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Properties of supercritical water oxygen mixtures Wang, Shuo 2001

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Properties of Supercritical Water Oxygen Mixtures By Shuo Wang B.Eng., Shandong University of Technology, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R M E N T S F O R T H E D E G R E E OF M A S T E R O F A P P L I E D SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T OF M E C H A N I C A L E N G I N E E R I N G We accept this thesis as conforming tojihe required^standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A September, 2001 © Shuo Wang, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T Supercritical water oxidation (SCWO) is a new process of waste treatment. The use of water and oxygen in SCWO makes it necessary to examine this mixture's thermodynamic properties, especially in the supercritical region around 25MPa. The Redlich-Kwong-Soave (RKS) and Hard-Sphere equations of state (EOS) are used to predict the fractions of the components and the constant pressure heat capac-ity of the supercritical water-oxygen mixture. The parameters in Hard Sphere EOS are adjusted for pressure of 25MPa while the parameters in the R K S EOS are taken from research papers (Abrams, D. S. and J. M . Prausnitz, AIChEJournal, Vol.21, N o . l , 116-128(1975); Dahl, S. and M . L . Michelsen, AIChE Journal, Vol.36, No.12, 1829-1836(1990)). Phase boundaries vary with pressure and temperature. For phase boundary calculations, the results of both models are better for lower pressure. At the pressure of 50MPa, the Hard Sphere EOS has more accurate results than the R K S EOS. The R K S EOS predicts the heat capacity more accurately than the Hard Sphere EOS at the pressures of 24MPa and 26MPa. At 24MPa for 2% oxygen mix-ture, the error of the peak value is about 15% for R K S EOS and about 40% for the Hard Sphere EOS. For the R K S EOS, the peak position differences with experimental results are within 2-3°C. Specific volumes are needed to verify the equations of state. A venturi was used to measure the density of the mixture. At a fixed mass flow rate, the pressure difference is proportional to specific volume. The diameter of the tube in the SCWO pilot plant is 6.2mm. The throat diameter is 3mm. Results show that the sound speed is much higher than the velocity at the throat when the mass flow rate of 0.0367kg/s is applied. The pressure difference between the inlet and the throat of the venturi can be measured by the transducer currently available in the lab. Measurements in the i i ( supercritical region are more stable than those in the subcritical region. Discharge coefficients were applied according to the water flow rates. The density predicted by the R K S EOS has better agreement to the experimental data in the supercritical region than that in the subcritical region. Excess volume is also calculated. It has a similar behavior to the heat capacity in that it reaches a maximum near the critical point. The peak varies with pressure and temperature. i i i A C K N O W L E D G E M E N T S I give my first thanks to my supervisor, Dr. Steven Rogak, for his support and guide in these two years. I appreciate he gave me the chance to work on supercritical water field. I learned a lot from him, which will make me more confident and stronger in my future life, personally and technically. I treasure the time spend together with all of the SCWO group members, for their giving me company in lab, for useful discussions and lots of jokes. I give my special thanks to Sanja Boskovic for giving me support in my work and life all the time. I had better time because of her. I thank Majid Bazargan, Mohammad Khan and Ivette Vera Perez for involving in my project. With their help and cooperation, I can complete my experiments. I also give thanks to Dr. Richard Branion for his help. I give sincere thanks to my committee members, Dr. W. Kendal Bushe, Dr. Dan Fraser, for their help. I appreciate their understanding and finishing my thesis in such a short time. My appreciation also goes to the technical support stuff in machine shop and electronic shop in the department. They were always so nice to me and gave me help whenever I wanted. I will never forget all of my friends in Vancouver, the time we spent together. iv T A B L E OF CONTENTS A B S T R A C T i i A C K N O W L E D G E M E N T S iv 1 I N T R O D U C T I O N 1 1.1 Supercritical water 1 1.2 Supercritical mixtures 2 1.3 Supercritical water oxidation 3 1.4 Objective and outline of thesis 5 2 P R E S S U R E - V O L U M E - T E M P E R A T U R E RELATIONS F R O M EQUATIONS OF STATE 9 2.1 Introduction 9 2.2 Literature review 11 2.2.1 Hard sphere theory 12 2.2.2 Square well potential 13 2.2.3 Lennard-Jones potential 15 2.2.4 Cubic EOS 16 2.2.5 Mixing rules for cubic EOS 17 2.3 The model of Redlich-Kwong-Soave EOS with M H V 2 mixing rule . . 19 v 2.3.1 Redlich-Kwong-Soave EOS 19 2.3.2 Mixing rules 20 2.4 Hard sphere equation 23 2.4.1 The repulsive term ( Prep ) 23 2.4.2 The attractive term 23 2.4.3 Mixing rule 24 3 C A L C U L A T I O N S OF P H A S E B O U N D A R Y A N D H E A T C A P A C I T Y 30 3.1 Phase boundary calculation 30 3.2 Enthalpy and constant pressure heat capacity calculation 33 3.3 Numerical implementation 35 3.4 Comparison of phase boundary from experiments and phase boundary from calculations 36 4 H E A T C A P A C I T Y A N D DENSITY M E A S U R E M E N T 45 4.1 U B C / N O R A M pilot plant 45 4.2 Instrumentation and data acquisition 47 4.2.1 Temperature measurement 47 4.2.2 Pressure measurement 48 4.2.3 Flow rate measurement • • • 49 4.2.4 Heat flux measurement 50 4.3 Heat capacity measurement 51 vi 4.3.1 Comparison of Cp from experiments to the results from calcu-lation 52 4.3.2 Temperature dependent parameters of the R K S EOS 54 4.4 Density measurement 55 4.4.1 Discharge coefficient of venturi 57 4.4.2 Cold water tests 59 4.4.3 Pure water density measurement 60 4.4.4 Water oxygen tests 63 4.4.5 Evaluation of the error of measurements 65 5 CONCLUSIONS 78 79 6 R E C O M M E N D A T I O N S 84 A C A L I B R A T I O N C H A R T S A N D T A B U L A T E D D A T A A . l Differential pressure calibration 84 A . 1.1 Test discription A.1.2 Raw data and correlation A. 1.3 Zero offset calibration A. 2 Oxygen flow calibration B E X P E R I M E N T A L D A T A B. l Test summaries B.2 Data files for the experimental runs vii B.3 Results of the experiments 102 C P R O G R A M 130 D F I L E LISTS 143 viii LIST OF FIGURES 1.1 Phase surface for a pure compound (Cansell et al., 1998) 6 1.2 Properties of pure supercritical water 6 1.3 Phase equilibrium of binary mixture (F=25MPa, calculated from R K S EOS (see Chapter 3)) 7 1.4 Schematic three-dimensional temperature-pressure mole fraction dia-gram at high temperatures for binary fluid systems like water-argon water-nitrogen etc. (Wu, et al., 1990) 7 1.5 The dependence of the molar volume VE on concentration for NaOH— H20 at 400°C (Franck, 2000) 8 1.6 SCWO system 8 2.1 Intermolecular forces 27 2.2 Square well potential 27 2.3 The fractions of water, at 25MPa with different Ao 2 keeping the other parameters constant (Hard Sphere EOS) 28 2.4 The fractions of water, at 25MPa with different ( keeping the other parameters constant (Hard Sphere EOS) 28 2.5 The fractions of water, at 25MPa with different £ keeping the other parameters constant (Hard Sphere EOS) 29 3.1 Program flow chart 39 3.2 Cp at 25MPa with 2% O2 using different temperature difference . . . 40 ix 3.3 The comparison of Cp using numerical and analytical methods using R K S EOS (2%0 2, 24MPa) 40 3.4 The comparison of Cp using numerical and derivative methods using Hard Sphere EOS (2%0 2, 24MPa) 41 3.5 Pressure-temperature curves of constant composition (isopleths) along the liquid-gas two phase equilibrium surface of the water-oxygen sys-tem with experimental points (Japas and Franck, 1985) 42 3.6 The fractions of water at 25MPa,30MPa,40MPa,50MPa (From R K S EOS) 43 3.7 The fractions of water at 25MPa 43 3.8 The fractions of water at the pressures of 23MPa and 40MPa 44 4.1 U B C / N O R A M Pilot Plant 67 4.2 Test section 68 4.3 Electrical heating schematic 68 4.4 Thermocouple distribution in test section (1) 69 4.5 Thermocouple distribution in test section (2) 69 4.6 Pressure and differential pressure measurement in test section . . . . 69 4.7 Cp of pure water at 26MPa 70 4.8 Cp at 26MPa with 2% 02 70 4.9 Cp at 24MPa with 5% 0 2 71 4.10 Cp at 24MPa and 26MPa with 2% and 5% 0 2 using RKS EOS . . . . 71 4.11 Cp at 24MPa and 26MPa with 2% and 5% 0 2 using Hard Sphere EOS 72 4.12 Cp with different exponent for r^. Elsewhere in this thesis, a' — 1 in the R K S model 72 x 4.13 Adiabatic exponent for the mixture 73 4.14 Sound speeds from R K S EOS 73 4.15 Venturi (dimension is in inches, 1 degree taper section is 2.0" long) . 74 4.16 Pressure drop for the venturi at the maximum flow (Cd =1) 74 4.17 Measurement of cold water (0.0123kg/s, 25.07MPa, 20°C) 75 4.18 Comparions of density for pure water under different flow rates (Cd = 1) 75 4.19 Comparions of density of water oxygen under different flow rates (3%0 2 ) Cd = 1, Runs 29 and 30) . . . 76 4.20 Comparions of density of water oxygen under different conditions, (Cd from Table 4.8, 0.0169kg/3, Runs 30,33 and 34) 76 4.21 Water-oxygen density measurement compared with R K S EOS predic-tion (25MPa, 8% 02, Cd from Table 4.8) 77 4.22 Excess volume of water oxygen mixtures 77 A . l Calibration data and data fit for D P T 429-5-44 91 A.2 Calibration data and data fit for transmitter of oxygen flow rate . . . 91 A . 3 Oxygen flow rate vs. voltage reading 92 B. l Temperature, pressure, pressure differential and oxygen flow rate (Run 12) 103 B.2 Cp measurement compared with R K S EOS prediction (Run 12) . . . 103 B.3 Temperature, pressure, pressure differential and oxygen flow rate (Run 13) 104 B.4 Cp measurement compared with R K S EOS prediction (Run 13) . . . 104 xi B.5 Temperature, pressure, pressure differential and oxygen flow rate (Run 14) 105 B.6 Cp measurement compared with R K S EOS prediction (Run 14) . . . 