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UBC Theses and Dissertations

The feasibility of using supercritical carbon dioxide as a coolant for the CANDU reactor Tom, Samsun Kwok-Sun 1978

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THE FEASIBILITY OF USING SUPERCRITICAL CARBON DIOXIDE AS A COOLANT FOR THE CANDU REACTOR B.A.Sc., The University of Br i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the. required standard by Samsun Kwok Sun/TOM THE UNIVERSITY OF BRITISH COLUMBIA January, 1978 Samsum Kwok Sun TOM; 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requ i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that 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 s tudy . 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 c o p y i n g o f 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r 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 a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 1 January 1978 ABSTRACT This study indicates the technical f e a s i b i l i t y of using super- c r i t i c a l carbon dioxide as a coolant for a CANDU-type reactor. A new con- cept of supercritical cooling loop i s proposed in this study. The reactor is cooled by a single phase coolant, which i s pumped at a high density liquid-like state. The supercritical-fluid-cooled reactor has the advan- tage of avoiding dryout as in gas-cooled reactors, and the advantage of low coolant-circulation power as for liquid-cooled reactors. As a result of eliminating dryout, the maximum operating tempera- ture of the fuel sheath can be increased to 1021°F (550°C) for existing Canadian fuel bundles. Accordingly, the coolant temperature in the case study of this work was calculated to be 855°F. This high temperature coolant can produce steam at a temperature and pressure comparable to that of conventional fo s s i l - f u e l plants. However, since the exit coolant tem- perature from the steam generator may be as low as 280°F, a portion of the supercritical carbon dioxide coolant i s used to produce low-pressure steam. A new dual-reheat cycle i s proposed to reduce the high degree of ir r e v e r s i b i l i t y in the steam generation process. In the new dual-reheat cycle, the coolant heats the low and high pressure feeds in a parallel manner instead of alternative heating as in dual-pressure cycles. The ideal overall plant efficiency of the new proposed dual-reheat cycle i s 33.02%, which i s comparable to that of the Pickering generating station. - i i - TABLE OF CONTENTS Page ABSTRACT .. i i TABLE OF CONTENTS i i i LIST OF TABLE v LIST OF FIGURES vi NOMENCLATURE ix ACKNOWLEDGEMENTS x i i 1. INTRODUCTION 1 2. CRITERIA FOR SELECTING ALTERNATE COOLANTS (FOR CANDU-TYPE REACTORS) 5 2.1 General . 5 2.2 Overall Cost and Performance Criteria 6 2.3 Nuclear Properties 7 2.4 Chemical Properties 9 3. HEAT TRANSFER TO A LIQUID COOLANT AND MAXIMUM FUEL SHEATH OPERATING TEMPERATURE 11 3.1 Dryout in Coolant Channels 11 3.2 Maximum Operating Sheath Temperature 13 4. PROPOSED SUPERCRITICAL C 0 2 COOLING LOOP FOR CANDU REACTORS 16 4.1 Use of the Supercritical Cycle in Reactor Power Generation .... 16 4.2 Supercritical CO2 Cooling Loop 18 5. HEAT TRANSFER TO SUPERCRITICAL «>2 COOLANT 21 5.1 Heat Transfer to Supercritical Fluids in Pipes 21 I 5.1.1 Peculiarities of Heat Transfer to Supercritical Fluids ..1 21 - i i i - Page 5.1.2 Previous Studies of Heat Transfer to Supercritical Fluids . 26 5.2 Engineering Calculation of Heat Transfer to Supercritical Fluids 31 5.2.1 Choice of Correlations 31 5.2.2 Examination of Chosen Correlation 34 5.3 Heat Transfer to Single Phase Fluids in Fuel Bundles 39 5.4 Heat Transfer to Supercritical CĈ  in Fuel Bundle 42 5.4.1 Specifications and Assumptions 42 5.4.2 Fuel Bundle Geometry and Subchannel Analysis 46 5.4.3 Predicted Heat Transfer to Supercritical Carbon Dioxide in 32-element Fuel Bundles 50 6. POSSIBLE STEAM CYCLES 54 6.1 General 54 6.2 Steam Cycle for the Supercritical Carbon Dioxide Cooled CANDU Reactor Plant 56 7. RESULTS AND DISCUSSION 59 7.1 Typical Heat Transfer in Bundles 59 7.2 Typical Steam Cycles 61 8. SUMMARY AND CONCLUSIONS 63 REFERENCES 117 APPENDIX I. Computing Procedure 124 APPENDIX II. Typical Calculation of Ideal Overall Efficiency 137 - iv - LIST OF TABLE Page TABLE 1 8 - v - LIST OF FIGURES Page FIGURE 1. Thermo-hydraulic Features of PHW CANDU Reactor Power Plant 66 FIGURE 2. Typical Property Variations i n Near-critical Carbon Dioxide 67 FIGURE 3. Process Path of Supercritical Cooling Loop on P-V Diagram 68 FIGURE 4. PHW CANDU Reactor Arrangement 69 FIGURE 5. Fuel Bundle 70 FIGURE 6. Thermal Hydraulic Regions in a Boiling Channel 71 FIGURE 7. Boiling Regions 72 FIGURE 8. Subchannel Temperature Rises in a 28-element Bundle Versus Reactor Channel Length (from [12]) , 73 FIGURE 9. High Sheath Temperature Operation vs. Time for Various Zirconium Alloy Fuel Elements (from [13]) 74 FIGURE 10. Proposed Supercritical C02 Cooling Loop 75 FIGURE 11. Heat Transfer Regions: I - Liquids, II - Subcritical Gases, III - Two-Phase, IV - Above-Critical Fluids, V - Near-Critical Fluids (from [18]) 76 FIGURE 12. List of Equations 77 FIGURE 13. Routine for Calculating Wall Temperature 79 FIGURE 14. Computer Program Flow Chart for Calculating Heat Transfer to Supercritical Flow in Heated Pipes 80 FIGURE 15. Prediction of Heat Transfer to Supercritical Flow, Experi- mental Results from [61] 81 FIGURE 16. Prediction of Heat Transfer to Supercritical Flow, Experi- mental Results from [49] 82 FIGURE 12. Prediction of Heat Transfer to Supercritical Flow, Experi- mental Results from [55] 83 FIGURE 18. Prediction of Heat Transfer to Supercritical Flow, Experi- mental Results from [42] 84 FIGURE 19. Equivalent Annulus Models for Rod-Bundles 85 - v i - Page FIGURE 20. Lumped Parameter Model for Rod-Bundles (from [71]) 86 FIGURE 21. Thermal Entrance Following a Grid Type Spacer (from [73]) 87 FIGURE 22. Comparison of Heat Transfer Data in a Uniformly Heated Seven-Rod Bundle (Pr = 0.7) with Wire-Wrap Spacers (from [74]) 88 FIGURE 23. Comparison of Friction Factors for Seven-Rod Bundle with Wire-Wrap Spacers (1/d = 12, 24, 36, °°) in a Scalloped Liner (p/d = 1.237) (from [74]) 89 FIGURE 24. Temperature Profiles in Coolant Channel with Difference Fuel Bundle Orientation 90 FIGURE 25. Effect of Ratio of Pitch to Rod Diameter on Nusselt Number and Friction Factor (from [78]) . 91 FIGURE 26. Cross Section of Fuel and Coolant Tube (from [12]) 92 FIGURE 27. Channel for Boiling Reactor (from [79]) 93 FIGURE 28. Tube-in-Shell Fuel Element (from [79]) 93 FIGURE 29. Maximum Operating Temperature Versus Inlet Coolant Temperature 94 FIGURE 30. Pressure Drop Versus Inlet Coolant Temperature 95 FIGURE 31. Selection of Minimum Equivalent Hydraulic Diameter 96 FIGURE 32. Cross Section of 32-Element Fuel Bundle 97 FIGURE 33. Radial Neutron Flux Distribution i n Fuel Bundle 98 FIGURE 34. Subchannel Coolant Temperature Rise i n 32-Element Fuel Bundles Versus Reactor Channel Length 99 FIGURE 35. Subchannel Sheath Temperature Rise in 32-Element Fuel Bundles Versus Reactor Channel Length 100 FIGURE 36. Pressure Drop Along Reactor Channel Length 101 FIGURE 37. Single Pressure Steam Generation 102 FIGURE 38. Dual-Pressure 600.0/70.3 (psia) Steam Generation 103 FIGURE 39. Turbine Circuit Heat Balance of Single Pressure Cycle 104 FIGURE 40. Turbine Circuit Heat Balance of Dual-Pressure 600/70.3 (psia) Cycle 105 - v i i - Page FIGURE 41. Dual-Pressure 1200.0/70.3 (psia) Steam Generation 106 FIGURE 42. Turbine Circuit Heat Balance of Dual-Pressure 1200.0/70.3 (psia) Cycle 107 FIGURE 43. Dual-Reheat 1200.0/70.3 (psia) Steam Generation 108 FIGURE 44. Turbine Circuit Heat Balance Of Dual-Reheat 1200/70.3 (psia) Cycle 109 FIGURE 45. Results of Heat Transfer to Supercritical CO Coolant i n 32-Element Fuel Bundles 110 FIGURE 46. Overall Performance of Supercritical Carbon Dioxide Coolant I l l FIGURE 47. Overall Dimensions of 32-Element Fuel Bundle 112 FIGURE 48. Summary of Steam Conditions of Thermal Cycles 113 FIGURE 49. Overall Performance of Thermal Cycles 114 FIGURE 50. Turbo-Generator Heat Balance (from [12]) 115 FIGURE 51. Expansion Line of CANDU Reactor Turbine Stages (from [82]) 116 - v i i i - NOMENCLATURE A experimental constant b experimental constant Cp specific heat at constant pressure Cp integrated specific heat, equation (13) CWP unheated wetted perimeter c experimental constant D equivalent hydraulic diameter, equation (25) d diameter eff efficiency F integrated f l u i d property F(T) f l u i d property as a function of temperature f normalized neutron flux f(x/d) correction factor, equation (24) Gr Grashof number h enthalpy K thermal conductivity k constant, 0.51504 M mass flow rate m experimental constant Nu Nusselt number n experimental constant . P pressure Pc ' c r i t i c a l pressure Pr Prandtl number • Dimensional quantities are evaluated in both S.I. and British units. - ix - Pr integrated Prandtl number, equation (15) p pitch distance Q heat flux q linear heat rate Re Reynolds number T temperature T integrated temperature, equation (21b) Tc c r i t i c a l temperature T(x) temperature as a function of distance TWP total wetted perimeter u velocity W work x distance from channel inlet y distance Z dimensionless parameter, equation (5) Subscripts b bulk in inlet m pseudo-critical temperature min minimum 0 constant property out outlet w wall or sheath z dimensionless parameter 1 location 1 2 location 2 - x - Greek Symbols T shear stress y viscosity p density £ m momentum eddy diffusivity thermal eddy diffusivity £ f r i c t i o n factor H length pu average product of pu o- p integrated density A differential change £ summation (j> constant, 0.25744 Cr i t i c a l Values_of_Carbon_Dioxide pc c r i t i c a l pressure, 73.82 bars or 1071.3 psia Tc c r i t i c a l temperature, 31.04°C or 87.9°F - x i - ACKNOWLEDGEMENTS I would like to express my sincere thanks and appreciation to my supervisor, Dr. E.G. Hauptmann for his help and guidance throughout the project. Additional thanks are due to Dr. P.G. H i l l for invaluable comments on phases of the research. I would also like to express my deep gratitude to my parents for encouraging me throughout the academic years. This work was sponsored by the National Research Council of Canada. - x i i - 1 1. INTRODUCTION It has become evident that the world's limited o i l and gas resources w i l l have to be supplemented by nuclear energy for electrical power generation. Nuclear power plants have already operated successfully for many years, and most public u t i l i t i e s are counting heavily on nuclear power to meet the bulk of ele c t r i c a l demands i n the next century. Power reactors can be classified according to types of coolant, moderator, and fuel. The most popular types are the pressurized water and boiling water reactors, which use enriched uranium as fuel and ordinary water as both coolant and moderator. Of special interest to Canada is the CANDU (CANadian JJeuterium Uranium) system, which uses natural uranium as fuel and heavy water (deuterium oxide) as coolant and moderator. Figure 1 shows the main thermo-hydraulic features of a PHW (Pressurized Heavy Water cooled) CANDU reactor power plant. The CANDU system is a successful design in terms of fuel economy and safety. Nevertheless, i t s t i l l has several weaknesses. One of these is that the pressurized heavy water coolant cannot be raised to a tempera- ture high enough to generate steam at a pressure and temperature comparable to that of conventional fossil-fuel stations. As a result, the practical overall plant efficiency of the CANDU system is relatively low (29.7%). Atomic Energy of Canada Limited (AECL) is pursuing many routes toward increasing the efficiency of the existing CANDU system. One such route i s looking for alternate coolants to heavy w a t e r ^ . In the development of the CANDU system, a prototype reactor using ordinary water instead of heavy water has been operating at Gentilly, 2 Quebec since 1971Li'1. A research reactor at Whiteshell, Manitoba has been operating since 1966, and i s cooled by terphenyl, an organic liquid that boils at a temperature above 700°F at atmospheric pressure . However, in both boiling and pressurized water reactors, operation at higher power levels may result in blanketing of the heat transfer surface with a layer of vapor. Such blanketing of the surface by vapor may result in "dryout" of the coolant with the possibility of fuel sheath failure. Thus, to avoid accidental rupture of the fuel sheath, i t s maximum temperature must be only slightly higher than the saturation temperature of the coolant used. Because the temperatures of both coolant and fuel sheath are constrained, the overall plant efficiency is limited. To obviate dryout and obtain higher coolant temperatures, the reactor can be cooled by gaseous coolants. Because gaseous coolants do not change phase throughout the operating temperature range, dryout due to phase change of coolant cannot occur. The fuel sheath can operate at i t s maximum temperature (based on metallurgical considerations), which i s usually much higher than the saturation temperature of most coolants. Compared with liquid coolants, however, gaseous coolants are characterized by low specific heat at constant pressure, low heat transfer coefficients, and low specific volume. For this reason, gas cooled reactors have large flow passages through [3] the reactor, and require higher pumping power for coolant circulation . To retain the^advantages and overcome the disadvantages of gaseous coolants, reactors may be cooled with "supercritical" coolants. These are fluids compressed above their c r i t i c a l pressure, and as a result always exist as a single phase. For a given pressure of a supercritical f l u i d , there is a temperature at which the maximum specific heat occurs, known as the pseudo- 3 c r i t i c a l or transposed c r i t i c a l temperature. The thermophysical properties of supercritical fluids change drastically across the pseudo-critical tem- perature. Figure 2 shows the thermal conductivity, viscosity, and density of supercritical CÔ  at 1100 psia. Above the pseudo-critical temperature, the properties of a supercritical f l u i d are somewhat like a normal gas. The thermal conductivity or dynamic viscosity i s not greatly affected by changes in temperature, and the density is low. In this region, the f l u i d is termed "vapor-like". Below the pseudo-critical temperature, the f l u i d i s termed "liquid-like", since i t has properties very much like a liquid. The thermal conductivity and dynamic viscosity decrease with increasing tempera- ture as i n ordinary liquids, and the density i s high. Therefore, a super- c r i t i c a l cooling loop which has i t s process path as shown in Figure 3 can be arranged so that the liquid i s pumped while in a liquid-like state, thereby greatly reducing pumping power. The purpose of the present work is to investigate not only the possibility of using supercritical carbon dioxide (CO^) as a coolant, but also the application of a supercritical cooling loop to the CANDU system. The major part of this work involves an analytical study of heat transport in the coolant channels and the thermodynamics of the CANDU reactor power station when supercritical CÔ  i s used as a coolant. Carbon dioxide was selected because of i t s availability at low cost, i t s inertness in contact with fuel and reactor materials, and i t s s t a b i l i t y at relative high tempera- ture. In addition, i t s low c r i t i c a l pressure makes application of a super- c r i t i c a l cooling loop particularly suitable to the CANDU reactor. In this investigation, the basic design of the CANDU reactor was assumed to be unchanged when supercritical CO2 was used as a coolant. However, the operating temperature of the fuel sheath was taken to be the maximum 4 allowable temperature (1021°F or 550°C). In the i n i t i a l part of this study, standard single and dual reactor steam cycles were analysed when using supercritical CO2 as a coolant. The results indicated the overall plant efficiencies would be lower than the current CANDU ideal plant efficiency of 33.94%. As a result, a new dual- reheat reactor steam cycle is proposed. In this new cycle, supercritical C02 generates high pressure steam at 1200 psia, 815°F and low pressure steam at 70.3 psia, 567°F simultaneously for high and low pressure turbines respec- tively. The expanded steam from the high pressure turbines is mixed with low pressure feed water to be evaporated in the steam generator before expansion through the low pressure turbines. This new steam cycle has a comparable ideal overall plant efficiency of ,33.02%. This indicates the possibility of increasing of the overall plant efficiency of CANDU reactor power sta- tions, i f the performance of supercritical CO^ coolant was optimized in this preliminary study. This study also points out other possible advantages of using supercritical CO^ as a coolant for the CANDU system: 1. lower coolant capital and upkeep costs; 2. compatibility with uranium carbide fuel having high uranium content and high thermal conductivity; 3. increased safety due to elimination of possible dryout. 5 2. CRITERIA FOR SELECTING ALTERNATE COOLANTS (FOR CANDU-TYPE REACTORS) 2.1 General Choosing the coolant i s a major consideration in the design of nuclear reactors. It strongly affects the performance of reactors, and eventually the cost of power generation. In selecting a good coolant for power reactors, several c r i t e r i a must be considered: 1. overall cost and performance: a) low capital and upkeep costs, b) availability; 2. nuclear properties of coolant: a) low neutron absorption cross-section, b) good radiation-stability, c) low induced-radioactivity; 3. chemical properties of coolant: a) stable at high temperature, b) compatible with fuel and reactor materials; 4. heat transport and flu i d properties of coolant: a) low pumping power, b) high density, c) high heat capacity, d) high heat conductivity, e) high saturation temperature (liquid coolant). Past studies have not identified a coolant which can adequately 6 meet a l l the above c r i t e r i a . Supercritical CO^ i s no exception. For example, at high temperature i t has poor heat transfer characteristics. However, when used in a supercritical cooling loop, the designer can s t i l l achieve good overall heat exchange by increasing the flow rate. The resulting high pressure losses do not result in excessive pumping power since the coolant is pumped in a liquid-like state. The economics of using supercritical CÔ  as a coolant for the CANDU system w i l l be discussed in this chapter. Nuclear and chemical properties of the coolant are also discussed, while heat transport c r i t e r i a are l e f t to a later chapter. 2.2 Overall Cost and Performance Criteria One of the major direct costs of a CANDU reactor power generation station i s the capital and upkeep costs of the coolant. For example, the total cost of building a 2 x 600 MWe CANDU-PHW (for Pressurized Heavy Water cooled) reactor power generation plant was about 300 million dollars in [41 1972 . The heavy water cost alone was 65 million dollars, with 37.4% of the heavy water used as coolant. The coolant upkeep cost of the Pickering [2] station was as high as 35% of the fueling cost . Because of the relatively low cost of CO2, the cost of power generation would be significantly reduced i f CÔ  were used as a substitute for heavy water in CANDU reactors. The capital.and upkeep costs of a coolant depend on the i n i t i a l inventory and rate of leakage of the coolant. The evaluation of the costs of using supercritical C 0 2 as a coolant for the CANDU system and the compari- son with heavy water can only be made after a detailed study for a particular design of power generating plant, which was beyond the scope of this i 7 study. 