA) ** .25D0 1208 X(1) = X(2) - .75D0 + (X(2) - TA) 1209 CALL NDINVTO. X. F. ACCEST. 5000. 5.D-04. FN4, 8.80) 1210 HOUTPL = HOUTP1 1211 OL = OLOSS(TA,X(1).HOUTPL,ASH) 1212 OS = (X(3) - ES) * ((AS*EMS)/RS) 1213 WRITE (6,70) X(2), X(1). OS. OL 1214 70 FORMAT (' '. 'MODIFIED ANALOG:(RADIAT ION 0NLY)'/T8, 1215 1 'EXPOSED WALL TEMPERATURE''. G12.5. ' (K)'/T8, 1216 2 'OUTER SHELL TEMPERATURE'', G12.5//T8, 1217 3 'HEAT RECEIVED BY SOLIDS''. G12.5. ' (W/M)'/T8. 1218 4 'HEAT LOSS TO SURROUNDINGS''. G12.5/'1') 1219 GO TO 100 1220 80 WRITE (6.90) 122 1 90 FORMAT ('1', 'SLOUTION TO MODIFIED ANALOG FAILED (NO CONV)') 1222 . 100 RETURN 1223 END ro ro cn 1224 c***«***«*« 1225 C 1226 C EXTERNAL SUBROUTINE FOR ANAL2 1227 C 1228 £**•*•***•» 1229 SUBROUTINE FN1(X, F) 1230 IMPLICIT REAL*8(A - H,0 - Z) 1231 REAL *8 K. KI. KP 1232 DIMENSION X(1). F(1). T(9) 1233 COMMON /BLK 1 / TA, TS. TG. AW. AS, EMW, EMG. EMS 1234 COMMON /BLK2/ RI. RO. DT, DR, RPM 1235 COMMON /BLK3/ KP, K. THETA, FD, ALPHA. ESH, KI, RF 1236 . COMMON /BLK4/ HEX. HCOV. HOUT, HSE, H0UT2 1237 ' COMMON /BLK16/ EG. ES. EA. ASH. HOUTP, HOUTPL 1238 SIGMA = 5.67D-08 1239 CALL TEMP(X. T) 1240 F(1) = (SIGMA*X(2)**4 - X(3)) / T(1) + (EG - X(3)) / T(3) + (X(4) 1241 1- X(3)) / T(4) 1242 F(2) = (EG - X(4)) / T(5) + (ES - X(4)) / T(6) + (X(3) - X(4)) / 1243 1T(4) 1244 F(3) = (HOUTP*ASH*SIGMA*(X(1)**4 - TA**4)) - (T(7)*(X(3) - EA)) 1245 F(4) = (2.*PI(O.)*KP*SIGMA*(X(2)**4 - X(1)**4)) / (DLOG(RO/RI)*T( 1246 19)) - (T(7)*(X(3) - EA)) 1247 RETURN 1248 END 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 C EXTERNAL SUBROUTINE FOR ANAL2 C EMW, EMG. EMS 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. COMMON /BLK2/ RI. RO, DT, DR, RPM COMMON /BLK3/ KP, K. THETA. FD. ALPHA, ESH 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)) 1- TA**4)) F(2) = (EG - SIGMA»X(2)**4) / T(3) + (ES -1 (SIGMA*(X( 1)**4 - X(2)**4)) / T(2) RETURN END KI , RF - (T(7)*SIGMA*(X(2)**4 SIGMA*X(2)**4) / T(4) + r o r o c n 1271 Qtt*t****** 1272 C 1273 C EXTERNAL SUBROUTINE FOR ANAL2 1274 C 1270 £****•*»••* 1276 SUBROUTINE FN3(X, F) 1277 IMPLICIT REAL '8(A - H.O - Z) 1278 REAL *8 K, K1 , KP 1279 DIMENSION X(1), F(1). T(9) 1280 COMMON /BLK 1 / TA, TS, TG, AW. AS. EMW. EMG. EMS 128 1 COMMON /BLK2/ RI. RO. DT. DR. RPM 1202 COMMON /BLK3/ KP. K, THETA. FD, ALPHA. ESH. K l . RF 1283 COMMON /BLK4/ HEX. HCOV. HOUT. HSE, HOUT2 1284 COMMON /BLK 16/ EG. ES. EA, ASH. HOUTP, HOUTPL 1285 SIGMA = 5.67D-08 1286 CALL TEMP(X, T) 1287 F(1) = ( (SIGMA*X( 1 )*M - X(3))/(T(1) + T(2))) + (EG - X(3)) / T(3) 1288 1 + (ES - X(3)) / T(4) 1289 F(2) = (HOUTP*ASH«SIGMA«(X(1)*»4 - TA'M)) - (T(7)'(X(3) - EA)) 1290 F(3) = ((2.00*PI(O.DO)*KP*SIGMA*(X(2)**4 - X(1)**4))/(OLOG(RO/RI)* 1291 1T(9))) - (T(7)*(X(3) - EA)) 1292 RETURN 1293 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* * * * * * * * * * * EMW, EMG, EMS Z) T(9) TG. AW, AS, DT. DR. RPM THETA. FD. ALPHA, ESH, Kl HCOV. HOUT, HSE, HOUT2 ES, EA. ASH. HOUTP, HOUTPL SUBROUTINE FN4(X, F) IMPLICIT REAL*8(A - H.O REAL*8 K, K1, KP DIMENSION X(1), F( 1) COMMON /BLK1/ TA. TS COMMON /BLK2/ RI. RO COMMON /BLK3/ KP, K. COMMON /BLK4/ HEX, COMMON /BLK16/ EG, SIGMA » 5.67D-08 • CALL TEMP(X. T) F( 1) » (EG - X(3)) / T(5) 1X(3)) / T(4) F(2) = (HOUTP*ASH*SIGMA*(X( 1)**4 -14 - EA) F(3) » (2.DO*PI(O.DO)*KP*SIGMA*(X(2)* 1*T(9)) - (T(7)*(SIGMA*X(2)**4 - EA)) RETURN END RF • (ES - X(3)) / T(6) + (SIGMA*X(2)**4 TA**4)) - T(7) • (SIGMA*X(2)< X( 1 )**4)) / (DLOG(RO/RI ) ro ro 1318 C • • » • * * * + * * 1319 C 1320 C SUBROUTINE TO CALCULATE NODAL TERMS FOR ANAL2 132 1 C 1322 C 1323 SUBROUTINE TEMP(X, T) 1324 IMPLICIT REAL*8(A - H,0 - Z) 1325 REAL*B K, K1, KP 1326 DIMENSION T( 1), X( 1) 1327 COMMON /BLK1/ TA. TS. TG, AW. AS. EMW. EMG. EMS 1328 COMMON /BLK2/ RI. RO. DT. DR, RPM 1329 COMMON /BLK3/ KP, K. THETA, FD. ALPHA, ESH, Kl , RF 1330 COMMON /BLK4/ HEX. HCOV, HOUT, HSE, HOUT2 133 1 COMMON /BLK 16/ EG, ES, EA, ASH. HOUTP, HOUTPL 1332 TR = 1.D0-EMG 1333 SIGMA = 5.67D-08 1334 EG = SIGMA * TG ** 4 1335 ES = SIGMA • TS •* 4 1336 EA = SIGMA • TA 4 1337 ASH = 2.DO • PI(O.DO) * RO 1338 T(1) = (1.D0-EMW) / (EMW*AW) 1339 T(9) = (X(1)**2 + X(2)»*2> * (X(1) + X(2)) * SIGMA 1340 T(2) = (DLOG(RO/RI)*T(9)) / (2.DO*PI(0.DO)* KP) 134 1 T(3) = 1.DO / (EMG*AW) 1342 T(4) = 1.DO / (TR * AS) 1343 T(5) = 1.DO / (EMG*AS ) 1344 T(6) = (1.DO-EMS) / (EMS*AS) 1345 CI - TA / X(1) 1346 C2 = 1.DO+C1 + C 1 * * 2 * C 1 * * 3 1347 HOUTPL = HOUT + (C2*SIGMA *X(1)* *3) 1348 HOUTP = HOUTPL / (((X(1)**2 • TA**2)*(X(1) + TA))* 1349 T(8) = 1.D0 / (HOUTP*ASH) 1350 T(7) = 1.00 / (T(1) + T(2) + T(8)) 1351 RETURN 1352 END ro ro CO 420 (K) 1060 298 "w 03 w w w cv cv, w->s out cv 0.8 0.8 0. 24 1.75 (m) 1.98 2 (RPM) .64 1. (W/m K) 800 (J/kgr K) 1800 (kgr/m3) 20 (W/m2 K) 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) 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 (M) ( M) (K ) THF. UNSTEADY STATE TEMPERATURE FIELD FOLLOWS: 1 2 3 4 5 6 7 8 9 t 035 . 06 19 845 . 05 15 055 . 9742 86 1 . 5882 863 . 298 1 862 . 5368 860. 4649 857 . 8585 855. 1468 2 840. 8499 845 . 2780 855. 2836 86 1 . 0801 862 . 9764 862 . 3660 860. 3951 857 . 8446 855. 1573 3 0 14 . 5 100 846 . 004 7 854 . 7361 860. 5726 862 . 6500 862 . 1888 860. 3195 857 . 8266 855. 1655 4 847 . 4 156 846 . 9513 854 . 3597 860. 0834 862 . 3 195 862 . 0054 860. 2381 857 . 8045 855. 1713 5 04 9 . 7555 847 . 9766 854 . 14 08 859 . 6296 86 1 . 9875 861 . 8161 860. 1512 857 . 7783 855 . 1746 6 85 1 . 7259 849 . 0111 854 . 0533 859. 2222 86 1 . 6583 86 1 . 62 15 860. 0588 857 . 7481 855. 1755 7 853'. 4 297 850. 02 12 854 . 0702 858. 8667 861 . 3364 86 1 . 4228 859. 9613 857. 7138 855. 1738 0 054 . 9331 850. 9922 854 . 1678 858 . 5640 861 . 0261 861 . 22 12 859. 8588 857. 6755 855 . 1696 9 856 . 2005 85 1 . 9188 854 . 3271 858 . 3124 860. 73 10 861 . 0184 859. 7518 857. 6333 855 . 1627 10 857 . 5026 852 . 8003 854 . 5326 858. 1086 860. 4536 860. 8160 859. ,6407 857. 5873 855. 1533 1 1 850 . 62 19 853 . 6300 854 . 7729 857 . 9486 860, . 1956 860. 6 158 859 . 5263 857 . 5376 855 . 14 12 12 , 051. 6549 854 .4346 855 , .0380 857 . 8279 859 . 9578 860. , 4 192 859. 4091 857 . 4843 855. 1265 13 060. 6 144 855 . 1928 855 . 3234 857 . 7422 859 . 7406 860. , 2275 859. , 2899 857 . 4278 855 . , 1092 14 OG 1 . 5 103 855 .9156 855. 62 14 857. .6876 859, .5439 860. ,0420 859. 1694 857. 3682 855 . 0893 15 862 . 3508 856 . 6055 855 . 9288 857 . 6605 859 , .3672 859. , 8635 859 . 0484 857. 3058 855 . 0669 16 863 . 1423 857 . 2653 856 . 2425 857 . 6573 859 , . 2097 859 , 6928 858 . ,9275 857 . 241 1 855 . C421 17 863. 8904 857 . 897 1 856 , 5599 857 . 6753 859, .0705 859 . 5302 858 . 8075 857. 1742 855. ,0150 10 864 5995 858 .5031 856 .8792 857 . 7119 858 .9488 859 , . 3762 858 , 6888 857 . , 1056 854 . 9856 19 865 . 2735 859 .0851 857 . 1990 857 . 7646 858 .8434 859 .2311 858 , .5721 857 . ,0356 854 . 9542 ?0 065 . 9 158 859 .64 50 857 .5182 857 .8315 858 .7535 859 .0948 858 .4578 856. ,9645 854 . 9209 2 1 8G6 . 5292 860 . 1843 057 . 8359 857 . 9 108 858 .6779 858 .9674 858 . 3462 856. .8927 854 . 8858 22 007 . 1 162 86D . 704 4 858 . 1513 858 .0010 858 .6 158 858 .8488 858 . 2377 856 . 8205 854 . 8491 23 867 . 6790 86 1 .2065 858 . 464 1 858 . 1006 858 .566 1 858 . 7389 858 . 1327 856. .7482 854 . 8109 0 24 868 .2194 861 .6918 858 . 7737 858 . 2085 858 .5280 858 .6374 858 .0313 856, .6761 854 , .77 15 25 868 . 7393 862 . 1615 859 .0800 858 . 3237 858 . 5007 858 . 5442 857 .9337 856 .6044 854 , . 7309 26 869 . 2400 862 .6163 859 . 3826 858 . 4452 858 .4833 858 . 4590 857 .8400 856 .5335 854 , .6895 27 869 .7230 863 .0574 859 . 68 14 858 . 5722 858 . 4752 858 .3816 857 .7504 856 .4634 854 .6472 28 870 '. 1895 863 . 4853 859 .9764 858 . 7039 858 .4756 858 .3116 857 .6650 856 .3944 854 .6044 29 870 . 6405 863 . 9010 860 . 2674 858 . 8398 858 .4840 858 .2487 857 . 5836 856 .3266 854 .561 1 30 07 1 .0772 8G4 . 3050 860 . 55-15 858 . 9793 858 .4997 858 . 1927 857 . 5065 856 .2603 854 .5175 3 1 87 1 . 5002 864 .698 1 860 . 8375 859 .12 19 858 .5221 858 . 1433 857 .4335 856 . 1955 854 .4737 32 87 1 . 9 106 865 .0807 86 1 . 1 166 859 . 2671 858 .5508 858 . 1002 857 .3647 856 . 1324 854 .4300 33 872 .3090 865 .4535 86 1 .3917 859 .4 146 858 .5854 858 .0630 857 .2999 856 .071 1 854 .3863 3 1 872 . 696 1 865 .8168 86 1 .6629 859 .5639 858 .6253 858 .0316 857 .2302 856 .01 16 854 .3428 35 873 .0725 866 . 1713 86 1 .9303 859 . 7 149 858 .6701 858 .0057 857 . 1825 855 .9540 854 .2996 36 87 3 .4389 866 .5172 062 . 1938 859 .8672 858 .7 196 857 .9850 857 . 1296 855 . 8985 854 .2569 37 873 . 7957 866 .8551 862 . 4536 860 .02Q6 858 . 7733 857 . 9692 857 .0806 855 .8449 854 .2147 30 074 . 1434 867 . 1853 862 . 7096 860 . 1748 858 .8309 857 .9581 857 .0353 855 . 7934 854 . 173 1 39 871 . 4025 867 .5081 862 . 962 1 860 . 3298 858 .8922 857 .9516 856 .9937 855 . 744 1 854 . 1321 10 074 .8 135 067 .8239 863 . 2 1 10 860 .4852 858 .9568 857 .9492 B56 .9556 855 .6968 854 .0919 ro 4 1 875 . 1367 868 . 1329 863 . 4564 860 . 6409 859 .0245 857 .9510 856 .9210 855 .6516 854 . 0525 co ro •12 875 . 1524 868 .4355 863 . 6985 860 . 7969 859 .0951 857 .9566 856 . 8897 855 . 6086 854 .0140 4 3 075 . 76 11 868 . 7319 063 . 9372 860 .9530 859 . 1683 857 . 9658 856 . 86 18 855 . 5677 853 .9763 4 4 87G . 0G30 869 .0224 864 . 1726 86 1 . 