105 B.7 Temperature, pressure, pressure differential and oxygen flow rate (Run 15) 106 B.8 Cp measurement compared with R K S EOS prediction (Run 15) . . . 106 B.9 Temperature, pressure, pressure differential and oxygen flow rate (Run 16) 107 B.10 Cp measurement compared with R K S EOS prediction (Run 16) . . . 107 B . l l Temperature, pressure, pressure differential and oxygen flow rate (Run 17) 108 B.12 Cp measurement compared with R K S EOS prediction (Run 17) . . . 108 B.13 Temperature, pressure, pressure differential and oxygen flow rate (Run 18) 109 B.14 Cp measurement compared with R K S EOS prediction (Run 18) . . . 109 B.15 History of temperature, pressure and pressure differential (Run 23) . 110 B.16 Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 23, Cd = 1 - 7A3Re^0^) 110 B.17 D P measurement compared with R K S EOS model (Run 23, Cd = 1) 111 B.18 Temperature, pressure, pressure differential and oxygen flow rate (Run 26) I l l B.19 Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 26, Cd = 1) 112 B.20 D P measurement compared with R K S EOS model (Run 26, Cd = 1) 112 B.21 Temperature, pressure, pressure differential and oxygen flow rate (Run 27) 113 xii B.22 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 27, Cd = 1) 113 B.23 D P measurement compared with R K S EOS prediction (Run 27, Cd - 1)114 B.24 History of temperature, pressure, pressure differential and oxygen flow rate (Run 28) 114 B.25 Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 28,Cd = 1 - 7.43/te<-°-5>) 115 B.26 D P measurement compared with R K S EOS model (Run 28, Cd = 1) 115 B.27 Temperature, pressure, pressure differential and oxygen flow rate (Run 29) 116 B.28 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 29, C d = l - 7A3Re^°^) 116 B.29 D P measurement compared with R K S EOS prediction (Run 29,Cd = 1)117 B.30 Cp measurement compared with R K S EOS prediction (Run 29) . . . 117 B.31 Temperature, pressure, pressure differential and oxygen flow rate (Run 30) 118 B.32 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 30, Cd = 0.697) 118 B.33 D P measurement compared with R K S EOS prediction (Run 30, Cd = 1)119 B.34 Cp measurement compared with R K S EOS prediction (Run 30) . . . 119 B.35 Surface temperature in test section (Run 30) 120 B.36 Temperature, pressure, pressure differential and oxygen flow rate (Run 31) . . . . : 120 B.37 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 31, Cd = 0.697) 121 B.38 D P measurement compared with R K S EOS prediction (Run 31, Cd = 1)121 xii i B.39 Cp measurement compared with R K S EOS prediction (Run 31) . . . 122 B.40 Surface temperature in test section (Run 31) 122 B.41 Temperature, pressure, pressure differential and oxygen flow rate (Run 33) 123 B.42 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 33, Cd = 0.697) 123 B.43 D P measurement compared with R K S EOS prediction (Run 33, Cd = 1)124 B.44 Cp measurement compared with R K S EOS prediction (Run 33) . . . 124 B.45 Surface temperature in test section (Run 33) 125 B.46 Temperature, pressure, pressure differential and oxygen flow rate (Run 34) 125 B.47 Water-oxygen density measurement compared with R K S EOS predic-tion (Run 34, Cd = 0.697) 126 B.48 D P measurement compared with R K S EOS prediction (Run 34, Cd = 1) 126 B.49 Cp measurement compared with R K S EOS prediction (Run 34) . . . 127 B.50 History of temperature, pressure, pressure differential and oxygen flow rate (Run 35) 127 B.51 Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 35, Cd = 0.697) 128 B.52 D P measurement compared with R K S EOS model (Run 35, Cd = 1) 128 B.53 Comparions of Cp under different conditions . 129 B.54 Discharge coefficients 129 xiv LIST O F T A B L E S 3.1 Fractions of water in the liquid phase at the temperature of 548K and 613K 38 3.2 Fractions of water at the pressure of 23MPa and 40MPa, for liquid phase and vapor (*) phases 38 4.1 Comparison of peaks obtained from R K S EOS and experiments . . . 53 4.2 Comparison of off-peak heat capacity obtained from R K S EOS and experiments 53 4.3 Comparison of discharge coefficient (Flow rate: 0.0171kg/s) 60 4.4 The standard deviation of pressure measurement for cold water at 25MPa 60 4.5 Densities measured at different flow rates (25MPa, 395°C, Cd=l) . . 62 4.6 Acuuracy of the pressure differential measurement for cold water (Flow rate: 0.0353kg/s, 25MPa) 62 4.7 Accuracy of the pressure differential measurement for cold water (Flow rate: 0.0212kg/s, 25MPa) 63 4.8 Discharge coefficients at different flow rates 63 4.9 Comparison of pressure drop and absolute pressure at 450°C . . . . 66 A . l Calibration data for differential pressure transducer D P T 429-5-44 . . 85 A.2 Zero offset for differential pressure transducer D P T 429-5-44 at differ-ent pressure 86 A.3 Zero offset obtained with different flow rate for differential pressure transducer D P T 429-5-44 86 xv A.4 Calibration data of oxygen flow rate at lOlkPa 88 A.5 Calibration data of oxygen flow rate at 22MPa 89 A. 6 Calibration data of oxygen flow rate at 27.2MPa . . . 90 B. l Channel configuration 99 B.2 List of the runs for water and water oxygen 100 B.3 Cold water tests (June 6, 2001) 101 xvi C H A P T E R 1 INTRODUCTION 1.1 Supercritical water The existence of the critical temperature was proven by Baron C. Cagniard around 170 years ago (Clifford and Bartle, 1996). At the critical point (CP), the difference between liquid and gas disappears. Also, the first and second derivatives of pressure P with respect to volume V vanish if the temperature is kept constant: {dP/dV)T = 0 (1.1) and (d2P/8V2)T = 0 (1.2) The critical point for water is 221bar and 674K. Above the critical point, there is a supercritical area [Figure 1.1]. The transition from liquid to gas is continuous and there is only one phase which is a state between liquid and gas. Much work has been done to investigate the properties of supercritical fluids (Clifford and Bartle, 1996; Zerres and Prausnitz, 1994; Franck, 2000). Changes of properties are pronounced in the region near the critical point. Molec-ular dynamics affect the properties, and the molecular interaction must be taken into 1 account. The properties for a supercritical fluid are very different from normal liquids and gases. It has gas- and liquid-like properties. Figure 1.2 shows the behavior of the properties of supercritical water. Mass transfer is rapid in supercritical fluids. The density is higher than that of normal gases, closer to liquid. Thus, supercritical fluids are also viewed as "dense gases". Near the critical point, the viscosity decreases significantly to gas-like values. Heat capacity reaches a maximum near the critical point. The heat transfer coefficients are high due to high heat capacity. Diffusivi-ties approach those of a gas phase: the diffusion coefficient for supercritical water is around ten times as that of liquid water. Ordinary binary mass diffusion was about fifteen times faster than that at 25° C and atmospheric pressure (Butenhoff et al., 1996). 1.2 Supercritical mixtures In one-component two-phase systems, vaporization or condensation occurs when the system temperature or pressure changes. The phase equilibrium of a two-component or multi-component system is more complicate because we need to consider the changes of the fractions of each component in the liquid and the vapor phases. At fixed pressure, we can get a T — x phase diagram [Figure 1.3]. The phase curve (ABCDE) separates liquid, vapor and liquid-vapor areas. Point C is the critical point. Curve AC represents the fraction in vapor phase, and curve CE represents the fraction in liquid phase. When a system is in two-phase area, if we suppose the fraction is at B in vapor phase at certain temperature, the fraction is at D in liquid phase. Increasing the temperature causes the disappearance of liquid phase. At the critical point C, the system is in single phase and the liquid-vapor difference has disappeared. A three dimensional figure can be used to represent the relation of pressure, tem-perature and composition of the components [Figure 1.4, (Wu, et al., 1990)]. The curve T0T1T2 is an isotherm. The curve PQPIP2 is an isobar. The curve XQXIX2 is the 2 P — T diagram at constant composition. The locus of the critical point is shown by the curve C1C2. For an water mixed with non-polar substances, the critical curve in the PTx digram begins at the critical point of water and proceeds to higher pressure without any interruption [Figure 1.4]. It has a minimum temperature point ( C ) . Mixing supercritical water with small amounts of other substances alters its prop-erties. The ideal mixture rules are usually not suitable. Excess properties are used to describe the deviation of the mixture from ideal conditions, ie. aB(P,T) = am(P,T) - 5>a° (F ,T) (1.3) i where is the excess property at pressure P and temperature T; am is the property of the mixture of fraction of each component i at P and T; and a°i is the property of the pure component at the same P and T. Figure 1.5 (Franck, 2000) shows the deviation of molar volume of a water-NaOH mixture from the ideal mixture behavior, which is called excess volumes VE. It becomes smaller with increasing pressure. Excess Gibbs' energy (GE) also deceases with pressure. Higher temperature decreases the deviation of excess enthalpy from ideal condition for some methanol mixtures (Larsen, et al., 1987). 