2.3 Nuclear Properties If supercritical CÔ  i s used as a coolant, i t must have good nuclear properties. Since these properties are usually functions of tem- perature but not pressure, supercritical CC^ has similar imicroscopic nuclear properties to gaseous CC^. Experience with British and French designed reactors has shown that gaseous CÔ  i s a good coolant in Jterms of nuclear properties. The most important requirement of a coolant from a nuclear point of view is that i t has low neutron absorption. The rate of neutron absorp- tion i s a function of the macroscopic absorption cross-section, which i s proportional to the product of the density and the microscopic absorption cross-section of the coolant molecules. Under present CANDU reactor operating conditions of 570°F and. 1450 psia, the macroscopic absorption _5 2 3 ' cross-section of heavy water is 2.7095 x 10 cm /cm . On the other hand, because of the low density of supercritical CÔ , i t s macroscopic cross- —6 2 3 section is only 5.9543 x 10 cm /cm which is much smaller than heavy water. The low macroscopic absorption cross-section usually means good radiation s t a b i l i t y and low induced-radioactivity, because the coolant has l i t t l e chance to be broken down or capture nuclear particles and become radioactive. Moreover, most of the possible neutron capture in CO2 results in further stable nuclides. Therefore, CÔ  i s stable in a radioactive environment. It would be unrealistic to consider coolant-grade C0 2 as a pure 8 substance. A typical coolant analysis might be 10 p.p.m. Ĥ O, 3 p.p.m. Ar, 10 p.p.m. Ĥ , 17 p.p.m. 0^, and 94 p.p.m. N 2 ^ . The reactions which are important in considering induced radioactivity of impurities are liste d in Table 1. Because of the shorter half-lives, the and 0 ^ nuclides decay 41 quickly. The Ar which has an intermediate h a l f - l i f e gives a low residual activity in the cooling c i r c u i t , but the activity level taken in coolant, leakage from the circuit as whole i s too small to constitute a hazard. REACTION CROSS-SECTION (barns) HALF-LIFE A r 4 0 ( n , Y ) A r 4 1 0.53 110 min. 0 1 8(n,Y)0 1 9 2.2 x 10"4 29.0 sec. 0 (n,p)N 16.0 x 10"6 7.5 sec. 0 1 7(n,a)C 1 4 0.4 5500 yrs. C 1 3(n,y)C 1 4 0.1 5500 yrs. N (n,p)C 1.76 5500 yrs. TABLE 1. 14 From a health physics point of view C i s an important long-lived consti- tuent of the coolant when i t i s released to atmosphere by leakage 14 or during blowdown. The C is expected from three main sources which are also listed in the Table 1. Although a proper estimate of radiation level or intensity from 14 C is beyond the scope of the present study, i t is clear that because i t s 14 long h a l f - l i f e , any amount of C production in the coolant w i l l require 9 careful measures to prevent i t s escape. This is parallel to the require- 3 ment of controlling tritium (T ) discharge while using heavy water as a coolant. 2.4 Chemical Properties The thermal st a b i l i t y of supercritical CÔ  i s quite satisfactory up to the temperature envisaged for most reactor operations. Carbon dioxide tends to dissociate into CO and 0^ as the temperature is increased, but even at 1000°K the equilibrium concentration of CO at a pressure as low as 1 0 atmospheres would be only 3 p.p.m. according to the data of Wagman et a l . t 6 ] . Mor eover, CO is relatively stable when irradiated so that the concentration of radiolysls products in supercritical CÔ  coolant should be low. This observation is confirmed by Bridge and N a r i n ^ . In the presence of impurities (e.g., NÔ ) that can react with small equilibrium r g 1 concentrations of radiolysis products, CO2 is progressively broken down . In British and French reactors, which use graphite as a moderator, a further reaction takes place between graphite and the chemically reactive species produced by the irradiation breakdown of CÔ , resulting in a problem of attrition of graphite. However, dissociation of CÔ  is not a serious problem in CANDU reactors which use heavy water instead of graphite as a moderator. When a reactor coolant i s selected, the choice of reactor fuel is restricted. Today, uranium dioxide (UO^) is used in CANDU reactors because of i t s compatibility with heavy water and light water. Uranium dioxide is an excellent fuel with a high melting point and high uranium content, but very low thermal conductivity. Uranium metal and uranium i I ! 10 carbide are also good fuels. They have high melting points, high uranium content and high thermal conductivity, but react vigorously with heavy or light water. For this reason, AECL has been looking for alternate coolants [91 which are compatible with uranium carbide fuel developed by AECL Using supercritical CO2 as a coolant w i l l allow use of uranium metal and uranium carbide as fuels. Although CÔ  reacts with uranium metal and uranium carbide, the rate of reaction is very slow. Measurements of the rate of change of weight of sintered uranium monocarbide specimens in dry C0 2 over the temperature range of 500°-830°C (931-1525°F) were made by A t i l l et a l . ^ ^ . The weight increased only by 120-170 mg/Kg/hr which was about 100 times as fast as the rate of change of weight of uranium metal under the same condition. The low rate of reaction between CO^ and either uranium metal or uranium carbide allows enough time for defective fuel to be recovered before i t becomes a serious problem. Wyatt has pointed out that when high integrity cans are used, i t should be possible to use uranium metal and uranium carbide fuels with C0„ coolant. 11 3. HEAT TRANSFER TO A LIQUID COOLANT AND MAXIMUM FUEL SHEATH OPERATING TEMPERATURE 3.1 Dryout In Coolant Channels AECL has pioneered in-reactor heat transfer testing with experi- mental and power reactors, and has gained a large amount of operating experience with effects of coolant flow on fuel sheath behavior. Figure 4 shows a PHW CANDU reactor arrangement, and Figure 5 shows the details of the fuel bundle. The coolant flows into the coolant channels and travels within subchannels between fuel rods. The phenomenon of "dryout" has been detected in coolant channels, resulting in sudden and significant increase in fuel sheath temperature. Figure 6 shows variation in sheath or wall tem- perature in a coolant channel, along with a notation of the types of flow regions believed to be present. Forced convection boiling in channels is an extremely complex phenomenon, but the main idea can be more easily understood by considering heat transfer from a small heating element submerged in a large stationary pool of liquid (pool boiling). Figure 7 shows the conventional representa- tion of the heat flux q versus the temperature difference AT between the heating, surface and liquid coolant bulk temperature. This curve can only be obtained i f the temperature of the heating surface is carefully con- trolled. In the region preceeding point "a" on the curve, the heating sur- face temperature is nearly at or only a few degrees above the liquid tem- perature. Thus, there is insufficient liquid superheat and no bubble for- mation. Heat transfer in this region is therefore by liquid natural con- vection only. As the heating surface temperature is increased, the heat 12 flux increases and the region "a-b" is reached. A small number of nuclea- tion centers become active, and a few bubbles are formed. These bubbles soon collapse as they move away from the surface. As AT is further increased between "b-c", the number of nucleation centers, and consequently the number of bubbles, increases rapidly with AT. The bubbles cause considerable agi- tation and turbulence of the liquid in the boundary layer and consequently a greater increase of the heat flux with AT. Region "b-c" is called the "nucleate boiling region". Point "c" is called the point of "cleparture from the nucleate boiling" (DNB), and the corresponding heat flux is called the "critical heat flux". The value of critical heat flux or the temperature of DNB depends on local coolant conditions. When region "c-d" is reached, the bubbles become so numerous that they begin to coalesce on the heating surface. In this case, a portion of the heating surface gets blanketed with vapor which acts as a heat insulator. This region is called "mixed nucleate-film boiling". In the region "d-e", a continuous blanket of vapor forms over the entire heating surface, and the heat transfer coefficient reaches its lowest value. This region is called "stable film boiling". In the region "e-f", called "film and radiation region", the temperature of the heating surface is so high that thermal radiation from the surface comes into play. The curve in Figure 7 could not be obtained i f the temperature of the heating surface was not closely controlled. In most practical cases, such as heat transfer in reactors, the heat flux is controlled. Referring to Figure 6, a sequence of events similar to that of pool boiling can now be imagined. The liquid enters as a single phase; the vapor bubbles start to form; small bubbles collapse together to form slugs, and a liquid annulus 13 forms. Finally, the local heat flux becomes c r i t i c a l to the local coolant condition, corresponding to the point "c" of the curve i n Figure 7. A further increase in heat flux or increase in vapor quality results in a sudden local temperature jump which corresponds to the jump from "c" to "c" 1 into the film-radiation region. The heating surface temperature at "c" 1 may be high enough to exceed the safe operating limit. The complete loss of coolant along with a sudden jump of temperature of the heating sur- face is called "dryout". Dryout can occur not only in a boiling channel where phase change takes place, but also at a point in a compressed liquid coolant channel because of increase in local heat flux or local coolant starvation. Dryout i s very dangerous i n reactor operation, since local temperature increases may soften the fuel elements. They may bend toward the coolant tube and cause poor heat transfer due to local coolant starvation, which further elevates the temperature. This may fi n a l l y result in fuel sheath failure. 3.2 Maximum Operating Sheath Temperature Since dryout is a c r i t i c a l design parameter, a l l coolant channel conditions must be controlled so that a significant margin of safety i s available to prevent dryout from occurring during normal operation. In other words, the operating temperature of the fuel sheath must be kept below the c r i t i c a l heat flux temperature, which i s approximately equal to the temperature of the reactor coolant. [121 . . . Figure 8 shows the calculated subchannel temperatures in a 28-element fuel bundle design with a given heat flux shape along the channel and given maximum linear rates for individual subchannels. The 14 heat flux shape and the maximum linear heat rates are based on data during steady state operation of a CANDU 500 reactor at the Pickering station . An inter-subchannel mixing rate was assumed in calculating the temperatures in Figure 8. The calculation agrees well with the measurement, and shows that the maximum local external sheath temperature exceeds the coolant saturation temperature by only about 5°F. Nucleate boiling would actually begin on the sheath surface at a temperature about 10°F above the coolant saturation temperature. The maximum operating sheath temperature in a CANDU reactor i s usually kept much below the maximum withstanding (metallurgical) tempera- ture of the fuel sheath. Zircaloy clad fuel elements (like those used in CANDU reactors) can actually operate at an elevated temperature for a certain period without failure. Operating l i f e i s assumed inversely proportional to the magnitude of the temperature. Some data from various [13] tests are summarized in Figure 9 which i s a semilog plot of time-to- defect versus temperature. It shows that zircaloy clad fuel elements can operate at a temperature below approximately 550°C without defect for a relatively long period. If supercritical CO^ were used as a coolant for a CANDU reactor, dryout could not happen since supercritical CO^ would remain in a single phase throughout the operating temperature range. Therefore, fuel sheath temperature jump would no longer be a cause of worry under normal operating conditions. The saturation temperature of the coolant is no longer a barrier to the upper limit of the maximum fuel sheath temperature, and the operating _ Pickering station i s the f i r s t commercial CANDU-PHW reactor power generating station, owned and operated by Ontario Hydro. 15 fuel sheath temperature can be as high as 550°C (1021°F). This implies that the maximum coolant temperature, and thus the efficiency of reactor power generation, can be higher than the corresponding liquid coolant. 16 4. PROPOSED SUPERCRITICAL C0 2 COOLING LOOP FOR CANDU REACTORS 4.1 Use of the Supercritical Cycle in Reactor Power Generation Supercritical fluids have gained much attention as coolant media in many applications. Currently, supercritical hydrogen i s used as the fuel and coolant in combustion chambers of large rocket engines. Super- c r i t i c a l water is used as a working f l u i d in fos s i l - f u e l power generating stations, and near-critical helium i s used as a coolant in super-conducting applications. Many other applications are being proposed and investigated. Among them i s an entirely supercritical turbine power cycle, with a nuclear reactor as a direct heat source (the entirely supercritical turbine power cycle w i l l be referred to as the supercritical cycle in this work). Although the supercritical cycle with a reactor as a direct heat source has not been previously investigated, both the supercritical cycle and the supercritical f l u i d cooled reactor have been separately studied. The performance of supercritical cycles has been analysed by many w o r k e r s ^ " * " 4 , ' , who pointed out the high efficiency of the cycle. In a case study of a supercritical CO^ power cy c l e ^ ^ ^ , the power plant has an overall efficiency as high as 48%. There are many advantages to the cycle: low volume-to-power ratio, no blade erosion in turbines, no cavita- tion in pumps and single stage turbine and pump operation. The SCOTT-R reactor system^^, which is cooled by using supercritical light water, i s currently under investigation. Reactors cooled by using supercritical fluids have the advantage of avoiding the limitations imposed by boiling heat transfer and also the advantage of high enthalpy rise through the core. As expected, direct application of the coolant as the working flu i d 17 of a supercritical cycle has the combined advantages as discussed above. The high operating pressure is the basic engineering problem when a supercritical f l u i d i s used not only as the working flu i d of the cycle, but also as the coolant of a reactor. The high operating pressure requires thicker structural parts of the system. The increase of reactor material results in an increase of neutron absorption, and hence, the decrease of the reactor neutron economy which directly determines the fueling cost in nuclear power generation. Therefore, the high efficiency of the supercritical cycle with a reactor as a direct heat source can be overshadowed by high fueling costs. The maximum operating pressure of the supercritical cycle depends on the c r i t i c a l pressure of the flui d used. For example, according to an analysis [14] of the supercritical CG^ cycle , the ratio of required turbine inlet pres- sure to pump inlet pressure i s equal to two for optimum performance, and pump inlet pressure should be maintained above the c r i t i c a l value. If that i s also true for the supercritical water cycle ( c r i t i c a l pressure of 3206 psia), the turbine inlet pressure must be approximately equal to 6500 psia. This pres- sure is too high to be practical in engineering designs. To solve the problem of high pressure operation, alternative fluids having a low c r i t i c a l pressure must be sought as substitutes for water. However, when the fl u i d i s used not only as a working flu i d but also as a coolant, the choice of the fluid i s restricted not only by the requirement of low c r i t i c a l pressure but also by coolant characteristics. Carbon dioxide has a low c r i t i c a l pressure (Pc = 1071.3 psia) and seems to meet almost a l l other necessary requirements. Supercritical CO 2 may make use of a super- c r i t i c a l cycle with a reactor as a direct heat source possible in the 18 future. 4.2 Supercritical CO Cooling Loop While the direct cycle described in the previous section appears attractive for future use, several major hardware development programs are required before a power plant using the cycle could be built. For example, power turbines which could operate with CÔ  would have to be developed and built. In addition, the present CANDU reactors would have to be modified. In order to be able to use present CANDU reactor technology and conventional steam turbine power generating equipment, an intermediate step of using a supercritical cooling loop is proposed in this work. A schematic diagram of the proposed loop is shown in Figure 10, along with i t s process path on a temperature-entropy diagram. The loop contains the following steps: (a)-(b). l i q u i d - l i k e supercritical CO is pumped from low pressure to high pressure; (b)-(c). high pressure C0 2 gains heat in the recuperator from low pressure CO leaving the steam generator; (c)-(d). supercritical CO is heated in the reactor; (d)-(e). hot supercritical CO generates steam in the steam generator; (e)-(f). low pressure CO from the steam generator gives up heat in the recuperator; 19 (f)-(a). supercritical CÔ  i s further cooled to become a high density liquid-like f l u i d . Recuperation is used in the loop to maintain high entry temperature of the coolant into the reactor. High entry temperature results in generation of higher pressure and temperature steam. The temperature of the coolant entering the reactor, i.e., the temperature of the coolant leaving the steam generator, affects the overall efficiency of power generation as w i l l be discussed in a later chapter. Supercritical CÔ  from the recuperator i s further cooled below the pseudo-critical temperature in order to reduce pumping power. Although gas-cooled reactors also have no limitations imposed by boiling heat transfer, the basic design concept of a supercritical CÔ cooled CANDU reactor i s entirely different, due to the drastic changes of thermophysical properties of supercritical CO^ across the pseudo-critical temperature. In the gas-cooled reactor, because the gaseous coolant i s circulated in a low density state, the flow areas are large and the speed of the coolant i s low in order to have minimum pressure drop and low pumping power. To cope with the problem of poor heat transfer to the slow moving coolant, fuel elements normally have fins added to their outer sur- faces. Hence, the flow area is further increased to deal with the problem of high pressure drop introduced by the fins. Therefore, gaseous coolants are not suitable for use in the CANDU reactor since: 1. large diameter coolant tubes would be required; 2. large amounts of coolant are required in the reactor, increasing the capital and upkeep costs; 20 3. poor heat transport characteristics of gaseous coolants may limit the coolant temperature. Because the pumping power would be low using supercritical (X^, the designer can afford higher pressure drops from high speed coolant flow in smaller cross-sectional area channels in order to create high heat trans- fer rates. For this reason, the supercritical CO^-cooled reactor does not have the previously discussed problems of gas-cooled reactors, despite the poor thermophysical characteristics of supercritical CÔ  at high temperature. Supercritical CÔ  cooling appears very practical for the CANDU reactor, which uses bundle-cluster or tube-in-shell fuel elements in small diameter coolant tubes. 