1090 859 . 2440 857 . 9786 856 .8369 855 . 5289 853 .9397 876 3584 869 . 3073 864 . 4019 46 876 . 6 177 869. 5866 864 . 634 1 4 7 876 . 9310 869. 8607 064 . 8602 4 8 877 . 2087 870. 1298 865. 0834 877 . 4809 870. 3940 865 . 3036 50 877 . 74 78 870. 6535 8G5 . 52 1 1 5 1 878 . 0098 870. 9085 865 . 7357 r.2 878 . 2669 87 1 . 1591 865 . 9477 5 3 878 . 5 193 87 1 . 4056 866 . 1570 54 878 . 7G73 87 1 . 6480 866 . 3638 55 879 . 0109 87 1 . 8865 866 . 5680 56 879 . 2504 872 . 12 12 866 . 7698 57 879 . 4 858 872 . 3522 866 . 9692 58 879 . 7 174 872 . 5797 867 . 1662 59 879 . 9452 872 . 8037 867 . 3609 CO 880. 1695 873 . 0244 867 . 5534 6 1 880. 3902 873 . 24 19 867 . 7 4 37 62 880. 6075 873 . 4563 867 . 9319 63 880. 82 15 873. 6677 868 . 1 179 64 . 88 1 .0324 873 8761 868 . 3019 65 88 1 . , 2401 874 . 0816 868 . 4839 66 88 1 . 4449 874 . 2844 868 . 6640 67 88 1 . 6468 874 . , 4844 868 . 842 1 68 88 1 . 8 159 874 , .6819 869 . 0184 69 882 .0422 874 8768 869. . 1928 70 882 . 2359 075 .0692 869 . 3654 7 1 882 . 427 1 875 . 2592 869 .5363 72 882 .6157 875 .4469 869 . 7054 7 3 882 .8019 875 .6322 869 . 8728 7 4 882 . 9857 875 . 8 154 870 . 0306 75 883 . 1672 875 .9964 870 . 2028 7C. 883 . 3465 876 . 1752 870 . 3654 77 883 . 5236 876 . 3520 870 .5264 78 883 .6985 876 . 5268 870 .6859 79 883 . 87 14 876 .6996 870 .8439 80 884 .0423 876 .8705 87 1 .0005 8 1 884 .2112 877 .0395 871 . 1556 82 884 . 3782 877 .2067 87 1 .3093 83 88 1 . 5 133 877 . 372 1 87 1 .4616 81 884 .7065 877 .5358 • 87 1 .6 126 85 884 . 8680 877 .6977 871 . 7622 86 885 .0277 877 .8580 87 1 .9 105 87 885 . 1858 878 .0167 872 .0576 88 885 . 342 1 878 . 1737 872 . 2033 89 885 . 1969 878 . 3292 872 . 3479 90 885 . 6500 878 .4832 872 . 49 12 9 1 885 .8016 878 .6357 872 .6333 92 885 .9517 878 . 7867 872 . 7743 93 886 . 1002 878 . 9363 872 .9141 94 886 .2474 879 .0845 873 .0528 861 . 2649 859. 3220 857 . 9946 861 . 4 206 859. 4020 858 . 0130 861 . 5760 859. 4839 858. 0359 861 . 7310 859. 5677 858. 0609 861 . 8856 859. 6530 858. 0886 862 . 0398 859. 7399 858. 1 188 862 . 1934 859. 8281 858 . 1514 862 . 3464 859. 9176 858. 1863 862 . 4989 B60. 0083 858 . 2234 862 . 6507 860. 1000 858. 2626 862 . 8018 860. 1927 858 . 3037 862 . 9522 860. 2863 858 . 3467 863 . 1019 860. 3807 858 . 39 15 863 . 2509 860. 4759 858 . 4379 863 . 399 1 860. 5717 858 . 4860 863 . 5466 860. 668 1 858 . 5355 863 . 6932 860. 7650 858. 5865 863 . 839 1 860. 8624 858. 6389 863 . 9842 860. 9603 858. 6925 864 . 1284 861 . 0586 858. 7474 864 . 27 19 861 . 1571 858 . 8034 864 . 4145 86 1 . .2560 858. 8605 864 . 5563 86 1 . .3552 858 . ,9186 864 .6972 861 . 4545 858 . ,9778 864 . 8374 86 1 .5540 859. .0379 864 .9767 861 .6537 859 . ,0988 865 . 1 152 861 . 7535 859 . 1606 865 .2528 86 1 .8534 859. . 2232 865 .3897 861 .9533 859 .2865 865 . 5257 862 .0533 859 .3506 865 . 6609 862 . 1533 859 .4 153 865 . 7952 862 . 2533 859 .4806 865 .9288 862 . 3532 859 .5466 866 .0616 862 .4531 859 .6131 866 . 1936 862 .5529 859 .6801 866 .3247 862 .6527 859 .7477 866 .4551 862 .7523 859 .8157 866 .5847 862 .8518 859 .8842 866 .7 135 862 .9512 859 .9530 866 .84 16 863.0505 860 .0223 866 .9688 863 . 1495 860 .0919 867 .0953 863 .2485 860 . 1619 867 .221 1 863 .3472 860 .2322 867 . 3461 863 .4458 860 . 3028 867 .4704 863 .544 1 860 .3737 867 .5939 863 .6423 860 .4448 867 .7167 863 .7402 860 .5162 867 .8388 863 .8379 860 .5878 867 .9602 863 .9354 860 .6596 868 .0808 864 .0327 860 .7317 856 . 8152 855. 4921 853. 9040 856 . 7964 855. 4575 853. 8694 856 . 7805 855. 4249 853. 8358 856. 7673 855. 3944 853. 8033 856 . 7569 855. 3659 853. 7719 856. 7491 855 . 3394 853 . 74 17 856 . 7438 855 . 3148 853. 7 126 856 . 7409 855. 2921 853 . 6846.. 856 . 7403 855. 2713 853 . 6578» 856. 742 1 855. 2524 853. 6022 856 . 7460 855. 2353 853 . 6078 856. 7520 855. 2200 853. 5845 056 . 760O 855 . 2064 853 . 5624 856 . 7700 855. 1945 853 . 54 14 856. 7820 855. 1843 853 . 52 16 856 . 7957 855. 1757 853 . 5030 856 . 8112 855. 1687 853. 4855 856. 8283 855 . 1633 853. 4692 856. 8472 855. 1594 853. 4540 856. 8675 855. 1570 853. 4399 856 . 8894 855. 1560 853 . 4269 856 . 9128 855 . 1564 853. . 4 150 856 . 9376 855. 1583 853 . 4042 856. .9637 855. , 1614 853. . 3945 856 .991 1 855 . 1659 853 . 3859 857 .0198 855. ,1716 853 . 3782 857 .0497 855 . 1786 853 . 37 16 857 .0808 855 . 1869 853 .3661 857 . 1 130 855 . 1962 853 .3615 857 . 1463 855 .2068 853 . 3579 857 . 1806 855 .2184 853 . 3552 857 . 2 159 855 . 23 12 853 . 3535 857 . 2522 855 .2450 853 .3528 857 .2895 855 .2598 853 .3530 857 .3276 855 .2757 853 . 3540 357 . 3666 855 .2925 853 .3560 857 . 4064 855 .3102 853 .3588 857 .4470 855 .3289 853 . 3625 857 . 4884 855 .3485 853 . 3670 857 .5305 855 .3690 853 .3724 857 .5733 855 .3903 853 .3785 857 .6169 855 .4125 853 .3855 857 .6610 855 .4354 853 .3932 857 . 7058 855 .4591 853 .4017 857 . 7513 855 .4836 853 . 4 109 857 . 7973 855 .5088 853 .4209 857 .8439 855 .5348 853 .4316 857 .8910 855 .5614 B53 .44 30 857 .9386 855 .5888 853 .4551 857 .9868 855 .6167 853 .4678 'IS- 0 86 TOO 079 . 23 13 073 . 1104 >in'i. *.'.) 7 3 079 . 3707 07 3 32G0 97 886 . 6803 879 . 5209 873 . 1623 98 885 . 8219 879. 6637 873. 5966 T l nm 962 1 879. 805 3 873. 7?9f) IOO 087 . 1011 879 : 9456 873 . 0622 10 1 007 . 2309 880. 0847 07 3 . 9935 102 007 . 3753 800. 2226 874 . 12 38 103 807 . 5 106 880. 3593 874 . 2531 104 807 . 6447 880. 4949 874 . 38 15 I05 887.. 7 776 880. 6294 8/4 . 5089 105 887'. 9094 880. 7627 874 . 6354 107 888 . 0401 88O. 8950 87 4 . 76 10 1O0 800 . 1697 80 1 . 026 1 874 . 8857 109 808 . 290 1 88 1 . 1562 875 . 0095 1 IO 000 . 4256 88 1 . 2853 075 . 1324 1 1 1 non. 55 in 80 1 . 4 1 34 075 . 2515 1 12 808 . 6773 88 1 . 5405 075 . 3758 1 13 888 . 8016 88 1 . .6665 875, , 4962 1 11 888 . 9250 88 1 . .7917 875 6 158 1 15 009 . 0474 88 1 . .9158 875 ,7345 1 16 089 1608 882 .039 1 075 . 0525 1 17 089 . 2833 882 . 1614 875 .9697 1 18 889 . 4009 882 . 2828 876 .0862 1 19 889 . 5276 882 .4033 876 .2018 120 889 6-154 802 .5230 876 . 3 168 1? 1 089 . 7623 802 . 64 18 076 . 4 309 122 809 . 8783 882 .7597 876 .5444 123 809 .9935 882 . 8769 876 .657 1 12 1 890 . 1079 882 .993 1 876 .7691 125 890 . 22 14 883 . 1086 876 .8805 126 890 . 334 2 883 . 2233 876 . 99 1 1 127 890 . 446 1 883 . 3372 877 . 101 1 128 890 . 557 3 883 . 4504 877 .2 103 129 882 . 500 1 883 .5627 877 .3190 130 876 .8 137 882 .6216 877 . 4269 13 1 872 . 379 1 80 1 . 1890 877 . 3972 132 068 . 7842 879 . 5483 877 .2016 133 865 .7450 877 .84 15 876 .8539 134 863 .0925 876 . 1383 876 . 3804 135 8 GO . 72 17 874 .4716 875 .8085 1 36 850 . 5652 872 . 8559 875 .16 17 137 056 . 5776 87 1 . 2960 874 .4595 138 854 . 7273 869 . 792 1 873 . 7 169 139 852 . 99 12 868 . 3423 872 .9457 140 85 1 . 3524 866 .9436 872 . 1548 14 1 849 . 7974 865 .5928 871 . 35 1 1 112 848 .3159 864 . 2867 870 . 5398 i n 846 . 8993 863 .0222 869 .7248 144 845 . 5408 86 1 . 7964 868 .9091 060 . 2 OOO 864 . 1297 060. 8O30 HGfl . 3200 864 . 2264 860. 8762 868 . 4386 864 . 3230 860. 9487 868,5565 864. 4192 861 .0213 868. 6737 864 . 5152 061. 094 1 868. 7903 864 . G109 861 . 16G9 868 . 9061 864 . 7064 861 . 2399 869. 02 14 864 . 8016 86 1 . 3 129 869. 1359 864 . 8965 861 . 3861 869 . 2499 864 . 991 1 861 . 4593 869 . 3632 865. 0855 86 1 . 5325 869. 4759 865. 1796 861 . 6058 869. 5879 865 . 2733 861 . 6791 869. 6993 865. 3668 861 . 7525 869 . 8 102 865 . ,4601 86 1 . 8258 869 . 9204 865 . ,5530 861 . ,8992 870. 03 OO 865 . , 6456 8G 1 . ,9726 870. 1390 865 , . 7379 862 , .0460 870. ,2475 865 .8300 862 . 1 194 870. , 3554 865. . 92 17 862 , 1928 870. . 4627 866 .0132 862 . 2661 870. . 5694 866 . 1043 862 . 3394 870. .6755 866 . 1952 862 .4 127 870 .781 1 866 . 2858 862 . 4860 870 .8862 866 .3760 862 . 5591 870 . 9907 866 . 4660 862 .6323 87 1 .0947 866 . 5557 862 .7054 87 1 . 1981 866 .6450 862 . 7784 87 1 . 3010 866 . 734 1 862 .8514 87 1 . 4034 866 .8229 862 .9243 87 1 .5053 866 .9114 862 .9971 87 1 .6066 866 .9995 863 .0699 87 1 . 7075 867 .0874 863 . 1425 87 1 .8078 867 . 1750 863 .2151 87 1 . 9077 867 . 2623 863 .2876 872 .0070 867 . 3493 863 . 3600 872 . 1059 867 . 4360 863 .4324 872 . 1865 867 .5225 863 .5046 872 .2318 867 .6063 863 .5767 872 .2310 867 .6836 863 .6484 872 . 1789 867 . 7499 863 .7190 872 .074 5 867 .8015 863 .7873 87 1 .9198 867 .8349 863 .8521 87 1 . 7 184 867 .8479 863 .9117 87 1 .4744 867 .8390 863 .9648 871 . 1924 867 .8076 864 .0101 870 .8769 867 .7536 864 .0465 870 . 53 18 867 .6773 864 .0730 870 .1611 867 .5795 864 .0890 869 . 7680 867 .4609 864 .0939 050 . 0354 855. 6454 853 . 48 13 858 . 0845 C55 . 6746 853 . 4953 858. 1340 855. 7045 853. 5100 853. 1839 855. 7349 853. 5254 858. 2343 855. 7G59 853. 5413 858 . 285 1 855. 7975 853. 5578 858 . 3362 855 . 8296 853 . 5749 858 . 3877 855 . 8622 853 . 5926 858. 4396 855. 8954 853. 6109 858 . 4917 855. 9290 853. 6297 858 . 5442 855. 9631 853. 6490 858. 5970 855. 9977 853. 6689 858 . 6501 856. 0327 853 . 6892 858. 7035 856. 0682 853 . 7 101 858 . 7572 856 . 104 1 853 . 73 15 858 . 8111 856. , 1404 853 . ,7533 858 . 0652 856. , 1772 853 . 7756 858 . 9196 856 . ,2143 853 . , 7984 858 . 9742 856. 2518 853. .8216 859 . 0291 856. 2897 853. , 8453 859 . ,084 1 856 . 3279 853 .8694 859. , 1393 856 . 3664 853 . 8939 859. , 1947 856. .4054 853 .9188 859 . 2503 856 .4446 853, . 