1.3 Supercritical water oxidation For a supercritical fluid mixture, solubilities change with pressure and temperature (Cansell et al., 1998). Supercritical water has a low dielectric constant allowing it to be miscible with non-polar compounds such as most organics and non-polar gases (Franck, 2000). An important application of supercritical water is Supercritical Water Oxidation 3 (SCWO). SCWO is a process to oxidize organic materials into carbon dioxide and inorganic acids. Figure 1.6 shows a SCWO reaction system. It consists of the wastewater and oxy-gen supply, preheater, reactor and cooling systems. Oxygen is mixed with wastewater before the reactor. In the reactor, the mixture is heated to supercritical state. The conversion of waste to in-organics occurs due to the high temperature. The product is separated into liquid and gas and discharged. SCWO is conducted at tempera-tures from 400°C to 650°C at 25MPa (Modell, 1990). This is an efficient process of waste treatment. The destruction efficiency can reach more than 99.999 %, allowing liquid effluents to be discharged without further treatment. Gaseous effluents can be discharged to the atmosphere. The properties of the mixture have effects on SCWO. Koo et al. (1997) tested the influence of running conditions on the oxidation of phenol. It is proved that the total organic carbon (TOC) in the liquid effluent decreases with reaction temperature, but increases with pressure. When temperature increases from 380 °C to 440 °C, the conversion of phenol increases from 11% to 99%. Higher concentrations of oxidant reduce the T O C in the effluent. The mixture of waste-water and oxygen has a weight ratio of about 95% waste and 5% oxygen. The aqueous wastes are dilute, so water and oxygen are the main parts of the mixture. The properties of the water-oxygen mixtures are essential for understanding the mechanisms of SCWO and the design of SCWO process. Also, as a new technology to treat the hazardous wastes, SCWO is required to be safe and stable. The reaction in SCWO can cause high temperatures. The oxidation reaction temperature cannot be too high, otherwise it causes system failure because of material limitations. To control the SCWO parameters of the system, we must be able to predict reaction rates of the oxidation process and the thermodynamic properties of the mixtures. Equations of state (EOS) can be used to calculate those properties. Equations of state represent the relation of pressure, density and temperature. This can be done from large amounts of experimental data. A lot of equations have 4 been proposed for different substances and systems at different pressure and temper-ature. Combined with mixing rules, an EOS can be used to calculate the density and i phase composition at a given pressure, temperature and mixture composition. From those properties, some other thermodynamic properties and transport properties can be calculated from suitable equations. A suitable EOS is needed for water-oxygen mixtures. 1.4 Objective and outline of thesis The work is focused on the simulation of the water-oxygen properties and measuring densities of water-oxygen mixtures. The constant heat capacity and the density of the mixtures were examined. Due to the use of water oxygen mixtures in SCWO, the study was mainly about finding suitable models and improving measurement in supercritical range around pressure of 25MPa. The development of equations of state is discussed in chapter 2. The available work in supercritical water oxygen mixture was reviewed. Two models of EOS, Redlich-Kwong-Soave EOS and Hard Sphere EOS, were used to calculate the heat capacity and density. Enthalpy was calculated from the equations of states. Heat capacity can be obtained from enthalpies. The numerical development is described in Chapter3. Experimental data were used to validate the two equations of state. The ex-periments of heat capacity and density were performed in the U B C / N O R A M pilot plant. The test facility is explained in Chapter 4. Test procedures and comparisons of experimental data with calculations are included in Chapter 4. In Chapter 5, the conclusions from the modeling and experiments are given. At the end, the recommendations for future work are discussed. 5 Pressure Temperature Molar volume Figure 1.1: Phase surface for a pure compound (Cansell et al., 1998) 140 120 100 80 60 20 i — Heat capacity ( kJ/kg/K ) — Specific volume ( m3/kg "10 ) - Viscosity ( Ns/m2 '1OE-6 ) Conductivity ( W/mk *0.01 ) ********if*t**i*»*tti***f ********* 200 250 300 350 400 450 Temperature (K) 500 550 600 Figure 1.2: Properties of pure supercritical water 6 700 r 1 1 1 1 1 1 r 680 -660 -W a t e r m o l e f rac t i on Figure 1.3: Phase equilibrium of binary mixture (P=25MPa, calculated from R K S EOS (see Chapter 3)) 0 Figure 1.4: Schematic three-dimensional temperature-pressure mole fraction diagram at high temperatures for binary fluid systems like water-argon water-nitrogen etc. (Wu, et al., 1990) 7 Oxygen C0 2 Water & Organics r Single Phase Reaction Heating Cooling Separation Water Ash Figure 1.6: SCWO system 8 CHAPTER 2 PRESSURE-VOLUME-TEMPERATURE RELATIONS FROM EQUATIONS OF STATE 2.1 Introduction An equation of state (EOS) represents the relation of pressure, molar volume and temperature. It can be developed from statistical mechanics or experimental data. The ratio of PV to RT is defined as compressibility factor (Z). For an ideal fluid, it equals to one. This leads to the ideal gas law: PV = RT (2.1) where R is the gas constant (8.314kJ/kmol/K). For real fluids, Z changes with temperature and pressure. It can be expressed using an infinite series (Sengers et al., 2000): B C D , o o X 1H h 1 h... (2.2) This is called a virial equation of state. In the equation, B is second virial coef-ficient, C the third virial coefficient, D the fourth virial coefficient and so on. They are temperature dependent and represent the molecular interactions. This equation 9 can be truncated for different uses. Virial coefficients can be derived from molecular theory and empirical equations. In statistical thermodynamics, the behavior of fluids is examined from a molecular view. Fluids consist of a large number of molecules. The molecules can be taken as spheres with a diameter a in continuous, random motion. The molecules travel and collide with each other. Most of the collisions are considered to involve two molecules. The average kinetic energy of the molecules doesn't change with time, but it is proportional to the temperature. At lower pressure, the distance between the molecules is much larger than the size of the molecules, so the volume of molecules is negligible. The forces of repulsion and attraction vary with the distance between molecules [Figure 2.1]. The attractive force is dominant when the distance is longer while the repulsive force is more important when two molecules are close. When the repulsive and attractive forces are equal, the potential net energy is at a minimum and the system is stable. Numerous models of molecular potential have been proposed empir-ically (Hirschfelder et al., 1964). A simple model is the Rigid Impenetrable Sphere. It gives a crude representation of the repulsive forces. It assumes that when the intermolecular distance is smaller than the diameter of the molecule, the potential is infinity; otherwise it is zero. The Square Well potential has been widely used for calculations. It represents rigid spheres of diameter surrounded by an attractive core of certain strength. An alternate model, the Lennard-Jones potential, represents both repulsive and attractive forces. In statistical thermodynamics, equations of state are obtained from the partition function and radial distribution function (Hirschfelder et al., 1964). The molecules are classified by energy levels. Level j has energy Ej. Each level has a different state (gj). The partition function is the sum of g ^ ~ E i l k T ^ at all energy levels, where T is the temperature and k is Boltzmann constant. The radial distribution function is defined as the number of molecules which are separated by a certain distance. The real reaction of the molecule is more complicated than it is assumed. For this reason, 10 empirical or semi-empirical equations are widely used for calculations. Section 2.2 reviews a selection of models in more detail. Two models were chosen to model the water-oxygen system, as described in sections 2.3 and 2.4. 2.2 Literature review It is essential to choose a proper model for a simulation. A suitable EOS should give us accurate predictions of densities and phase diagrams. It should also be used to calculate the derived thermodynamic properties and transport properties. To choose an EOS, we need to consider the composition of the mixture, the pressure and temperature. Fewer adjustable parameters will make the simulation easier and the EOS more understandable. A n EOS applied for similar conditions should be considered first because it usually has similar behavior for similar systems. For high pressure and temperature mixtures, the molecular interaction is more complicated and is no longer negligible. This makes it more difficult for modeling. The available equations and mixing rules need to be modified. Little work has been done on supercritical water-oxygen mixtures. For Vapor-Liquid Equilibrium (VLE) , a commonly used way is a cubic EOS with suitable mixing rules. The simplest cubic EOS is the van der Waals EOS. Both repulsive forces and attractive forces are considered. Other cubic equations are obtained based on this idea. A series of hard sphere-based EOS's with different potential models have been developed for high pressure systems. The hard sphere equation is considered to be a convenient model to study the behavior of dense gases and liquids (Thiele, 1963). Christoforakos and Franck (1986) developed a new EOS for supercritical water mixtures. They considered two terms of the equation, repulsive and attractive terms: P — Prep + Pattr (2-3) 11 The repulsive term is based on the hard sphere theory of the Carnahan-Starling EOS and the attractive term is taken from the square well potential. This model has been tested for binary water systems with non-polar components including water-nitrogen, water-xenon and water-carbon dioxide systems, etc. The calculations were performed under pressures lower than 250MPa. The deviation was within the mea-surement error. The water-oxygen system should be very similar to the water-nitrogen system, so this model is a logical choice for the present thesis. 