21 5. HEAT TRANSFER TO SUPERCRITICAL C0 2 COOLANT Heat transfer to supercritical fluids in fuel bundles has not been studied as far as the author is aware. However, the separate cases of heat transfer to supercritical fluids in heated pipes and to single phase fluids in fuel bundles are currently under investigation. This chapter w i l l dis- cuss the previous investigations of these cases individually. Finally, based on these separate studies, heat transfer to supercritical CÔ  in fuel bundles is analysed. 5.1 Heat Transfer to Supercritical Fluids in Pipes 5.1.1 P££uii£Eiti?f?_0.f_5£2*: l£§B5l e E_^2_§yP£E£ riti££ i_l!iyi^ This section discusses the peculiarities of heat transfer to supercritical fluids which must be taken into account i n actual design of heat exchanger equipment. These peculiarities strongly affect the per- formance of a supercritical cooling loop. Heat transfer problems are classified according to the fl u i d T181 states as shown in Figure 11 . The five regions are the liquid, gas, two-phase, above-critical, and near-critical regions. Regions I and II refer to single phase liquid and gas in which heat transfer rates can be predicted by the usual forced convection equations. The fl u i d in region III is a.two-phase fluid and heat transfer occurs by boiling. Region IV is the above-critical region where the mechanism for heat transfer is not well understood. Region V is the near-critical region where special heat transfer phenomena s t i l l require a great deal of study. It i s very d i f f i c u l t to precisely define the boundaries of the 22 regions, especially the boundaries which separate the near-critical region from i t s adjacent regions. There are two important reasons for this: the transition is not abrupt and sharp, and the extent of the influence of the c r i t i c a l region on heat transfer is a function of the process or path by which the flu i d approaches the c r i t i c a l point. For example, a hysteresis [19] loop in density near the c r i t i c a l point could be obtained by f i r s t heating and then cooling along an isobar. The supercritical CÔ  cooling loop investigated in this study uses a f l u i d which passes through the above-critical and near-critical regions. Interest in the study of heat transfer to supercritical fluids was initiated by the work of Schmidt in the early 1930's. He pointed out that because the specific heat at constant pressure and the compressi- b i l i t y both grow very large at the c r i t i c a l point, the heat transfer coefficient, which is: a function of the Prandtl and Grashof numbers (Pr, Gr), should be very large. For supercritical ammonia, Schmidt later found an apparent thermal conductivity (the thermal conductivity required of a solid bar of the same dimension as the test chamber to transfer the experimentally observed amount of heat for a given temperature gradient) as large as 4000 times that of pure copper over a narrow temperature range. However, until the mid-1950's, very few investigations into super- c r i t i c a l heat transfer were reported. Because of the increase of applica- tions of heat transfer to supercritical fluids, investigations i n this region have been greatly increased in recent years. Although some ["22 23 2A 231 workers ' ' ' reported that the heat transfer coefficient was T 26 27 28 291 enhanced as proposed previously by Schmidt, many other workers ' ' ' reported that heat transfer in the near-critical region actually approached a minimum value. 23 In his survey, Petukov^^"' classified the heat transfer to super- c r i t i c a l fluids into three regimes: normal, reduced, and improved. In the normal heat transfer regime, there i s significant variation in physical properties of the supercritical f l u i d across the flow, but the dependence of Nusselt number (Nu) on Reynolds number (Re) and Pr is approximately the same as in the case of heat transfer to constant property flows. Therefore, relationships which govern the normal heat transfer can usually be derived. In the reduced heat transfer regime, the heat transfer coefficient and wall temperature do not satisfy the same relationships which govern normal heat transfer in the supercritical region. The heat transfer c o e f f i - cient depends not only on Re and Pr but also on the heat flux, and i s less than the value calculated from the normal heat transfer relationships. The reduced heat transfer condition usually occurs when the mean fl u i d temperature is less than the pseudo-critical temperature at a particular combination of bulk velocity and heat flux, and the wall temperature is usually, but not necessarily, much greater than the pseudo-critical tempera- ture. The improved heat transfer regime occurs only at a high heat flux with the bulk temperature less than the pseudo-critical and the wall tempera- ture slightly greater than the pseudo-critical temperature. The heat transfer rate increases with increasing bulk density and velocity, but decreases with increasing bulk temperature. As in the reduced heat transfer regime, the heat transfer coefficient depends on Re, Pr, and also the heat flux. [371 Hsu explained the increases and decreases of heat transfer rates by comparison with the boiling mechanisms of a liquid, discussed in Chapter 3. He pointed out that when the temperature difference is small, 24 the mechanism can be linked to nucleate b o i l i n g , a regime of very good heat transfer, thus the maximum- When the temperature d i f f e r e n c e i s large, the mechanism can be compared to f i l m b o i l i n g , a regime of poor heat t r a n s f e r , [31 32 331 thus the minimum. Many other workers ' ' strongly supported t h i s explanation of a pseudo-two-phase f l u i d . However, when a s u p e r c r i t i c a l f l u i d i s i n equilibrium, i t must e x i s t as a s i n g l e phase, which opposes [34] such a pseudo-two-phase f l u i d hypothesis. Hauptmann showed that " b o i l i n g " does not e x i s t during heat transfer to s u p e r c r i t i c a l f l u i d s , and concluded that a l l of the unusual r e s u l t s could be explained i n s i n g l e phase [35] terms. Kafengauz explained the ph y s i c a l nature of the unusual r e s u l t s of heat transfer at s u p e r c r i t i c a l pressure. He suggested b o i l i n g could be observed i n some experiments because fl u c t u a t i o n s i n pressure caused bulk pressure to f a l l below the c r i t i c a l . In h i s work, a t h e o r e t i c a l approach was developed which was v e r i f i e d by experimental data with d i i s o p r o p y l cyclohexane. Disregarding the possible existence of b o i l i n g , P e t u k h o v ^ ^ described reduction and improvements i n heat transfer to s u p e r c r i t i c a l f l u i d s with a rather c l e a r p i c t u r e . The onset of the regime with reduced heat transfer i s apparently associated with a change i n the hydrodynamics of the flow and, i n p a r t i c u l a r , with turbulent transfer processes due to the great v a r i a t i o n i n thermophysical properties of the medium over the F361 cross section of the flow. For example, Wood found M-shape v e l o c i t y p r o f i l e s during s u p e r c r i t i c a l f l u i d flow i n pipes. This was very d i f f e r e n t from the U-shape v e l o c i t y p r o f i l e s of constant property f l u i d i n pipes. Free convection due to the large density gradient present may also have a s i g n i f i c a n t e f f e c t on the nature of the flow. This point i s confirmed by the fact that reduced heat transfer occurs only i n a heated tube with 25 ascending flow but not descending flow. In the l a t t e r case, normal heat transfer i s observed. This i s probably because of the more intense mixing due to opposing d i r e c t i o n s of forced and free convection near the w a l l . Petukhov^"^ explained that improvement i n heat transfer at high heat f l u x i s also apparently due to the very pronounced changes i n thermophysical properties of the medium (mainly density and s p e c i f i c heat at constant pressure) over the cross section of the flow. The density of the medium at the wall i s several times le s s and the s p e c i f i c heat i s several times more than i n the core of the flow. F l u i d which reaches the hot wall from the core, owing to turbulent transport, has a r e l a t i v e l y high thermal conductivity and low s p e c i f i c heat. I f there are large d i f f e r e n c e s i n temperature between the f l u i d next to the w a l l and the f l u i d a r r i v i n g from the core, the f l u i d picks up heat very r a p i d l y and expands i n an explosive manner. This process must r e s u l t i n more intense mixing of the f l u i d i n the layer next to the w a l l , and hence, i n improved heat t r a n s f e r . These s p e c i a l features of v a r i a b l e property turbulent heat transfer are not taken into account i n e x i s t i n g theories, and therefore, present theories cannot s a t i s f a c t o r i l y describe the improved heat transfer regime. Although dryout due to b o i l i n g would not be a l i m i t a t i o n i n the design of a s u p e r c r i t i c a l cooling loop, other l i m i t a t i o n s due to the p e c u l i a r i t i e s of heat transfer to s u p e r c r i t i c a l f l u i d s must be taken into account. The reduced heat transfer regime could occur i n the recuperator of the .proposed s u p e r c r i t i c a l cooling loop, where the coolant i s at a temperature near the c r i t i c a l value. However, the reduced heat transfer regime can be avoided by c a r e f u l design of the heat transfer surface and the f l u i d v e l o c i t y , etc., for a given heat f l u x d i s t r i b u t i o n . On the 26 other hand, a compact and efficient recuperator can be designed based on knowledge of the improved heat transfer characteristics of supercritical fluids. 5.1.2 Previous_Studies of_Heat Transfer to Supercritical Fluids Heat transfer mechanisms in supercritical fluids are not yet well understood. However, many investigations are currently underway in the near-critical region where most of the heat transfer peculiarities take place. Heat transfer i n the above-critical region, where variations i n thermophysical properties of the f l u i d across the flow affect heat transfer to a lesser extent, i s usually thought of as a special case of the near- c r i t i c a l region. Therefore, the mechanisms of heat transfer in the near- c r i t i c a l region may also apply to the above-critical region. Theoretical analyses of heat transfer in the supercritical region with allowance for the temperature dependence of thermophysical properties f381 T391 of the f l u i d were f i r s t made by Deissler and Goldmann . In these investigations, fluids were assumed to be incompressible, the effect of body forces were negligible, the dissipation of kinetic energy and fluctua- tion of the thermophysical properties were ignored, and the pressure was assumed to be constant over the cross section of the flow. In some later [40 41] investigations ' , the effect of f l u i d inertia and buoyant forces were also taken into account. . Theoretical analyses which are based on the continuity, energy, and Navier-Stokes equations usually employ the Prandtl mixing length con- cept. The basic idea is to simplify the shear stress term of the Navier- Stokes equation and the heat conduction term of the energy equation. Then 27 the shear stress and heat flux can be expressed in the simpler forms T = ( y + p£*? 1̂  ' ( 1 ) q = - (K + p Cpe h) |^ . (2) Most types of eddy d i f f u s i v i t i e s can be broken into two categories: (1) continuous, and (2) multiple-part. In each category, the turbulent momentum transport can be further considered as a function of local thermo- physical properties. Numerical solutions are usually presented as a part of theoretical analyses. However, correlation of predictions and experiments are success- ful only for the particular experiments used to provide the modified eddy diffussiv i t i e s . For this reason, theoretical methods are not widely used for designing heat transfer equipment using supercritical fluids. Experimental studies have been a major part of the research on heat transfer to supercritical fluids. The experimental works can be classified as either free or forced convection. Since free convection i s minimal in the supercritical cooling loop of this study, i t is not discussed further. Forced convection experiments can be further divided into two major categories: experiments which attempt to explain the mechanism of heat transfer to supercritical fluids, and conventional heated pipe experi- ments which are used to determine dimensionless correlations. The second category is directly related to the application of a supercritical CO^ cooling loop and is the only one discussed here. 28 Many heated tube experiments have been carried out in England, Japan, U.S.A., and U.S.S.R. A l l of these experiments were very similar in design and used mainly water and CC^ as experimental fluids. Hydrogen, * oxygen, and Freon have also been studied by some authors. The results of most of these efforts f a l l into the range of the near-critical region, where heat transfer differs most from normal heat transfer as governed by single phase flu i d correlations. The earliest form of correlation was in the form of the Dittus- n i - [42,43,44] Boelter equation : Nu = A ( R e z ) b ( P r w ) C , (3) where A, b, and c are experimental constants. The Reynolds number Re is z evaluated at the reference temperature T^ for a given pressure, which is expressed in the form: T = T + Z(T -T, ) . (4) z b w b The parameter Z i s a function of the dimensionless temperature: Z = (T -T )/(T -T ) , (5) m b w b and Pr^ is the Prandtl number evaluated at the wall temperature. The sub- scripts represent the conditions of flu i d bulk, wall, and pseudocritical, respectively. Another early correlation was proposed by Miropolski and [451 Shitsman . It has a form similar to the Dittus-Boelter equation: Freon: dichlorofluqr methane (12). 29 Nu = 0.023(ReJ°' 8 Pr . (6) b mm where Pr . is the minimum Prandtl number defined as: mm /•PrL ; Pr, < Pr * b b w Pr . =< mm (7) *Pr ; Pr < Pr w w b This simple modification, however, produces some remarkable results. It correlates some experimental data at the near-critical region with great [54] accuracy Correlation factors were later introduced for-bulk temperature^^'' and thermophysical property variations '"^. The correlations can be expressed as: Nu = Nu (T. /T ) n , (8) o b w Nu = Nu Q(y w/y b) a ( K / K / ( p w / p b ) c (C P/Cp b) d , (9) where a, b, c, d, and n are experimental constants. The Nusselt number NUq is evaluated from constant property fl u i d flow correlations: Nu = 0.023(Re, )°' 8 ( P r J 0 , 4 , (10) o b b , [51] or by : (5/8.0) Reb Pr b Nu = Tjo 2/1 ' ° 12.7[? /8.0] ± / Z [(Pr, T1* - 1.0] + 1.07 o b 30 £ = [1.82 Log (ReJ - 1.64] 2 , 0 . (12) o b The term Cp in equation (9) i s called the integrated specific heat, defined as: Cp" =' (h -h, )/(T -T ) . .... (13) w b w b The integrated specific heat has been proven by experiments to be [23] an efficient term for correlating data. Therefore, Swenson et a l . generalized their experimental data for heat transfer by: n n n , c n tT) N0.923 ,-—.0.63 , , .0.231 Nu = 0.00459 (Re ) (Pr ) (p /p, ) , (14) w w w b where Pr i s called the integrated Prandtl number and defined as: w Pr = Cp (u /K ) (15) w W W The concept of using not only integrated specific heat but a l l physical properties on an integrated average basis was suggested by [52] Hiroharu et a l . for supercritical fluids. They suggested the normal correlations for heat transfer to single phase fluids be used as correla- tions for supercritical fluids. However, the Nusselt number is evaluated at integrated property values which would be expressed as: T w b „ •* F = _ T i  F ( T ) d T • ( 1 6 ) Tb As with theoretical analyses, most of the experimentally determined correlations must be modified to f i t the data of particular experiments. 31 However, these correlations usually predict heat transfer with greater accuracy than the theoretical analyses, especially in the experimental data range from which the correlations have been derived. Some correla- tions cover a rather wide range of experimental data, and have sufficient accuracy for engineering designs. Therefore, i t i s possible to study some applications of heat transfer to supercritical fluids simply by chosing an appropriate correlation. 5.2 Engineering Calculation of Heat Transfer to Supercritical Fluids 5.2.1 Choice_of_Correlations The correlation chosen for the supercritical CG^ cooling loop of this work must be: (1) accurate in correlating heat transfer in the operating range of the proposed supercritical cooling loop, and (2) easy to use in computing programs. In this section, several types of correla- tion are discussed. Although reference temperature correlations were the earliest ones developed, there are drawbacks in their use. F i r s t , they f a i l to pre- dict heat transfer coefficients accurately in the temperature range near the pseudo-critical temperature; a small difference of the reference temperature value T z can cause a large variation in the predicted heat transfer co e f f i - cient because of the extremely sharp property variations near the pseudo- c r i t i c a l temperature. Second, this type of correlation is very inconvenient when heat flux is specified and the wall temperature i s to be calculated. In contrast to the reference temperature correlations, the minimum Prandtl number correlations produce accurate results i n the near-critical 3 2 region. Although these types of correlation were derived from supercritical water heat transfer data, they successfully correlated oxygen data of Powell and CÔ  data of Bringer and Smith . However, they failed to predict heat transfer to supercritical hydrogen. More important is the fact that these correlations can account for the maximum heat transfer reported by Powell'" 2^, as well as Dickson and Welch^4''. Therefore, the minimum Prandtl number correlations should be considered for use in ca l - culating heat transfer to supercritical CO . The correction factor types of correlation are also popular, although the temperature correction factor i s not very accurate. For [ 4 7 1 example, the correlation Nu = Nu (T /T ) " 0 , 2 7 , (17) o w b adequately predicts heat transfer to supercritical fluids at high tempera- ture but breaks down when the heat flux i s increased. This i s mainly due to the fact that the dependence of f l u i d properties on temperature is different in different temperature ranges. Therefore, correction factors must be used on fl u i d properties which strongly affect the f l u i d dynamics and heat transfer of the flow. The f i r s t correlation with property factors [ 2 2 ] was introduced by Petukhov et a l . : Nu = Nu (u. /U ) 0 , 1 1 (K, /K )"°' 3 3 (Cp/Cp K) 0' 3 5 (18) O D W D W D The key feature of this correlation i s the use of the integrated specific heat Cp. This correlation covers a rather wide range of experimental data for water and CO^. However, i t f a i l s to predict heat transfer rates correctly at very high heat flux, possibly because at high temperature drops (high heat flux) the density difference becomes important. The sig- nificance of the density correction factor was pointed out by Kutateladze and Leontev'"48'', and also by Swenson et a l . ^ " ^ . A correlation modified by a density ratio correction factor was [491 f i r s t introduced by Kranshchekov and Protopopov , based on CÔ  data. This correlation was later developed for calculation of heat transfer to supercritical CO^ at a normalized pressure (P/Pc) = 1.02 - 5.25^"^ as: Nu = NU q (Cp/Cp b) n ( P w / P b ) m , .... (19) where: m = 0.35 - 0.5(P/Pc), and n = 0.4 at T /T < 1.0 or I /T > 1.2 , w m - b m - and, n = n, = 0.22 + 0.18 (T /T ) I w m at 1.0 < T /T < 2.5 , - w m - and, n = n. + (5nn - 2) (1 - T./T ) I I D m at 1.0 < T./T < 1.2 . - b m - The correlation was later tested by Kransnoshchekov et a l . ^ " ^ with data from heat transfer to supercritical CO^ at temperature differences up to 805°C. Experimental data agreed with this correlation within an accuracy T561 of + 15%. Petukhov et a l . also tested this correlation with their experimental data, and gained satisfactory agreement. This correlation was further used to calculate local heat transfer rates to supercritical CO2 in annular channels with internal heating by Glushcheznko and Gandzyuk^ T581 and in a rectangular channel heated from one side by Protopopov Despite the fact that the correlation (19) was derived from heat transfer data for a circular heated tube, in both the previous cases the calculated 34 wall temperatures agreed well within the experimental data range of T, > T and T, < T . Heat transfer rates were most unpredictable at b m b m r T, = T with heat transfer coefficients about 30-50% too low. b m An integrated Prandtl number correlation has been developed for f59l studying heat transfer to supercritical water coolant in reactors However, this type of correlation was based on supercritical water data and gave poor results when used to predict heat transfer to supercritical [52] Hiroharu et a l . confirmed experimentally the accuracy of the types of correlation using a l l properties as integrated properties. However, a l l their experiments were carried out with low temperature differences (low heat flux), and the correlations have not been tested by other authors. This suggests that these types of correlations have no guarantee of accuracy for predicting heat transfer rates at high tempera- ture differences. The above discussion shows that the minimum Prandtl number and integrated specific heat types of correlations are preferable for calcula- tion of heat transfer for the supercritical CO^ cooling loop of this work. The correlation given in equation (19) was f i n a l l y chosen instead of the minimum Prandtl number correlation because not only does i t meet a l l requirements pointed out at the beginning of this section, but because i t has been widely tested with experimental heat transfer data for supercriti- cal co'2. 