9442 859, .3061 856 . 484 1 853 .9699 859 . 3620 856 .5240 853 .9960 859 . 4 180 856 .5642 854 .0225 859 .4743 856 .6046 854 .0493 859 .5306 856 .6453 854 .0765 859 .5871 856 .6864 854 . 104 1 859 .6437 856 .7276 854 . 1320 859 .7004 856 .7691 854 . 1603 859 .7572 856 .8109 854 . 1888 859 .8142 85G .8529 854 . 2 177 859 .8712 856 .8952 854 .2470 859 .9283 856 .9376 854 .2765 859 .9855 856 .9803 854 .3063 860 .0428 857 .0232 854 .3364 860 . 1001 857 .0663 854 .3668 860 . 1575 857 . 1097 854 .3975 860 .2150 857 . 1532 854 .4284 860 . 2723 857 . 1968 854 . 4597 860 . 3293 857 . 2407 854 . 49 1 1 860 . 3856 857 .2846 854 .5229 860 .4407 857 .3286 854 .5548 ^ 860 .4941 857 .3724 854 .5870 00 860 .5452 857 .4160 854 .6194 860 .5934 857 .4591 854 .65 19 860 .6381 857 .5014 854 .6845 86O .6788 857 .5429 854 . 7 170 145 146 1.17 M a 149 150 151 152 153 154 155 156 157 150 159 160 16 1 044 . 842 . 04 1 . 040. 839 . 830 . 037 . 836 . 835 . 83 1 . 833 . 832 . 83 1 . 830. 829 828 835 2345 9757 7600 5840 4444 3306 264 1 2 189 2009 . 2084 2400 29 12 . 3G97 . 4655 . 5803 . 7 133 . 86 19 860. 059 . 858. 857 . 056 . 855 . 854 . 853. 852 . 85 1 . 850. 849 . 84 8 . 84 7 . 846 . 845 . 845 6068 4009 .3266 ,2319 1650 1242 100 1 . 1 153 1446 . 1947 . 2647 3536 . 4604 . 5844 7247 8806 .05 15 868 867 866 860 064 064 063 862 06 1 86 1 860 859 858 858 857 856 855 .0951 . 2816 .4787 .6787 .8852 .0908 . 3200 . 5489 . 7858 .0309 . 2842 . 5456 . 8 152 .0930 . 3788 . 6726 . 9742 869 . 868 . 860 . 868 , 867 . 867 . 866 . 866. 865 865 864 864 063 863 862 862 86 1 3557 9268 4837 0284 5628 0883 6065 . 1 185 .6254 . 128 1 .6275 . 1242 .6 189 . 1 122 .6046 .0965 . 5882 867 . 867 . 866. 866. 866. 866 . 866. 865. 865. 865 . 865. 864 . 864 . 864 . 863 863 863 3227 1659 9918 .8010 .5961 3767 . 1443 .9000 .6447 3792 . 1043 8209 5296 2311 .9260 .6 148 . 298 1 064 864 864 863 863 863 863 863 863 863 863 863 863 863 862 862 862 .0874 .0693 .0395 .9982 .9455 .8815 .0067 .7212 .6255 .5199 .4049 .2807 . 1478 .0066 .8574 . 7007 .5368 860. 860. 860. 860. 860. 860. 860. 860. 860. 860. 860. 860. 860 860 860 860 8G0 7149 7460 7716 7916 8054 .8 130 .8142 .8088 .7968 . 7782 .7529 .7210 .6825 .6376 .5862 .5286 .4649 10 1 1 12 13 14 15 16 1 852 . 5 172 850. 0206 847 . 6454 845. 3608 843 . 1351 840. 9436 838 . 7691 2 052 . 5343 850. 0362 84 7 . 6570 845 . 3683 843. 1394 840. 9457 838 . 7697 3 852 . 5504 850. 0515 847 . 6686 845 . 3759 843 . 1438 840. 9478 838 . 7705 4 852. 5653 850. 0664 847. 6802 845 . 3836 843 . 1484 840. 9501 838 . 77 13 5 852 . 579 1 850. 0809 847 . 69 17 845 . 39 15 843. 1530 840. 9525 838 . 7722 6 852 . 59 16 850. 0949 847 . 7031 84 5 . 3993 843 . 1578 840. 9549 830 . 7732 7 852 . 6028 850. 1004 047 . 7 14 4 84 5 . 4073 843 . 1627 840. 9575 838 . 7742 8 852 . 6126 850. 1213 847 . 7255 845. 4152 843 . 1676 840. ,9602 838. 7754 9 852 . 6209 850. 1335 847 . 7365 845. 4232 843. 1727 840. ,9630 838. 7766 10 852 6278 850. 145 1 847 . , 7472 845 . 4312 843. 1779 840. 9659 838. ,7779 1 1 852 . 6332 850. 1560 847 . 7577 845 . ,4392 843 . 1831 840. .9688 838. ,7793 12 852 . 6370 850 . , 1661 847 . 7679 845 . 4472 843 . 1884 840. .9719 838 , 7808 13 852 .6393 850. , 1754 847 .7778 845 . , 455 1 843. 1937 840, .9750 838 , .7823 14 852 . 6399 850 . 1838 847 . 7874 845 .4629 843. 199 1 840 .9783 838 , .7840 15 852 . 6388 850. . 19 14 847 . 7966 845 . 4706 843. 2045 840 .9816 838 .7857 16 852 .6361 850 .1981 847 . 8054 845 .4783 84 3. 2100 840 .9849 838 . 7874 17 852 .63 18 850 .2038 847 .8137 845 .4858 843 . , 2 154 840 . 9883 838 . 7893 18 852 . 6258 850 . 2086 847 . 82 17 845 . 493 1 843 . ,2209 840 .9918 838 . 79 12 19 852 .6181 850 . 2 123 847 .829 1 845 . 5003 843 , 2264 840 .9954 838 . 7932 20 852 .6089 850 .2151 847 .8360 845 .507 2 843 , 2318 840 .9990 838 . 7952 2 1 852 .590 1 850 .2168 • 847 .8424 845 .5140 843, .2372 841 .0026 838 .7973 22 852 .5857 850 . 2 175 847 . 8482 845 .5205 843 , . 2425 84 1 .0062 838 .7995 23 852 . 57 19 850 . 2 172 847 . 8535 845 . 5268 84 3 . 2478 84 1 .0099 838 .8017 24 052 . 5567 850 . 2 159 847 . 858 1 845 .5329 843, . 2530 84 1 .0136 838 .8039 25 852 . 5401 850 . 2 135 847 . 8622 845 .5386 843 . 258 1 84 1 .0173 838 .8062 26 852 . 5222 850 .2101 847 .8656 845 .5440 843 .2631 84 1 .02 10 838 .8085 27 052 . 5032 850 . 2057 84 7 . 8684 845 .5492 843 .2680 841 .0247 838 .8109 28 852 .4031 850 . 2004 847 .8706 845 .5539 843 . 2727 84 1 .0284 838 .8133 29 852 .4619 850 . 194 1 847 .8721 845 .5584 843 .2774 84 1 .0320 838 .8157 30 052 . 4398 850 . 1868 847 . 8730 845 .5625 .843 .2818 84 1 .0356 838 .8182 3 1 052 . 4 168 850 . 1707 84 7 .8733 84 5 .5662 843 . 2862 84 1 .0392 833 . 8207 857 857 857 857 857 857 857 857 857 857 857 857 857 857 857 857 857 .5830 .62 16 .6585 .6932 ,7255 .7553 .7822 .8061 .8267 .8439 .8575 .8674 .8736 .8758 .8740 .8683 .0585 836 . 836. 836 . 836 . 836 . 836 . 836 . 836 . 836. 836 . 836 . 836 . 836 . 836 836 836 836 836 836 836 836 836 836 836 836 836 836 836 836 836 036 17 6008 6007 6006 6006 6007 6008 6010 6013 .6016 6019 6024 6028 6034 .6040 .6046 .6054 .6061 .6070 .6078 .6088 .6098 .6108 .6119 .6131 .6143 .6155 .6168 .6182 .6196 .6210 .6224 854 . 854 . 854 . 854 . 854 . 854 . 854 . 854 . 854. 855. 855. 855. 855. 855 855 855 855 7495 78 17 8136 8451 8759 9060 9352 9633. . 9902' .0158 .0400 .0625 .0832 . 1022 .119 1 . 1340 . 1468 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 . 834 834 834 834 834 834 834 834 834 834 834 834 834 834 18 4329 4325 432 1 43 18 43 15 4312 4 309 4307 4305 4304 4303 4303 4 302 4303 4 303 4 304 4 306 .4308 .4310 .4313 .4316 .4319 .4323 .4328 .4332 .4337 .4343 .4349 .4355 .4362 .4369 ro co cn 32 052 . 393 1 850. 1698 847 . 8729 33 852 . 3686 850. 1600 847 . 87 19 34 852 . 3436 850. 1495 847 . 8703 30 852 . 3101 050. 1383 84 7 . 8680 30 852 . 292 1 850. 1263 847 . 8652 37 852 . 2658 850. 1 137 84 7 . 86 18 3n 852 . 2392 850. 1006 847 . 8578 39 852 . 2 124 850. 0868 847 . 8532 40 052 . 1854 850. 0726 847 . 8482 4 1 052 1584 850. 0579 847 . 84 26 4 2 052 . 13 14 850. 04 28 847 . 8366 4 3 052*. 10 14 850. 0273 84 7 . 8300 4 4 052 . 0776 850. 01 15 847 . 823 1 15 052 . 0509 819 . 9954 84 7 . 8 157 4 0 052 . 02 44 849 . 9791 84 7 . 8079 4 7 05 1 9902 849 . 9625 84 7 . 7998 4 0 05 1 9722 049 . 94 50 047. 79 13 49 85 1 .9467 84 9 . 9290 84 7 7825 SO 85 1 .92 15 8 49 9121 84 7 . 7734 5 t 85 1 .8967 849 . 895 1 847 . 7640 52 85 1 . 8724 849 . 878 1 847 . 7543 53 85 1 .8485 849 . 861 1 847 . 7445 54 85 1 .8252 849 . 844 1 847 .7344 55 85 1 . 8024 849 . 8272 847 . 7242 50 85 1 . 7801 849 . 8 104 847 .7 138 5 7 05 1 . 7505 84 9 . 7937 84 7 . 7033 SO OS 1 . 7174 049 . 7772 04 7 . 0920 59 85 1 . 7 169 849 . 7609 847 .68 19 O'J 05 1 . 09 / 1 849 . 7447 04 7 . 67 1 1 0 1 85 1 . 6779 849 . 7288 847 . 6602 02 05 1 . 6593 849 .7 131 847 .6493 03 05 1 .64 14 849 .6976 847 .6384 64 85 1 .6242 849 .6824 847 .6275 65 85 1 .6076 849 .6676 847 .6 166 66 85 1 . 59 18 849 . 6530 84 7 .6058 0 7 05 1 . 5706 84 9 .6307 04 7 . 5950 G8 85 1 . 562 1 84 9 .6248 047 . 5843 on 05 1 . 5403 849 .6112 047 .5737 70 851 .5352 849 . 5980 047 . 563 1 7 1 851 .5228 849 .5851 847 .5527 72 85 1 .5111 849 .5726 847 . 5424 73 85 1 . 5001 849 .5605 847 . 5323 74 05 1 . -1098 0 19 . 5488 04 7 . 5222 75 85 1 . 4801 849 . 5375 847 .5124 7 0 05 1 .4712 0 19 .5266 047 .502 7 77 85 1 .4630 8 19 .5162 847 .4932 78 85 1 . 4554 849 .5061 847 .4839 79 85 1 . 4485 849 . 4965 847 . 4748 OO 85 1 . 4423 849 . 4873 847 . 4059 0 1 85 1 . 4 308 849 . 4785 847 . 4 57 2 845 . 5696 843. 2903 84 1 . 0428 845. 5725 843. 2943 841 . 0462 845. 5751 843. 2980 84 1 . 0496 8 15. 5773 843. 3016 84 1 . 0530 845. 579 1 843. 3050 84 1 . 0562 845. 5805 843 . 3081 84 1 . 0594 845 . 5015 843 . 3 1 10 84 1 . 0625 845. 5821 843. 3137 841 . 0654 845. 5823 843. 3162 84 1 . 0683 845. 582 1 843 . 3 184 84 1 . 07 1 1 845. 58 15 843 . 3203 84 1 . 0737 815. 5805 843. 322 1 84 1 . 0762 845 . 5792 843. 3235 84 1 . .0786 845. 5775 843. 3248 84 1 . ,0808 845 . 5754 843 . 3257 84 1 , ,0829 845 . 5730 843 . 3265 84 1 .0848 845 . 5703 84 3 . 3269 84 1 . 0866 845. 5672 843. 3272 84 1 .0883 845. 5638 843 . 3272 84 1 .0898 845 . 5601 843. 3269 84 1 .0912 845. 5561 843. 3264 84 1 .0923 845. 5519 843 . 3257 841 .0934 845. .5473 843. . 3247 84 1 .0943 84 5 . .5425 843. .3235 84 1 .0950 845. .5375 843 . 3221 84 1 .0956 845 .5323 843. , 3205 84 1 .0960 045 . 52G8 843 . 3 107 84 1 .0962 845 .521 1 843 .3 166 84 1 .0963 845 .5153 843 .3 144 84 1 .0963 845 . 5092 843 .3120 84 1 .0960 845 . 503 1 843 . 3094 84 1 . 0957 845 .4967 843 . 3066 841 .0952 845 .4903 843 .3037 84 1 .0945 845 .4037 843 . 3006 84 1 .0937 845 .4770 843 . 2973 84 1 .0928 045 . 4702 84 3 . 2939 84 1 . 09 1 7 845 . 4634 843 . 2904 84 1 .0905 045 . 4564 843 . 2867 84 1 .0092 845 . 4494 843 . 2829 84 1 .0878 845 .4424 843 .2791 84 1 .0862 845 .4353 843 .2750 84 1 .0845 845 . 4283 843 .2709 84 1 .0827 845 .42 12 843 . 2668 84 1 .0808 845 .4141 843 . 2625 84 1 .0788 045 . 40/0 84 3 .2081 84 1 .0767 845 .3999 843 .2537 841 .0744 845 . 3929 843 . 2493 841 .072 1 845 . 3859 843 .2447 84 1 .0698 845 . 3789 843 . 2402 84 1 .0673 045 . 3720 843 . 2356 84 1 .0648 838. 8232 836 . 6239 834 . 4376 838. 8256 836. 6254 834 . 4384 838 . 8281 836. 6270 834 . 4392 838. 8306 836. 6286 834. 4401 838 . 8331 836. 