2.2.1 Hard sphere theory The hard sphere model only considers the repulsive forces (Sengers et al., 2000). From this approximation, the compressibility factor becomes Z = l + Ayg{r) (2.4) where g(r) is radial distribution function. It is a function of the molecular distance r. Parameter y stands for the packing fraction of molecules. V = g — (2-5) where JV"o is Avogadro number, a is the diameter of the molecules and V is the volume of the fluid. Taking the Percus-Yevick approximation for the radial distribution, Thiele (1963) obtained an EOS: Z = PV/RT = (1 + y + y2)/(l - yf (2.6) 12 The virial equation can be written as a polynomial function of y Z = l + B2y + B3y 2 + B4y 3 (2.7) Carnahan and Starling (1969) postulated that all the coefficients are integers Bi = (i2 + i — 2) (i > 1). Then the virial equation becomes: Z = 1 + 4y + 1<V + 18y3 + 284 + 40y° + (2.8) Which gives a new equation: Z = PV/RT = (l + y + y 2- y 3)/(l - y) (2.9) 2.2.2 Square well potential The square well potential is one of the simplest forms for molecular potential (Hirschfelder et al., 1964): 4>{r) = oo r < a e a<r<ACT 0 r > a (2.10) where e is the strength of the attractive core surrounding the spherical molecule and A is the width of the square well. The distance between the molecules is r [Figure 2.2]. This model takes into account both attractive and repulsive forces. It has a 13 good agreement in the description of gases with complex molecules. From the square well potential, the second and third virial coefficient are given by: JB = 4 / 5 [ l - ( A 3 - l ) ( e £ / f c T - l ) ] (2.11) C = 2 / 5 2 [ 5 - ( A 6 - 1 8 A 4 + 3 2 A 3 - 1 5 ) ( e e / f c r - l ) - ( 2 A 6 - 3 6 A 4 + 3 2 A 3 + 1 8 A 2 - 1 6 ) ( e e / f c T - l ) 2 -(6A6 - 18A4 + 18A2 - 6)(e£/*T - 1)3](2.12) where k is the Boltzmann constant. Christoforakos and Franck (1984) used the second virial coefficient for some binary supercritical aqueous mixtures such as water-nitrogen mixture. Shmonov et al. (1993) took the second and third virial coefficients which are calculated from the square well potential for the binary system of water-methane up to 723K and 200MPa. P a U T = RT{Vl-VmC/B) ( 2 - 1 3 ) Parameter 8 is: P = lN0a3 (2.14) where N0 is the Avogadro number and a is the diameter of the molecules. This model was also used as attractive term combined with the model of hard sphere as repulsive term by Heilig and Franck (1990) for ternary water systems. The agreement with the experimental data was satisfactory. 14 2.2.3 Lennard-Jones potential The Lennaxd-Jones (LJ) potential model gives good predictions for non-polar molecules (Hirschfelder et al., 1964). It is a function of the molecular distance r: <i> = M & 2 - & ) } (2.15) where e is the maximum energy of attraction at r = 2 1 / , 6 a. The first term in the equation represents the repulsive energy, the second the attractive energy. The second virial coefficient is given as: B = ABB* (2.16) where B* is a function of (e/kT) and [r/a). It was tabulated by Hirschfelder et al. (1964). Saur et al. (1993) used L J potential to simulate the counterflow diffusion flames up to 3000MPa. The second virial coefficient (B) of the Lennard-Jones potential was taken in the attractive term: Pattr = BT^—f- (2.17) *m Thermodynamic properties of heat capacity, density, viscosity, thermal conduc-tivity and diffusion were calculated. The profiles along the frame of the properties, the temperature and concentration of each component were given. The results were reasonable for a diffusion flame. Because of the simplicity of the square well potential model, it is used for the attractive term in the hard sphere model in this thesis. 15 2.2.4 Cubic EOS Cubic equations are widely used because of their simplicity. A well known EOS is the van der Waals equation. Van der Waals proposed the equation based on the assumption that the space of molecular motion was reduced because of the volumes of the molecules. It is modeled by the constant parameter b (Sengers et al., 2000). Also, it uses a correction using parameter a for the decrease of pressure caused by intermolecular interactions. v — 0 vz where parameters a and b are determined from the relations (Reid, et al., 1987): ( 2 - 1 9 ) where T c is critical temperature and Pc is critical pressure. a is dependent on the attractive force between molecules which b is dependent on the repulsive forces. Based on van der Waals' idea, a series of cubic equations have been developed, including the Redlich-Kwong (RK) equation, which is more accurate than the van der Waals EOS. The equation of state proposed by Peng and Robinson has been found to be useful for both liquids and real gases. 16 2.2.5 Mixing rules for cubic EOS To extend an EOS to mixtures, we need to consider the change of properties caused by the fractions of each component. A mixing rule relates a mixture's parameters to the compositions and pure components' parameters. For an ideal case, we have Qm = T,XiQi (2.21) i where Qm is the mixture parameter, Q{ is the pure component parameter and X j is the fraction of species i. For all two-parameter cubic equations, the mixing rules recommended by Reid et al. are (1987): am = £ £ X i X ^ a ^ i l - kiS) (2.22) i j bm = J2xibi (2-23) i where aj, b{ are parameters for pure component and kij are binary interaction coefficients. For strongly polar compounds, this rule is no longer adequate. Equation 2.23 is extended to a quadratic to get a higher flexibility: &m = £ £ W > u (2-24) « i with bu = bi and bij = bji = (1 - hij)(bi + bj)/2, where /iy is a empirical binary constant. 17 Vidal and Huron (Dahl and Michelsen, 1990) gave a mixing rule by introducing the general expression of excess Gibbs energy by matching the excess Gibbs energy at infinite pressure. The Huron-Vidal (HV) mixing rules can be expressed as: c ln2 RT (2.25) i=i where a is the dimensionless parameter a = a/bRT, X j the mole fraction of component i and GE the excess Gibbs energy. Based on the modification of the H V mixing rule by Michelsen, Dahl and Michelsen (1990) proposed a new model (MHV2) for mixtures for high pressure systems. Com-bined with the Redlich-Kwong-Soave equation, M H V 2 mixing rule was investigated for binary aqueous systems. In the calculations, both of the universal quasi-chemical (UNIQUAC) and the universal functional activity coefficient (UNIFAC) model of ex-cess Gibb's energy were applied. The phase boundary was calculated at constant temperatures (Dahl and Michelsen, 1990). The temperature range is from 473K to 623K. In the calculations, a single set of parameters taken from low pressure and low temperature was used. This gave a good agreement for water methanol and water acetone systems. The only deviation observed was at 523K for water-ethanol mixture (around a 7% error in the H2O fraction at the phase boundaries). The two models of RKS-EOS with MHV2 mixing rule and Hard Sphere EOS with Square Well potential were used for the calculations of water oxygen mixtures. 18 2.3 The model of Redlich-Kwong-Soave EOS with MHV2 mixing rule 2.3.1 Redlich-Kwong-Soave EOS The Redlich-Kwong-Soave equation of state (RKS EOS) has been widely used. Soave improved the R K EOS to both liquids and gases, polar and non-polar compounds (Soave, 1984). with b = 0 .08664^. The attraction parameter a is modified in such a way that the vapor pressures are reproduced: RT a (2.26) P = v — b v(v + b) a = 0 . 0 4 2 7 4 8 ^ ( / (T r ) ) 2 ' with where y = -\f% + 1 and TT is reduced temperature TT = T/TC. The coefficients C 1 , C 2 , C 3 are given below (Dahl et al., 1991): For water, C, = 1.0873, C 2 = -0.6377, C 3 = 0.6345 For oxygen, Ci = 0.8252, C 2 = 0.2515, C3 = -0.7039 19 2.3.2 Mixing rules M H V 2 mixing rules were used for the calculation of water-oxygen mixture. With a reference pressure of zero to match the excess Gibbs energy, the two parameters a and b are calculated from: 6 = E ^ (2-27) «=i c c c QE qi(oLmix - ^XiOi) + g 2 (a4a - 5Zs<ai) = J2xM-r) + (2-28) i = i i = i i = i b ' ^ The coefficients 91,^2 are -0.478 and -0.0047 respectively (Dahl and Michelsen, 1990). amiX can be found by solving the above equation. Substituting amiX into the EOS, we can get the volume if P,T and x are given. Any appropriate model for the excess Gibbs energy can be used for the mixing rule. Here the U N I Q U A C model is used. In the U N I Q U A C model, the excess Gibbs function is calculated as a sum of a combinatorial term and a residual term (Abrams and Prausnitz, 1975): GE = G f + Gf (2.29) The combinatorial part basically accounts for non-ideal mixture which is caused by the differences of size and shape of constituent molecular species, while the residual part considers the difference between intermolecular interaction energies. G? = ^ £ * M * / s « ) (2-30) K 1 i=i 20 ^ t x 4 q ) M 6 i i / 9 i ) ( 2 - 3 1 ) The parameters u and 9 involve the "structure" parameters r and q (the van der Waals volume and area of the molecule relative to those of a standard segment, respectively), as explained below. The mixture is assumed to be represented by a three dimensional lattice. A molecule is represented by a number of bonded segments. Each segment occupies a lattice site in a three-dimensional lattice model. The number of segments per molecule is r. zq is the number of external nearest neighbors where z is the coordination number of the lattice. For pure components, r and \q can be calculated from: r = Vw/15.17 (2.32) Z-q = 4»/(2.5 1 Q 9 ) ( 2 - 3 3 ) where Vw and Aw are the van der Waals volumes and area of the molecules. For some components, q and r have been tabulated (Dahl et al., 1991; Abrams and Prausnitz, 1975). In this calculation, r becomes 0.9200 for water and 1.7640 for oxygen; | ^ is 1.40 for water and 1.91 for oxygen. The combinatorial factor u is the number of ways that the molecules can be arranged in space. It depends on the molecular configuration of the mixture. Abrams and Prausnitz (1975) related it to the local area and q. Larsen et al. (1991) took the modified form from Kikic: 21 where 9 is surface area factor (Abrams and Prausnitz, 1975); 9u is the fraction of local area of component i occupied by component i; and % is the fraction of local area of component i occupied by component j. Thus we have: 9u + % = 1 (2.35) Assuming all interaction energy between molecules is zero, we obtain: 0S)=ei = ((^g)ixi)/(£(^g)jxj) (2.36) 0ji=(9jTij)/(J2^mTmi) (2.37) m where the Boltzman factor r is obtained from: The interaction parameter is a function of temperature with three coefficients: T dBij - aBij,\ + aBij,2(T - T0) + a e i j ^ ^ n — + T - T 0 ) (2.38) where T 0 is the reference temperature, 298.15K. Dahl et al. (1991) took the first two terms for various binary systems up to 589K and proved that high temperature predictions are acceptable using low temperature 22 parameters. Thus the form ClBij = + ClBij,2(T — T 0) (2.39) can be used for water oxygen mixtures. For a water-oxygen mixture, O B H 2 O , 0 2 , I = 0-BO2,H2O,I = 688.7 and aBH2o,o2,2 = aB02,H2o,2 = -0.9018 (Dahl et al., 1991). 