5.2.2 Examination of Chosen Correlation When the correlation given by equation (19) was f i r s t developed, 35 i t was based on the data range: 1.01 < P/Pc < 1.33; 0.6 < T,/T < 1.2; — - - b m - 0.6 < T /T < 2.6; 2 x 10 4 < Re < 8 x 10 5; 0.85 < Pr,_ < 55.0; - w m - - b - - b - 0.09 < P w / P b < 1.0; 0.02 < Cp/Cp < 4.0; 2.3 x 10 4 < < 2.6 x 10 6 W/cm2; and H/d > 15.0. This correlation predicted heat transfer coefficients in a heated circular pipe within the accuracy of + 15.0%. Later, this corre- lation was further developed to cover data up to P/Pc =5.25. It was par- t i c u l a r l y interesting that this correlation predicted the pronounced peak of wall temperatures in experiments within the suggested accuracy up to the data range of Re, = (0.6 - 1.2) x 10 6 and T /T < 3.6. b w m - In equation (19), NU q is calculated from equation (11). This formula was developed by Petukhov and Kirillov'"^''"'' from a theoretical analysis of fu l l y developed heat transfer to a single phase, constant property flow in a circular, heated pipe. This formula predicts heat transfer coefficients for constant property fluids within the accuracy of + 10.0% for the range: 0.5 < Pr^ < 2000.0 and 1.0 x 10 4 < Rev < 5.0 x 10 6. - - b - - b - It must be noted that in contrast to the conventionally used formula with constant exponents for Re^ and P^, this formula more accurately corresponds to the dependence of heat transfer on Re^ and Pr^. This is particularly apparent in the case of Re, > 1.0 x 10^ and Pr. > 10.0. b - ta- in the regions more remote from the c r i t i c a l state, the properties of a supercritical f l u i d become approximately constant over a relatively wide range of temperature and the correlation in equation (19) reduces to [22] equation (11). Petukhov et a l . showed that the formula in equation (11) also agrees with experimental data in the near c r i t i c a l region under the condition of small variation of properties over the cross section; i.e., at small temperature drops. Thus, the correlation i s a very general 36 expression for heat transfer in the near-critical region as well as in regions remote from the c r i t i c a l state. In order to calculate the temperature distribution along a heated channel, the pressure distribution of the supercritical f l u i d along the channel must be known since the thermophysical properties which deter- mine heat transfer are a function of both temperature and pressure. Many works are devoted to the investigation of heat transfer at supercritical pressure. However, studies of flow and pressure drop are scant. Tarsova and Leontez developed one of the f i r s t engineering correlations for the calculation of f r i c t i o n coefficients. Later, r 621 Kuraeva and Protopopov further developed this correlation by including the effects of possible free convection. The improved correlation was based [ 4 9 ] on the data of Kranoshchekov et a l . . This correlation can be put into the following form: € = K = t (UA>°'22 f o r Gr/Re2 < 5 x 10 _ 4 s o w D b — 2 0 * 1 -4 2 -1 E = E 2.15 (Gr/Ref) for 5 x 10 *< Gr/Ref < 3 x 10 1 , s b b — (20) where EQ i s calculated from equation (12). The viscosity u b i s evaluated at the temperature T^ which corresponds to the average pressure and inte- grated density: Ail I " 1 n = -1- f dx P ML J p(x) I ••• ( 2 1 a ) 37 and the viscosity u i s evaluated at the temperature T which is the wall w w temperature of the tube: A J Tw " M I T . - ( x ) d x • • • • <21W w The pressure drop is then calculated from the formula: .2 A p = , l £ u L _ M + ( p u ) 2 ( ^ _ _ 2pd pout P i n Thus, when the equation (22) is applied outside the near-critical region and the physical properties of the flow do not change drastically, the equation can reduce to the form: Q AP = £ (pu) 2 A£/2pd . (23) - - Based on the correlation in equation (19) for heat transfer, the formula in equation (22) for pressure drop, the energy equation, and the continuity equation, a computer program was designed to calculate the wall temperatures, bulk fl u i d temperatures, and pressure drop along a heated pipe with a given heat flux as a function of distance from the entrance. The basic assumptions in the program are: (1) the pipe is hydrauli- cally smooth; (2) the heat transfer coefficient i s a function of local condi- tions only; (3) the flow is turbulent and f u l l y developed; and (4) pressure is constant at a cross section. An additional factor f(x/d) allows for the 38 increase in heat transfer in the i n i t i a l portion of the p i p e ^ ; f(x/d) = 0.95 + 0.95(x/d)" 0 , 8 . (24) A l l required equations for the calculation are list e d in Figure 12. The flow chart for the program i s shown in Figures 13 and 14, and details of the procedure are given in Appendix I. The program was f i r s t tested against the experimental data of Jackson and Evans-Lutterodt . As shown in Figure 15, the program accurately predicts the wall and bulk temperatures along the heated pipe for the case in which buoyancy forces do not exist. Both the bulk and wall tem- peratures along the pipe are below the pseudo-critical temperature. This program was also used to predict heat transfer to super- c r i t i c a l flow in pipes under other conditions as shown in Figures 16, 17, and 18. Figure 16 compares predicted values with the experiments of [49] Kranoshchekov et a l . . In this case, both the bulk and wall temperatures along the pipe are above the pseudo-critical temperature. The difference between the calculated wall and bulk temperature profiles i s within + 10% of the experimental values. Figure 17 shows data from Kranoshchekov et a l . ^ " ^ in which the bulk temperatures along the pipe are less than the pseudo-critical temperature and the wall temperatures are greater than the pseudo-critical temperature. The computed results correspond amazingly well with the experimental data. The heat flux and wall temperatures in Figure- 17 are beyond the range of limits f i r s t suggested for the correla- [42] tion. Figure 18 shows the special case of Schnurr , in which the fl u i d bulk temperature spans the pseudo-critical temperature. However, the calcu- lated results s t i l l match well with the experimental data. 39 In general, the program predicts heat transfer rates within the expected accuracy of the correlation. Accuracy increases in regions further away from the c r i t i c a l state, but calculated temperatures are higher than the actual at an intermediate distance from the c r i t i c a l state. Under the latter condition, the correlation factor in equation (19) may s t i l l be less than a unity. In conclusion, the program based on the correlation in equation (19) and the conservation equations is suitable for analyses of the supercritical CO^ cooling loop in this study. 5.3 Heat Transfer to Single Phase Fluids in Fuel Bundles Heat transfer in fuel bundles (see Figure 4) can be classified as either single or two-phase. The work in each class is further divided into theoretical and experimental studies. In this section, only previous studies on the prediction of heat transfer to single phase fluids in bundles are discussed. The single phase fluids of concern are compressed liquids and subcritical gases. The theoretical works mainly involve prediction of performance characteristics of clustered rod bundles under conditions of f u l l y developed heat transfer to axially uniform heat flux, together with the related hydro- dynamic problems for longitudinal flows. The coolant can be in laminar, turbulent, or slug flow. There are three methods for analysing heat transfer in bundles currently in use: fundamental, equivalent annulus, and lumped parameter. The fundamental method "for' theoretical prediction of heat transfer to single phase fluids in bundles is based on solution of the momentum and energy equations. This approach has gained certain success for laminar 40 flow, where analytical solutions have been found for the velocity fields of: (a) i n f i n i t e rod clusters, (b) semi-infinite rod clusters, and (c) some special geometry f i n i t e rod clusters. Based on the known velocity fields, heat transfer rates can be determined for laminar longitudinal flow in an i n f i n i t e rod cluster. In the case of turbulent flows, heat transfer has also been successfully analysed for i n f i n i t e rod clusters. There are three approaches in these analyses: (1) both velocity and temperature fields are computed by iterative procedures; (2) the velocity f i e l d i s f i r s t determined graphically and used in the differential form of the energy balance, which is numerically integrated by f i n i t e difference procedures; (3) both the differential forms of the momentum and energy balance are integrated numerically by f i n i t e difference procedures to obtain the velocity and temperature fields. Despite the f a i r l y accurate predictions made for simple geometries, the fundamental method is not so easy to apply to more complex flow geometries. In the equivalent annulus method, the complicated rod cluster i s divided into many annuli. An equivalent model can be used to describe the T671 entire array of rods in the bundle as shown in Figure 19a . The bundle is approximately divided into annuli, with some being heated from both sides and others from one side only. The basic solution for the concentric annulus is then applied to each region. This method is mainly used by designers to adjust rod spacing to obtain the desired flow s p l i t between regions, and has been found to be quite satisfactory for relatively widely spaced bundles. Another equivalent annulus model considers individual rods in T681 parallel rod arrays. As seen in Figure 10b , the hexagonal flow passage defined by zero shear planes around a rod may be approximated by a circular 41 tube of equivalent area. This flow passage (bounded by the rod and the zero shear planes) i s assumed to be equivalent to the region between the center tube and the zero shear plane of an annulus. The advantage of the equivalent annulus model i s that heat trans- fer rates can be easily calculated by applying available solutions for the concentric circular tube annulus. The.principal limitation, of course, is how well the equivalent annulus represents the actual flow. The experimen- tal data shows that the equivalent annulus method is only roughly c o r r e c t ' . The analogy tends to underestimate the Nusselt number at high Reynolds number, and breaks down completely at close rod spacing (i.e., Pitch/Rod Diameter ^ 1.1) because the circular shear plane i s no longer a good approximation to the actual zero shear planes. In the lumped parameter type of analysis, the flow in the array i s divided into many subchannels as shown in Figure 2 0 ^ ^ . The characteristic dimension of each subchannel is expressed as the equivalent hydraulic diameter: n = 4 x Cross-sectional Area . . h Wetted Perimeter ' ^ 2 5 ) The axial pressure gradient in each subchannel i s then determined on the basis of the equivalent hydraulic diameter and a f r i c t i o n factor which i s calculated from a given correlation. The flow is balanced for a l l sub- channels in the array by equating the pressure drops. The f l u i d bulk tem- perature in each subchannel is calculated from the known heat flux d i s t r i - bution and inter-subchannel mixing rates. Based on the equivalent hydraulic diameter, subchannel flow rate and local f l u i d bulk temperature, the local heat transfer coefficient and wall temperature can be calculated from a 42 given correlation. The lumped-parameter method can deal with large and complicated geometry rod arrays incorporating non-uniform heating from rod to rod as well as in the axial direction, and easily accommodates f l u i d property changes. Many computer programs have been developed based on this approach. The work of O l d a k e r ± s a typical example of computer pro- grams developed by AECL to study the performance of Canadian fuel bundles. However, the lumped-parameter method requires judgement in defining sub- channels and determining f r i c t i o n and heat transfer correlations. The application of this method must be incorporated with experiments, since experiments are needed to determine mixing rates between subchannels. Besides mixing rate experiments, many other experiments have been carried out in order to predict heat transfer in bundles and supple- ment the analytical works, especially in the early state of the develop- ment of nuclear power reactor designs. Experiments have been conducted not only...on the effects of rod cluster geometry on heat transfer but also the effects of rod spacers, bearing pads, and end-plates of fuel bundles. Today, the accumulated and combined knowledge from the analytical and experimental works i s adequate for predicting the performance of fuel in preliminary designs. However, the actual performance of a new fuel bundle design must s t i l l be checked by experiments. 5.4 Heat Transfer to Supercritical C09 in Fuel Bundle 5.4.1 Specifications_and Assumptions Heat transfer to supercritical carbon dioxide in fuel bundles has r 43 not been studied before. However, reasonable estimates can be based on the separate studies of heat transfer to supercritical carbon dioxide in heated pipes and heat transfer to single phase fluids in bundles. This section w i l l discuss the approach and the assumptions of the analysis of heat transfer to supercritical carbon dioxide in CANDU-type fuel bundles. Because of the complicated geometry of CANDU fuel bundles, the fundamental method discussed in the previous section i s not useful. Because of the small ratio of pitch to rod diameter for CANDU fuel bundles, the equivalent annulus method is also not suitable. Therefore, the lumped- parameter method was the only method used for analysis of heat transfer to supercritical carbon dioxide in bundles. Certain assumptions were made in order to put the method into use. Because there is no available experi- mental data for supercritical carbon dioxide mixing rates between sub- channels of fuel bundles, the f i r s t assumption was that of zero mixing between subchannels. The mixing can be caused by turbulent interchange, [72] diversion cross-flow, flow scattering, and flow sweeping . Turbulent interchange results from eddy diffusion between subchannels and can be characterized by the eddy di f f u s i v i t y of momentum. Diversion cross-flow is the direct flow caused by radial pressure gradients between adjacent subchannels. These gradients may be induced by gross differences between the subchannel heat flux distributions, and the differences in subchannel equivalent hydraulic diameters. Flow scattering refers to the non- directional mixing effects associated with spacers, bearing pads, and end- plates. Flow sweeping is the direct cross-flow effect Associated with mechanisms designed to create mixing between subchannels. These mechanisms are mainly wire wrap spacers, helical fins, contoured grids, and mixing vanes, but they are not used in present CANDU fuel bundles. Therefore, 44 i f supercritical carbon dioxide were used as a coolant for the CANDU reactor, mixing due to flow sweeping does not exist, but strong mixing due to the effects of the other three causes 1can be expected. Mixing can balance the enthalpies and average the flui d bulk temperatures of subchannels. If there is no mixing, the bulk temperature of the coolant in the particular subchannel which has the maximum linear heat rate can be over-predicted. Also, because the local wall temperature depends on the local f l u i d bulk temperature for a given flux, the wall tem- perature i s then over-predicted. Evidently, the zero mixing assumption i s not only untrue but also a very serious assumption. It w i l l cause under- estimation of the performance of a fuel bundle in an analysis when maximum withstanding temperature of the fuel bundle i s used as the upper limit for design. The second assumption is that the rod bundle i s smooth. There- fore, turbulence created by spacers, bearing pads, and end-plates is not taken into account in this analysis. The structural arrangements in fuel bundles create not only mixing between subchannels, but also high eddy diffusivity of momentum and heat i n each subchannel. The eddy dif f u s i v i t y is closely related to heat transfer. As the degree of dif f u s i v i t y increases, [731 the heat transfer coefficient increases. Hoffman et a l . found that spacers improve the local heat transfer by tripping the boundary layer. This results in starting a new thermal entry length as shown in Figure 21. Other mechanisms also can improve the general heat transfer throughout the r 731 entire bundle as well as local heat transfer. For example, Figure 22 shows that the Nusselt numbers in a fuel bundle with wire wrap spacers are greater than Nusselt numbers calculated from the correlation for single 45 phase fluids flowing in circular smooth tubes. Hence, the second assump- tion w i l l over-predict the wall temperature profiles in subchannels. In other words, the "smooth bundle" assumption w i l l under-estimate the per- formance of fuel bundles in term of maximum withstanding operating tempera- ture. The smooth bundle assumption may also under-predict pressure drop through the bundle. The increase of turbulence created by the fuel bundle structure increases the skin f r i c t i o n along the fuel bundle. In addition, the direct drag due to the structure can cause further pressure drop. [741 On the other hand, Figure 23 shows that the fr i c t i o n factor in a closely packed bundle (pitch to rod diameter, p/d = 1.141) may actually be less than that of a smooth circular tube. This phenomenon is apparently due to the effect of close-packing. The decrease of f r i c t i o n factor in a closely packed bundle (such as a CANDU fuel bundle) may counter balance the increase of fr i c t i o n factor due to the effect of end plates and other parts of the fuel bundle structure. Therefore, as a f i r s t approximation, the fric t i o n factor used in this work was calculated from a correlation for supercritical carbon dioxide flowing in a smooth tube. The last assumption in applying the lumped-parameter method i s that subchannels are continuous throughout the reactor core. In other words, i t was assumed that the relative orientation of bundles in the reactor core does not affect heat transfer or hydrodynamics of coolant flow in subchannels. Fortunately, experiments^"' '^•1 confirm that although there may be discontinuities in subchannels between fuel bundles, the relative orientation of fuel bundles has l i t t l e effect on average heat transfer. Figure 2 4 ^ ^ shows calculated results of wall temperature 46 discontinuities due to change in orientation of fuel bundles, and the general shape of the wall temperature profile when the subchannel is assumed to be continuous. Similar to the analysis of heat transfer to single phase fluids in circular pipes, heat transfer coefficients for supercritical carbon dioxide flow in fuel bundles are assumed to depend only on the local con- ditions. Also, the coolant flow in subchannels i s assumed to be ful l y developed (but with a factor f(x/d) as in equation (23) allowing for increase in heat transfer in the i n i t i a l portion of the subchannel). In addition, the effect of the ratio p/d can be taken into account for heat * transfer by using Figure 25 . From the above discussion, despite the many assumptions, an analysis of heat transfer to supercritical carbon dioxide i n the CANDU- type fuel bundles can be made. Moreover, the result of this analysis should be on the conservative side, that i s the maximum calculated sheath temperatures should be larger than actual expected values. 