6302 834 . 4409 838 . 8356 836 . 6318 834 . 44 18 838 . 8380 836 . 6334 834 . 4428 838 . 8404 836.6351 834 . 4437 838. 8428 836. 6367 834 . 4447 838 . 8452 836 . 6384 834 . 4457 838. 8475 836. 6401 834. 4467 838. 8498 836. 6417 834 . 4477 838 . 8520 836 . 6434 834 . 4488 838 . 8542 836. 6451 834 . 4499 838 . 8563 836 . 6467 834 . 45 10 838 . 8584 836 . 6484 834 . 4520 830 . 8603 836 . 6500 034 . 4532 838 . 8622 836 . 6516 834 . 4543 838 . 864 1 836. 6532 834 . 4554 838 . 8658 836 . ,6547 834 . 4565 838. 8675 836 . ,6562 834 .4576 838 . 8691 836. 6577 834. .4587 838 .8706 836. .6592 834 .4598 838 .872 1 836. .6606 834, .4609 838 .8734 836. .6620 834 , .4620 838 . 8746 836 .6633 834 .4631 838 .8758 836 . 6G4G 834 . 4G4 2 838 .8768 836 .6658 834 .4652 838 .8778 836 .6670 834 .4663 838 .8786 836 .6682 834 .4673 838 .8793 836 . 6692 834 .4683 838 .8800 836 .6703 834 .4693 838 .8805 836 .6712 834 .4702 838 .88 10 836 .6722 834 .4712 838 .8813 836 .6730 834 .4 721 838 . 00 16 836 .6738 834 . 4730 838 .88 17 836 .6746 834 .4738 830 .80 10 836 .6752 034 .4746 838 .80 17 836 .6758 834 . 4754 838 .8816 836 .6764 834 .4761 838 .8813 836 .6769 834 .4768 838 .8810 836 .6773 834 .4775 830 . 8806 836 .6776 034 . 4782 838 .8801 836 .6779 834 .4788 838 .8795 83G .6782 034 .4793 838 .8788 836 .6783 834 .4709 838 .8780 836 .6784 834 .4803 838 .8771 836 .6784 834 .4808 838 .8762 836 .6784 834 .40 12 838 .875 1 836 .6783 834 . 48 15 C o c n 02 05 1 . 43 19 849 . 4701 04 7 . 4487 03 85 1 . 4277 849. 4622 847 . 4405 84 851 . 4242 849. 4547 84 7. 4325 05 851 . 42 13 849. 4477 847 . 4247 86 851 . 4 190 849 . 44 1 1 847 . 4172 07 85 1 . 4 174 849 . 4350 847 . 4099 80 85 1 . 4 164 849 . 4293 047 . 4029 89 05 1 . 4 160 849 . 4240 847 . 3962 90 05 1 . 4 162 849 . 4 192 847 . 3897 9 1 05 1 . 117 1 849 . 4 14 8 847 . 3835 92 85 1 . 4 105 849 . 4 109 84 7 . 3776 9 3 05 1 . 4 2Q6 049 . 4074 847 . 37 19 34 85 1 . 4232 849 . 4043 847 . 3666 95 85 1 . .4264 849 . 4017 847 . 36 15 96 851 4 302 849 . 3996 847 . 3567 97 851 4345 849 . 3978 847 . 3522 98 05 1 .4 394 849 . 3965 847 3480 99 851 .4449 849 . 3956 847 . 3440 100 05 1 . 4509 849 . 3952 847 , . 3404 lOI 85 1 . 4574 849 3951 847 . 337 1 102 851 . 4644 849 . 3955 847 . 3340 103 851 . 4720 849 . 3963 847 . 33 13 104 851 .4001 849 . 3976 847 . 3289 105 85 1 . 4007 849 . 3992 847 . 3267 106 851 .4977 849 .4013 847 . 3249 107 05 1 . 507 3 849 . 4037 847 . 3233 100 85 1 .5173 849 . 4066 847 . 322 1 103 05 1 . 5279 849 .4098 847 . 32 1 1 1 10 85 1 . 5388 849 .4 135 847 .3204 1 1 1 05 1 . 5503 849 .4175 847 . 3201 1 12 85 1 . 5622 849 . 4220 847 . 3200 1 13 051 . 5745 849 .4268 847 . 3203 1 14 85 1 .5073 849 .4320 847 . 3208 1 15 85 1 .6005 849 . 4376 84 7 . 32 16 1 16 851 .6141 849 . 4435 847 . 3227 1 17 851 .6282 849 . 4499 847 . 324 1 1 18 851 .6426 849 .4566 847 .3258 1 19 851 .6575 849 .4636 847 . 3278 120 85 1 .6728 849 .4710 847 .3301 121 851 .6884 849 .4788 847 .3327 122 85 1 . 7044 849 . 4069 847 . 3355 123 05 1 . 7209 849 .4954 847 . 3387 124 85 1 . 7376 849 . 504 2 847 . 34 2 1 125 851 .7548 849 .5133 847 . 3458 126 85 1 .7723 849 .5228 84 7 .3498 127 85 1 . 7901 849 .5326 847 . 3540 128 85 1 .0083 849 . 5427 847 . 3586 129 85 1 . 8269 849 .5532 847 . 3634 130 85 1 .8458 849 . 5639 847 . 3685 13 1 851 . 8650 849 .5750 847 . 3738 045 . 3652 843 . 2309 84 1 . 0622 845. 3584 843 . 2263 841.0595 845. 3517 843. 2216 84 1 . 0567 845. 3452 843. 2169 841 ,0539 845. 3387 843.2122 84 1 . 051 1 845 . 3323 843 . 2076 84 1 . 0482 845. 3260 843 . 2029 84 1 . 0453 845. 3 198 843 . 1982 841 . 0423 845 . 3138 843 . 1936 84 1 . 0393 845 . 3079 843 . 1890 84 1 . 0362 845 . 302 1 843 . 1844 84 1 . 0332 845 . 2964 843 . 1799 84 1 . 0301 84 5 . 2909 843 . 1754 84 1 . 0270 845 . 2056 843 . 17 10 84 1 . 0239 845. 2804 843 . 1666 84 1 . 0207 845 . 2754 843 . 1623 84 1 . 0176 845 . 2705 843 . 1580 84 1 . 0145 845. 2658 843 . 1538 84 1 . 0114 845 . 2612 843 . 1497 84 1 . 0083 845 . 2569 843 . 1456 84 1. 0052 845. . 2527 843 . 14 17 84 1 . 0021 845. . 2487 843 . 1378 840. .9990 845 . , 2448 843 . 1340 840. ,9960 845. .24 12 843. . 1303 840. .9930 845 . 2377 843 , . 1267 840, ,9900 8 15 . 2345 843 , 1232 840 .9871 845 . 23 14 843 . 1 198 840 .9842 84 5 . 2285 843 . 1 165 840 .9813 845 . 2258 843 . 1 133 840 .9785 845 . 2233 843 . 1 103 840 .9757 845 .2211 843 . 1073 840 .9730 845 . 2 190 843 . 1045 840 .9703 845 .2171 843 . 1018 840 .9677 845 . 2 154 84 3 .0992 840 .9652 845 . 2 140 843 .0967 840 .9627 84 5 .2127 843 .0944 840 .9602 845 .21 16 843 .092 1 840 .9579 845 .2108 843 .0901 840 .9556 845 .2102 843 .0881 840 .9534 845 . 2097 843 .0863 840 .9512 845 . 2095 843 .0846 840 .9491 845 . 2095 843 .083 1 840 .9471 845 . 2097 843 .0817 840 .9452 845 .2101 843 .0804 840 . 9434 845 .2107 843 .0793 840 .94 16 845 .21 16 843 .0783 840 .9400 845 . 2 126 843 .0775 840 .9384 845 .2139 843 .0768 840 .9369 845 .2153 843 .0763 840 .9355 845 . 2 170 843 .0759 840 .9342 830 . 8740 036 . 6782 834 . 48 19 838 . 8729 836. 6779 834 . 482 1 838. 8716 836. 6777 834 . 4824 838. 8703 836.6773 834. 4826 838 . 8689 836. 6769 834 . 4827 838 . 8675 836. 6765 834 . 4828 838 . 8660 836 . 6760 834 . 4829 838. 8644 836. 6754 834 , 4829 838. 8628 836. 6748 834. 4829 838 . 8612 836. 674 1 834 . 4829 838 . 8594 836. 6734 834 . 4828 838 . 8577 836 . 6727 834 . 4826 838. 8559 836 . 6718 834 . 4825 838 . 8540 836 . 67 10 834 . 4822 838. 8521 836. 6701 834 . 4820 838 . 8502 836 . 6691 834 . 4817 838 . 8483 836 . 6682 834 . ,4814 838 . 8463 836. ,667 1 834 . ,4810 838. ,8443 836. 6661 834 . 4806 838 . ,8423 836 . 6650 834 , .4802 838 . 8403 836 . 6638 834 , .4797 838 .8382 836. ,6627 834 , .4792 838 .836 1 836 , 6615 834 . 4787 838. .8340 836. 6603 834 .4781 838, .8320 836, ,6590 834 .4775 838 .8299 836. .6577 834 .4769 838 .8278 836 .6564 834 .4763 838 .8257 836 .6551 834 .4756 838 .8236 836 .6538 834 .4749 838 .8215 836 .6524 834 .4742 838 .8194 836 .651 1 834 .4734 838 .8173 836 .6497 834 .4726 838 .8153 836 .6483 834 .47 18 838 .8132 836 .6469 834 .47 10 838 .8112 836 .6455 834 .4702 838 .8092 836 .644 1 834 .4693 838 . B072 836 .6427 834 .4685 838 .8052 836 .6412 834 .4676 838 .8033 836 .6398 834 .4667 838 .8014 836 .6384 834 .4658 838 .7995 836 .6370 834 .4649 838 .7977 836 .6356 834 .4639 838 . 7959 836 .6342 834 .4630 838 .7941 836 .6328 834 .4621 838 . 7924 836 .6314 834 .461 1 838 . 7907 836 .6300 834 .4602 838 .7891 836 .6287 834 . 4592 838 . 7875 836 .6273 834 . 4582 838 . 7859 836 .6260 834 . 4573 838 . 7844 836 .6247 834 .4563 (32 85 1 . 8845 849. 5064 84 7 . 3794 845 . 2 189 133 85 1 . 9044 849 . 598 1 847 . 3853 845. 2210 134 85 1 . 9245 849. 6101 847 . 3915 845. 2233 135 85 1 . 9450 849 . 6224 847 . 3979 845. 2258 136 85 1 . 9658 84 9. 6350 847 . 404 5 845 . 2285 137 85 1 . 9868 849 . 6479 847 . 4 1 15 845. 2315 138 852 . 008 2 849 . 661 1 847 . 4 187 845 . 2346 139 852 . 0298 849 . 6745 847 . 4 261 845 . 2379 140 852 . 05 17 849 . 6883 847 . 4338 845 . 24 15 14 1 852 . 07 39 849 . 7023 847 . 44 17 845 . 2452 112 852.. 0964 84 9 . 7 166 047 . 4 4 99 845 . 2492 113 852 . 119 1 849 . 731 1 847 . 4583 84 5 . 2533 144 852 . 1420 849 . 74 60 84 7 . 4670 845 . 2577 145 852 . 165 1 849 . 76 10 847 . 4759 845 . 2623 146 852 . 1084 849 . 7763 847 . 4850 845. 2670 147 852 . 2118 849 . 7919 84 7 4944 845 . 2720 148 852 . 2353 849 . 8077 84 7 . . 5040 845 . 2 7 7 1 1 19 852 . 2588 849 . 8236 847 .5138 845 . 2825 150 852 . 2823 049 . 8398 847 . 5239 845 . 2880 151 852 . 3057 84 9 . 856 1 847 . 534 1 845 . 2938 152 852 . 3290 849 . 8726 847 5446 845. 2997 153 852 .3519 849 . 889 1 847 .5552 845. . 3058 154 852 . 37 4 6 849 . 9057 8 17 . 566 1 845 3 12 1 155 852 . 3968 849 . 9224 847 . 5770 845 .3186 156 852 . 4 186 849 , . 9390 847 . 5082 845 . 3252 157 852 . 1398 849 . 9556 847 . 599 1 845 . 3320 158 852 . 1604 849 .9721 847 .6 108 845 .3390 159 852 . 4802 849 . 9884 847 .6223 845 .3461 160 852 . 4991 850 .0046 847 . 6338 845 .3534 10 1 852 . 5 172 850 .0206 847 .6454 845 . 3608 19 20 2 1 22 1 832 . 263 1 830 .0908 827 .9 164 825 . 7407 2 832 . 2626 830 .0904 827 .9 160 825 .7405 3 832 . 2622 830 .0899 827 . 9 157 825 . 7402 4 832 . 26 17 830 .0895 827 .9 154 825 . 7400 5 832 . 26 13 830 .089 1 827 .9151 825 . 7398 6 832 . 2609 830 .0887 827 .9147 825 . 7 396 7 832 . 2605 830 .0883 827 . 9 144 825 .7394 8 832 . 2601 830 .0879 827 .914 1 825 .7391 9 832 . 2597 830 .0875 827 .9 138 825 . 7389 IO 832 . 2594 830 .0872 827 . 9 135 825 . 7387 1 1 832 . 259 1 830 .0868 827 .9 132 825 . 7384 12 832 . 2588 8 30 .0865 827 .9129 825 .7382 13 832 . 2585 830 .0862 827 .9126 825 .7380 14 0 32 . 2503 030 .0008 027 .9 123 825 .7378 15 832 . 258 1 830 .0855 827 .9120 825 .7375 16 832 . 2579 830 .0852 827 .9117 825 .7373 1 7 032 . 2577 03O .0850 827 .9 114 825 .737 1 18 832 . 2576 830 .0847 827 .9112 825 . 7369 843 . 0757 840. 9330 838 . 7830 836. 6234 834 . 4553 843 . 0756 840. 9319 838. 7816 836. 6221 834 . 4544 843 . 0757 840.9309 838 . 7 002 836. 6209 834 . 4534 843. 0759 840. 9299 838. 7789 836. 6197 834 . 4525 843. 0762 840. 9291 838 . 7777 836. 6185 834 . 4515 843 . 0767 840. 9284 838. 7765 836. 6173 834 . 4506 843 . 0774 840. 9278 838 . 7754 836. 6162 034 . 4496 843 . 0782 840. 9273 838. 7744 836. 6151 834 . 448.7 843. 0792 840. 9269 838 . 