2.4 Hard sphere equation 2.4.1 The repulsive term ( Prep ) Taking Canarhan-Starling form, Christoforakos and Franck (1986) developed the re-pulsive term for a single phase as: where Vm is the volume of the mixtures, B can be calculated from equation 2.14. 2.4.2 The attractive term From the second virial coefficient of the square well potential, the attractive term of the equation is: Prep = RT(Vm + PK + ~ P3)V-l(Vm - B) - 3 (2.40) 23 Pattr = - ^ ( \ 3 - l ) [ e x p ( - ^ - l ) } (2.41) with e/k = T e /n ( l + ) where k is the Boltzmann constant, e is the strength of the attractive core sur-rounding the sphere molecule and A is the width of the square well. 2.4.3 Mixing rule The new parameters OAB^AB and XAB are introduced for mixtures. (TAB = ({a A + <TB)/2 (2.42) e = Z{eA-eByi2 (2.43) XAB = max(XA, XB) (2.44) Reliable parameters can give a better simulation of phase equilibrium. Those parameters (A,C,£, and m) can be fitted to available experimental data. Also, an empirical equation can also be used to obtain the values of the parameters. Some of data of those parameters have been tabulated. The square well width A decreases with the molecular polarities (Haar and Shenker, 24 1971). Hirschfelder (1964) suggested fitting the second virial coefficient to get A and determine the values of the parameters in Hard-Sphere equation model. For water, XH2O is 1.199 (Christoforakos and Franck, 1986). For oxygen A 0 a has been changed from 1.35 to 1.8. From Figure 2.3, we can see that decreasing Ao 2 causes an increase in the two-phase area. Ao 2 of 1.5 is taken in the model for oxygen. This value gives better phase boundary prediction at 25MPa. At high pressure, the repulsive term becomes important. A temperature depen-dent sphere diameter is used to modify the steepness of the vertical ascent of the repulsive term of the square well potential. Consider the potential of: where m is the characteristic constant for the fluids (Haar and Shenker, 1971). The influence of m on the equations varies with temperature. We can determine m by comparing the calculations from the equation of state and the experimental data at a low temperature and at densities about twice as the critical one. At high temperatures the equation of state is less dependent on m. Christoforakos and Franck (1986) used m=10. Results show that m has very small effect on the values. Thus, m = 10 is used in this thesis. We also use £ and £ by considering the interactions between the molecules. They can be calculated from: a{T2) = a{Tl){Tl/T2f1'^ (2.45) This results in: 8(T2) = BCTJiTjTifM (2.46) C = oj -1/6 (2.47) 'ra 25 (2.48) with am is given by (Hirschfelder et al., 1964): a m = [l + 0 . 8 9 2 ^ f ( ^ ) 1 / 2 ] (2.49) The subscript n is for non-polar components and the subscript p is for polar (b0)n is expressed in cubic centimeter per mole (52.26 for oxygen). Parameter t* is a function of the reduced dipole moment (1.2 for water). From equation 2.49, we have am = 1.06. In Christoforakos' new equation, a" 1 / 6 and are taken as two separate adjustable parameters, C and £, for high pressure system. Figure 2.4 shows that the phase diagram is shifted and it has a bigger effect on liquid phase. For this thesis, ( was taken as 1.095. £ doesn't change the phase diagram a lot [Figure 2.5]. It is taken as 0.65 arbitrarily. component. an is the polarizability of the non-polar component (1.60A for oxygen). 26 Distance Figure 2.1: Intermolecular forces 4 ) r Figure 2.2: Square well potential 27 0.8 5 is S 0.4 0.2 yy y 0.8 1.0 1.2 y ' y' s / s' -f y y s' y ' y y' 500 550 600 650 Temperature (K) Figure 2.3: The fractions of water, at 25MPa with different A 0 2 keeping the other parameters constant (Hard Sphere EOS) 0.8 •S 0.6 -2 0.4 r , ——"7^7 • * yy/ y y ^y •*y y 0.8 1.0 - _ 1? y, ss y * y * / s / / / / / s y s y . y y y / , s ^y s jy y y^ y \ y ^ y'/ 0 I — 1 ' ' 500 550 600 650 Temperature (K) Figure 2.4: The fractions of water, at 25MPa with different C keeping the other parameters constant (Hard Sphere EOS) 28 Student Version otMATLAB 0 I 1 1 500 550 600 650 Temperature (K) Figure 2.5: The fractions of water, at 25MPa with different £ keeping the other parameters constant (Hard Sphere EOS) Student Version of MATLAB 29 CHAPTER 3 CALCULATIONS OF PHASE BOUNDARY AND HEAT CAPACITY 3.1 Phase boundary calculation The fugacity coefficient is often used in phase equilibrium calculations. The fractions of each phase were obtained from the fugacity values of the pure component in each phase. When two phases of a fluid are in equilibrium, the molar Gibbs energies for each phase are the same. In the case of vapor liquid equilibrium, this means that GV = GL From this, the fugacity has the relation: fV = fL For a water-oxygen mixture below its critical point, there is a liquid and a vapor phase. In each phase, the composition varies with pressure and temperature. The equilibrium of the liquid-vapor mixture can be calculated using the fugacity coefficient combined with an equation of state. The fugacity coefficient is defined as *i = ^ (3-1) XiP 30 where Xi is the fraction of the component for each phase. For mixtures, each species has the same chemical potential and the same fugacity in the liquid phase as those in the vapor phase: tf = tf This gives f?=ft In an ideal mixture, the chemical potential of component i is (Hirschfelder et al., 1964) m (T, P) = tf] (T) + RTlnP + RTlnXi (3.2) where /x-°^(T) is the chemical potential of pure component i at the temperature of the mixture. From the Maxwell relations, we obtain dfii/dP = dV/drii (3.3) for a mixture, where rij is the number of moles of species i. Thus, the chemical potential of component i in a mixture of real gases is Pi{T, P) = tf\T) + RTlniPxJ + jf - — ]dP (3.4) We integrate this equation at constant temperature. In terms of fugacity, this equation becomes 31 1 f.p a y T>rr lnf.(T, P) = te(T, P) - tf(T))/RT = InP + lux, + _ j f [ _ - — ]dP (3.5) The last term is equivalent to integrating from zero to a relative pressure PQ, and then from P0 to pressure P. PQ is low enough that the mixture can be assumed an ideal gas. At this state, the volume is V^. The process going from zero to PQ can be taken to be an ideal process and the volume at PQ is infinity. ,fi(T,P)^ 1 fPdV RT ' n ( n ^ ) = ^ / ^ - - p ] d P ( 3- 6 ) where is the fugacity coefficient fa of component i. For a constant mixture composition, the change of pressure dP is (dP/dV)dV rv°o dP RT RTlnfa = Jv [-V(-^-)T,v,nj - -y-}dV - RTlnZ (3.7) Z is the compressibility factor (Equation 2.1). Reid et al. (1987) summarized the fugacity coefficients for liquid and vapor: RTlntf = f^l-V^TV,*, - ~ R T l n Z L (3-8) RTlntf = j^[-V(^-)T,v,nj - ^}dV - RTlnZv (3.9) with ZL = q£ and, 32 7v _ PVV " RT Those equations can be transformed into: RTlntf = j^i-V^U + (1 - Xi)(^-)V/RT - ± - - RTlnZL (3.10) izr/T^r=Jv~i- y(%u+(! - x ^ d £ ) v / R T - v - i r ] d v - R T l n Z V ( 3 J 1 ) The effect of pressure on phase properties is significant at high pressure. The fu-gacity coefficient is dependent on pressure at constant composition and temperature. At high pressures the critical and supercritical phenomena are difficult to represent using a simple equation. This makes the calculation of vapor liquid equilibrium diffi-cult at high pressure compared to that at low pressures (Reid et al., 1987). A suitable EOS for high pressures is needed to find the fugacity coefficients. 3.2 Enthalpy and constant pressure heat capacity calculation From the Maxwell relations, the enthalpy for simple compressible substances can be derived(van Wylen and Sonntag, 1978): dh = Cp°dT +{v- T(^f)P]dP (3.12) This derivative can be transformed to: 33 BP BP dh = Cp°dT + [„(|-)T + T(—)v]dv (3.13) The absolute enthalpy can be obtained by integrating the equation (3.13) from absolute zero to the state of given pressure and temperature. h^[cPUT + [ H ^ ) T + T { % W v (3.14) The first term is integrated from zero to the state of the given temperature and the second term is integrated from infinite specific volume to the volume at given condition. This can be taken to be an ideal gas mixture. f T Cp°dT = Y Xi f T CptdT (3.15) Jo Jo The specific heat capacities for ideal gas of water and oxygen are taken from van Wylen et al. (1993). The derivatives were calculated from the equations of state, thus, the enthalpy for each component can be found. rT „ rv°° dP dP h = }ZxiJQ CPidT-Jv [v(fo)T + T{w)v)dv (3-16) From the definition of constant pressure heat capacity (Cp): CP=(§)P (3-17) the constant heat capacity can be calculated numerically: 34 dh hi — h2 (3.18) dr Ti - r 2 where hi, h2 are enthalpies at temperatures of 7\ and T2 respectively. The constant heat capacity calculated from this equation is the average of the heat capacities between 7\ and T2. The temperature difference affects the value of Cp. So the numerical error must be considered. Saur et al. (1993) applied the equation to the mixtures to get the heat capacities. The first term is the heat capacity of ideal mixtures. The rest of the equation is the residual heat capacity which shows the deviation of heat capacity of a real gas from that of an ideal gas. The difference between the numerical (Equation 3.18) and analytical (Equation 3.19) methods are discussed later. 