5.4.2 Fuel Bundle_Geometry and Subchannel Analysis As discussed previously, fuel bundle design must be based on balance of subchannel enthalpies for optimum performance. This enthalpy balance can minimize temperature differences between subchannels, and reduce the axial increase of bulk temperature in a particular subchannel which has the maximum heat rate. Figure 25: Kay summarized the effect of the ratio p/d on Nusselt number for fully developed, turbulent flow parallel to a bank of smooth circular rods. 47 Enthalpy balance can be controlled by creating mixing with specially designed mechanisms such as wire-wrap spacers. Mixing also can be created by properly arranging the geometry of the fuel bundles. [12] Figure 26 shows the geometry of an existing fuel bundle used in the CANDU 500 reactor at Pickering. Because of the different cross-sectional areas of subchannels, subchannel flow rates and velocities are different. This can result in cross-sectional pressure gradients, hence, subchannel mixing. Figure 8 shows the subchannel temperature profiles for a given axial and radial heat flux distribution. Because of the enthalpy balance due to mixing, the temperature differences between flows are very small. However, this same bundle cannot achieve sufficient enthalpy balance when supercritical carbon dioxide i s used as a coolant. Because of the low density of supercritical carbon dioxide at high temperature and resulting high flow speed, the resistance in subchannel 4 does not allow adequate flow for a given heat output. Therefore, despite the mixing between subchannels, the enthalpy, bulk temperature, and hence the wall temperature in this subchannel w i l l increase rapidly. The wall tempera- ture may eventually exceed the limiting value. A similar problem has been encountered in the development of the CANDU boiling reactor, fog reactor, and once-through reactor, in which low [79] density light water is used as a coolant . In order to solve the problem of rapid axial increase of temperature due to enthalpy unbalance, f i l l e r s (or blockers, see Figure 27) are introduced to block subchannel passages which have too large a flow area. Tube-in-shell fuel elements (see Figure 28) have also been suggested. The geometries of these fuel elements allow coolant distribution according to subchannel heat rate, and resulting 48 good enthalpy balance and small temperature differences between subchannels. supercritical carbon dioxide cooled CANDU reactor, the subchannel dimensions must be determined. If necessary, f i l l e r s have to be introduced into the fuel bundle to assist subchannel flow balance. The subchannel dimensions are governed by heat transfer rates and a relationship must be found between the coolant flow rate and the maximum subchannel wall temperature with a given operating condition. versus inlet coolant temperatures for a given axial heat flux distribution and a given outlet coolant temperature. The heat flux distribution i s the average axial value for the CANDU reactor at Pickering. Each curve represents the relationship between the maximum wall temperature and the inlet coolant temperature for a constant equivalent hydraulic diameter. The inlet temperature can be interpreted as the coolant flow rate which can be expressed as: In order to find a suitable fuel bundle geometry for the proposed Figure 29 i s a plot of subchannel maximum wall temperatures (Axial Heat Flux Distribution) dl M = 0 (26) h(T out ) - h(T. ) in where: £ is the length of the subchannel^ h(T ) i s the outlet coolant enthalpy, which i s a function of the outlet coolant temperature, T out' h(T. ) i s the inlet coolant enthalpy, which i s a function of the inlet coolant temperature, T. i n 49 When the axial heat flux distribution and the outlet temperature i s fixed, the flow rate i s approximately proportional to the inlet coolant tempera- ture. . The outlet temperature was f i r s t chosen to be 775°K. However, when the equivalent diameter i s larger than 1.4 Do (7.4 cm), the maximum wall temperature exceeds the safety limit of 550°C which has been dis- temperature of 750°K cussed previously. Therefore, a lower coolant outlet! I was chosen. A similar plot i s also used to show the relationship between the inlet temperature, i.e. the flow rate, and pressure drop as shown in Figure 30. Figures 29 and 30 show that for a given equivalent hydraulic diameter of a subchannel, the total pressure drop increases, and the maximum wall temperature decreases with increase of the inlet coolant temperature; i.e., the flow rate or velocity in a subchannel. For a very small equivalent diameter subchannel, the f i n a l pressure drops below the c r i t i c a l pressure. Therefore, the coolant cannot be maintained in a super- c r i t i c a l state throughout the cooling loop. For a large equivalent hydraulic diameter subchannel, the pressure drops less quickly, which allows higher inlet coolant temperature; i.e., higher flow rates. The high coolant flow rate w i l l produce high heat transfer rates and guarantees a maximum wall temperature below the safety limit. High inlet coolant temperature is also very desirable for steam generation as w i l l be dis- cussed in a later chapter. However, use of large subchannel equivalent hydraulic diameters w i l l require large diameters for fuel bundles and coolant channels. Coolant channels which have large diameters and fixed wall thickness cannot withstand higher pressures, and f a i l to meet the 50 c r i t e r i a of compactness for reactor design. Today, only small diameter coolant channels are used in the CANDU reactor. Because this work is only a preliminary study, without any attempt being made to optimize the coolant performance in bundles in terms of power station efficiency, an equivalent subchannel hydraulic diameter of 0.75 cm = 1.44 Do and a coolant inlet temperature of 400°K were chosen for this study. These values were chosen on the basis of maximum wall temperatures and reasonable flow rates and therefore, relatively low pressure drop. As a result, the fuel bundles can be operated safely, and the carbon dioxide coolant can be kept in a supercritical state throughout the cooling loop. For given inlet and outlet coolant temperatures, and a specified heat flux distribution, the relationship between the equivalent hydraulic diameter and maximum wall temperature, and also the fi n a l pressure for a given inlet pressure can be studied. Figure 31 shows that for a constant flow rate, the equivalent hydraulic diameter increases with maximum wall temperature and fi n a l pressure. The upper limit of equivalent hydraulic diameter occurs when 1.02 cm = 2.3 Do and the maximum wall temperature reaches the safety limit of 550°C. The lower limit to achieve compact fuel bundles i s approximately 0.75 cm = 1.44 Do, since the fi n a l pressure drops drastically from this point. An equivalent hydraulic diameter of 0.75 cm was chosen to make the design as safe and compact as possible. 5.4.3 5££dicted_Heat_Transfer_to_Supe Fuel Bundles An equivalent subchannel hydraulic diameter of 0 .75 cm was chosen 51 coolant. However, specifying the equivalent hydraulic diameter does not uniquely determine the exact geometry of subchannels since equivalent hydraulic diameter depends on flow area and wetted perimeter in a sub- channel. Final design selections must also take into account overall compactness of the bundle. When the subchannel equivalent hydraulic diameter is chosen to be 0.75 cm and with 1.52 cm diameter fuel rods (the same as those at Pickering), the most compact subchannel geometry i s the square array. Figure 32 shows the geometry of a 32-element fuel bundle with a square array of fuel rods, along with pertinent dimensions. The overall diameter of this 32-element fuel bundle is approximately equal to the existing fuel bundle used in CANDU reactors (Pickering type). This ensures that the basic design of the CANDU reactor would remain the same i f supercritical carbon dioxide were used as a coolant. This type of fuel bundle i s not the best when considering the coolant enthalpy balance. In the CANDU reactor, the heat flux distribution within a fuel bundle increases away from the center of the bundle. Therefore, the proposed fuel bundle f a i l s to distribute coolant flow according to the radial heat flux distribution. However, blockers can be used to reduce the large flow area subchannels which are near the coolant tube wall. Heat transfer performance of this 32-element fuel bundle was analysed in a typical case when using supercritical carbon dioxide as a coolant for the CANDU reactor. The heat flux distribution was based on [12] data obtained during steady operation of the Pickering station . The axial heat flux distribution i s : Q(x) = q(n) • sin (kx + (j)) , (27) 52 where: Q(x) is the axial heat flux distribution for a subchannel as a function of distance from the inlet, in KW/M, q(n) i s the maximum linear heat rate of a subchannel n, in KW/M, x is the distance from the inlet of a subchannel, in M, k i s a constant, 0.51504, <j) is a constant, 0.25744. The maximum linear heat rate q(n) is based on the radial neutron flux dis- r o o i tribution in the Pickering 28-element bundle shown in Figure 33 . The formula for the calculation i s : , v f TWP - CWP , c Q , O O N q ( n ) = F . 9 6 " 4. 775 ' 3 5 , 8 > ....(28) where: f i s the normalized neutron flux at the radial distance of the center of the subchannel, TWP i s the total wetted perimeter in cm, CWP is the cold (unheated) wetted perimeter in cm. Based on the assumptions and lumped parameter method discussed previously, heat transfer to supercritical carbon dioxide in the proposed 32-element bundle was studied in detail. The computer program used was similar to that already described for circular tubes, with the exception that subchannel flow rates were balanced to produce equal subchannel pressure drops. Typical results are shown in Figures 34 to 36. The coolant temperature profiles in individual subchannels are shown in Figure 34. Wall temperature profiles are shown in Figure' 35. Pressure drop along the coolant tube is shown in Figure 36. When the inlet tempera- ture of supercritical coolant was 400°K (260°F), the f i n a l bulk temperature at the outlet of the coolant channel was found to be 730.7°K (855.6°F). 53 These results w i l l be summarized and discussed in Chapter 7. 54 6. POSSIBLE STEAM CYCLES 6.1 General The ultimate goal of this study i s to study ways to improve the overall efficiency of the CANDU reactor. Accordingly, i t i s important to investigate potential steam cycles in association with the CANDU reactor when supercritical carbon dioxide i s used as a coolant. The coolant temperature is one of the most significant para- meters in steam generation. The higher the coolant temperature, the higher w i l l be the thermal efficiency of the cycle, and hence, the lower w i l l be the fueling cost. Coolant temperatures fix the steam temperature and the associated saturation pressure as shown in Figure 37. Line "e-g" in Figure 37 represents the state points of the CO^ as i t cools down in the steam generator from "e" to "g". This i s approximately a straight line since dH = M • Cp ••: dT for high temperature supercritical carbon dioxide and the specific heat at constant pressure varies only slightly with temperature. The line "a-b-c-d" represents constant pressure heating of the water-steam system. Because of the second law of thermodynamics and i t s limitation on direction of heat flow, the saturation temperature of the water must be lower than the temperature of the CO^, approximately at the point where the coolant line intercepts the saturation curve on the enthalpy versus temperature diagram. Therefore, the cycle pressure is also fixed .according to the saturation temperature. Iii other words, although only the entry temperature of the coolant into the steam generator deter- mines the maximum temperature of the steam, both the entry and exit tem- peratures determine the steam pressure. Because the maximum temperature 55 as well as the pressure of the steam cycle determine the efficiency of the reactor power plant, high coolant entry and exit temperatures are desirable. The problem of low coolant exit temperature i s often encountered in nuclear reactor power generation. In order to solve the problem of low [80] steam pressure due to the low coolant exit temperature, Wooten et a l . suggested a dual-pressure steam cycle. Figure 38 shows the paths of the different processes with dual-pressure steam generation. It i s necessarily drawn on a different basis from that of Figure 37 which has enthalpy of water as an abscissa, and instead, the abscissa of Figure 38 i s the dis- tance along the steam generator from the entrance to the exit. In the dual-pressure steam generation cycle, feedwater i s separated into two branches where i t is pumped by two feedwater pumps to two different pressures. The temperatures of the feedwater entering the steam generator are equal in both branches. As shown in Figure 38, the low pressure feedwater is further heated in the economizer from "a" to "b", evaporated at a particular constant pressure and temperature from "b" to "c", superheated in the superheating section from "c" to "d", and then fed to a low pressure turbine stage. The high pressure water i s heated in a second economizer from "a"' to "b , M, in which i t is subjected to the same downstream coolant as the low pressure feedwater economizer. It is further heated in a third economizer from "b"1 to "c" 1, evaporated from "c" 1 to "d"', then super- heated from "d"' to "e" 1 by the incoming coolant. Finally, i t i s fed to the high pressure turbine inlet. In this process of dual-pressure steam generation, the coolant heats the high and low pressure steam alternatively. 56 The dual-reheat steam generation cycle was also discussed by T781 Wooten et a l . . Dual-reheat steam generation is similar to dual- pressure steam generation, except that the expanded high pressure steam from the high pressure turbine is joined with the low pressure feedwater and superheated together before expansion through the low pressure turbine stage. The dual-pressure steam cycle can result in higher thermal efficiencies than the single-pressure steam cycle. Of course, the more complicated dual-pressure steam cycle circuit increases the capital cost of the plant. The dual-reheat cycle has the additional advantage of lower moisture content at turbine exhaust and higher work output per unit mass of working f l u i d . However, the advantages of the dual-pressure steam sys- tem are reduced when the coolant exit temperature increases and approaches the temperature of coolant entering the steam generator. 6.2 Steam Cycle for the Supercritical Carbon Dioxide Cooled CANDU Reactor Plant This section discusses the possible steam cycles with supercritical carbon dioxide coolant conditions based on the previous calculated results of heat transfer in the 32-element bundle. The previously calculated outlet temperature of the supercritical carbon dioxide coolant i s 855.6°F (730.7°K), and the inlet temperature was chosen to be 260°F (400°K). For.some heat loss the coolant entry temperature into the steam generator was chosen to be 845°F. The coolant temperature leaving the steam generator was chosen to be 280°F, which assures heat exchange in the recuperator. 57 In analyzing steam cycle performance, only pressure drop across the reactor and turbine stages was taken into account. The pressure losses in piping, recuperator, and steam generator were not considered. The latter pressure losses depend on structural details, but with good engineering design, these pressure drops can be kept small. For example, the pressure drop for gaseous carbon dioxide flowing through the steam r g-j^ i generator at the Hinkley Point gas-cooled power plant is only 1.6 psia The single-pressure steam cycle was analyzed f i r s t . From the temperature versus enthalpy diagram in Figure 37, when the entry and exit temperatures are 845°F and 280°F respectively, the generated steam has a pressure of 130 psia and a maximum temperature of 815°F. Based on the turbine cir c u i t , shown in Figure 39, the ideal overall plant efficiency is only 26.73%. The low efficiency i s mainly due to the low steam pressure, which in turn is due to the low exit temperature. The dual-pressure cycle was also studied as a potential energy- conversion cycle. Figure 38 shows the supercritical carbon dioxide coolant and steam paths in the steam generator for the case of high to low pressure ratio of 600/70.3 (psia). Figure 40 shows the corresponding turbine cir c u i t . This cycle gives an ideal overall plant efficiency of 29,42%; higher than the single-pressure cycle. To study the effect of high to low pressure ratio on the e f f i - ciency of the cycle, the case of the dual-pressure cycle of 1200/70.3 (psia) was also analyzed. Figures 41 and 42 show steam generation and enthalpy balance for the turbine circuit respectively. The ideal overall plant efficiency for this case i s 29.87%, which i s about the same as the previous dual-pressure cycle. In the dual-pressure cycle, the ratio of flow rates 58 of high pressure to low pressure steam decreases when the pressure ratio increases. Therefore, the ideal overall efficiency of the dual-pressure cycle remains approximately constant under given coolant conditions. To study the further possible improvement of efficiency with given coolant conditions, a new dual-reheat steam cycle having a pressure ratio of 1200/70.3 (psia) is proposed. Figure 43 shows the steam genera- tion loop. Figure 44 shows the heat balance for the turbine c i r c u i t . In this cycle, the expanded high pressure steam is joined with low pressure feedwater and evaporated together in the steam generator to become the low pressure steam. In this case, the high and low pressure feeds are heated in a parallel fashion instead of alternatively as in the dual-pressure (non-reheat) cases. This reheating method i s somewhat different from the [80] method suggested by Wooten et a l . . Because of parallel heating, the large temperature difference between the coolant and the feeds is eliminated. This decreases the i r r e v e r s i b i l i t y of the steam generation process. There- fore, the ideal overall plant efficiency of the new dual-reheat cycle i s increased for a given coolant condition. The ideal overall plant efficiency of the dual-reheat cycle of this case was calculated to be 33.02%. The results of cycle performance are summarized and discussed in the next chapter. Comparisons are also made with the performance of the existing Pickering station. 