7734 836. ,6140 834 . 4478 843 . 0803 840. 9265 838 . ,7724 836. ,6129 334 . ,4469 843. 08 16 840. 9263 838 . , 77 16 836, .6119 G34 . 4460 843 . 083 1 840. 9263 838 . 7708 836 .6110 834 . 445 1 843 . 0847 840. ,9263 838 , .7701 836 .6100 834 ,4443 843 . 0864 840. ,9264 838 , . 7694 836 .609 1 834 . 4434 843 . 0883 840. ,9266 838 . 7688 836 .6083 834 .4426 043 . 0904 840. ,9270 838 . 7683 836 .6074 834 .44 18 843 . 0926 840. .9274 838 . 7678 836 . 6066 834 .44 10 843 . 0949 840, ,9280 838 . 7675 836 .6059 834 .4402 843 .0975 840. .9287 838 . 7672 836 .6052 834 .4395 843 . 1001 840, .9295 838 .7670 836 .6046 834 .4388 843 . 1030 840 .9304 838 . 7668 836 .6040 834 .4381 843 . 1059 840, .9314 838 .7667 836 .6034 834 .4374 843 . 1091 840 .9325 838 . 7668 836 .6029 834 .4367 843 . 1 124 840 .9338 838 . 7668 836 .6024 834 .4361 843 . 1 158 840 .9351 838 .7670 836 .6020 834 .4355 843 . 1 194 840 .9366 838 . 7673 836 .6017 834 . 4349 843 . 1231 840 .9382 838 .7676 836 .6014 834 .4344 843 . 1270 840 .9399 838 . 7680 836 .601 1 834 .4339 843 . 1310 840 .9417 838 .7685 836 .6009 334 .4334 843 .1351 840 .9436 838 .7691 836 .6007 834 .4329 23 24 823 . 5646 82 1 .3887 823 . 5644 82 1 . 3887 823 .5643 82 1 . 3886 823 .5642 82 1 . 3885 823 . 5640 82 1 . 3885 823 .5639 82 1 . 3884 823 .5638 821 .3883 823 .5636 82 I .3883 823 .5635 821 .3882 823 . 5633 82 1 . 3882 823 .5632 82 1 .3881 823 .5631 821 .3880 823 .5629 821 .3079 023 .5628 82 1 .3879 823 .5626 821 . 3878 823 .5625 82 1 . 3877 023 .5623 82 1 . 3877 023 .5622 82 1 .3876 19 832 . 2574 830. 0845 827 . 9 109 20 832 . 2574 830. 0842 827. 9 107 2 1 832 . 2573 830. 0840 827 . 9 t04 22 832 . 2573 830. 0B38 827. 9102 23 832 . 2573 830. 0836 827 . 9 100 2 1 832 . 2573 830. 0835 827 . 9097 25 832 . 2573 830. 0833 827 . 9095 20 832 . 2574 830. 0832 827 . 9093 27 832 . 2575 830. 0831 827. 9092 28 832 . 2577 830. 0830 827 . 9090 29 832 . 2578 830. 0829 827 . 9088 30 832 . 2580 830. 0829 827 . 9087 3 1 832 . 2583 830. 0828 827 . 9085 32 832 . 2585 830. OB28 827 . 9084 33 832 . 2588 830. 0828 827 . 9083 3 4 832 . 2591 830. 0828 827 . 9082 35 832 . 2595 830. 0829 827 . 908 1 3G 032 . 2598 830. 0830 827. 9080 37 832 . 2602 830. 0830 827 . 9079 38 832 2606 830. 0832 827 . 9079 39 832 . 26 1 1 830. 0833 827 . 9078 •40 832 . 26 15 830. .0834 827 . 9078 .1 1 832 . 2620 830. .0836 827 .9078 4 2 032 2626 830 0838 827 .9070 43 832 .2631 830. .0840 827 .9078 4 4 032 . 2637 030, .084 2 827 . 9070 45 832 . 2642 8 30 .0844 027 .9079 46 832 . 2648 830 .0847 827 . 9079 4 7 832 . 2655 830 .0850 827 .9080 4 8 832 . 266 1 830 .0853 827 .908 1 49 832 . 2667 830 .0856 827 .9082 50 832 . 2674 830 .0859 827 .9083 51 832 . 268 1 830 .0863 827 .9084 52 832 . 2688 830 .0866 827 . 9086 53 832 . 2605 830 .0870 827 . 9087 54 832 . 2702 830 .0874 827 .9089 55 832 . 2709 830 .0878 827 .9091 56 832 . 27 17 830 .0882 827 .9093 57 832 . 2724 830 .0886 827 .9095 58 832 .2731 830 .0891 • 827 .9097 59 832 . 2739 830 .0895 827 .9099 GO 832 . 2746 830 .0900 827 .9102 6 1 832 . 2 754 830 .0905 827 . 9 104 6 2 832 . 276 1 830 .0909 827 .9107 63 832 . 2769 830 .0914 827 .9109 64 832 . 2776 830 .09 19 827 .9112 65 832 . 2784 830 .0924 827 .9115 66 832 . 279 1 830 .0929 827 .9118 67 832 .2798 830 .0934 827 .912 1 68 832 . 2805 830 .0939 827 .9124 825. 7367 823 . 5620 821 . 3875 825. 7365 823. 56 19 821 . 3874 825. 7363 823. 5617 821 , 3874 825. 7361 023. 5616 821 . 38/3 825. 7309 823. 5615 821 . 3872 825 . 7357 823 . 5613 82 1 . 3872 825 . 7355 823 . 5612 82 1 . 3871 825. 7353 823. 5610 821 . 3870 825. 7351 823. 5609 82 1 . 3869 825. 7349 823. 5608 82 1 . 3869 825. 7347 823. 5607 821 . 3868 825 . 7346 823. 5605 821 . 3867 825. 7344 823 . 5604 82 1 . 3867 825 . 7343 823 . 5603 82 1 . 3866 825. 734 1 823. 5602 821 . 3865 825. 7340 823. 5600 821 . 3865 825 . 7339 823. 5599 82 1 . 3864 825 . 7337 823 . 5598 82 1 . 3864 825. 7336 823 . 5597 82 1 . . 3863 825. 7335 823. 5596 82 1 . . 3862 825 . 7334 823 . 5595 82 1 . 3862 825. 7333 823. .5594 821 .3861 825. 7332 823 . 5594 82 1 . 386 1 825 .7332 823 . 5593 82 i . 3860 825. .7331 823. .5592 821 . 3860 825 . 7330 823 .5591 82 1 . 3859 825 .7330 823 .5591 821 . 3859 825 . 7330 823 . 5590 821 . 3859 825 .7329 823 .5590 821 . 3858 825 . 7329 823 . 5589 821 . 3858 825 . 7329 823 .5589 821 . 3858 825 .7329 823 . 5588 821 .3857 825 .7329 823 . 5588 82 1 . 3857 825 .7329 823 . 5588 82 1 . 3857 825 . 7330 823 .5588 821 . 3857 825 .7330 823 .5587 821 .3856 825 .7331 823 .5587 821 .3856 825 .7331 823 .5587 82 1 .3856 825 .7332 823 .5587 821 .3856 825 .7333 823 .5588 821 .3856 825 .7334 823 .5588 821 . 3856 825 .7335 823 . 5588 821 . 3856 825 . 7336 823 .5588 821 . 3856 825 . 7337 823 .5589 82 1 . 3856 825 . 7338 823 .5589 821 . 3856 825 . 7339 823 .5590 821 .3856 825 . 734 1 823 . 5590 82 1 . 3856 825 .7342 823 .5591 82 1 . 3857 825 .7344 823 .5591 821 . 3857 825 .7345 823 . 5592 821 . 3857 69 83? . 28 12 830. 094 4 027 . 9127 825 . 7347 823. 5593 821 . 3857 70 832 . 28 19 830. 0949 827 . 9130 825 . 7349 823 . 5594 821 . 3858 7 1 832 . 2826 830. 0954 827 . 9134 825. 7351 823. 5595 821 . 3858 72 832 . 2033 830. 0959 827. 9137 825. 7353 823. 5596 821 . 3858 73 832 . 2839 830. 0964 827 . 9140 825. 7354 823. 5597 821 . 3859 74 832 . 2846 830. 0969 827 . 9 144 825. 7356 823. 5598 821 . 3859 75 832 . 2852 830. 0974 827 . 9 147 825 . 7359 823. 5599 821 . 3860 76 832 . 2858 830. 0979 827 . 9150 825 . 7361 823. 5600 821 . 3860 77 832 . 2864 830. 0984 827 . 9 154 825 . 7363 823 . 5601 82 1 . 386 1 78 832 . 2869 830. 0989 827 . 9157 825 . 7365 823. 5602 821 . 3861 79 832 . 2875 830. 0994 827 . 9161 825 . 7367 823. 5603 821 . , 3862 80 832 ! 2880 830. 0998 027 . 9 164 825. 7370 823. 5605 821 . ,3862 8 1 832 . 2885 830. 1003 827 . 9 168 825 . 7372 823 . 5606 82 1 . , 3863 82 832 . 2889 830. 1007 827 . 9171 825 . 7374 823. 5607 821 . 3863 83 832 . 2094 830. 1012 827 . 9 175 825 . 7377 823 . 5609 82 1 , .3864 R-l 032 . 2098 03O. 1016 027 . 9 t78 825 . 7379 023 . 5610 82 1 . 3065 05 " 03 2 . 2902 030. 1020 027 . 9 182 825 , 730 1 023 . 56 12 82 1 . 3065 86 832 . , 2906 830. , 1024 827 . 9 185 825 . 7384 823 . ,5613 82 1 . 3866 07 032 , .29 10 830. . 1020 827 9 100 825. . 7 306 823. 5615 821 . 3867 88 832 . 29 13 830. . 1032 027 . 9192 825 . 7388 823. .5616 821 . 3867 89 832 . 29 16 830 . 1036 827 .9195 825 . 7391 823 ,5618 821 .3868 90 832 . 29 19 830 . 1039 827 .9 198 825, .7393 823. .5619 821 .3869 91 832 . 292 1 830 .1043 827 .9201 825 . 7396 823 .5621 821 .3870 92 832 . 2923 830 . 1046 827 .9204 825 .7398 823 . 5622 82 1 .3870 93 832 . 2925 830. . 1049 827 .9207 825 .7400 823 .5624 821 .3871 94 832 . 2927 830 . 1052 827 .92 10 825 .7403 823 .5625 821 .3872 95 832 . 2928 830 . 1054 827 .92 13 825 . 7405 823 .5627 821 . 3873 96 832 . 2929 830 . 1057 827 . 92 16 825 . 7407 823 .5628 82 1 . 3873 97 832 . 2930 830 . 1059 827 .9218 825 . 7409 823 .5630 821 .3874 98 832 . 293 1 830 . 1062 827 .922 1 825 . 74 12 823 .5631 821 .3875 99 832 . 293 1 830 . 1064 827 .9223 825 .7414 823 .5633 821 .3876 100 832 . 293 1 830 . 1066 827 .9226 825 . 74 16 823 .5634 82 1 . 3876 101 832 . 293 1 830 . 1067 827 .9228 825 . 74 18 823 . 5636 821 .3877 102 832 . 2930 830 . 1069 827 .9230 825 . 7420 823 .5637 821 .3878 103 032 . 2929 030 . 1070 827 .9232 825 .7422 823 . 5639 82 1 . 3879 104 832 . 2920 830 . 107 1 827 .9234 825 . 7424 823 .5640 821 . 3879 105 832 . 2927 830 . 1072 827 .9236 825 .7426 82.3 . 5642 82 1 . 3800 106 832 . 2925 830 . 1073 827 .9238 825 . 7428 823 .5643 821 .3881 107 832 . 2923 830 . 1074 827 .9240 825 .7429 823 . 5644 821 . 3882 103 832 . 292 1 830 . 1074 827 .924 1 825 .7431 823 .5646 821 .3882 109 832 . 29 19 830 . 1074 827 .924 2 825 . 7433 823 .5647 821 .3883 1 to 832 . 29 16 830 . 1074 827 . 9244 825 . 7434 823 .5648 821 . 3804 1 t 1 832 . 29 13 830 . 1074 827 .9245 825 . 7436 823 . 5649 82 1 . 3884 1 12 832 .2910 830 . 1074 027 .9246 825 . 7437 823 .5651 821 .3885 1 13 832 . 2907 830 . 1073 827 .9247 825 .7438 823 .5652 821 .3886 1 14 832 . 2904 830 . 1073 827 .9248 825 . 7440 823 .5653 821 . 3886 1 15 832 . 2900 830 . 1072 827 .9248 825 . 744 1 823 .5654 821 . 3887 1 16 832 . 2096 830 . 1071 827 .9249 825 . 7442 823 .5655 821 .3887 1 17 832 . 2092 830 . 1070 827 . 9249 825 . 7443 823 . 5656 821 . 3888 1 18 832 . 2008 830 . 1068 827 .9249 825 . 7444 823 .5657 82 1 . 3888 11^ 832 . 2383 830. 1067 827 . 9249 825 . 7444 823 . 5657 821 . 3089 i?o 832 20/8 830. 1065 827 . 9249 825 . 7445 823. 5658 82 1 . 3989 1? 1 832 2874 830. 1063 827 . 9249 825. 7-146 823 . 5659 82 1 . 3890 12? R32 . 2R69 830. 1061 827. 9249 825. 7446 823 . 5660 321 . 3890 123 832 . 2863 830. 1059 827 . 9249 825. 7447 823. 5660 821 . 3891 12 1 832 . 2050 830. 1057 827 . 9240 825 . 7447 823. 5661 821 . 389 1 125 832 . 2053 830. 1054 827 . 9248 825. 7448 823 . 5662 821 . 3892 1 26 0<2 . 20 17 830. 1051 827 . 9247 825 . 7448 823. 5662 82 1 . 3892 127 832 . 284 1 830. 10-19 827 • 9246 825, 7448 823. 5662 82 1 . 3892 128 832 . 2036 830. 1046 027 . 9240 825 . 7448 823. 5663 821 . 3893 120 832 . 2830 830. 104 3 827 . 9244 825. 7448 823 . 5663 82 1 . . 3893 130 832 . 2824 830. 1040 827 . 9243 825. 7.448 823. 5663 821 . 3893 13 1 832 . 28 17 830. 1036 827 . 924 1 825 . 7448 823 . 5664 82 1 . . 3893 132 832 . 28 1 1 830. 1033 827 . 