3.3 Numerical implementation The initial program written by Dr. S. Rogak was based on the Redlich-Kwong EOS and the simple mixing rules. This program is modified to R K S EOS model with MHV2 mixing rule and Hard Sphere EOS. The program is contained in Appendix C. The flow chart of the program is given in Figure 3.1. In order to calculate the fractions of each component, it is convenient to introduce a factor of K for high pressure vapor liquid system. Cp = Cp°(T)-R + T (3.19) 35 where yi is the mole fraction of component i in the vapor phase and Xi is the mole fraction of component i in the liquid phase. <^ f is the fugacity coefficient of component i in liquid phase and <j>Y the fugacity coefficient of component i in vapor phase. Ki is obtained by solving equations 3.11 and 3.12 for the fugacity coefficients. Then the new fractions of water (xnew, ynew) calculated from Kj are compared to the old ones to judge whether the iteration should be stopped or not [Figure 3.1]. Cp values are dependent on the temperature difference when the numerical method is applied. The lower temperature difference gives more accurate results, but requires additional computation. When the temperature difference decreases, the influence is smaller. From Figure 3.2, we can see that the peak values are almost the same when temperature differences are 0.5K and 1.0K. For Cp calculations, the temperature difference is 1.0K. Equation 3.18 and 3.19 are both used. Figure 3.3 and Figure 3.4 compare the results of these two equations. The two curves are almost the same. For the mixture with 2% oxygen the peak difference is only 2kg/kJ /K. Equation 3.18 was used in all calculations for heat capacity because it is easier to implement. 3.4 Comparison of phase boundary from experi-ments and phase boundary from calculations The phase boundary calculated from the two models was compared with the experi-mental data taken from Franck's group. 36 Japas and Franck (1985) conducted the experiments to measure the composition and specific volume of water-oxygen mixtures. The apparatus for the experiments is a high pressure autoclave with a sapphire window. The "synthetic method" was used to determine the fraction in vapor phase as described below. Water is injected to the autoclave which has been filled with pressurized oxygen. The temperature was increased. The results of the simultaneously recorded temperature and pressure showed a break point at the transition from a two-phase region to a single phase. This break point gave a value for the PTx curve. The relation of specific volume to pressure and temperature was obtained by further increasing the system pressure. By decreasing the temperature and observing the disappearance of single phase, the break points can also determined. To obtain the fraction in liquid phase, an "analytic method" was applied. Samples of the liquid phase were extracted and cooled to 77.3K. The amount of water was determined by weighting the samples. The results of this experiment cover a temperature range from 500K to 663K and a pressure range from 20MPa to 280MPa [Figure 3.5]. The specific volumes were also given when the fraction of water is lower than 0.93. Excess volume was calculated at 673K. Figure 3.6 shows the change of phase diagram with the pressure. The phase boundary only changes with pressure and temperature. The critical point predicted by the R K S EOS changes from 644K at 25MPa to 627K at 50MPa. For the Hard Sphere EOS, it changes from 648K at 25MPa to 646K at 50MPa. The comparison of phase diagram is given in Figure 3.7 and Figure 3.8 at 25MPa and 50MPa. The phase boundary predicted by the R K S EOS and the Hard Sphere EOS at 25MPa is very close to the experimental data [Figure 3.7]. Considering the measure-ment error, the calculation is acceptable. The difference in the fractions of water in vapor phase between the data from the models and experiments is bigger than that for liquid phase [Table 3.1]. When the pressure is higher, the accuracy of the model decreases. But the Hard Sphere EOS has better results [Figure 3.8]. At 40MPa and 600K, the difference is 11% for Hard Sphere EOS and 18.5% for R K S EOS. Table 3.2 shows the effects of temperature calculation. The increasing of the 37 temperature also decreases the accuracy of the calculation. This could be caused by the temperature dependent parameters. Before concluding whether or not the model performance is acceptable, predictions of density and heat capacity are considered. Temperature (K) Pressure (MPa) Experimental data R K S EOS Hard Sphere EOS 548 28.0 0.99 0.983 0.99 548 59.0 0.98 0.96 0.99 613 23.0 0.99 0.981 0.997 613 30.0 0.98 0.924 0.993 613 39.0 0.97 0.946 0.99 613 87.3 0.99 0.983 0.99 Table 3.1: Fractions of water in the liquid phase at the temperature of 548K and 613K Temperature (K) Pressure (MPa) Experimental data R K S EOS Hard Sphere EOS 623 23.0 0.99 0.98 0.994 613 23.8 0.99 0.98 0.995 593 23.0 0.99 0.98 0.997 550* 23.0 0.358 0.412 0.31 578* 40.0 0.374 0.451 0.297 600* 40.0 0.481 0.570 0.429 611.5* 40.0 0.585 0.65 0.52 Table 3.2: Fractions of water at the pressure of 23MPa and 40MPa, for liquid phase and vapor (*) phases 38 Guess component fraction x, y 1 Calculate specific volume from e qu ati on s of st at e * Calculate new fractions x ^ , y r I Calculate enthalpy, constant heat capacity, sound speed etc. Figure 3.1: Program flow chart 39 50 40 30 3 20 10 + DT=0.5K + DT=1.0K 0 DT=5.0K i * > o F X 1 1 i * + + 1 1 J V mmmm* i i 0 550 600 650 Temperature (K) 700 Figure 3.2: Cp at 25MPa with 2% 0 2 using different temperature difference 80 70 60 ~ 50 & n 40 320 340 360 380 400 Temperature (C) • Analytical * Numerical 440 Figure 3.3: The comparison of Cp using numerical and analytical methods using R K S E O S (2%0 2, 24MPa) 40 50 40 30 .... 1 r 4 <a < Numerical + Derivative ++ <« + <,.< 1 < £ + Jf1 1 1 1 _J I I 1 1 540 580 620 660 700 Temperature (K) Figure 3.4: The comparison of Cp using numerical and derivative methods using Hard Sphere EOS (2%0 2, 24MPa) 41 Figure 3.5: Pressure-temperature curves of constant composition (isopleths) along the liquid-gas two phase equilibrium surface of the water-oxygen system with exper-imental points (Japas and Franck, 1985) 42 0 500 Water Mole Fraction Temperature (K) Figure 3.6: The fractions of water at 25MPa,30MPa,40MPa,50MPa (From RKS EOS) 0.2 H H RKS EOS Hard-Sphere EOS + Experimental s / y ' \ ' s s 1 0 I 1 1 — ' 500 550 600 650 Temperature (K) Figure 3.7: The fractions of water at 25MPa 43 500 550 600 650 Temperature (K) Figure 3.8: The fractions of water at the pressures of 23MPa and 40MPa 44 CHAPTER 4 HEAT CAPACITY AND DENSITY MEASUREMENT 4.1 UBC/NORAM pilot plant The U B C / N O R A M test facility is used for heat capacity and density measurements. The pilot plant was designed for high temperature and high pressure [Figure 4.1]. It can reach temperatures of over 600°C. It consists of a regenerative heat exchanger (6.096m long), preheater 1 (4.572m), preheater 2 (4.572m), a test section (3.658m), a reactor (113m), outlet of the reactor to the heat exchanger (5.791m) and a process cooler (6.096m). A water tank is used to supply the system with distilled water. Water is pumped into the system by a high pressure, triplex positive displacement plunger pump (Gi-ant Model P 57) from the tank. The pump can provide flow rates from 0.61/min to 2.251/min. The flow rate is controlled by the pump speed which is controlled by a Re-liance Electric variable frequency drive (VFD). Since the pump can cause fluctuations in flow and pressure, a pulsation damper is used to reduce these effects. The system pressure is maintained by throttling the flow after the process cooler through a back pressure regulator (Tescom # 54-2462T24S) and a needle valve (Belimo NS24-SRS). The flow first flows through a regenerator heat exchanger. The regenerator heat exchanger is a counter-current flow, tube in tube heat exchanger with 12.7mm I.D. (Schedule 80) seamless pipes. The hot fluid flows through the outer annulus. It can 45 transfer up to 30kW to the cold flow at a design flow rate of l l /min . The Praxair PGS-45 oxygen container is used to supply liquid oxygen at pressures of 1.38-3.45MPa. Vaporized oxygen gas is compressed to about 27.58MPa in three stages in a Haskell Inc. air-driven compression system. The compression system con-sists of two boosters in series. Oxygen flow to the reactor is controlled by maintaining a constant pressure differential across needle valve. Flow is adjusted by adjusting the needle valve. Oxygen is mixed with water before the heat exchanger. The fluid is heated in the two preheaters which are electrically heated by run-ning current directly through the tube walls. A l l the data for the experiments were extracted from the test section. It consists of two longer sections (1.524m each), a short entrance part and a short exit part (0.305m each) [Figure 4.2]. In density mea-surement, the exit part was replaced with a venturi. The sections are separated by five high-pressure fittings. Heat is supplied to the two longer parts. There are some Inconel thermocouples to measure the wall temperature and bulk temperature. The reactor is composed of sixteen tubes (6.096m), a length (3.353m) connecting the test section to the reactor and a length (5.791m) connecting the reactor to the regenerative heat exchanger. The reactor can also be heated when it is necessary. Two electrical heating systems are designed to supply heat to the SCWO facility [Figure 4.3]. System 1 supplies power to the two preheaters and test section. Silicon Controlled Rectifiers (SCR) are used to regulated the power for heating. The power goes from the SCR through two step-down single-phase transformers(240/24 V A C , Hammond manufacturing ) for each of the preheaters or the test section. Each paired transformer is wired in series from the SCR panel and parallel to each heater. 