59 7. RESULTS AND DISCUSSION 7.1 Typical Heat Transfer in Bundles The f l u i d dynamic and heat transfer conditions of supercritical carbon dioxide in the subchannels of the 32-element fuel bundles are summarized in Figure 45. This figure shows that the highest subchannel maximum wall temperature occurs in subchannel 6, which has the highest enthalpy among a l l the subchannels. The lowest subchannel maximum wall temperature i s only 409.0°C, which i s 137.0°C lower than the highest sub- channel maximum wall temperature. The great difference between the sub- channel wall temperatures and between the subchannel coolant temperatures shows that the assumption of zero mixing between subchannels has a major effect on subchannel enthalpy balance. Therefore, the maximum operating temperature of the 32-element fuel bundles, i.e. the highest subchannel maximum wall temperature, is over predicted. The overall performance of supercritical carbon dioxide coolant in the 32-element fuel bundles i s summarized i n Figure 46. For the purpose of comparison, the typical overall performance of heavy water coolant in the 28-eiement fuel bundles for Pickering Station is also l i s t e d . Because of the high heat extraction per unit mass for supercritical carbon dioxide coolant, the total mass flow rate i s much lower than for heavy water coolant at a given channel power. Since there is no dryout when using supercritical carbon dioxide as a coolant, the allowable operating tempera- ture of the fuel bundle i s much higher. As a result, the coolant temperature is much higher than the existing case of heavy water coolant. However, the pressure drop in the case of supercritical carbon dioxide coolant may be 60 underestimated as pointed out in the previous discussion. The detail sizes of the proposed 32-element fuel bundle and the existing 28-element Pickering fuel bundle are shown in Figure 47 for compari- son. The overall dimensions of these two fuel bundles are approximately the same; this satisfies the design restraint that the basic structure and reactor physics of the CANDU reactor remain the same. The 32-element fuel bundle has a slightly larger radius but contains more fuel, thereby reducing the required number of fuel channels for a given reactor power. Thus, the larger radius and thicker coolant tubes may not increase the total weight of structural material in the reactor core. To balance the excess flow in the outer subchannels, flow restricting blockers are used in.the 32-element fuel bundle. This results in an increase in reactor structural material and hence a possible decrease of neutron economy, which determines the fueling cost. However, this can be minimized by selecting blockers which are f a i r l y transparent to neutrons although impermeable to coolant flow (possibly high temperature plastic or ceramics). Optimum design of fuel bundle geometry would allow for subchannel enthalpy balance. In other words, the subchannel flow areas should allow for sub- channel flow rates proportional to the subchannel linear heat rates. If blockers are not used in the 32-element fuel bundle and subchannel enthalpies are balanced, the average flow area of each subchannel would be larger than proposed here. This allows higher flow rate of the coolant, and hence, higher inlet coolant temperature into the reactor; i.e., higher exit temperature from the steam generator. Therefore, the plant efficiency can be increased. 61 7.2 Typical Steam Cycles The steam conditions for the thermal cycles which were analysed previously are summarized in Figure 48. For purposes of comparison, the [121 corresponding steam conditions of the Pickering generating station are also listed. The performance of these thermal cycles is summarized in the table of Figure 49. The ideal overall plant efficiency of the Pickering generating station i s 33.94%, which is based on the turbo-generation heat [121 balance as shown in Figure 50 , and the actual plant efficiency i s only 29.1%. One of the conditions which strongly effects the ideal results and is not usually noted, is the turbine efficiencies assumed in the calculation. The efficiency of the low pressure turbines in both cases of the dual- pressure cycle i s assumed to be 88.5%. This i s based on the existing efficiency of the low pressure turbines of the Pickering Station. For the low pressure turbine of the dual-reheat cycle and the high pressure turbines of a l l cycles using supercritical carbon dioxide, turbine efficiencies are assumed to be 90%. This is j u s t i f i e d on the basis of higher inlet-steam pressures and temperatures and lower outlet moisture content. The steam condition when using supercritical carbon dioxide as a coolant is very much different from the steam condition of the Pickering generating station because of the different coolant conditions. However, the steam condition also depends on the method of generation. Therefore, despite similar coolant conditions in the three cases of the single pressure, dual-pressure, and dual-reheat cycles, the efficiencies are different. The calculated ideal overall plant efficiency of the dual-reheat 62 cycle is 33.02% which is competitive with the calculated ideal overall plant efficiency of the Pickering generating station. This shows that the use of supercritical carbon dioxide coolant may improve the existing thermal efficiency of CANDU reactor power plants when the performance of super- c r i t i c a l carbon dioxide coolant is optimized further from the design of this study. Another positive result of this study shows the advantage of the concept of pumping in the supercritical cooling loop; the pumping power required by the supercritical carbon dioxide i s small. Despite that supercritical carbon dioxide is assumed to be pumped at a temperature of 90°F, which is slightly lower than the pseudo-critical temperature of 99.5°F at 1240 psia, the ideal pump work i s only slightly larger than for heavy water coolant of the Pickering Station. The annual average tempera- ture of lake or river water in Southern Canada is approximately 48°F. Therefore, the estimate of pumping power for the supercritical carbon dioxide coolant i s very conservative. 63 8. SUMMARY AND CONCLUSIONS This study indicates the technical f e a s i b i l i t y of using super- c r i t i c a l carbon dioxide as a coolant for a CANDU-type reactor. The reactor is cooled by a single phase coolant, which is pumped at a high density liquid-like state. The supercritical-fluid-cooled reactor may have the combined advantage of avoiding dryout as i n gas-cooled reactors, and the advantage of low coolant circulation power as for liquid-cooled reactors. As a result of eliminating dryout, the maximum operating tem- perature of the fuel sheath can be increased to 550°C (1021°F) for existing Canadian fuel bundles. The coolant temperature in the case study of this work was calculated to be 855°F. This high temperature coolant can pro- duce steam at a temperature and pressure comparable to that of conven- tional fossil-fuel plants. However, since the exit coolant temperature from the steam generator may be as low as 280°F, a portion of the super- c r i t i c a l carbon dioxide coolant is used to produce low pressure steam. In the cases of single pressure and dual-pressure cycles, the steam generation process i r r e v e r s i b i l i t y i s high, and as a result, the useful energy extracted from the supercritical coolant is low. The ideal overall plant efficiencies of the single pressure, dual-pressure 600/70.3 (psia), and dual-pressure 1200/70.3 (psia) cycles are 26.73%, 29.42%, and 29.87% respectively, compared to 33.94% for the Pickering generating station. A new dual-reheat cycle i s proposed to reduce the high degree of 64 i r r e v e r s i b i l i t y in the steam generation process. In the new dual-reheat cycle, the coolant heats the low and high pressure feeds in a parallel manner instead of alternative heating as in dual-pressure cycles. The ideal overall plant efficiency of the new proposed dual-reheat cycle i s 33.02%, which is comparable to that of the Pickering generating station. The present result i s conservative since the performance of supercritical carbon dioxide has been underestimated because of the assump- tions of: (1) no mixing between subchannels, and (2) smooth and continuous subchannels. Mixing can improve heat transfer coefficients and also minimize the coolant and wall temperature rise in the particular subchannel having the maximum linear heat rate. For a given maximum fuel bundle sheath temperature, the no-mixing assumption underpredicts the inlet and outlet temperature of the reactor (i.e., the exit and entry temperature of the steam generator respectively). The smooth and continuous subchannel assump- tion also contributes to the underprediction of the performance of super- c r i t i c a l carbon dioxide by underestimating the heat transfer coefficient and overestimating the difference between wall and bulk temperatures. Therefore, the calculated ideal overall plant efficiency of the dual- reheat cycle is a conservative estimate. The performance of supercritical carbon dioxide has not been optimized in this work. If fuel bundle geometry was designed to achieve subchannel enthalpy balance without using blockers, the net flow area in a given diameter coolant channel could be larger. A larger net flow area allows higher flow rates, i.e. higher reactor inlet coolant temperatures without higher pressure drops. The optimization of fuel bundle geometry should be included in future research. 65 While the technical f e a s i b i l i t y of using supercritical carbon dioxide as a reactor coolant has been demonstrated in the present work, a great number of other factors must be considered in the design of an actual power station. Logistics of coolants supply and storage, safety, supporting hardware designs and overall cost must be considered. The complexity of financing a modern electrical generating station would demand serious study in i t s own right. It was not possible to consider a l l of these features i n a preliminary study lik e the present. It might be noted, however, that because of the great cost involved in heavy water production, the use of carbon dioxide may also offer an economic advantage. Evidently, a great deal of research on this matter is required. , Steam pressure turns the turbine. 4. Turbine shaft turns the generator rotor to generate electricity. TURBINE W o, Power lines take the electric power to communities. , 'Heavy water' coolant transfers the heat from the fuel to the boiler where ordinary water is turned into steam. f o c W o o Hb o 1. Heat is produced by fissioning uranium fuel in the reactor. CONDENSER 7. Water is pumped back into the boiler. 6. Lake or river water cools the used steam to condense it into water. Figure 1. Thermo-Hydraulic Features of PHW CANDU Reactor Power Plant TEMPERATURE, °F igure 2. T y p i c a l Property V a r i a t i o n s i n N e a r - C r i t i c a l Carbon Dioxide 68 VOLUME Figure 3. Process Path of S u p e r c r i t i c a l Cooling Loop on P-V Diagram 1, Thsfue! pellet becci radioactive in the reactor. __ PELLET FUEL BUNDLE 3. The fuel bundle is immersed in coolant flowing through a zirconium tube. The coolant is continuously monitored to detect radioactivity Figure 4. PHW CANDU Reactor Arrangement 4, The tubes are fixed reactor vessel of stainless steel.  TEMPERATURE QUALITY Steam FLOW REGIMES \ Heated W a l l \ ^ Single Phase Dry Out — ^ . Liquid Deficient Wall Temperature Steam Quality'"' \ •1 Annular Slug Bubble 1 t $. Low Flux — 0 / Single Phase Temperature wt % Steam Water Figure 6. Thermal Hydraulic Regions in a Boiling Channel 1000 Figure 7. Boiling Regions 10 12 DISTANCE FROM INLET |*T| Figure 8 . SUBCHANNEL TEMPERATURE RISES IN A 28 ELEMENT BUNDLE VERSUS REACTOR CHANNEL LENGTH (FROM [12]) < 4 . J 7 J 0 0 - 4 ms 74 1 10,000-, Figure 9. High Sheath Temperature Operation vs Time for Various Zirconium Alloy Fuel Elements (from [13]) 75 ENTROPY Figure 10. Proposed Supercritical C0„ Cooling Loop 76 Figure 11. Heat Transfer Regions: I - Liquids, II - Subcritical Gases, III - Two-Phase, IV - Above-Critical Fluids, V - Near-Critical Fluids (from [18]) 77 'Ah M, u " , =- -1 '2 - AP-4 1. Continuity M = Constant. 2. Energy Conservation Equation: h 2 = h± + Mh E = [1.82 log (Re, ) - 1.64]"2. O D 4. Heat_Transfer_Correlation for_Constant_Property_Flow: a/8) Re. Pr x, o b b  Nu = ——- 7yj\ • ° 12.7 JTW - 1) + 1.07 O D 5. Pressure Drop Equation for Supercritical_Flow: A P - C ^ t ^ W ( i - i ) , 2Pd W2 H l E = E = E (y /U, ) ° ' 2 2 for Gr/Re 2 < 5 x 10 _ 4 , s o w b b - E = E 2.15 (Gr/Re, for 5 x 10 _ 4 < Gr/Re 2 < 3 x 10 - 1 , s b b - p,.yb evaluated at P = P^ + AP, T = ( T b l + T b 2)/2 , U evaluated at P = P. + \ AP, T = (T . + T „)/2 . w 1 2 wl w2 (FIGURE 12 continued on the following page.) 78 6. Heat_Transfer_Correlation_f2£_§upercritical_Flow: Nu = NU q (P w/P b)°' 3 (C?/Cp b) n , n = 0.4 at T./T < 1 or T./T > 1.2 , b m - b m - n = n = 0.22 + 0.18 (T /T ) at 1 < T /T < 2.5 , 1 w m - w m - n = n, + (511, - 2) (1 - T./T ) at 1 < T./T < 1.2 . 1 1 b m - b m - Figure 12. List of Equations CALCULATE LOCAL WALL TEMPERATURE T ' w FOR CONSTANT PROPERTY FLOW CALCULATE HEAT FLUX Q' = f(Nu, T b, Tw', Aj) FIND PROPERTIES j AT I WALL TEMPERATURE T ' 1 w 1 ITERATION T ' = T ' + AT w w IF |Q - Q'|/Q > 0.001 IF |Q - 0'|/0 < 0.001 CALCUALTE Nu FROM o CONSTANT PROPERTY CORRELATION BASED ON BULK PROPERTIES CALCULATE N.u "FROM SUPERCRITICAL FLOW CORRELATION BASED WALL AND BULK PROPERTIES OUTPUT T = T ' w w Figure 13. Routine for Calculating Wall Temperature INPUT INFORMATION: T>h, WP, A1, G, Q, P r T. b l FIND WALL TEMPERATURE AT POINT 1, [SEE FIGURE 13] FIND BULK AND WALL PROPERTIES AT POINT 1 ) AP' = AP P 2* = P - AP' h 2 * = h 2' CALCULATE PRESSURE DROP AP FROM SUPERCRITICAL P™*" FLOW CORRELATION IF |AP - AP'| > 0.01 1 IF |AP - AP'| < oToi j = P T = T 2' b l b2 CALCULATE PRESSURE DROP AP' FROM CONSTANT PROPERTY FLOW CORRELATION FIND AVERAGE BULK AND WALL TEMPERATURES, AND PRESSURE OF POINT 1 AND 2 I FIND OTHER PROPERTIES BASED ON AVERAGE BULK AND WALL TEMPERATURE, AND PRESSURE OUTPUT P, Figure 14. Computer Program Flow Chart for Calculating Heat Transfer to Supercritical Flow in Heated Pipes FLOW RATE = 0.159 Kg/Sec WALL HEAT FLUX =1.01 W/cm2 PIPE DIAMETER = 18.97 mm PRESSURE = 1100 Psia T E M P E R S ! DOWNFLOW, EXPERIMENT UPFLOW, EXPERIMENT PREDICTION 10 20 30 40 50 60 70 80 90 100 DISTANCE ALONG TEST SECTION FROM START OF HEATING, DIAS. 110 120 130 Figure 15. Prediction of Heat Transfer to Supercritical Flow, Experimental Results from [60] DISTANCE ALONG TEST SECTION FROM START OF HEATING, DIAS. Figure 16. Prediction of Heat Transfer to Supercritical Flow, Experimental Results from [49] CO 100 E 3 H H 75 50 25 10 20 30 DISTANCE ALONG TEST SECTION FROM START OF HEATING, DIAS. Figure 17. Prediction of Heat Transfer to Supercritical Flow, Experimental Results from [55] CO DISTANCE ALONG TEST SECTION FROM START OF HEATING, In Figure 18. Prediction of Heat Transfer to Supercritical Flow, Experimental Results from [42] (b) Single Rod Equivalent Annulus [68] Figure 19. Equivalent Annulus Models for Rod-Bundles 86 Wire wrap is used to increase mixing between subchannels which are divided at the minimum spacings between fuel-rods. The. sub- channels are classified and designated with numbers, according to their geometry and mixing condition. Figure 20. Lumped Parameter Model for Rod Bundles (from [71]) 300 2 00- o • o SPACER O O o TUBE NO. 4 • TUBE NO. 7 88 ° o 0 ° o o 0 n I —.8S8888S8WS 100 0 « i 0 10 20 30 40 50 AXIAL POSITION, L/d Figure 21. Thermal Entrance Following a Grid Type Spacer (from [73]) oo • I l l 1 I I I I I 1 1 1 1 ' r REYNOLDS NUMBER Figure 22. Comparison of Heat Transfer Data in a Uniformly Heated Seven-Rod Bundle (Pr = 0.7) with Wire-Wrap Spacers (f r 0m [74]) 0.03 | \ F " " f ^ ^ » ^ « ~ — ^ • ^ ^ ^ ^ ^ • ^ — ^ ^ ^ ^ Figure 23. Comparison of Friction Factors for Seven-Rod Bundle with Wire-Wrap Spacers (1/d = 12, 24 36, oo) in a Scalloped Liner (p/d = 1.237) (from [74]) oo 90 1500 400 0 3.0 6.0 9.0 12.0 15.0 DISTANCE FROM CORE INLET, In. Figure 24. Temperature Profiles in Coolant Channel with Different Fuel Bundle Orientation(from [77]) 0.5; 1.1 1.2 1.4 1.6 1.8 2.0 . s/r 0 NUSSELT NUMBER OF ROD BUNDLES NUSSELT; NUMBER OF.CIRCULAR TUBES FRICTION FACTOR OF ROD BUNDLES FRICTION FACTOR OF CIRCULAR TUBES 1.0 Nu Nu f f c i r c i r Figure 25. Effect of Ratio of Pitch to Rod Diameter Nusselt Number and Friction Factor (from [78]) 92 2 8 E L E M E N T B U N D L E FULL S C A L E C R O S S S E C T I O N O F FUEL A N D C O O L A N T TUBE F i g u r e 26. CROSS SECT ION OF FUEL A N D C O O L A N T TUBE 4<.3750o.3 (FROM [12]) R C V . ' l . . 