9240 825. 7447 823 . 5664 82 1 . . 3893 133 832 . 2805 830. 1029 827 . 9238 825 . 7447 823. 5664 82 1 . 3894 13 1 032 . 2798 8 30. 1026 827 . 9237 825 . 7446 823 . 5664 82 1 . 3894 1 35 ' 832 . 2792 830. 1022 827 . 92 35 825. 7446 823 . 5664 821 . 3894 1 30 032 . 2786 830. 1018 827 9233 825 . 7445 823 . 5664 821 . 3894 137 032 . 2779 830. 1014 827 .9231 825 . 7444 823 . 5664 82 1 . 3894 138 032 2772 0 30. 1010 827 . 9229 825 . 7444 823 , 5664 82 1 . 3894 139 832 . 2766 830. 1006 827 . 9227 825. 7443 823 . 5663 82 1 .3894 1 10 832 . 2759 830. 1002 827 .9225 825 . 7442 823 , .5663 821 .3894 1-1 1 832 . 2753 830. 0998 827 . 9222 825 , .7441 - 823 . 5663 821 . 3894 1'I2 832 . 2746 830. 0993 827 . 9220 825 . 7440 823 . 5662 821 .3894 143 832 .2739 830. 0989 827 . 92 17 825. . 7438 823 .5662 82 1 . 3894 14 4 032 . 2733 830. 0985 827 . 92 15 825 .7437 823 .5661 821 . 3893 145 832 . 2726 830. 0980 827 . 92 12 825 . 7436 823 . 566 1 82 1 . 3893 146 832 .2720 830. 0976 827 .9210 825 .7434 823 .5660 821 . 3893 147 832 .2713 830. .0971 827 .9207 825 . 7433 823 .5659 821 .3893 148 832 . 2707 830. 0967 827 . 9204 825 .7431 823 .5659 821 . 3893 149 832 . 2700 830. .0962 827 . 9201 825 . 7430 823 . 5658 821 . 3892 150 832 . 2694 830. .0958 827 . 9 198 825 .7428 823 .5657 821 .3892 15 1 832 . 2688 830 .0953 827 .9195 825 . 7426 823 .5656 82 1 . 3892 152 832 . 2682 830 .0948 827 .9192 825 .7425 823 .5655 821 .3891 153 832 . 2676 830 .0944 827 . 9 189 825 . 7423 823 .5654 821 .3891 154 832 . 2670 830 .0939 827 . 9 186 825 .7421 823 .5653 821 .3891 155 832 . 2664 830 .0935 827 . 9 1B3 825 . 74 19 823 .5652 . 82 1 . 3890 156 832 .2658 830 .0930 827 .9180 825 .7417 823 .5651 821 .3890 157 832 . 2652 830 .0926 827 .9176 825 .7415 823 .5650 82 1 . 3889 . 158 832 . 2647 830 .0921 827 9173 825 .7413 823 .5649 821 .3889 159 832 . 2642 830 .09 17 827 .9170 825 .74 1 1 823 .5648 82 1 . 3888 160 832 . 2636 830 .0912 827 . 9 167 825 .7409 823 .5647 821 .3888 16 1 032 .2631 830 .0908 827 .9164 825 . 7407 823 . 5646 821 . 3887 ro THF r> T F At) / STATE TEMPERATURE FIELD FOLLOWS: 1 87 1 .6430 869.444 1 3 867.2465 865.0501 85 1 10 .8982 1 1 849.7 106 12 847.5242 13 845.3391 832 19 2542 20 830.0777 827 2 1 .9024 22 825.7284 8 12 28 7099 29 8 10.5441 808 30 3802 31 806.2 172 793 37 2644 38 79 1 . 1098 788 39 9565 40 786.8043 773 4G .9166 4 7 77 1 .7728 769 48 .6303 49 767.4889 754 55 . 6656 56 752.5325 750 57 4006 58 748.2699 735 64 . 5 103 65 733.3879 731 66 . 2665 67 729.1464 7 16 73 4500 74 7 14. 3380 7 12 75 .227 1 76 710.1174 697 82 . 4835 83 695.3819 693 84 28 14 85 691 . 1821 678 9 1 .6101 92 676.5 187 674 93 4285 94 672.3394 659 100 8287 101 657.7476 655 102 .6675 103 653.5886 6 1 1 109 . 1386 1 10 639.0675 636 1 1 1 9975 112 634.9286 622 1 18 5387 1 19 620.4776 618 120 4176 12 1 616.3587 127 604.0283 128 601.9771 129 599.9270 130 597.8779 862 5 ,8550 860.66 12 858.4686 8 856.2772 854.0871 843 14 1552 15 840.9725 16 838.7911 17 836.6109 18 834.4319 23 823.5556 804 32 .0553 24 82 1 .3840 33 801.8947 25 819.2137 34 799.7353 26 817.0445 35 797.5771 27 814.8766 36 795.4202 784 4 1 .6534 42 782.5036 43 780.3551 44 778.2077 45 776.0616 765 50 . 3487 51 763.2097 52 761.0719 53 758.9353 54 756.7998 746 59 1404 60 744.0120 61 74 I .8848 62 739.7588 63 737.6340 727 68 .0274 69 724.9096 70 722.7930 71 720.6775 72 718.5631 708 77 0089 78 705.9015 79 703.7953 80 701.6902 81 699.5863 689 86 .0839 87 686.9869 88 684.8910 89 682.7962 90 680.7026 670 95 2515 96 668.1647 97 666.0790 98 663.9944 99 661 .9110 651 104 .5108 105 649.4341 106 647 .3585 107 645.2e41 108 643.2108 632 6 1 4 595 1 13 .8609 122 .3009 131 .8300 1 14 630.7942 123 612.2442 132 593.7831 1 15 628.7287 124 610.1886 133 591.7373 1 16 626.6643 125 608.1341 134 589.6926 1 17 624.6009 126 606.0806 135 587.6490 ro ro 136 585.6065 137 583.5651 145 567 . 2724 146 565.2407 154 549.0253 155 547.0032 163 530.8642 164 528 .8516 172 512 .7885 173 510.7853 18 1 •in4 . 797 2 182 492.8033 190 476.0096 191 474.9050 199 459.0650 200 457.0895 208 4 1 .3225 209 4 39.3561 2 17 423 . 66 14 218 42 1.7040 581 138 , 5247 139 579.4854 563 147 2 100 148 561.1804 156 544 . 9821 157 542.9621 526 165 . O'lO 1 166 524.8295 508 174 7831 175 506.7820 4 90 183 8 105 184 488.0 187 472 192 . 92 14 193 470.9389 455 201 1 151 202 453.14 17 437 210 . 390R 2 1 1 435 . 42G4 4 19 2 19 7477 220 417.7923 226 406.08 IO 227 404. 1326 228 402.1851 229 400.2386 140 577.4473 575. 14 1 4101 573 142 ,3741 143 571.3391 144 569.3053 149 559.1519 557 150 1245 55E 151 .098 1 152 553.0727 153 551.0485 158 540.9431 538 159 .9252 536 160 .9084 161 534.8926 162 532.8779 167 522.8201 520 168 8 117 518 169 ,8043 170 516.7980 171 514.7927 176 504.7820 502 177 7829 500 178 .7849 179 498.7880 180 496.792 1 185 48G . 820O 484 186 .8303 482 187 . 8496 188 480.8619 189 478.8752 194 468.9574 466 195 .9768 464 196 .9973 197 463.0189 198 461 .0414 203 45 1 . 1693 449 204 . 1979 447 205 .2276 206 445.2582 207 443.2898 212 4 33.463 1 431 2 13 . 5008 429 2 14 .5394 215 427.5791 216 425.6197 221 4 15.8380 413 222 .8846 41 1 223 .9322 224 409.9808 225 408.0304 230 398.2932 ro co AT FI.OW MOULFS AVERAGE EXPOSED WALL TEMPERATURES (KI: 1-ZONE= 877.70 4-ZONE/M* 860.95 *2 = 877.11 #3 = 884.04 04= 888.70 AVLRARE CUVF.REO WALL J E MP ERA I URE ; 047.84 CALCUIATFO HFAT FLOWS (W/M): INTEGRA T ED EXPOSED WALL : HAD I A I 1 ON CONVECTION TOTAL 39219. 32072. 7 1290. 1-ZONE 39219. 3207 2. 71290. COVERED WALL: CONVICTION •4 704 3. -4 704 3. ",01. IDS: RADIATION *•»*.• CONVECTION TOTAL ...... 68853. 65832 . O.13468E+06 THROUGH WALL: DIFFERENCE PROF ILE 24247. 24 185. 24247 . -ZONE 39093. 32072 . 71165. -47043 . 6892 1 . 65832 . 0. 13475E+06 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 (W/M) HEAT LOSS TO SURROUNDINGS' 24246. MODIFIED ANALOG:(RADIATION ONLY) EXPOSED WALL TEMPERATURE' 904.76 (K) OUTER SHELL TEMPERATURE' 401.40 HEAT RECEIVED BY SOLIDS' 73805. (W/M) HfcAF LOSS TO SURROUNDINGS' 25613 A p p e n d i x A8 FORTRAN SOURCE L IST ING AND SAMPLE OUTPUT FOR K ILN FLAME MODEL FORTRAN SOURCE L IST ING KILN FLAME MODEL 1 2 c 3 c 4 c FLAME MODEL FOR ROTARY KILN 5 c 6 c J.P. GOROG 7 c NOV. 27. 1981 8 c 9 c 10 1 1 IMPLICIT REAL*8(A - H.O - Z) 12 DIMENSION X(500), T(500). TW(500). 0L(500) 13 DIMENSION X1(4). F(4), ACCE5T(4) 14 DIMENSION TSH(500). 0FS(50O). QFW(500). 0FSH(500) 15 REAL'8 L, MWF, MF. MAT. MAP. MO. MS. MEN, KP 16 REAL *8 M(500), MOP. MWCP. MST. MAEP 17 EXTERNAL FN2. FN3 18 . COMMON /BLK1/ ASAT. D 19 COMMON /BLK2/ R 20 COMMON /BLK3/ EMG, EMS. EMW 2 1 COMMON /BLK4/ HOUT, HCOV. HSE. HEX 2.2 COMMON /BLK5/ KP, TS. TA. RO 23 COMMON /BLK6/ TF. OLOSS. TW1, TSA 24 COMMON /BLK7/ FFS. DR. AF1. AS. AW. FSF. FSW, FFW 25 COMMON /BLK8/ X, M. GHV. AF, 11 26 COMMON /BLK9/ SPHEAT, H2L0 27 COMMON /BLK10/ OSOLID. QWALL, QSHELL 28 COMMON /BLK 11/ MF. MO 29 LOGICAL* 1 NAME(20) 30 LOGICAL LZ1 3 1 CALL SETLIO('6 '. '-A ') 32 CALL SETLIO('8 '. '*SINK+ ') 33 CALL CMD('$EMPTY -A OK '. 13) 34 c 35 c READ INPUT AS FOLOWS: 36 c 37 c 38 c NAME = Fuel type 39 c 40 c ID = Run I d e n t i f i c a t i o n number 4 1 c 42 c 0 = K i l n diameter (m) 43 c L = K i l n length (m) 44 c BTH = Thickness of k i l n l i n i n g (m) 45 c RES = Residence time of s o l i d s (hr) 46 c OS = Mass flowrate of s o l i d s (kgr/s) 47 c FR = F i r i n g rate of fuel (d/kgr s o l i d ) 48 c 49 c TF = Inlet fuel temperature (K) 50 c TSA * Secondary a i r temperature (K) 51 c TS " Solids temperature (K) ro 00 52 C TA = Ambient temperature 53 C 54 C EMW = Wall e m i s s i v i t y 55 C EMS = sol Ids emlsslv!ty 5G C EMG = Flame e m i s s i v i t y 57 C o Combined HTC at outer s h e l l (W/m**2 K) 58 C HOUT 59 c HCOV = Convective HTC at covered wall (W/m**2 K) 60 c HSE = Convective HTC at s o l i d s surface (W/m»*2 K) 6 1 c HEX = Convective HTC at exposed wall (W/m**2 K) 62 c KP = Thermal co n d u c t i v i t y of l i n i n g (W/m K) 63 c 64 c PPA ; = Percent of stoichiometric a i r as primary 65 c PPO = Percent of primary a i r as oxygen 66 c PTA = Percent of stoichiometric a i r to t a l 67 c = Air to fuel r a t i o for I n i t i a l fuel (kgr/kgr 68 c AF 69 c DO = Equivalent diameter of burner (m) 70 c H2L0 = Hydrogen loss 7 1 c GHV = Gross heating value of fuel (J/kgr) 72 c ROJET = Jet density (kgr/m**3) 73 c 74 READ (5.20) (NAME(I).1=1.20) 75 READ (5.30) ID 76 READ (5.10) D, L, BTH. RES. OS 77 READ (5.10) FR. TF. TSA. TS. TA 78 READ (5,10) EMW, EMS. EMG 79 READ (5.10) HOUT. HCOV. HSE. HEX. KP 80 READ (5.10) PPA, PPO. PTA 8 1 READ (5,10) AF, DO, H2L0, GHV. ROJET 82 SPHEAT = 1500.00 83 RHOS = 600.DOO 84 10 FORMAT (5G12.5) 85 20 FORMAT (20A1) 86 30 FORMAT (13) 87 TIF = 1850.DO 88 AFO = AF 89 TFO = TF 90 0S1 = (OS + 100.DO/56.DO'QS) / 2.DO 91 RES = RES * 3600.DO 92 ASF » (0S1*RES) / (L*RH0S) 93 ASAT = ASF / (PI(O.DO)*(D/2.DO)**2) 94 RO = D / 2.D0+BTH 95 MF = OS * FR * (1.DO/GHV) 96 MAT = MF * AF 97 MAP = (1.DO-PP0) * PPA * MAT 98 MOP = PPO * PPA • MAT 99 MAEP = 4.31DO * MOP 100 MEN = MAT - MAP - MAEP 101 MO = MF + MOP + MAP 102 ROAIR = 28.82DO / (0.08205D0*TSA) 103 ROCP = 0.20D0 ro vo 104 TERM = (R0JET/R0CP) ** 0.500 * (ROJET/ROAIR) *• 0.5D0 105 FL = (1.DO+MEN/MO) * DO • 6.DO * TERM 106 BL = FL - 6.00 • DO * TERM 107 DDXX = .25D0 108 N = IDINT(BL/DDXX) + 1 109 NI * N * 1 1 10 X( 1 ) = 6 . DO * 00 • TERM 111 DO 40 I = 2, N 112 40 X ( I ) = X ( I - 1 ) + DDXX 113 X(N1) = FL 114 M(1) = MO 115 DO 50 I • 2. N1 116 50 M(I) = ((1.D0/6.D0)*X(I)«M0) / DO * (1.DO/TERM) 117 AF = MEN / MO 1 18 T(1) - TF 1 19 CALL VIEW 120 QTOTS = O.DO 121 QTOTW = O.DO 122 0T0T5H = O.DO 123 QTOT =0.00 124 DO 80 I = 1. N 125 TR « 1.DO-EMG 126 RS = 1.DO-EMS 127 SIGMA = 5.67D-08 128 ES = SIGMA * TS ** 4' 129 EF = SIGMA * T(I) * * 4 130 EA = SIGMA * TA * * 4 13 1 T1 = AW * EMG 132 T2 = AS * TR 133 T3 = AS * EMG 134 T4 = (EMS*AS) / RS 135 G = T3 + T2 + T4 136 EAD = (T1 *EF + T2*((T3»EF + T4*ES)/G)) / (Tl + T2 - (T2**2)/G) 137 IF ( I .GT. 1) GO TO 60 138 X1(4) = ES 139 X1(3) • EAD 140 X1(2) = ((X1(3) - .25D0*(X(3) - EA))/SIGMA) •* .25DO 141 X1(1) = ((X1(3) - .75DO*(X(3) - EA))/SIGMA) ** .2500 142 60 CONTINUE 143 TF * T(I) 144 CALL NDINVT(4, X1. F, ACCEST. 