2.5cm thick copper cables are used to connect the transformers to the tubes by cable clamps. The cable clamps are bolted to stainless steel rods which are silver-soldered to the tube. The grounded cable is attached at the ends. In the preheaters, a double connector is attached in the middle (24volts), while in the test section, two single cable clamps are in the middle. Each transformer can deliver 24 volts at 450amps. 46 Power to Preheater 1 and the test section is controlled manually from the SCR panel. Preheater 2 can be controlled either manually or from the feedback of the temperature. System 2 supplies power to the reactor. It is wired in a similar fashion as system 1. After the reactor, the fluid flows through the cold side of the reactor and then a process cooler. The fluid is cooled down to 40-50°C and depressurized to atmo-spheric pressure with a back pressure regulator (Tescom # 54-2162T24S). The fluid is separated into gas and liquid and recycled. A hazard analysis has been completed. A main problem is over-pressurization. Three relieve valves provide overpressure protection to the system. Each of them is set to trip at 27.6MPa and release the full pump flow rate. The system can be depressurized very quickly by dumping it to the depressurization stack via a ball valve in an emergency situation. Seven surface temperatures along the systems are monitored to provide overheating protection. The temperatures are displayed in Control Panel 2. The set point of the temperature alarms can be adjusted manually and the heat will be shut off when the alarms trip. On Control Box 1, a heater stop button can cut the power to the heater and stop the oxygen flow when it is needed and the emergency stop button will stop the pump, the power and oxygen flow. The system is enclosed with 18 gauge steel sheet to prevent high temperature sprays in the event of a leak. 4.2 Instrumentation and data acquisition 4.2.1 Temperature measurement In the SCWO test facility, all the thermocouples are K-type Chromel-Alumel with twisted shielded extension wire. A l l bulk thermocouples are sheathed and ungrounded. Initially, five bulk ther-47 mocouples were placed in the ports of the unions in the test section (B1-B5). Only three of them were used in the heat capacity and density measurements. At first, temperatures at B2, B3 and B4 were measured in Cp tests [Figure 4.4]. In the den-sity measurements, B2, B3 and B5 were used [Figure 4.5]. The bulk temperatures in preheater two were measured at the inlet and outlet of the preheater. The sampling ports also have thermocouples for bulk temperatures. A l l the thermocouples used for alarms are Inconel-sheathed, ungrounded probes and clamped onto the tube with the steel straps. In the test section, 20 top surface thermocouples and 10 bottom ones were spot welded by welding the two wires as the junction for the thermocouple. The positions of the thermocouples were given in Appendix B. 4.2.2 Pressure measurement The system pressure was monitored by an absolute pressure transducer which is located at the inlet of the test section. The pressure range of the transducer is 0 to 34.48MPa and it gives a 0 to 10 volts reading correspondingly. It was calibrated with a digital calibrator (0-52MPa) with water as a working fluid. The system was pressurized with needle valve and then depressurized. The pressure of the system and the output reading from the computer were recorded. A linear fit was used to correlate absolute pressure to voltage. The correlation obtained from the data fit is: P = 6.8119V+ 0.0444 (4.1) where P is the system pressure (MPa) and V is the voltage reading (V). This relation gives an error of 2.1% (0.7MPa of full scale). The pressure drop through the venturi was measure by a pressure differential transducer (DPT429 Validyne DP303-44) [Figure 4.6]. The range of the output sig-48 nal is from 0 to 10 volts. It can measure a pressure drop of up to 220kPa. A bypass valve (Ball valve V429) was used to protect the diaphragm of the transducer. The calibration of the transducer was performed with a hand pump (See details in Ap-pendix A . l ) . The correlation between the output signal and the pressure differential was obtained as: A P = 22.1293V (4.2) where A P is the differential pressure drop (kPa) and V is the voltage reading (V). The accuracy of the transducer measurement is +0.6%. In the operation of the experiments, fluctuations in the pressure drop were observed. Cold water tests were run to analyze the errors (These are discussed in section 4.4.2). 4.2.3 Flow rate measurement After the system is pressurized, the water flow rate is verified by measuring the flow rate at the discharge of the system using a graduate cylinder and a stopwatch before the heating and oxygen flow start. A n orifice flow meter contained in a transmitter (Foxboro E13DH-ISAM2) was used to measure the oxygen flow rate. The diameter of the orifice is 0.86mm. The pressure drop produced by the orifice can be measured by the transmitter. The span of the pressure differential transmitter is between 20 and 2O5mi/20. The output signal ranges from 4mA to 20mA. Since the data acquisition system can only accept a voltage signal, a 500f2 resistor was connected with the output of the transmitter to provide 2 to 10 volts signal. The orifice was calibrated using oxygen at a pressure of 27.2MPa (See Appendix A.2 for details). The correlation of the mass flow rate of oxygen and the voltages measured by the transmitter is expressed as: 49 m = 4 . 1 6 ( V - V 0 ) 0.5 (4.3) where m is the mass flow rate of oxygen (kg/h), V is the voltage reading (V) and V 0 is the zero offset (V). The error of the measurement is +5%. The maximum flow rate it can measure is about 12kg/h. Overloading of the transmitter can cause a shift in the zero offset and give a wrong reading of the flow rate. To avoid overload, a ball valve is used as a bypass for the transmitter. The error related to the zero offset is discussed in appendix A.2. 4.2.4 Heat flux measurement The voltage supplied to the preheaters and reactor is displayed on the SCR control panel. The voltage can only be used to approximate the heat flux on each section. In test section, the heat flux was calibrated by calculating the difference of the inlet and outlet enthalpies. The correlation of the heat flux and the voltage is expressed as (UBC SCWO System Calibration Review, 1999): where Q is the heat flux supplied in the test section (kW) and V is the voltage reading from the SCR panel (V). Q = 0.00010683794V2 - 0.01355V + 0.57175 (4.4) 50 4.3 Heat capacity measurement Heat capacity measurements were taken in U B C / N O R A M pilot plant by Dr. Rogak and S. Bosvokic. Heat capacities were examined at 24, 25, and 26MPa for water-oxygen mixtures with 2%, 3%, 5% and 8% oxygen. Constant heat flux was supplied in test sections 1 and 2. The flow is maintained in steady state. In this condition, the change in enthalpy of the fluid from the inlet to outlet of the test section can be calculated from the heat supplied to the test section. Heat losses were neglected. The differential enthalpy is the heat flux. It is assumed that the heat flux in test section 1 is the same as that in section 2. Bulk temperatures were measured. The temperature of the fluid is taken as the average of the inlet and outlet temperatures. Then the enthalpy can be calculated from with Cp being the heat capacity at the average temperature of 7\ and T 2 ; Q being the heat flux supplied to test section; Ti the inlet temperature and T 2 the outlet temperature. The results of these heat capacity measurements have been discussed by Rogak (2000) and Boskovic (2001). Heat capacities reached a maximum value at the critical point. The peak value and position are affected by oxygen concentration and pressure. The increase of oxygen reduces the peak value and shifts the peak position to the left. At higher pressures, the peak value is also lower but the peak position moves to the right. 51 4.3.1 Comparison of Cp from experiments to the results from calculation Heat capacities were calculated using the two models at the same conditions as those of the experiments. Figure 4.7 compared the heat capacity of pure water obtained from R K S EOS to the real value. The calculated peak is 1°C left of the experimental peak, but it is 24% lower. The deviation from the measured value is higher in subcritical region (30% at 370°C) than that in the supercritical region (16% at 420°C). For water oxygen mixture, the peak positions calculated from the R K S EOS are to the left of those from the experiments except for the experiments done on Dec. 8, 1999 (Runs 17 and 18). The possible reason could be the oscillation of the oxygen flow rate. The difference is within 2-3°C [Table 4.1]. The peak values are lower. At temperatures lower than the critical point, the calculated heat capacity is higher than those measured. In supercritical region, heat capacity is lower [Table 4.2]. More results are given in Appendix B. Note that Rogak [2000] reported a consistent bias in the U B C temperature measurement of about 2.5°C, but results in this thesis have not been corrected. For the Hard Sphere EOS, the heat capacities are higher than experimental data in the sub-critical region and lower in the supercritical region [Figures 4.8 and 4.9]. For the R K S EOS, the calculated results have a better agreement with the experimental data at temperatures below the critical point. The R K S EOS shows the same effects of oxygen and pressure on heat capacity as it is measured [Figure 4.10], but for the Hard Sphere EOS, oxygen increases the peak at the same pressure [Figure 4.11]. 