9 6 6 93 CALANDRIA TUBE AIR GAP PRESSURE TUBE OXIDE FUEL ROD SHEATH: ZIRCALOY-2 FILLER(OR BLOCKER) MINIMUM SPACING: 0.02 In Figure 27. Channel for Boiling Reactor (from [79]) AIR GAP CALANDRIA TUBE PRESSURE TUBE U02 FUEL ZIRCALOY SHEATH COOLANT FLOW Figure 28. Tube-in-Shell Fuel Element (from [79]) D h = P ° , T _ = 775 ° K MAXIMUM WITHSTANDING TEMPERATURE OF FUEL SHEATH 550 w OS S3 .J w 500 P ± n = 100.0 .Bar REACTOR CORE LENGTH = 594 cm SUBCHANNEL POWER = 92.3 KW HEAT FLUX DISTRIBUTION: sin(kx + <f>) WHERE k = 0.51504 i = 0.02574 x = DISTANCE Do = 1 UNIT OF EQUIVALENT HYDRAULIC DIAMETER = 0.521 cm 450 320, 340 360 380 400 INLET TEMPERATURE OF COOLANT, K 420 440 Figure 29. Maximum Operating Temperature Versus Inlet Coolant Temperature Figure 30. Pressure Drop Versus Inlet Coolant Temperature vo i I r 1 1 r 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 EQUIVALENT HYDRAULIC DIAMETER, cm Figure 31. Selection of Minimum Equivalent Hydraulic Diameter (I.D. 4.457 in.) FULL SCALE CROSS SECTION OF FUEL BUNDLE AND COOLANT TUBE Figure 32. Cross Section of 32-Element Fuel Bundle 98 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 o H oi w fe o Pi fe- I EXPERIMENTAL DATA FROM REFERENCE [83] -j AECL REPORT 2772 (AUG. 1967) 1 2 3 4 RADIAL DISTANCE FROM BUNDLE CENTERLINE, cm Figure 33. Radial Neutron Flux Distribution in Fuel Bundle 0 1.0 2.0 3.0 4.0 5.0 6.0 DISTANCE FROM INLET, Meters A O vo Figure 34. Subchannel Coolant Temperature Rise in 32-Element Fuel Bundles Versus Reactor Channel Length 600 0 1.0 2.0 3.0 4.0 5.0 6.0 DISTANCE FROM INLET, Meters Figure 35. Subchannel Sheath Temperature Rise in 32-Element Fuel Bundles Versus Reactor Channel Length o o 80 I I _ L _ I I I _ J 0 1.0 2.0 3.0 4 .0 5.0 6.0 DISTANCE FROM INLET, Meters Figure 36. Pressure Drop Along Reactor Channel Length i—1 o  1000 REPRESENTATIVE POSITION IN STEAM GENERATOR Figure 38. Dual-Pressure 600.0/70.3 (psia) Steam Generation 104 FROM REACTOR SUPERCRITICAL C0 2 3.4905X107 Lbm/Hr 845 °F 272.7 Btu/Lbm HEAT EXCHANGER 200.0 Btu/Lbm Figure 39. Turbine Circuit Heat Balance of Single Pressure Cycle 105 FROM REACTOR SUPERCRITICAL CO 3.4905X107 Lbm/Hr 845 °F 272.7 Btu/Lbm LEGEND: g - FLOW RATE, Lbm/Hr p - PRESSURE, Psia h - ENTHALPY, Btu/Lbm x - STEAM QUALITY, % s - STEAM w - WATER TO RECUPERATOR SUPERCRITICAL C0 2 280 °F 11 £..05 i'f.. /T h,~. REPRESENTATIVE POSITION IN STEAM GENERATOR Figure 41. Dual-^Pressure 1200.0/70.3 (psia) Steam Generation 107 FROM REACTOR SUPERCRITICAL CO, 3.4905X107 g 1 845 °F, 272.7 h POINT A , 2.4547X10° g 1200 p, 815 °F 1389.03 h LEGEND: g - FLOW RATE, Lbm/Hr p - PRESSURE, Psia h - ENTHALPY, Btu/Lbm x - QUALITY OF STEAM s - STEAM w - VJATER POINT D 96680 gw 225833 gs 3.8 p 120.56 hw 1087.06 hs Figure 42. Turbine Circuit Heat Balance of Dual-Pressure 1200.0/70.3 (psia) Cycle REPRESENTATIVE POSITION IN STEAM GENERATOR Figure 43. Dual-Reheat 1200.0/70.3 (psia) Steam Generation o 00 109 FROM REACTOR SUPERCRITICAL CO, 3.4905X107 g 845 °F 272.7 h LEGEND: g - FLOW, Lbm/Hr h - ENTHALPY, Btu/Lbm 1200 Psia 815 °F 1389.03 h 70.3 Psia 567.19 °F 1315.4 h 116.05 h Figure 44. Turbine Circuit Heat Balance of Dual-Reheat 1200/70.3 (psia) Cycle - NO. OF FLOW TOTAL UNHEATED HEAT MAX. FLOW FINAL ENTHALPY SUB- SUB- AREA WET. PER. WET. PER. RATE POWER TEMP. . RATE ENTHALPY CHANGE CHANNEL CHANNELS cm ̂ cm cm Q, KW/M , KW °C , Kg/Sec . KJ/Kg KJ/Kg 1 1 0.8947 4.775 35.8 143.0 409.0 0.4651 545.8 298.3 2 4 0.8947 4.775 37.8 151.0 426.0 0.4638 563.7 316.2 3 4 0.8947 4.775 44.5 177.8 505.0 0.4238 650.5 403.0 4 4 0.8947 4.775 39.7 158.6 457.0 0.4394 598.2 350.7 5 12 0.6571 3.777 1.116 31.4 125.6 532.0 0.3018 651.7 404.2 6 8 0.8947 4.775 48.2 192.7 546.0 0.4178 695.3 411.8 7 4 1.3872 8.921 3.773 60.8 242.9 538.0 0.5839 651.7 404.2 ASSUMED.HEAT INPUT DISTRIBUTION: q =Q-sin(kx WHERE k = 0.51504 A = 0.02547 x = DISTANCE INLET CONDITION: PRESSURE = 100 Bar = 1450 Psia TEMPERATURE = 126.85 °C = 260 °F ENTHALPY = 247.5 KJ/Kg = 110.26 Btu/Lbm 4) TOTAL COOLANT CHANNEL POWER: 6113 MW TOTAL COOLANT CHANNEL MASS FLOW RATE: 15.091 Kg/Sec OUTLET CONDITION: PRESSURE = 85.5 Bar = 1240 Psia TEMPERATURE = 457.8 °C = 855.6 °F ENTHALPY =640.9 KJ/Kg = 275.60 Btu/Lbm Figure 45. Results of Heat Transfer to Supercritical CO Coolant in 32-Element Fuel Bundles I l l "* COOLANT CHARACTERISTICS — — 1 HEAVY WATER CARBON DIOXIDE INLET PRESSURE (Psia) j 1 1443 1450 OUTLET PRESSURE (Psia) 1254 1240 PRESSURE DROP (Psia) 189 210 MAXIMUM SHEATH TEMPERATURE (°C) 304 550 COOLANT INLET TEMPERATURE (°C) 249 127 COOLANT OUTLET TEMPERATURE (°C) 293 458 COOLANT TUBE INNER DIA. (In.) | j 4.070 4.458 J FUEL ELEMENT DIA. (In.) 0.598 0.598 § NO. OF FUEL ELEMENTS IN A BUNDLE 28 32 1 MAXIMUM CHANNEL POWER(MW) | j 5.125 6.113 | MAXIMUM CHANNEL MASS FLOW RATE | j 22.2 Kg/Sec 15.1 Kg/Sec 1 POSSIBILITY OF DRYOUT | [ YES NO | COOLANT COST | ] HIGH LOW j | FUEL USED | U°2 U 02 U C i Figure 46. Overall Performance of Supercritical Carbon Dioxide Coolant 112 PICKERING NEW DESIGN I COOLANT HEAVY WATER SUPERCRITICAL C0 2| CORE LENGTH 19 Ft. 6:In. 19 Ft. 6 In. I MAXIMUM CHANNEL POWER 5.125 MW 6.113 MW NO. OF FUEL RODS 28 32 0. D. OF SHEATH 0.598 In. 0.598 In. 0. D. OF FUEL BUNDLE 4.03 In. 4.358 In. INTER-ELEMENT SPACING (MIN.) 0.05 In 0.496 In. COOLANT TUBE X-SECTIONAL AREA 13.010. Sq. In. 15.605 Sq. In. B BLOCKER X-SECTIONAL AREA NIL 1.622 Sq. In. FUEL BUNDLE X-SECTIONAL AREA 7.864 Sq. In. 8.988 Sq. In. COOLANT FLOW AREA 5.146 Sq. In. 4.995 Sq. In. COOLANT TUBE I. D. 1 4.070 In. 1 4.457 In. j Figure 47. Overall Dimensions of 32-Element Fuel Bundle PICKERING SINGLE PRES. CYCLE DUAL PRESSURE CYCLE 600/70 DUAL PRESSURE CYCLE 1200/70 DUAL-REHEAT CYCLE 1200/70 TURBINE EFFICIENCY H.P. 72.6% L.P. 88.5% 90,0% H.P. 90.0% L.P. 88.5% H.P. 90.0% L.P. 88.5% H.P. 90.0% L.P. 90.0% STEAM PRESSURE, Psia H.P. 585 L.P. 70.3 130 H.P. 600 L.P. 70.3 H.P. 1200 L.P. 70.3 H.P. 1200 LIP. 70.3 MAXIMUM STEAM TEMPERATURE, °F H.P. 483;5 L.P. 435.0 815.0 H.P. 815.0 L.P. 435.0 H.P. 815.0 L.P. 435.0 H.P. 815.0 L.P. 567.2 PRESSURE RATIO, H.P./L.P. 8.321 8.535 17.07 17.07 STEAM FLOW RATE I Lbm/Hr H.P. 5969862 L.P. 5550833 400600 H.P. 3036500 L.P. 4246800 H.P. 2454700 L.P. 4452600 H.P. 2946300 L.P. 4045800 STEAM FLOW RATE RATIO, H.P./L.P 1.075 0.7150 0.5513 0.7282 EXHAUST STEAM WETNESS Hi P. 11% L.P. 1Q% 5.4% H.P. Super. L.P. 9% H.P. 3.9% L.P. 9% H.P. 3:9% L.P. 9% EXHAUST STEAM PRESSURE, Psia H.P. 70.3 Psia L.P. 1.5 In Hg 1.5 In Hg H.P. 75.0 Psia L.P. 1.5 In Hg H.P. 75.0 Psia L.P. 1.5 In Hg H.P. 75.0 Psia L.P. 1.5 In Hg TURBINE CYCLE EFFICIENCY 34.60% 27.6% 30.3% 30.7% 33.8% Figure 48. Summary of Steam Conditions of Thermal Cycles PICKERING SINGLE PRES. CYCLE DUAL PRESSURE ' CYCLE 600/70.3 DUAL PRESSURE CYCLE 1200/70 DUAL-REHEAT CYCLE 1200/70 HEAT TRANSFERRED TO SECONDARY CIRCUIT, Btu/Hr 5.67 x 10 9 5.67 x 10 9 5.67 x 10 9 5.67 x 10 9 5.67 x 10 9 SHAFT WORK OUTPUT, Btu/Hr 1.9760 x 10 9 1.5640 x 10 9 1.7212 x 10 9 1.7504 x 10 9 1.9263 x 10 9 COOLANT FLOW RATE, Lbm/Hr 6.1300 x 10 7 3.4905 x 10 7 3.4905 x 10 7 3.4905 x 10 7 3.4905 x 10 7 PRIMARY PUMP WORK, Btu/ Hr 3.9365 x 10 7 4.3170 x 10 7 4.3170 x 10 7 4.3170 x 10 7 4.3170 x 10 7 FEED WATER FLOW RATE, Lbm/Hr H.P. 5,969,362 DRAIN 489,525 4,006,600 H.P. 3,036,500 L.P. 1,210,300 H.P. 2,454,700 L.P. 1,997,900 H.P. 2,946,300 L.P. 1,099,500 SECONDARY PUMP .WORK, Btu/Hr 1.2229 x 10 7 2.0030 x 10 6 6.7061 x 10 6 1.0443 x 10 7 1.0714 x 10 7 NET WORK OUTPUT, Btu/Hr 1.9224 x 10 9 1.5147 x 10 9 1.6672 x 10 9 1.6927 x 10 9 1.8724 x 10 9 IDEAL OVERALL EFFICIENCY 33.94% 26.73% 29.42% 29.87% 33.02% Figure 49. Overall Performance of Thermal Cycles 1 8 9 5 2 5 l h / h r MECHANICAL LOSSES 13X7 KW ELECTRICAL LOSSES 8162 KW NET GENERATOR OUTPUT 543320 LEGEND KW O Ib/hr BUILDING HCATING ' I I 1 ^ L J II A t O H /" — W l f A T E R r . , . 0 4 I MAIN STM. CHESTS nln n J—I |T0 CONDENSER g . Flow in Ib/hr a - Pressure in PSIA f - Temperature in degrees F h . Enthalpy b l u / l b x - Quality of steam in percent Ib/hr • Flow ol steam lb/, rw. Flow of water s • Enthalpy of steam . btu/lb S 9 8 9 B 6 2 Ib/hr 3 1 2 . 0 5 h 3 3 9 . 9 8 F BOILER LIVE STEAM REHEATER DRAINS 48952S l» /h r 468.37 h » ; L -i - - C — y — * • — r — • 11 •*-*-*• 4—4- »• » —* ' i 1 * ,\ *\> *• i A A ' ! » *. \\ *.—*. .'««.-. u. < ' j•* * •«•— J | [WATER EXTRACTION BE.LT 319.42 h H I T . 2 « h * 91503 Ib/S^-W 3 I S 0 3 6 Ib/hr 124.45 PS IA 160637 Ib/hr f f 272.90 h 1175.78 h i J65486 Ik/hrtfi IM>IZ3 IWhr ' 66.79 PSIA 1275.13 h [ 3 0 5 7 6 r J i 0 3 3 * ! b / > » / NO. 4 DE AERATOR HEATER 38 0 PSIA 2M.IT F 120.56 h '087.06 h . 9 6 6 8 0 Ib/hrW 2 Z 5 8 3 9 16/hr 3.8 PSIA 2 0 8 0 Ib/h, 1201.8 h i 70 . o r h J 102.10 V ^ 676613 NO. 4 HEATER NO. 5 HEAT5S NO.3 HEATE* NO. 2 H'ATSS NO. 1 M?ATE1! Figure 50. T U R 3 0 G E N E R A T O R H E A T B A U N C c (FROM [12]) 116 .0 1.1 1.2 1.3 1.A 1.5 1.6 ENTROPY, Btu/Lbm Figure 51.; Expansion Line of CANDU reactor Turbine Stages (from [ 8 2 ] ) 117 REFERENCES 1. 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Sastry, V.S. and Schnurr, N.M.; "An Analytical Investigation of Forced Convection Heat Transfer to Fluids Near the Thermodynamic C r i t i c a l Point", ASME Transactions, J. Heat Transfer, Vol. 97, No. 5, Pg. 226, 1975. 45. Miropolski, L. and Shitsman, M.E.; "Heat Transfer to Water and Steam at Variable Specific Heat in Near-Critical Region", Soviet Phy.-Tech. Phys., Vol. 2, No. 10, Pg. 2196, 1957. 46. Colburn, A.P.; "A Method of Correlating Forced Convection Heat Transfer Data", AICHE Journal, Vol. 29, Pg. 174, 1933. 47. Boure, P.J., Pulling, D.J., G i l l , L.E., and Denton, W.H.; "Forced Convection Heat Transfer to Turbulent CO2 in the Supercritical Region", Int. J. Heat Mass. Transfer, Vol. 13, Pg. 1339, 1968. 48. Kutateladze, S.S. and Leontiev, A.I.; TURBULENT BOUNDARY LAYER IN COMPRESSIBLE GASES, Arnold, London, 1964. 49. Kranoshchekov, E.A. and Protopopov, V.S.; "Experimental Study of Heat Exchange in Carbon: Dioxide in the Supercritical Range at High Temperature Drops", High Temperature, USSR, Vol. 4, Pg. 375, 1965. 50. Kranoshchekov, E.A. and Protopov, V.S.; "A Generalized Relationship for Calculation of Heat Transfer to Carbon Dioxide at Supercritical Pressure", High Temperature, U.S.S.R., Vol. 9, Pg. 125, 1971. 51. Petukhov, B.S. and K i r i l l o v , V.V.; "On the Question of Heat Transfer to a Turbulent Flow of Fluids in Pipes", Teploenergetika, No. 4, Pg. 63, 1958. 52. Hiroharu, K., Niichi, H., and Masaru, H.; "Studies on the Heat Transfer of Fluid at a Supercritical Pressure", Bulletin of JSME, Pg. 654, 1967. 53. Shiralkar, B.S. and G r i f f i t h , P.; "Deterioration in Heat Transfer to Fluids at Supercritical Pressure and High Heat Fluxes", ASME Transactions, J. Heat Transfer, Vol. 91, No. 2, Pg. 27, 1969. 54. Dickinson, N.L. and Welch, O.P.; "Heat Transfer to Supercritical Water", ASME Transactions,;J. Heat Transfer, Vol. 80, No. 4, Pg. 746, 1958. 55. Kranoshchekov, E.A., Protopopov, V.S., Parhovhik, I.A., and S i l i n , V.A.; "Some Results of an Experimental Investigation of Heat Transfer to Carbon Dioxide at Supercritical Pressure and Temperature Heads up to 850°C", High Temperature, USSR, Vol. 19, No. 5, Pg. 992, 1971. 121 56. Petukhov, B.S., Grigorev, V.S., and Polyakov, A.F.; "Experimental Investigation of Heat Transfer in Single-Phase Near-Critical Region with Heat Input Varying Along the Length of the Pipe", Toploenergetika, Vol. 20, No. 3, Pg. 71, 1973. 57. Glushchenko, L.F. and Gandzyuk, O.F.; "Temperature Conditions at the Wall of an Annular Channel with Internal Heating at Supercritical Pressure", High Temperature, USSR, Vol. 10, No. 4, Pg. 734, 1972. 58. Protopopov, V.S. and Igamberdyev, A.T.; "Results of an Experimental Investigation of Local Heat Transfer at Supercritical Pressure in a Rectangular Channel Heated From One Side", High Temperature, USSR, Vol. 10, No. 6, Pg. 116, 1972. 59. Kellogg, H.B. and Brown, G.B.; "Seventh Quarterly Progress Report for 1000 MWE Supercritical Pressure Nuclear Reactor Plant Study", WCAP-2525, October, 1963. 60. Jackson, J.D. and Evans-Lutterodt, K.; "Impairment of Turbulent Forced Convection Heat Transfer to Supercritical Pressure CO2 Caused by Buoyancy Forces", Research Report N.E. 2, University of Manchester, Department of Nuclear Engineering in the Faculty of Science, March, 1968. 61. Tarasova, N.V. and Leontz, V.A.I.; "Hydraulic Resistance During Flow of Water in Heated Pipes at Supercritical Pressure", High Temperature, USSR, Vol. 6, No. 4, Pg. 721, 1968. 62. Kuraeva, I.V. and Protopopov, V.S.; "Mean Friction Coefficient for Turbulent Flow of Liquid at a Supercritical Pressure in Horizontal Circular Tubes", High Temperature, USSR, Vol. 12, No. 1, Pg. 194, 1974. 63. Sastry, V.; "Thermophysical Properties of CO2 in the Near C r i t i c a l Region". 64. Vargaftirk, N.B.; TABLES ON THE THERMOPHYSICAL PROPERTIES OF LIQUIDS AND GASES, Second Edition, Hemisphere Publishing Corporation, Washington, London, 1974. 65. Altunin, V.V. and Sakhabetdinov, M.A.; "The Viscosity of Liquid and Gaseous Carbon Dioxide at Temperatures of 220 - 1300°K and Pressure up to 1200 Bar", Thermal Engineering, Vol. 19, No. 8, Pg. 124, 1972. 66. Altunin, V.V. and Sakhabetdinov, M.A.; "The Thermal Conductivity of Liquid and Gaseous Carbon Dioxide in the 220 - 1300°K Temperature Range at Pressure up to 1200 Bar", Thermal Engineering, Vol. 20, No. 5, Pg. 121, 1973. 67. Brown, G., Packman, G. and Greenough, G.B.; "Fuel Elements", J. Brit. Nucl. Energy Society, Vol. 2, No. 2, Pg. 186, 1963. 68. Sutherland, W.A.; "Experimental Heat Transfer in Rod Bundles", Heat Transfer in Rod Bundles, ASME, New York, 1968. 122 69. Maresca, M.W. and Dwyer, D.E.; "Heat Transfer to Mercury Flowing in Line Through a Bundle of Circular Rods", ASME Trans., J. Heat Transfer, Vol. 86, No. 3, Pg. 180, 1964. 70. Friedland, A.J., Dwyer, D.E., and Bonilla, C.F.; "Heat Transfer to Mercury in Parallel Flow Through Bundles of Circular Rods", International Development in Heat Transfer, International Heat Transfer Conference, Colorado, USA, Paper 62, 1961. 71. Oldaker, I.E.; "Sheath Temperature, a G-20 Computer Program for Fuel Bundles", AECL-2269, May, 1965. 72. Rogers, J.T. and Todreas, H.E.; "Coolant Interchannel Mixing in Reactor Fuel Rod Bundles Single Phase Coolants", Heat Transfer in Rod Bundles, ASME, New York, 1968. 73. Hoffman, H.W., Wentland, J.L., and Stelzman, W.J.; "Heat Transfer with Axial Flow in Rod Clusters", Int. Development in Heat Transfer, International Heat Transfer Conference, Colorado, USA, Paper 65, 1961. 74. Hoffman, H.J., Miller, C.W., Sozei, G.L., and Sutherland, W.A.; "Heat Transfer in Seven Rod Clusters - Influence of Linear and Spacer Geometry on Sheath Fuel Performance", GEAP-5289, General Electric Co., San Jose, California, October, 1966. 75. Midvioy, W.I.; "An Investigation Into the CHP Performance of Horizontal and Vertical 37-Element Assemblies Cooled by Freon", Proceedings of the 6th Western Canada Heat Transfer Conference, May, 1976. 76. Kidd, G.J., Jr., Stelzman, W.J., and Hoffman, H.W.; "The Temperature Structure and Heat Transfer Characteristics of an El e c t r i c a l l y Heated Model of Seven Rod Cluster Fuel Element", Heat Transfer in Rod Bundles, ASME, New York, 1968. 77. Samuels, G.; "Design and Analysis of the Experimental Gas-Cooled Reactor Fuel Assemblies", Nuclear Science and Engineering, Vol. 14, Pg. 37, 1962. 78. Kays, W.M.; CONVECTIVE AND MASS TRANSFER, McGraw-Hill, New York, 1966. 79. Pon, G.A.; "Light Water-Cooled Heavy-Water-Moderated Natural-Uranium Power Reactor", The Ninth AECL Symposium on Atomic Power, Toronto, Ontario, AECL-1807, 1963. 80. Wooten, W.R.A., Taylor, A.J., and Worley, N.E.; "Steam Cycles for Gas-Cooled Reactors", Proc. Second United Nations, International Conf. on Peaceful Uses of Atomic Energy, Geneva, Vol. 7, Paper P/273UK, 1958. 81. Fraas, A.P. and Ozisik, M.N.; HEAT EXCHANGER DESIGN, John Wiley and Sons, Inc., New York, 1965. 82. Renshaw, R.H.; "Heat Transport", AECL Nuclear Power Symposium, Lecture 6, 1971. 123 83. Serdual, K.J. and Green, R.E.; "Lattice Measurements with 28-Elements Natural UO2 Fuel Assemblies, Part II: Relative Total Neutron Densities and Hyperfine Activation Distributions in a Lattice Cell", AECL-2772 Chalk River, Ontario, August 1967. 124 APPENDIX I. Computing Procedure The basic approach of the computer program for heat transfer to supercritical carbon dioxide is iteration. Figure 13 shows the routine for calculating a local wall temperature when a local f l u i d bulk tempera- ture, pressure, heat flux, flow rate, and geometry are given. The super- c r i t i c a l flow is f i r s t assumed to have constant properties. The local heat transfer coefficient and wall temperature are calculated from a correlation for constant property flow, i.e. equation (11). Then, an assumed new value of the wall temperature is increased or decreased from the calculated value in an iterative manner, and the corresponding heat transfer coefficient i s , therefore, calculated from the correlation for supercritical flow; i.e. equation (19). This iteration continues until the heat flux (evaluated from the given local conditions, assumed wall temperature, and corresponding heat transfer coefficient) is equal to the given heat flux. The fin a l value of the assumed wall temperature is the result. Thermophysical properties of carbon dioxide near the c r i t i c a l T 631 point in the calculation are taken from Sastry . Thermal properties of carbon dioxide in the above c r i t i c a l region are taken from the data of Vargaftik^ 4"', and physical properties are calculated from the formulas * M . • [65,66] of Altunin et a l . Figure 14 shows the routine for calculating pressure drop from one increment to the next along a heated pipe. An assumed pressure drop is f i r s t calculated from the correlation of constant property flow; i.e. equation (23). Then, the assumed pressure of the second increment is calculated from P 0 = P1 - dP. The flu i d bulk enthalpy is calculated from 125 the given enthalpy and input heat rate of the f i r s t increment, i.e. h, 9 - h, n + dh. Based on the assumed pressure and calculated enthalpy, other fl u i d properties and also the wall temperature of the second increment can be evaluated. Assuming linear temperature and pressure variations between increments, a new pressure drop is calculated from the correlation for supercritical flow, i.e. equation (22). Based on the new pressure drop and known enthalpy, the properties and wall temperature of the second increment are evaluated again for the calculation of another new pressure drop. These steps are repeated unt i l the calculated pressure drop is con- verged. increment to another until the f l u i d bulk temperatures, wall temperatures, and pressure drop along the entire pipe are calculated. The computer program is listed in this Appendix. The procedure based on the above routines i s carried out from one 126 1 C * * MAIN PROGRAM * * 2 REAL P ( 3 0 ) . TM( 30) T ( 3 0 , 100) , MD(30 ,100 ) , H ( 3 0 , 1 0 0 ) , CP (30 ,100 ) 3 REAL V K 3 0 . 100 I , M 3 0 . 1 0 0 ) 4 REAL L ( IOOO) , T P ( 1 0 0 0 ) , TP( IOOO), PP(IOOO) 5 REAL MOP, MOW, MORX, MDWX, MOB A, MOWA 6 REAL KB, KW, KBX, KWX, KRA, KWA, MFR 7 INTEGER INC 8 1=0 9 10 1=1+1 10 RFA0<8,81) P ( I ) , TM(I ) 11 81 FORMAT(F10.5, F9 .2 ) 12 IF(TM(TI . E O . 0.0) GO TO 20 13 DO 11 J-lt 100 14 REftD(8,82) T ( I , J > , MD( I , J I , H ( I , J ) , C P ( I , J ) 15 82 FORMAT(F 10 .2 , F 1 4 . 3 , F 1 3 . 2 , F12 .2 ) 16 I F ( T f I , J ) . E O . 0 .0 ) GO TO 10 17 MD( I , J ) = 1 C 0 0 . 0 / M D U , J I 18 11 CONTINUE 19 20 1=9 20 21 1=1*1 21 REA0(7,91) P ( I ) , TMt-1) 22 91 F C R M A T ( F 9 . 5 , F10 .2 ) 23 IF{TM{I| . E O . 0.0) GO TO 30 2* DO 66 J = l , 100 '"• 25 RFAD<7,92> T ( I , J ) , M O ( I , J ) , H ( I , J » , C P U . J I , V I ( I , J ) , K ( I , J I 26 92 F0RMAT(F10.2, F 1 4 . 3 , F 1 3 . 2 , F 1 2 . 2 , E 1 6 . 4 , E15 .4 ) 27 I F ( T ( I , J » . E O . 0.0) GO TQ 21 28 MD( I , JJ=100C.0 /MD( I , J ) 29 66 CONTINUE 30 30 READ<5,31> D, MFR, 0 , PG, TB, BTU 31 31 FORMAT(F10.6, 5F10 .1 ) 32 RFAD(5,83) INC, CONT 33 83 FORMA T{I 10, F10. I ) 34 IF(CC.NT . E O . 0.0) GO TO 24 35 WRITF<6,26) 36 26 PORMAT (*.-' , 'THIS CALCULATION IS AT CONSTANT PRESSURE' ) 37 24 IF(3TU .'EO. 0.0) GO TO 77 38 WRITE(6,36I 39 36 FORM AT(* - • f 'GIVEN DIAMETER, MASS FLOW RATE, HEAT FLUX, PRESSURE•, 40 I • , 9ULK-TEMPPRATURE, * * * * * BRITISH UNITS * * * * * •) 41 W R I T E ( 6 ' » 1 7 ) D, MFR, 0, PG, T B , BTU 42 0=0*0.0254 43 MFR=MFR*0.45359 44 0=0*0.003154 45 P G = P G * 1 . 0 1 3 2 5 / 1 4 . 7 46 TB= (TB+460 .0 ) *5 .0 /9 .0 47 77 MFR=MFR/3600.0 48 I F { ( O G . G E . 75 .0 ) .AND. (PG . L E . 100.0) ) GO TO 61 49 W P ! T F ( 6 , 6 2 1 PG 50 62 F ORM AT ( • - • , F 1 0 . 2 . ' INPUT PG IS OUT OF DATA RANGE') 51 GC TO 13 52 61 IF ( ( TP . G E . 273.151 .AND. (TB . L E . 1500.0 ) ) GO TO 40 53 WRITE(6,63) TB 54 63 FORMAT{• - ' , F 1 0 . 2 , ' TNPUT TB IS OUT OF DATA RANGE') 55 GO TO 13 56 40 N=10 57 I F ( (TB .C,T. 400 .0 ) .OR. (PG .GT . 8 2 . 7 0 ) ) N= 1 58 CALL PROP(PG,TB,TM,P,T,H,MO,VI ,CP,K,TMC,HB,MCB,V IB.CPB,KB,N) 127 5 9 I F ( N . N E . 1 ) GO TO ,51 6 0 C A L L V I S C J P G , T B , MOB, V I B ) 61 C A L L C O N D ( P G , T B, MDB, KB I 6 2 5 1 W R I T E ( 6 , 3 2 ) 6 3 3 2 F O R M A T ( ' l ' , " G I V E N O I A M F T E R , MASS FLOW R A T E , H E A T F L U X , P R E S S U R E • , 6 4 1 •, B U L K - T E M P F R A T U R E , E N T H A L P Y , NO. OF I N C R E M E N T S ' ) 6 5 W R I T E ( 6 , 1 7 ) D, MFR, 0, PG, T B , HB, I N C 6 6 17 FORMAT t• ' , 6 E 1 7 . 4 , 1 1 8 ) 6 7 H I = H B 6 8 DL= 0 6 9 A = 3 . 1 4 1 5 9 2 6 * 0 * 0 / 4 . 