800, .1, FN2. 6210) 145 TFL = O.DO 146 TFU = 1.D05 147 E l l = 5.0-05 148 1 1 = 1 149 CALL 2ER01(TFL. TFU, FN3, E l l . LZ1) 150 IF ( .NOT. LZ1) GO TO 230 ^ 151 T(I + 1 ) = TFL en 152 IF (I .GT. 1) GO TO 70 ° 153 TW(1) ' X1(2) 154 TSH(1) • X1(1) 155 70 T(I + 1 ) » TFL 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 80 90 100 1 10 120 130 140 150 160 TW(I + 1) = X1(2) TSH(I + 1) = X1(1) OFS(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) -OTOTS = OTOTS + (X(I + 1) QTOTW = OTOTW + (X( I X ( D ) • - X(I)) - X ( D ) QTOTSH + (X(I 1) + 1) - X(I)) OLOSS * OSOLID * OWALL OTOTSH CONTINUE OCALC * QS * 100.DO / 56.DO * 1.637D06 FRN = FR * (1 .DO-H2L0) CALL HEADER WRITE (6.90) (NAMEM ) . 1 = 1 OSHELL , 20) , FORMAT 1f fuel 2 3 4 5 WRITE FORMAT 1 2 3 4 5 ( GHV. 1X . H2L0, AFO, FR. FRN T5, 'Fuel type.'. I , 20A1/T5. 'Gross heating value =', 1X, E12.5. 1X, '(d/kgr)'/T5. 'Hydrogen loss ='. 1X. E12.5/T5. 'Air to fuel r a t i o of fuel ='. IX, E12.5, IX, '(kgr/kgr)'/T5, 'Gross f i r i n g rate of fuel ='. 1X, E12.5, IX. '(J/kgr sol1d)'/T5. 'Net f i r i n g rate of fuel =' E12.5. 1X. '(J/kgr s o l i d ) ' / / ) (6.100) D. L, DO, OS, BTH, ASAT (' '. T5. ' K i l n diameter ='. 1X. 'K1ln length ='. IX, E12.5. IX. 'Equivalent burner diameter ='. 'Mass flowrate of s o l i d s =', 1X, •Lining thickness ='. IX. E12.5, 'Percent s o l i d s loading ='. IX, IX , E12.5. IX, '(m)'/T5, '(m)'/T5. 1X, E12.5, 1X, '(m)'/T5, E12.5, 1X, '(kgr/s)'/T5, 1X. '(m)'/T5. E12.5, //) WRITE (6, FORMAT (' 110) EMS, ' . T5, ' EMW, Wal 1 EMG em Iss1vIty ' Wa11 em 1ss1v1ty =' IX, E12.5//) 1X, E12 ', IX, E12.5/T5, 5/T5, 'Flame e m i s s i v i t y =' WRITE FORMAT TSA, TA, PPA, PPO. FL I n i t i a l fuel temperature =', 1X, E12.5. IX, Secondary a i r tmeperature =', IX, E12.5, IX, Ambient a i r temperature =', 1X, E12.5, IX, Percent of stoichiometric a i r as primary =', a i r as oxygen «=' , 1X, E12.5// ', IX, E12.5. 1X. '(m)'/) WRITE FORMAT (6.120) TFO ( ' ' . T5. '(K)'/T5. '(K)'/T5. '(K)'/T5. E12.5/T5. T5. 'Calculated (6.130) ('1', T47, ^Temperature (K) 'Sol Ids', T38, 'Wal1'. T52, 'Ambient'/) 1X, 'Percent of f 1 ame prImary length « 1 2 DO 140 I = 1. NI WRITE (6, 150) X(I) FORMAT (' ', T4, E 1 E12.5, T79, WRITE (6,160) FORMAT ('1'. T31, /T5. 'Distance (m)', T22, 'Flame', T68. 'Shel1', T82, , TS. TW(I). T(I) 12.5. T19. E12.5, E12.5) , TSH(I), T34. E12. TA 5. T49. E12.5. T64. ro on Heat flu x 1 DO 170 'Sol Ids'. I = 1 . N T39, 'Wal 1 (W/m , T54 **2)'/T5, 'Shel I ' 'Distance (m)'. T22, /) 208 209 2 IO 2 I 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), OFSH( I ) FORMAT (' '. T4. E12.5. T19. E12.5. T34. £12 5. T50. E12.5) QTOTW - OTOTSH MO * SPHEAT • (TFO - 298.DO) MEN •* SPHEAT * (TSA - 298. DO) FR • OS HINSF + HINSA + HINFL (HINSF/TOTIN) (HINSA/TOT IN) (HINFL/TOTIN) PI(O.DO) * (D .00 N OREGN HINSF HINSA HINFL TOTIN PHINSF = PHINSA = PHINFL = ASHELL = HOTS = 0 HOTSH = 0.00 HOTCP = M(N1) DO 190 1 = 1 . HOTS » HOTS + HOTSH = HOTSH CONTINUE H0TH2 = HINFL * H2L0 TOTOUT = HOTCP + HOTS + PHOTCP = (HOTCP/TOTOUT) PHOTS = (HOTS/TOTOUT) * PHOTSH = (HOTSH/TOTOUT) PH0TH2 = (H0TH2/T0T0UT) PTHEO = (HOTS/OCALC) • WRITE (6.200) HINSF 1H0TCP. PHOTCP. HOTS 2PTHE0 FORMAT ( ' 1 ', T5. 'Heat 1 2 = 100.DO 100.DO 100.DO 2.DO'BTH) SPHEAT • (T(N1) - 298.DO) (<X(I + + <<X(I 1 ) - X(I))*AS*QFS(I)) + 1) - X(I))*ASHELL*OFSH(I)) HOTSH + H0TH2 * 100.DO 100.D0 * 100.00 * 100.DO * 100.DO PHINSF. HINSA. PHINSA, PHOTS. HOTSH, PHOTSH. HINFL H0TH2, , PHI NFL PH0TH2. TOTIN. TOTOUT, 3 4 5 6 7 8 9 * 1 balance on f 1 ame'///T5 , 'Heat In (W):', T59, 'Percent of total:'/T5, 'Sensible heat of fuel mixture T40. E12.5, T63, F6.2/T5, 'Sensible haet of secondary a i r =', T40. E12.5. T63, F6.2/T5, 'Chemical energy of fuel ='. T40, E12.5, T63, F6.2//T15. 'Total =', T40. E12.5///T5, 'Heat out (W):'/T5. 'Sensible heat of combustion prod =', T40. E12.5. T63. F6.2/T5. 'Heat received by s o l i d s ='. T40, E12.5, T63. F6.2/T5. 'Heat loss from s h e l l =', T40. E12.5. T63, F6.2/T5, 'Hydrogen loss ='. T40. E12.5. T63. F6.2//T15. 'Total ='. T40. E12.5////T5. '% of th e o r e t i c a l c a l c i n a t i o n energy to s o l i d s =', 1X, F6.2/'<'/'4'/) P9700(ID) CALL STOP WRITE (6.220) FORMAT ('1'. GO TO 250 WRITE (6.240) FORMAT ('1', STOP END ERROR RETURN FORM NDINVT') ERROR RETURN FROM FLAME TEMP SOLUTION') ro ro 257 c********** 258 C 259 C SUBROUTINE TO CALCULATE SOLIDS FILLRATE 260 C 261 £*+*••***** 262 SUBROUTINE FIIL(ASAT, FD. PHI. 0) 263 IMPLICIT REAL*8(A - H.O - Z) 264 EXTERNAL FN 1 265 LOGICAL LZ 266 COMMON /BLK2/ R 267 X = O.DO 268 Y = 1.002 269 E1 * 5.D-04 270 CALL ZEROKX. Y. FN 1 . E l . LZ) 271 IF ( .NOT. LZ) GO TO 10 272 PHI = X / 2.DO 273 R » D / 2.DO 274 FD = (R - R*DCOS(PHI)) / D 275 RETURN 276 10 WRITE (6,20) 277 20 FORMAT ('1', 'SOLUTION CALL FROM FILL FAILED') 278 CALL EXIT 279 STOP 280 END 28 1 £*•**«*+*** 282 C 283 C EXTERNAL ROUTINE FOR FILL 284 C 285 £«***•***»* 286 FUNCTION FN1(X) 287 IMPLICIT REAL*8(A - H.O - Z) 288 COMMON /BLK2/ R 289 COMMON /BLK 1/ ASAT. D 290 FN1 * ASAT * PI(O.DO) * R * * 2 - R * * 2 / 2.DO * (X - DSIN(X)) 291 RETURN 292 END ro cn co 293 £•***•*»•*** 294 C 295 C SUBROUTINE TO CALCULATE VIEW FACTORS 296 C 297 £•*****•»•** 298 SUBROUTINE VIEW 299 IMPLICIT REAL+8(A - H.O - Z) 300 COMMON /BLK1/ ASAT. D 301 COMMON /BLK2/ R 302 COMMON /BLK7/ FFS, DR, AF. AS. AW, FSF. FSW, FFW 303 R = D / 2.DO 304 CALL FILL( ASAT, FD. PHI. D) 305 FFS = (2.D0*PHI) / Pl(O.DO) 306 DR = D / 3.DO 307 AF = PI(O.DO) * DR 308 AS = D * DSIN(PHI) 309 AW = D * (PI(O.DO) - PHI) 310 FSF = (AF * FFS) / AS 3 11 FSW " 1.DO-FSF 312 FFW = 1 .DO-FFS 313 RETURN 314 END 315 C*»*»****•» 316 C 317 C EXTENAL ROUTINE TO CALCULATE FLAME TEMPERATURE 318 C 319 C********** 320 SUBROUTINE FN3(X1) 321 IMPLICIT1 REAL*8(A - H.O - Z) 322 REAL*8 X(500), M(500). MF. MO 323 COMMON /BLK6/ TF, OLOSS. TW, TSA 324 COMMON /BLK8/ X, M, GHV, AF, 11 325 COMMON /BLK9/ SPHEAT, H2L0 326 COMMON /BLK11/ MF, MO ' 327 H2L01 = 1.00-H2L0 328 TERM2 » MF / MO • 329 FN 1 = M(I1) * SPHEAT * (TF - 298.DO) + TERM2 • (M(I1 + 1) 330 1 * (1.D0/AF) * GHV * H2L01 + (M(I1 + 1) - M(I1)) * SPHEAT 331 2 298.) - M(I1 + 1) * SPHEAT * (XI - 298.00) - (X(I1 * 1) -332 3* OLOSS 333 RETURN 334 END - M i l l ) ) * (TSA -X(I1)) ro cn 335 (;******•**• 336 C 337 C EXTERNAL FUNCTION TO EVALUATE HEAT FLOW CIRCUIT 338 C 339 £*»*•***•*« 340 SUBROUTINE FN2(X. F) 341 IMPLICIT REAL*8(A - H.O - Z) 342 REAL*8 KP 343 DIMENSION X( 1 ). F(1) 344 COMMON /BLK1/ ASAT. D v 345 COMMON /BLK2/ R . ' 346. COMMON /BLK3/ EMG. EMS, EMW 347 COMMON /BLK4/ HOUT, HCOV, HSE. HEX 348 COMMON /BLK5/ KP. TS, TA. RO 349 COMMON /BLK6/ TF, OLOSS. TW. TSA 350 COMMON /BLK7/ FFS. DR. AF. AS, AW. FSF, FSW. FFW 35 1 COMMON /BLK10/ OSOLID, OWALL. QSHELL 352 TR = 1.D0-EMG 353 RS = 1.DO-EMS o 354 RW = 1.DO-EMW 355 SIGMA = 5.67D-08 356 ES = SIGMA * TS *» 4 357 EF = SIGMA * TF ** 4 358 EA = SIGMA • TA ** 4 359 T1 = (1.DO-EMW) / (EMW*AW) 360 T2 = (DL0G(R0/R)*(X( 1) + X( 2))* (X ( 1)•*2 + X(2 ) **2)/(2.D0*PI(O.DO)* 36 1 1KP) ) • SIGMA 362 T3 = 1.D0 / (FFW*AF*EMG) 363 T4 = 1.DO / (AS*FSW) 364 T5 = 1.D0 / ( F S F * A S * E MG) 365 T6 = (1.DO-EMS) / (EMS*AS) 366 CI = TA / X(1) 367 C2 • 1.D0+C1 + C1 ** 2 + C1 •* 3 368 HO » HOUT + (C2•SIGMA*X( 1 ) •*3) 369 HOP = HO / (((X(1) + TA)*(X(1)**2 + TA**2))*SIGMA) 370 HCOVP = HCOV / (((X(2) + TS)*(X(2)**2 + TS**2))*SIGMA) 371 HSEP •= HSE / (((TF + TS)*(TF**2 • TS**2) )*SIGMA) 372 HEXP * HEX / (((TF • X(2))*(TF**2 + X(2 ) **2 )) *SIGMA) 373 ACOV = (2.DO*PI(O.DO)*R) - AW 374 ASH = 2.DO * PI(O.DO) * RO 375 T8 = 1.DO / (HOP*ASH) 376 T9 = 1.D0 / (HCOVP'ACOV) 377 T10 = 1.D0 / (HSEP*AS) 378 T i l - 1.D0 / (HEXP*AW) 379 T12 = 1.00 / (FSF *AS*TR) 380 RE = (T4*T12) / (T4 + T12) 301 T13 = SIGMA * X(2) ** 4 382 F(1) « (EF - X(3)) / T3 + (T13 - X(3)) / Tl + (X(4) - X(3)) / RE tn 383 T14 = SIGMA * X(1) ** 4 ^ 384 F(2) = (X(3) - T13) / Tl + (T14 - T13) / T2 * (EF - T13) / T11 + ( 385 1ES - T13) / T9 386 F{3) = (EF - X(4)) / T5 + (ES - X(4)) / TG + (X(3) - X(4)) / RE 387 T20 = T2 + T8 388 F(4) = (TI3 - EA) / T20 - (T14 - EA) / T8 389 10 FORMAT (12(G12.5.2X)) 390 OLOSS = (EF - X(3)) / T3 + (EF - T13) / T i l + (EF - ES) / TIO + ( 391 1 EF - X(4)) / T5 392 OSOLID = (X(4) - ES) / T6 + (EF - ES) / T10 + (T13 - ES) / T9 393 T30 = SIGMA * X(2) ** 4 394 T31 = SIGMA * X(1) ** 4 395 QWALL = (X(3) - T30) / Tl + (EF - T30) / T i l 396 OSHELL = (T31 - EA) / T8 397 RETURN 398 . END 399 c * * * * * *»»»* 400 C 401 C SUBROUTINE TO PRINT HEADER 402 C 403 404 SUBROUTINE HEADER 405 LOGICAL*1 STARS(34) /34*'*'/ 406 WRITE (6.10) STARS. STARS 407 10 FORMAT (' ', T48. 34A1/T48, '* UNIVERSITY OF BRITISH COLUMBIA *'/ 408 1 T48. '* METALLURGICAL ENGINEERING *'/T48. 409 2 '* FLAME.MODEL VERSION 1 *'/T48, 34A1//) 4 10 RETURN 411 END ro cn cn 412 C* ****•*»•* 4 13 C 4 14 C SUBROUTINE TO SET 9700 OUTPUT 4 15 C 4 16 (;*•***••*•* 4 17 SUBROUTINE P97OO(I0) 418 LOGICAL * 1 A(132) 4 19 INTEGER*4 CNT. PG 420 INTEGER*2 LEN 421 REWIND 6 422 CALL GETLSTC'6 '. CNT) 423 CNT = CNT / 1000 424 PG = 0 425 NUMB = 1 426 DO 50 I = 1. CNT 427 CALL READ(A, LEN. 0. LNUMB. 6) 428 IF (PG .EQ. O) GO TO IO 429 L = O 430 CALL FINOST(A. 2, '1'. NUMB. L + 1, L. »40) 431 IF (PG .GE. 1) GO TO 30 432 10 WRITE (8.20) 433 20 FORMAT ('1'//////////) 434 PG = PG • 1 435 LNCK = 10 436 GO TO 40 437 30 CONTINUE 438 CALL PAGE(LNCK, ID. PG, A) 439 40 LNCK = LNCK + 1 440 IF ((LNCK - 60) EQ. O) CALL PAGE(LNCK, ID, PG. A) 441 CALL WRITE(A, LEN, O, LNUMB, 8) 442 50 CONTINUE 443 RETURN 444 END ro 4«15 £**•*****•• 446 C 447 C SUBROUTINE TO PRODUCE PAGE NUMBERS 448 C 449 450 SUBROUTINE PAGE(LNCK, 10, PG, A) 451 LOGICAL*! A(132). I BLANK, FMT(41), Fl(3) 452 INTEGERM PG 453 DATA I BLANK /' '/. FI /ZF 1 , ZF2, ZF3/ 454 DATA FMT /Z4D, Z7D, Z4C, Z7D, Z6B, ZE3. ZF1, ZF2, ZF1, Z6B. Z7D, 455 1 ZD7. ZC1, ZC7, ZC5, Z40, Z7D, Z6B, ZC9. ZF3, Z6B, Z7D, Z60, 456 2 Z7D, Z6B, ZC9. ZF2. Z61. Z7D, ZF1. Z7D. Z61, Z61. Z61 , Z61, 457 . 3 Z61, Z6 1, Z61, Z6 1. Z61, Z5D/ 458 IF (PG .LE. 9) FMT(27) = FI(1) 459 IF (PG .GE. 10 .AND. PG .LE. 99) FMT(27) • FI(2) 460 IF (PG .GE. 100) FMT(27) = FI(3) 461 WRITE (8.FMT) ID. PG 462 A(1) = IBLANK 463 PG * PG • 1 464 LNCK = 10 465 RETURN 466 END ro cn oo SAMPLE INPUT DATA f u e l t y p e : n a t u r a l gas Rj = 1.52 (m) L = 80 R Q = 1.60 Q g r o s s f i r i n g r a t e = 1 . 10 . x 10 ( J / k g r s o l i d ) i n l e t f u e l t e m p e r a t u r e = 298 (K ) s e c o n d a r y a i r t e m p e r a t u r e = 298 T s = 1100 (K) T = 298 a £w = °' 8 e s = 0 , 8 e g = 0 . 25 p e r c e n t o f 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 = 20 p e r c e n t oxygen e n r i c h m e n t = 0 A i r / f u e l r a t i o = 16 .97 ( k g r a i r / k g r f u e l ) h y d r o g e n l o s s = 9 . 82 p e r c e n t e q u i v a l e n t b u r n e r d i a m e t e r = .17 (m) g r o s s h e a t i n g v a l u e o f f u e l = 5.52 x 10^ ( J / k g r ) e q u i v a l e n t j e t d e n s i t y = 1.03 (kgr/m ) 260 SAMPLE OUTPUT + UNIVERSITY OF BRITISH COLUMBIA * * METALLURGICAL ENGINEERING * * FLAME.MODEL VERSION 1 * Fuel type: NATURAL GAS Gross heating value of fuel = 0.55200E+08 (J/kgr) Hydrogen loss = 0.98200E-01 Ai r to fuel r a t i o of fuel = 0.16970E+02 (kgr/kgr) Gross f i r i n g rate of fuel = O.11050E+O8 (.J/kgr s o l i d ) Net f i r i n g rate of fuel = 0.99649E+07 (J/kgr s o l i d ) K i l n diameter = 0.30480E+01 (m) K i l n length = 0.80000E+02 (m) Equivalent burner diameter = 0.16930E+00 (m) Mass flowrate of s o l i d s = 0.13130E+01 (kgr/s) L i n i n g thickness = 0.15200E+00 (m) Percent s o l i d s loading = 0.56394E-01 Wall e m i s s i v i t y = 0.80000E+00 W-Tll e m i s s i v i t y = 0.80000E<00 flame e m i s s i v i t y = 0.25000E+00 I n i t i a l fuel temperature = 0.29800E+03 (K) Secondary a i r tmeperature = 0.29800E+03 (K) Ambient a i r temperature = 0.29800E+O3 (K) Percent of s t o i c h i o m e t r i c a i r as primary = 0.20000E+00 Percent of primary a i r as oxygen = 0.0 Ca l c u l a t e d 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» | O X i X i X i X i X i X i X i X i X i X i X i X i X - i - X . X i X i X i X i X i X i X i X i X i X i X i X i 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 O O O O O o O O O O O O o O O O O O O O O ID CD -4 cn cn Xi & & Xi ii Xi Xi CJ CO CO co CJ CO CJ CO ro ro ro ro —. O O cn - J Xi Xi co IO ro o o CO OO ^ 1 cn cn Xi ro o CO cn CO O cn ro - * O o CO Ol cn SI CO CJ to u 1^ CJ cn cn cn CJ O *. c, cn ro cn CO cn Xi Xi -J O CO ro ro —' co O -j o ~j 00 CO CD ~j ii CO CO ro -o cs CJ cn ro cn CO -* cn -* O Tt CO co CD —' m m m m m m m m m m m m rr? m m m m rn m m m m m m m m m m + + + + + +• 4- + + +• + + i- + -— + + 4- + + + '+ + + +• + + O O O O O o o O O O O 9 O O O O O 6 O O O O O O O O O x. u ii Xi x. Xi Xi Xi ii X. x- — Xi Xi Xi ii Xi Xi CO to CO CO CO 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 to - j Ul IO 03 09 CO co CO CD 00 CO -o -J -j 1^ C7> 31 cn cn cn Xi Xi CO ro - i cn CO - J CO cn cn cn x> CO ro O CD oo cn Ul CO o CO cn ~j to O ro ro CD Xi Ol CO O ~j Xi cn 00 00 co Ul ro -j O O 00 cn cn CD Xi Xi - j -A - i —* - J O CO CO . ro cn ro ~1 IO CJ oo cn CO cn O CD CO CO co O Xi ro O ro -o - J ro O m m m m m m m m m m m m m m rr. m m m m m m m m m m m m m + + + + + + + + + + •*- + — + + •f- -i + + + + + + + + • O O O O o O O O O O O O O O C O O O O O O O O O O O O O Xi X* Xi Xi i i Xi Xi Xi Xi Xi i i Xi Xk x> — i i Xi — i i Xi Xi i i Xi Xi CO CO CO CO IS 3 T3 CD -n i — fi) 0) ** 3 C co ^  a> 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 cn (Tl cn cn Ol Ol Ol Ol Ul Ul Ul cn Ul Ul L- cn Ul Ul cn Ul Ul Ul Ul cn Xi Xi Xi Xi O o O O o O o o CO CD CO CD CO co CO -0 Ol Ul Xi CJ ro - i CO ov ro to C/J -j cn Ul Xi Xi CO o CO ~4 cn CO . CO " _* Ol cn ~J CO oo cn o O Ul ~ j - J Ul -j CO o O CO Ol cn Ol Ul o ro IC _» 01 Xi to co o Ul CO ro -J o cn cn 00 ro CD -j Ol o ro ~ l CO Ul ro Cl CJ c* CJ 1^ ro ro cn CD Ol CO -0 Cl cn —* —. —• m m m m m m m m m m m m m m rr. m m m m m m m m m m m —• + +• + + + + + + + + •f + + - *• + + + + + + + + + + + O O O O O O o O O O O O o o c o O O O O O O O O o o o O CO CO CO CO CJ CO CO CO CO CO CO CO CO CO CJ CO CO CO CO CO CO CO CO CO CJ CO CO CO 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 HMMMMUMMMMMMMUBUUrOIOIOMMMMUUrJIO > CDCDCOCOCDCOCOCOCDCDCOCOCDCOC^ C O C O O U J C O C D C O t O l D C D C D l o 3 cocococococDOoasoooocDcooocoxoacocDcococDCococDCDCocoai cT 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 £5 m n i m r n r n n i m r n f T i n i r T i r i i i i r n r r T i r n r r i r n r n n i r n n i i T i r n r n r n r n 3 + + + + + + + + + + + + + + --T--I--I--I- + + + + + + + + + rr 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 UUUUUUUCJUUUUUUUUUUUUUUCJUUUUU 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 ro •Jl CDooco-o^-o^iff)ffifficTiinuicncnXiXi : i X i c o c j cnxi-.cocnx.-i . L p c n x . — cp cn x. — cp cn x. — co ff> ai O 01 O on O 01 O ui O tn O cn O in O ci O in O cn O u ( B i f l o i f j a B t o c m i i i f l i s i o a i f i o f f l o a f f l i o i i ) ' " " m m m m m m m m m m m m m r n m m m m m m r n i 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 to SJ — CO Cl £. Ul o cn O X. Xi Xi Xi 10 CO CO 10 m m m m + o Ul a> 3 fi 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 to ro ro to to to to ro CO -j Ul CO CO — CJ cn CO CO CO to to ro O O to CO CO ~4 Cl Ul CJ to o —. to ro o O Ul CO co ~J 01 to CO & O Ul CO CO 10 to _* O -J CO CO o Xi ro CO Xi CJ CO Xi to -o o CO & Xi CO CD cn CO O <j CO cn ffl O to O Ul — to Ul CO O CO m CO CO Cl cn & en cn cn to O Ul •o cn cn CO X. O Xi ~1 co O -* CD O -0 m m m m m m m m m m m m m m m m m m m m m m m m m m m 4- +• 4-4-M + •f + •1- + + + + 4- •f + •(- + + + + •i + 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 01 Ul cn cn cn cn cn ffl cn ffl cn ffl ffl ffl ffl ffl ffl cn Ul Ul cn Xi Ul cn Ul Ul cn C/l 0 a Ul 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 ~j ffl Xi ro 1^ cn cn oi cn ffl cn cn 01 Ul cn Ul cn Xi Xi CO CO to _* -* O - i CD X. CD CO c o CO CO cn Ul CO ro o CD Ul CO o cn to CO to ffl CD to -0 to 00 O CO O £ X CO CO CO CO -j Xi f£ ro Xi CJ CO CO ro •^j co ro CO CD CO co Ul -* ~ I CO CO o to o CO -0 CJ Xi CO Xi Ul CO -j CO — Xi CD CJ CO Xi O 01 Xi to CO cn 00 m. m rn m m m m m m rn m m m m m m m m m m m m m m m m —' •f. + + + + 4. + + + + •i- + + <r +• + + + + +• + •*- + + '— o o O O O O 6 O O o O O O O O o O o o O O O O O O O O 3 cn cn in Ul Ul Ul cn Ul Ul cn Ul Ul cn Ul cn cn Ul cn Ul cn Ul Xi Xi Xi Xi Xi CO • 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 cj co to Ul - i 01 - i - i ffl m m m + + + O O O Ul Ul Ul O O O to — — — ffl O -j CJ to m m m + + + O O O Ul Ul Ul - i C O C o c o c o c o c o c o c o C D O o c D c o - o ~ J c n c n u i X i C o r o O c o c o ~ j f f i u i X i r o O C D f f i c o O c n t o - i - ' U i ~ i c a ~ j O u i ~ J ~ J O i X i O X i O i u i o c o - i X i - i f o o i - i C n - o x i co^rocn-^cococoOOCDtoroco-~icocncDcouvxi X i-oroco— - o o u i c n f f i c D^j - i-.OCoxi-iUifficocn m m m m m m m m m m m m m r n r n r r r m m r n . m m •(. + -I- + + + + + + + + + + T + + + + + -I- + O O O O O O O O O O O O O O O O O O O O O U l X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i X i Heat balance on flame Heat In (W): Sen s i b l e heat of fuel mixture • 0. Sen s i b l e haet of secondary a i r = 0. Chemical energy of fuel = 0. 0 0 14509E+08 Percent of t o t a l 0.0 0.0 100.00 Total - 0.14509E+08 Heat out '(W) : Sen s i b l e heat of combustion prod = 0. 1 1072E»08 76 32 Heat r e c e i v e d by s o l i d s = 0. 14255E+07 9 .83 Heat loss from s h e l l « 0. 58610E+06 4 . 04 Hydrogen loss = 0. 14247E+07 9 . 82 Total = 0. 14509E+08 y. of t h e o r e t i c a l c a l c i n a t i o n energy to s o l i d s = 37.14 ro CTl 

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