0 7 0 A L = 3 . 1 4 1 5 9 2 6 * D * D L 7 1 T L = F L O A T ( I N C ) * O L 72 VM=MFR/A 7 3 G = 9 . 8 1 7 4 9S=l.O 7 5 M=l 7 6 C A L L H F A T I D , D L , M, Q, A H ) 7 7 AHB= A H/MFR 7 8 C A L L F T N O ( D , M F R , A H , P G , H B , T M , P , H , T , M D , V I , C P , K , T W , F A I L , D S , M ) 7 9 I F ( F A I L .NE . 0 . 0 ) GO TO 4 5 8 0 N=10 81 I F U T W . G T . 4 C 0 . 0 ) .OR. I P G ,GT. 8 2 . 7 ) ) N = l 82 C A L L P R O P ( P G , T W , T M , P , T , H , M O , V I ,CP,K,TMW,HW,MOW,VIW,CPW,KW,N) 8 3 I F ( N . N E . 1 ) GO TO 3 3 84 C A L L V I S C ( P G , T W , V D W , V I W ) 85 C A L L CONO(PG,TW,MDW,KW) 86 3 3 T P N T I N U E 8 7 L ( M ) = ( F L O A T I M ) - 0 . 5 > * D L * l 0 0 0 . 0 88 T F ( M ) = T B 8 9 T P ( M ) = T M 9 0 P P ( M ) = P G 91 I F ( M . G E . INC,) GO TO 9 8 9 2 M=M+1 9 3 HBX=HB+AHB 9 4 I F U H P X . G E . - 9 5 . 0 > . AMD. (HBX . L E . 1 6 0 0 . 0 ) ) GO TO 1 8 95 W R I T E ( 6 , 1 9 ) HBX 9 6 1 9 F O R M A T ( • — ' , F 1 0 . 2 , • HBX I S C U T OF OATA R A N G E ' ) 9 7 G C T O 4 5 9 8 1 8 C O N T I N U E 9 9 C A L L HE A T I 0, O L , M, 0, A H X ) 1 0 0 AHBX=AHX/MFR 1 0 1 I F I C O N T . E O . 0 . 0 ) GO TO 2 7 1 0 2 P X = P G 1 0 3 P F = P G 1 0 4 GO TO 1 5 1 0 5 27 R F = V M * D / V T B 1 0 6 F = 1 . 0 / ( I 1 . 8 2 * A L 0 G 1 0 ( R E ) - 1 . 6 4 ? * * 2 1 1 0 7 I F I R E . G T . 1 C C O O 0 . O ) F = 0 . 0 1 7 9 6 8 9 4 1 0 8 E V = E * ( ( V I W / V I R ) * * 0 . 2 2 ) 1 0 9 P 0 = F V * { V M * * 2 ) * D L * 2 . 0 / ( * M D B * D ) 1 1 0 P D = P D * 0 . 0 0 0 0 1 1 1 1 P X = P G - P O 1 1 2 TEST= 0 . 0 1 1 3 NM=0 1 1 4 4 4 NN-NM+l 1 1 5 I F I N M . L E . 2 5 ) GO TO 5 4 1 1 6 W R I T E ( 6 , 5 5 ) M 128 117 55 F O R M A T P - ' , * * * * * * * * ITERATION OF PRESSURE DROP DOES NOT COVERGE * * * * * * 118 GO TO 45 119 54 CONTINUE 120 I F ( ( P X . G E . 75 .0 ) .AND. ( P X . L E . 100.0 ) ) GO TO 15 121 WPITE(6,16I PX 12? 16 FORMAT(•- * t F l 0 . 2 t • PX IS OUT OF DATA RANGE') 123 GO TO 45 124 15 N=10 125 I F U H B X . G T . 285 .0 ) .OR. (PX .GT . 82 .7 ) ) N=l 126 CALL PROP<"X,HBX ,T M , P, H,T , MD , V I , CP , K , TMC i T BX , MC BX, V IBX ,C PB X, KBX, N ) 127 IFJN .NE. 1) GO TO 46 128 CALL V ISC(PX, TSX, MCBX, VIBX) 129 CALL C0N0(PX, TBX, MDBX. KBX) 130 46 CALL F IND ( 0 , MFR , AH X , PX ,HBX ,TM, P, H, T , MD , VI , CP ,K ,TWX , F AI I, DS ,M) 131 T F ( F A IL . N E . 0 .0) GO TO 45 132 TFJCONT . N E . C O ) GC TO 88 133 N=10 134 IFUTWX . G T . 400.01 .OR. ( p X . G T . 82.70)1 N=l 135 CALL PROP(PX,TWX,TM,P,T,H,MD,Vl ,CP,K,TMCtHWX,MDWX,VIWX.CPWX,KWX,N) 136 IF(N . N E . 1) GO TO 49 137 CALL V ISC(PX, TUX» MQWX, VIWX) 138 CALL C.OND(PX, TWX, MDWX, KWX) 139 49 TPA=(TB+TBX ) /2.0 140 TWA=(TW+-TWX)/2.0 141 PA=(PG+PX) /2 .0 142 N=10 143 IF({TBA . G T . 400.0 ) .OR. (PA . G T . 82.70) ) N=l 144 CALL PROP(PA , T P A , T M , P , T , H , M D , V I ,CP,K,TMC,HBA,MOBA,VIBA,CPBA,KB A,N) 145 IF(N . N E . 1) GO TO 65 146 CALL VTSCCPA, T3A, MDBA, VIBA) 147 CALL CCND(.P.A, TB A, MCPA, KBA) 148 65 CONTINUE 149 N=10 150 IFUTWA . G T . 4 0 0 . 0 ) .OR. (PA . GT . 82 .70 ) ) N= 1 151 CALL PROP(PA,TWA,TM,P,T,H,MD,Vt,CP,K,TMC,HWA,MOWA,VIWA,CPWA,KWA,N) 152 IF(N .NE. 1) GO TO 67 153 CALL V ISC (PA, TWA, MCWA, V TWA) 154 CALL CONDI ".'A, TWA, MDWA, KWA) 155 67 REA=VM*D/V IB 156 PAR=(1.0—MDWA/MDBA )*G*C*(MDBA**2 )/(VM**2) 157 IF(PAR . G T . C.0005) GO TO 72 158 FACTOR=l .0 159 GO TO 75 160 7? IF( (PAP . G T . 0,0005) .AND. (PAR . L E . 0 .3 ) ) GO TO 73 161 WRITE(6,74) PAR 162 74 FORMAT!•-• , F 1 0 . 6 , • PAR IS TOC LARGE ' ) 163 GO TO 45 164 73 FACT0P=2 .15* (PAR* *0 .1 ) 165 75 E A = 1 . 0 / ( ( 1 . 8 2 * A L 0 G 1 0 ( R E A ) - l . 6 4 ) * * 2 ) 166 EF=EA* ( (V IWA/V IBA) * *0 .22) 167 EF=FF*FACTOR 168 PDN=(VM**2)* ( (EF*DL/ (2.0*MDBA*D) ) + (MCB-MDBX)/(KDB*MDBX )) 169 PDN=PDN*0.00001 170 PF=PG-PDN 171 CHECK=PX-PF 172 t FT(TEST . G F . 0 .0 ) .AND. (CHECK . G E . 0 . 0 ) ) GO TO 53 173 WRITE(6,97) PG,PX,PA,PF.CHECK,TEST 174 97 FORMAT(• ' , •PGPXPAPF • , 4 F 1 0 . 4 , ' C H E C K , T E S T • , 2 F 1 5 . 6 ) 129 175 IF(ABS(CHFCK) . L E . 0 .1 ) GO TO 88 176 IF (ABS(TEST) . L E . 0 .1 ) GO TO 88 177 PX=PG-(TEST+CHECK) 178 TEST=CHEGK 179 GO TO 44 180 53 TFST=CHECK 181 OIF=ABS(PX-PF) 182 IF{OIF . L T . C.OOi ) GO TO 88 183 PX=PF 184 GO TO 44 185 88 PG=(PX+PF)/2.0 186 TB=TBX 187 TW=TWX 188 HB=HBX 189 VIB=VIBX 190 GO TO 33 191 98 WRITE(6,22) 192 22 «=ORMAT(«-«, « * * * R F S U L T * * * J , L ( J ) , T F ( J ) , T P < J ) , P P ( J ) ' ) 193 ! F ( B T U . EO. C O ) GO TO 78 194 WRTTE<6,93) 195 93 FGRMATC ' t 'RESULTS APE IN BR TT ISH UN ITS ' ) 196 78 00 34 J = l . M 197 IFJRTU . E O . 0.0) GO TO 79 198 L ( J ) = L ( J ) / 2 5 . 4 199 T F ( J ) = T F ( J ) * 9 . 0/5.0 - 4 6 0 . 0 200 T P ( J ) = T P ( J ) * 9 . 0 / 5 . 0 - 4 6 0 . 0 201 P P U » =PP{ J ) * 1 4 . 7/1.01325 202 GO TO 39 203 79 TF ( J) =T.F ( J ) -273 .15 2 04 T P ( J | = T P { J ) - 2 7 3 . 1 5 205 39 WPITE(6,14) J , U J ) , T F ( J ) , T P ( J ) , PP (J ) 206 14 FCRMAT(' ' , I 10 , F12 .2 , • *****>', 2F 10 . 2, ' • * * * * ' , F I 5.2> 207 34 CONTINUE 208 ENTH=HB-HI 209 WATT=ENTH*MFR 210 ' WRITE16.35) ENTH 211 35 F O R M A T ( * - ' , 'ENTHALPY CHANGE ( K J / K G ) ' , F20 .2 ) 212 WRITE('6,37) WATT 213 37 FORMAT(* - * , "HEAT INPUT RATE ( K J / S E C ) ' , F20 .2 ) 214 T P d ) - T F ( J ) 215 CALL SCALE IL , M, 1 2 . 0 , LMIN, OL, I ) 216 CALL SCALECTP, M, 1 0 . 0 , TPMIN, OTP , I) 217 DC 99 1=1, M 218 TF{ I )^.(TF( I ) -TPMIN)/OTP 219 99 CONTINUE 220 CALL AX ISJO.O, 0 . 0 , 6HLENGTH, - 6 , 12 .0 , 0 . 0 , LMIN, DL) 221 CALL AX !S (0 .0 ,0 .0 ,11HTEMPERATURE,11 ,10 .0 ,90 .0 ,TPM IN ,OTP) 222 CALL LIMF(L , T P , M, 1) 223 CALL LTMEfL , TF, M, 1) 224 CALL PLOTND 225 GO TC 13 226 45 M=M-1 227 IFIM .GT . 0) GO TO 98 228 WRITE 16,25) ; 229 25 FORMAT(• — ' , ' INPUT CONOITIGN IS OUT OF RANGE. ' ) ?30 13 WRITE(6,23) 231 23 FORMA T ( • 1 ' » • •) 232 STOP 233 END 130 1 C * * SUBROUTINF FOP CALCULATING LOCAL WALL TEMPERATURE * * 2 SUBROUTINE F IND(D,MFR,AH,PG,HR,TM,P,H, T ,MD,V I ,CP ,K ,TW,FA IL ,DS ,M) 3 REAL P ( 3 0 ) , TM(30 ) , MD(30,100) , H130 .100 ) , C P ( 3 0 , 1 0 0 ) 4 REAL T ( 3 C , 1 0 C I , V H 3 0 . 1 0 0 ) , K(30,100) 5 REAL MFR, MDB, MOW, KB, KW 6 RFAL PF, PR, NUO, NUL 7 LOGICAL TRY1, TRY2, TRY3, TRY4, TRY5, TRY6 8 IF(AH . C E . 0 .0 ) GO TO 77 9 WRITE(6,79) 10 79 F O R M A T ! • - ' , ' * * * T H I S SUBROUTINE FOP THE CALCULATICN OF T H E ' , 11 1 •• HEATING PROCESS C N L Y * * * * ) 12 GO TO 13 13 77 FAIL=0.0 14 DL=D 15 A=3 .1415926*0*0 /4 .0 16 AL=3.1415926*0*DL 17 I F U P G . L E . 100.0) . AND. (PG . G E . 7 5 . 0 ) ) GO TO 74 18 WRITEI6.34) PG 19 34 FORMAT(•-•, F 1 0 . 3 , • PG IS OUT CF DATA RANGE. ' ) 20 GO TO 13 21 74 I F K H R . L E . 1600.0) .AND. (HB .GE . - 9 5 . 0 ) ) GC TO 75 22 WRITE<6,36) HB 23 36 F O R M A T C - ' , F 1 0 . 3 , • HB IS CUT CF DATA RANGE*' ) 24 GO TO 13 25 75 N=10 26 I F ( ( H « . G T . 285.0) .CR. IPG . G T . 8 2 . 7 0 ) ) N=l 27 CALL PROP(i>G,HB,TM,P,H,T,MO,VI ,CP , K , TMB , TB , MPB ,VI5 ,C.PB ,KB,N) 28 I F ( N . F O . 10) GO TO 66 29 CALL V ISC IPG, TB, MCB, VIB) 30 CALL CCMniPG, T B , MCB, KB) 31 66 R F = 4 . 0 * M F R / ( 3 . 1 4 1 5 9 2 6 * D * V I 8 ) 32 IFIRE . G T . 50CO0.O)GO TO 63 33 WRITE(6,51) RE 34 51 FORMAT! . ' - ' , F 1 0 . 0 , ' RE IS OUT CF OPERATION RANGF' ) 35 GO TO 13 36 63 PR=CPR*VIB/Ke 37 ' T P Y l = . F A L S E . 38 TRY 2 = . F A L S E . 39 TRY3=.FALSE. 40 TRY4=.FALSE. 41 TRY5=.FALSE. 42 TRY6=.FALSE. 43 FP={ 1 . 0 / 8 . 0 ) / ( ( 1.8 2*AL0Gn<RE ) - l . 64 ) * *2 ) 44 NUO=FF*RF*PR/(12.7*S0RT(FP » * ( P R * * ( 2 . 0 / 3 . 0 ) - l . 0 1 + 1.07) 45 X=OL*(PLOAT(M)-0.5) 46 F A C T O P = 0 . 9 5 + 0 . 9 5 * 1 I 0 / X » * * 0 . 8 ) 47 COO=FACTO D*NUO*KB/D 48 TW=TR+4H/(C0C*AL*DS) 49 73 I F U T W - T B t . L E . O.C) TW=TB+0.5 50 IF((TW . L E . 1500.0) .AND. (TW .GE . 275 .0 ) ) GO TO 78 51 WRITE(6,38) TW 52 38 F O R M A T ! ' - ' , F l O . 3 , • TW IS OUT OF DATA RANGE*) 53 GO TO 13 54 78 N-10 55 I FUTW . G T . 400.0) .OR. (PG . G T . 82 .70 ) ) N=l 56 CALL PROP(PG,TW,TM,P,T,H,MD,VI ,CP, K,TMW,HW,MDW,VIW,CPW,KW,N) 57 IF (N . N E . II GO TO 88 58 CALL V ISC IPG, TW, MOW, VIW) 131 59 CALL COND(PG, TW, MOW, KW) 60 88 RTB=TR/TMB 61 RTW=TVi/TMW 62 PN=0.0 63 IF IRTB . G E . 1.2) GO TO 59 64 TFJRTW . L F . 1.0) GO TO 59 65 I F U P R . G E . 0 . 8 5 ) .AND. (PR . L E . 6 5 . 0 ) ) GO TO 61 66 WRITE(6,53) PR 67 53 FORMAT( • » F 1 0 . 2 , • PR IS OUT OF OPERATION RANGE' ) 68 GC TO 13 69 61 I F ( ( ( P T B . G T . 0 . 0 9 ) .AND. (RTB . L T . 1.2) ) . A N D . 70 1 URTW . G T . 0 .09) .AND. (RTW . L T . 4 . 0 ) ) ) GO TO 59 71 WRITE(6,58) RTB, RTW 72 58 FORMAT(• 2 F 1 0 . 1 , • RTB OR RTW IS OUT OPERATIONAL RANGE . ' ) 73 GO TO 13 74 59 IFI I RT Vi . G T . 1.0) .AND. (RTW . L E . 4 .0 (1 PN = 0.2 2+0. 18*RTW 75 IF ( (RTB . G E . 1.0) .AND. (RTB . L E . 1.21) PN=PN+(5.0*PN-2.0)* (1-RTB) 76 IF ( (RTB . G E . 1.2) .OR. (RTW - L E . 1.0)) PN=0.4 77 IF (PN . N E . 0 .0) GO TO 57 78 W I T E ( 6 , 5 6 ) PN 79 56 F O R M A T ! ' - ' , 'PN IS NOT SPEC I F I ED ' , F10.4 J 80 GO TO 13 fll 57 CONTINUE 82 CPA=!HW-HB)/(TW-TB) 83 P.CP=CPA/CPB 84 I F U R C P . G E . 0 .02 ) .AND. ( RCP . L E . 4 .0 ) ) GO TO 41 85 WRITE(6,20) RCP 86 20 F O R M A T ! • - ' , F 1 0 . 2 , • RCP IS OUT CF OPERATICN RANGE' ) 87 GO TO 13 88 41 RMD=MDW/M08 89 IF(RMD .GT• 0.00) GO TO 43 90 WRITF(6,22) RMD 91 22 F O R M A T ! ' - ' , F 1 0 . 2 , ' RMD IS SMALLER THAN 0 . 0 9 . ' ) 92 GO TO 13 93 43 EP=(RMD**0.3)* (RCP**PN) 94 HA=COO*FR*AL+(TW-TBI 95 DF=ABS(HA-AH)/AH 96 IF (DE . L T . 0 .00005) GO TO 99 97 IF(TRY5 .OR. TRY6) GO TO 93 98 IF ITRY3 .OR. TRY4) GO TO 92 99 IF( (HA-AH) . G T . 0 .0) GO TO 81 100 TRY1=.TRUE. 101 IF{TRY2I GO TO 92 102 TW=TW+10.0 103 GO TO 82 104 81 TRY2=.TRUE. 105 I F (TRY ! ) GO TO 92 106 TW=TW-10.0 107 82 I F ! . N O T . TRYI .OR. .NOT. TRY2) GO TO 73 108 92 TF!1HA-AH) . G T . 0 .0 ) GO TO 83 109 TRY3=.TRUE. 110 IF ITRY4) GO TO 93 111 TW=TW+5.0 112 GO TO 84 113 83 TRY4=.TRUE. 114 IF1TRY3) GO TO 9 3 115 TW=TW-5.0 116 84 I F ! . N O T . . TRY3 .OR. .NOT. TRY4) GO TO 73 117 93 I F U H A - A H ) . G T . 0.0) GO TO 85 ' 118 TH=TWfl.0 119 TRY5=.TRUE. 120 GO TO 86 1 121 85 TW=TW-1.0 122 TRY6=.TRUE. 123 86 I F ( .NOT. TRY5 .OR. .NOT. TRY6) GO 124 GO TO 99 125 13 F A I L M 3 . 0 126 ER= 1.0 127 99 DS=ER 128 RETURN 129 END 133 1 C * * SUBROUTINE FOR INTERPOLATING TH FR MO PHYSIC A L PROPERTIES FROM THF TABLE * 2 C * * THE SUBROUTINE S fiI NT USED HERE IS AVAILABLE IN THE UBC COMPUTER LIBRARY 3 SUBROUTINF P " O P I P G , H G » T M f P , H , T , M D , V I , C P , K , T M C , T C , M D C , V TC.CPC,KC,N) 4 REAL P ( 3 0 ) , H I 3 0 , 1 0 0 ) , T ( 3 0 , l O O ) , TM(30) 5 RF AL MC (30 ,100 ) , V I ( 3 0 , 1 0 0 ) , CPC30, 1001. K ( 3 0 , 100l 6 REAL X ( 6 0 ) , Y ( 6 0 ) , 0(13) 7 RFAL T I , MOI, V I I , C P I , KI 8 REAL T F , MDF, V I F , CPF, KF 9 REAL TC , MOC, V IC , CPC, KC, TMC 10 p=4 11 IF(N . EO . 10) M=2 12 NN=35 13 IF(N . N E . 10) NN=58 14 DO 33 !=N, 30 15 CP=PG-P(I) 16 IF(OP . EO . 0 .0 ) TMC = TMI 11 17 IF(DP . L T . 0.0) TMC=( ( P G - P ( I - l ) ) / ( P ( I ) - P ! I - l ) ) ) * (TM( I ) -TM( I - l ) ) + 18 1 T M ( I - l ) 19 ! F ( D P . L E . 0 .0 ) GO TO 22 20 33 CONTINUE 21 22 DO 10 J = l , NN 22 X ( J I =H ( I , J ) 23 Y ( J ) = T ( I , J ) 24 10 CCNTINUE 25 TF= SAI NT(NN , X, Y, HG, M, 0) 26 DO 20 J = l , NN 27 Y I J ) = « 0 ( T , J ) 20 20 CONTINUF. 29 MDF= SAINT(NN, X, V, HG, M, 0) 30 DO 30 J = l , NN 31 Y ( J I = C P ( I , J ) 32 30 CONTINUE 33 CPF=SAINT(NN, X, Y, HG, M, 0) 34 IF(OP . N E . 0.0) GO TO 81 35 TC=TF 36 * MDC=MDF 37 CPC=CPF 38 IF(N .NE . 10) GO TO 88 39 81 IF (N .NE . 10) GO TO 99 40 DO 40 J = l . NN 41 Y ( J ) = V I ( I , J ) 42 40 CONTINUF 43 VIF=SAINT(NN, X, Y, HG, M, 01 44 DO 50 J = l , NN 45 Y ( J ) = K ( I , J ) 46 50 CONTINUF 47 KF=SAINT(NN, X, Y , HG, M, 0) 48 IFIOP . N E . 0.0) GO TC 99 49 VIC=VTF 50 KC=KF 51 GO TO 73 52 99 00 15 J = l , NN 53 X ( J ) = H ( ( I - 1 ) , J ) 54 Y ( J ) = T ( ( I - l ) , J ) 55 15 CONTINUE 56 TI = SAI NT(NN, X, Y , HG, M, 0) 57 DO 25 J = l , NN 58 Y ( J ) = M D ( ( I - l ) , J ) 134 59 25 CONTINUE 60 MDI = SA I NT ( NN » X, Y v HG t M, 0» 61 DO 35 J = l , NN 62 Y ( J J = C P ( ( T - l l , J ) 63 35 CONTINUE 64 CPI=SAINTIMN, X, Y, HG, M, 0) 65 IP(N . N E . 10) GO TO 66 66 DO 45 J = l , NN 67 Y ( J ) = V I ( ( I - l ) , J ) 68 45 CCNTINUE 69 VI I=SAINT(NN, X, Y, HG, M, 0) 70 DO 55 J = l , NN 71 Y ( J ) = K ( ( I — 1 ) , J ) 72 55 CONTINUE 73 KI=SATNT(NN, X, Y, HG, M, Q) 74 66 F A C T O R = ( P G - P I I - l ) » / ( P ( I ) - P ( I - l ) ) 75 TC=FACTOR*(TF-TI )+TI 76 MPC, = FACTOR* (VCF-MO II+M0I 77 CPC=F6CT0R*(CPF-CPI)+CPI 78 IF (N .NF . 10) GO TO 88 79 V IC=FACTOR*(V IF-V I I )+VI I 80 KC=FACTOR*(KF-KI)+KI 81 GO TO 73 82 88 VIC=0.0 83 KC=0.0 84 73 RETURN 85 END 135 1 C * * SUBROUTINE FOR CALCULATING VISCOSITY * * 2 SUBROUTINE VISCCPG, TG» OG, U» j 3 REAL A(5,5 I | 4 A<I,1)=0.24856612 5 A( I ,2 )=0.004894942 6 A I 2 t I) =- 0. 3 733C066 7 A (2 ,2 )= l . 22753488 8 A<3,1,=0.363854523 9 A (3 ,2 )= -C .774229021 10 Al.4,1 )=-0. 0639070755 11 A(4,21=0.142507049 12 TC=TG/304.2 13 DC=DC/468.0 14 U T = l S O R T ( T C I > * ( 1 . 0 E - 8 > * ( 2 7 2 2 . 4 6 4 6 1 - 1 6 6 3 . 4 6 0 6 8 * ( 1 . O / T C t • 15 1 4 6 6 . 9 2 0 5 5 6 * { 1 . 0 / ( T C * T C ) ) ) 16 X=0.0 17 DO 71 M=l, 4 j 18 DO 71 N=l , 2 19 X=X+A(M,N)*(DC**MI/ (TC**JFLOAT(N-t . I I I 20 71 CONTINUE 1 21 Y=EXP(X) 22 U=Y*UT 23 RETURN 24 END 136 1 C * * SUBROUTINE FOR CALCUALTING THERMAL CONDUCTIVITY * * 2 SUBROUTINE CONDIPG, TG , DGt K) 3 REAL A (5 ,5 ) 4 REAL K 5 A (1 ,1 )=1.18763738 6 A ( l ,2>=-2 .73693975 7 A(1,31=2.52042816 8 A ( 2 , l ) = - 2 . 3 0 7 7 8 4 1 4 9 A(2 , 2 » = 4 . 4 1 9 9 4 8 7 2 10 A (2 ,3 )= -0 .0915667463 11 AC3. 1 ) = 2. 61294395 12 A (3 ,2 )= -4 .0C329344 13 A ( 3 , 3 )=- i .3 5 3 4 5 3 2 4 14 Al.4,1 , = - 1 . 2 8 3 2 5 5 9 15 A(4,2)=2.13659771 16 A (4 ,3 )=0.376570783 17 A(5,11=0.219542368 18 A (5 ,2 )= -0 .402133782 19 A « 5 , 3 ) = 0 . 0 0 0 20 TC=TG/304.2 21 DC=DG/468.0 22 EXK><1.OE-6J*(57 .2860124-78.14 35192* (1 .O/TC)+ 23 I 49.18 7 1 1 8 4 * U . 0 / ( T C * T C ) I —11 .509 43 47 *( I . 0 / ( T C * * 3 .0) ) >*SQRT(TC, 24 X=0.0 25 DC 71 M=l, 5 26 DO 71 N= l , 3 27 X = X + A ( M , N i * ( C C * * M , / ( T C * * ( F L O A T t N - l ) l J 28 71 CONTINUE 29 Y = E X P ( X » 30 K=Y*EXK 31 RETURN 32 END 137 APPENDIX II. Typical Calculation of Ideal Overall Efficiency 1. Ideal Overall Efficiency of Duel-reheat Cycle Data required for calculation;: Water 1200 psia Temperature, °F 815 567.19 (saturation) 245 140.6 Water 70.3 psia Temperature, °F 567.19 302.93 (saturation) 245 140.6 Carbon Dioxide 85.5 Bar Temperature, °F 845 597.19 333.0 280.0 260.0 Enthalpy, Btu/£bm 1389.03 571.85 (liquid) 215.00 110.59 Enthalpy, Btu/£bm 1315.4 272.74 (liquid) 215.00 110.59 Enthalpy, Btu/£bm 272.70 203.72 131.18 116.05 110.26 a) Steam Generation_(steam produced by 1.0 &bm carbon dioxide) A l l carbon dioxide in temperature range of 597.19°F to 845°F is required for producing H.P. steam (see steam generation process path as shown in Figure 43). Therefore, the enthalpy balance is (1.0)(272.7 - 203.72) = x (1389.03 - 571.85) , x = 0.08441 bm H.P. steam . If H.P. steam were expanded isotropically from 1200 psia to 75.0 psia, the ideal f i n a l state enthalpy i s 1117 Btu/&bm (from Steam Enthalpy 138 Versus Entropy diagram). Therefore, for a turbine efficience of 90.0%, the actual final state enthalpy i s h f = 1389.03 - (0.9)(1389.03 - 1117) = 1144.2 Btu/£bm . Enthalpy balance for heating H.P. and L.P. steam from 302.93°F to 567.19°? by carbon dioxide i s (203.72 - 131.18) = 0.08441[(571.85 - 272.74) + (1315.4 - 1144.2)] + y(1315.4 - 272.74) , y = 0.0315 £bm L.P. steam. Enthalpy of feed water which is heated by 280°F to 333°F carbon dioxide i s calculated from (131.18 - 116.05) = 0.08441(272.74 - h ±) + 0.03150(272.74 - tu) , h. = 142.21 Btu/£bm for feed water, l [12] Heat transferred to coolant loop i s given by 1661.30 x 10 6 watt = 5.67 x 10 9 Btu/hr . Therefore, mass flow rate of supercritical carbon dioxide coolant i s MC0 2 = 5.67 x 109/(272.7 - 110.26) = 3.4905 x 10 7 Jtbm/Hr. H.P. steam flow rate i s 139 "HP = ° - 0 8 4 4 1 M c o 2 = 2.9463 x 10 6 £bm/Hr. L.P. steam flow rate i s \v = ° - 0 3 1 5 0 M c o 2 = 1.0995 x 10 6 &bm/Hr. Total L.P. steam flow rate which includes reheated steam i s = 2.9463 x 10" + 1.0995 x 10* 4.0458 x 10 6 &bm/Hr. Recuperation enthalpy balance (see Figure 44) i s (1.0995 x 10 6 + 2.9643 x 106)(142.21 - 59.71) = 1^(1105 - 70.0) , = 322500 JLbm/Hr recuperating steam. b) Ideal_Overall_Efficiency_^calcu ) From the flow rates at the inlets and outlets of the tubines, and their corresponding enthalpies, shaft work output can be calculated as WSHAFT = ^ ( h ' M ) INLET " I <h * M ) OUTLET = 2.9463 x 106(1389.03 - 1144.2) + 4.0458 x 106(1315.4) - 322500(1105) - (4.0458 x 10 6 - 322500)(1010.0) = 1.9263 x 10 9 Btu/Hr. For 100% pump efficiency, pump work for feed water circulation i s 140 W = AP • M/p = [(1200 x .144)(2.9463 x 106) + (70.3 x 144)(1.0995 x 106)]/62.4 = 8.3374 x 10 9 Ft-Abf/Hr = 1.0714 x 10 7 Btu/Hr. Therefore, turbine efficiency i s eff = (1.9263 x 10 9 - 1.0714 x 107)/5.67 x 10 9 = 33.80%. For 100% pump efficiency, pump work for coolant circulation i s W c = AP • M/p = (210 x 144)(3.4905 x 107)/31.419 =3.3595 x 10 1 0 Ft-£bf/Hr = 4.317 x 10 7 Btu/Hr. Therefore, ideal overall efficiency i s (1.9263 x 10 - 1.0714 x 10 - 4.317 x 10 ) e f f = — . 5.67 x 10 = 33.02%. 2. Ideal Overall Efficiency of Pickering Station The calculation i s based on the turbo-generator heat balance of ri21 T821 Figure 50 , and the expansion lines shown in Figure 51 . The mass flow rate at the inlet of the H.P. turbine is 5964622 Jlbm/Hr, and the flow rates at the outlet are 550833, 91503, 318836, 190, and 210 bm/Hr. 141 There i s 3050 &bm/Hr drain loss. The drain loss i s expected because of high wetness of the H.P. steam. However, i t is a small percentage of the total flow rate (0.05%). From the flow rates and their corresponding enthalpies, the power output of the H.P. turbine can be calculated and is WTTT1 = 7(h • M) . - Y(h • M) HP L^ 'inlet L outlet W__ = 1201.8(5964622 - 3050) HIT - 1083.8(5550833) - 319.42(91503) - 1117.28(318836) - 1201.8(210 + 190) = 7.6269 x 10 8 But/Hr. Similarly, the power output of the L.P. turbine stage can be calculated. The inlet flow rate is 4819224 £bm/Hr, and the outlet flow rates are (2080 - 210 - 190), 11816, 33860, 96680, 225833, 180707, 173393, and 180637 £bm/Hr. The main exhaust flow rate i s calculated to be 3914738 Jtbm/Hr based on the "no mass loss" assumption. The enthalpy of the main exhaust steam is interpolated from the expansion line at the \82 1 pressure of 1.5 in Hg (see Figure 51 . The associated steam wetness and enthalpy i s 10% and 1000 Btu/£bm respectively. Therefore, the output power of the L.P. turbine stage is equal to WLP = £ ( h * M ) i n l e t " £ ( h • M ) o u t l e t W = 1250.28(4819224) jLilr - 1201.80(2080 - 210 - 190) - 1068.37(11816) 142 - 100.79(33860) - 120.56(96680) - 1087.06(225833) - 1116.08(180707) - 1163.65(173393) - 1210.71(180637) - 1000.00(3914738) = 1.2133 x 10 9 Btu/Hr. The total shaft work i s wi™ + W T , = 1-9760 x 10 9 . For 100% pump efficiency, the pump work for feed water circulation i s Wfeed = £ ( A P ' M / p ) = (38.0 - 1.5 x 0.491)(144)(4647277)/62.4 + (700 - 38)(144)(5969862)/62.4 = 9.5197 x 10 9 Ft-Jlbf/Hr = 1.2236 x 10 7 Btu/Hr. Therefore, the turbine cycle efficiency i s 1.9760 x 10 9 - 1.2236 x 10 7 eff = 5.67 x 10 9 = 34.61%. The heavy water coolant i s circulated at a rate of 61300000 ilbm/hr, and [12] the pressure drop is (1443 - 1254) psia . For 100% pump efficiency, the pump work for heavy water coolant circulation i s 143 W = AP • M/o COOLANT ' M = [(1443 - 1254) x 144] • (61300000) 54.476 = 30625 x 10 1 0 Ft-Jibm/Hr = 3.9365 x 10 7 Btu/Hr. Finally, the overall efficiency i s (7.629 x 10 8 -f 1.2133 x 109) - (1.2236 x 10 7 + 3.9365 x 107) eff = . 5.67 x 10 = 33.94%.

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