52 Run No. Running conditions Temperature at peak (°C) Cp at peak (kJ/kg/°C) Oxygen Pressure Experimental R K S EOS Experimental R K S EOS concentration (%) (MPa) 12 2 23.9 378.8 375.9 75.0 62.3 13 5 23.9 372.7 371.9 64.5 53.4 14 5 26.1 380.7 378.9 37.2 30.6 15 2 26.1 387.0 383.9 54.1 32.1 20 2 25.2 374.9 379.9 59.0 40.4 21 5 25.2 370.7 372.9 44.9 34.3 32 3 25.3 - 378.9 - 37.2 33 3 25.3 380.2 378.9 70.4 37.2 34 7.8 25.3 371.6 368.9 31.8 39.1 35 7.85 25.3 371.1 368.9 43.3 39.1 37 7.85 23.97 369.5 366.9 65.7 48.8 Table 4.1: Comparison of peaks obtained from R K S EOS and experiments Run No. Running conditions T (°C) Cp (kJ/kg/°C) T (°C) Cp (kJ/kg/°C) o2 (%) P (MPa) Experimental R K S EOS Experimental R K S EOS 12 2 23.9 419.9 7.0 5.2 325.9 6.5 7.5 13 5 23.9 425.9 5.8 4.5 320.9 6.7 7.6 14 5 26.1 425.9 7.0 5.4 320.9 6.3 6.9 15 2 26.1 422.9 7.8 6.0 318.9 6.0 6.7 20 2 25.2 416.9 7.9 5.4 320.9 7.4 7.4 21 5 25.2 417.9 7.7 5.9 311.9 7.5 6.4 32 3 25.3 420.9 9.7 5.6 326.9 7.9 7.3 33 3 25.3 419.9 8.7 5.6 301.9 6.0 6.1 34 7.85 25.3 400.9 9.6 6.1 340.9 10.6 10.7 36 7.85 25.3 417.9 6.5 4.8 341.9 8.6 9.0 37 7.85 24.0 411.9 6.9 4.7 333.9 9.4 10.1 Table 4.2: Comparison of off-peak heat capacity obtained from R K S EOS and exper-iments 53 4.3.2 Temperature dependent parameters of the R K S EOS The temperature dependent parameters used in the R K S EOS were for lower tem-perature. To get better results for higher temperature, the parameters were changed to fit the results to the experimental data. The model contains the temperature parameter a^ - and parameters r (see Chapter 2). The first method is to increase the parameter in equation 2.29. Another form of asij was used: T dBij = aBij,\ + aBij,2(T - T 0 ) + aBij>3(Tln~) (4.6) A more effective way is to alter the expression of r^: where a' is bigger than 1. A l l three ways have similar effects. The Cp curve shifts to the right. Heat capacity is lower in the sub-critical region and it is higher in the supercritical region. This makes it closer to the measurements. But the peak value was reduced [Figure 4.12]. To get better results in most cases, a' is 1.14 for equation 4.7. Since this parameter was fitted for higher temperature, it resulted in a deviation of the phase boundary from the experimental values. In the liquid phase, the fraction of water decreased. Correspondingly, the water fraction in the vapor phase increased. Because adjustment of Tij did not give a clear improvement, the RKS model is run with a' = 1 in all other graphs and tables. In other words, no parameters have been adjusted in the R K S model, except for the run shown in Figure 4.12. 54 4.4 Density measurement An experimental measure of density is needed to compare with the results from cal-culations. A venturi was used to measure the density of the mixture. The pressure differential of the inlet and throat pressures of the venturi is a function of the flow velocity. Applying Bernoulli equation P + pgz + = constant ( 4 . 8 ) to the venturi, density is found by measuring the pressure difference across the venturi. z is constant because the level doesn't change. Assuming constant density, we obtain: A P = P , - P 2 = ^ - ^ (4.9) where AP is the pressure differential, Pi is the pressure at the inlet, P 2 is the pressure at the throat, Ux is the velocity at the inlet, and U2 is the velocity at the throat. From continuity rh = pUA = constant ( 4 -10) the velocity at the inlet and the throat can be obtained by knowing the flow rate m. The temperature at the inlet and the throat doesn't change significantly and we can assume the temperature is constant inside the venturi. Compared to the absolute 55 pressure, the pressure drop is small. Thus we can assume that the density of the flow is constant. The pressure drop can be measured by a DP cell. Considering the losses along the venturi, a discharge coefficient is introduced to correct the Bernouli's Equation (White, 1999) where rh is the mass flow rate, Cd is the dimensionless discharge coefficient, A2 is the area at the throat of the venturi, Ax is the inlet area, and p is the density. In the SCWO plant, the experiments are run at constant pressure and constant mass flow rate. The diameter of the tube in the SCWO pilot plant is 6.2mm. In the experiments, we first pressurize the system to the set pressure and then heat the water-oxygen mixture to reach the supercritical state. Thus, the mixture will change from two-phase (gas-liquid) state to single phase (gas) state gradually. The venturi was installed in the part after test section 2 without heat flux applied. The back pressure and the exit pressure are known as the system pressure. If the velocity at the throat reaches the sound speed, a shock might form. This will make the pressure inside the venturi unstable. Thus, the velocity at the throat should not be higher than sound speed. The sound speed is given (White, 1999): P = (4.11) 1/2 (4.12) c a n b e derived by: (4.13) 56 ( § £ ) T can be calculated from the equation of state. 7 is the adiabatic exponent. It is defined as 7 = 9l (4.14) 1 Cv 7 varies with the temperature. Around the critical point, it changes significantly. It can be calculate from C p - C v = -T({^)P(^)v (4.15) Figure 4.13 shows the variation of the adiabatic exponent with temperature. There are two peaks around the critical area. The peak value is high. Like the adiabatic exponent, the sound speed also has large variations in the critical area [Figure 4.14]. The lowest sound speed is in the critical area, but it is still high enough to make the flow sub-sonic in the throat of the venturi. The throat diameter was chosen as 3mm to keep the flow sub-sonic. The venturi pressure drop is higher at lower pressure [Figure 4.16] and higher temperature because the density is lower. The pressure drops are almost the same with increasing oxygen fraction except that there is a slight difference around the critical point. The pressure drop results were used to choose appropriate transducer. 4.4.1 Discharge coefficient of venturi The discharge coefficient has a great effect on the density measurement. It is af-fected by different factors, such as Reynold number, pressure taps, flow rate, venturi shapes etc (Upadhyay, 1993). It has already been proven that the discharge coefficient doesn't change with inlet pressure. As the inlet diameter increases, the discharge co-57 efficient also increases and becomes asymptotic. Correlations were given to describe the relationship between these factors and discharge coefficient. A simple relation of Cd as a function of the dimension of a venturi is given as (White, 1999): Cd = 0.9858 - 0.196^ i M . 5 (4.16) where B' is the ratio of the throat diameter to the inlet diameter. The correlation is valid in the Reynolds number range of 1.5 x 105 to 2 x 106. The calculated discharge coefficient of the venturi is 0.978. The discharge coefficient changes rapidly in laminar flow while it becomes nearly constant when flow is turbulent. Hall (Upadhyay, 1993) gave the equations of Cd only related to the Reynolds number at the throat. Shapiro (Upadhyay, 1993) introduced the length parameter of the venturi V = L+Leg, where L is the length of the cylindrical portion and Leq is the equivalent length of the conical cylindrical portion. The discharge coefficient can be approximated by: Cd = 1 - G.92Red05(laminar) (4.17) Cd = 1 - 0.184JReJ0"2{turbulent) (4.18) Cd = 1 (4.19) where JAL is friction factor 13.74 (4.20) 58 and D is the inlet diameter. Considering other parameters like roughness and geometrical parameters, Upad-hyay developed another equation for discharge coefficient CA \ *%f* + (1 - B'4) + £*£^ (4.21) with friction factor ^eq ~ \lnn ( e' >i 4 - 5.74 1 2 (4.22) where 9 is inclination of venturi from horizontal (0 for this case), D is the inlet area, D 2 is the throat diamter, Deg is the equivalent diameter of the conical portion, X is the distance from first tapping to down stream, U2 is velocity at the throat, and e' is roughness (0.002mm, typical value for new stainless steel). Table 4.3 compares the discharge coefficent calculated using the equations for pure water. The form Cd = 1 — aReb is used in this thesis (Section 4.4.3). 4.4.2 Cold water tests Testing the deviation of transducer measurement The fluctuation of the flow rate causes a shift in the transducer reading. Cold water tests were performed to get the noise level of the transducer. The flow rate was kept constant at a pressure of 25MPa. The temperature of the fluid was 20°C. To reduce the fluctuations in the flow, the pressure after the pump was increased to AOOOpsi by partly closing the valve right after the pump in the water feeding line. This was done 59 Temperature (°C) Reynolds Number (x lO 4 ) Hall Upadhyay Shapiro 20 0.0723 0.960 0.990 0.537 300 0.767 0.975 0.994 0.754 310 0.795 0.975 0.995 0.757 320 0.826 0.976 0.995 0.760 330 0.863 0.976 0.995 0.763 340 0.905 0.976 0.996 0.767 350 0.955 0.976 0.996 0.771 360 1.02 0.977 0.997 0.777 370 1.12 0.977 0.997 0.784 380 1.34 0.978 0.998 0.797 390 2.08 0.980 0.999 0.827 400 2.25 0.980 0.9997 0.832 410 2.31 0.980 0.9998 0.834 420 2.33 0.980 0.9998 0.835 430 2.34 0.980 0.9999 0.835 440 2.34 0.980 0.9999 0.835 450 2.33 0.980 1.000 0.835 Table 4.3: Comparison of discharge coefficient (Flow rate: 0.0171kg/s) at different flow rates. Flow rate (kg/s) Pressure (MPa) Standard deviation of pressure (%) 0.0373 24.88 14.6 0.0123 25.07 24.0 Table 4.4: The standard deviation of pressure measurement for cold water at 25MPa In both cases, the oscillation of the pressure differential is very significant [Figure 4.17]. When the velocity is higher, the accuracy of the measurement becomes higher. 60 4.4.3 Pure water density measurement Experiment procedure Pure water tests were performed to calibrate the venturi. During the test, the distilled water was heated to get the needed bulk temperature at the outlet of the test section. It was assumed that it has the same heat capacity along the test section. Since the heat flux was the same in test section 1 as that in test section 2, the temperature difference in test section 1 and test section 2 were assumed to be the same. Also, we assume it is isothermal in the unheated part. That is, the temperature measured by the thermocouple at the end of test section (B5) is used as the temperature of the fluid through the venturi. The experiments were run at a flow rate of 0.0353kg/s to 0.0169kg/s. Heat fluxes were applied to the preheaters, the test section and the reactors to bring the temper-ature of the fluid in the venturi up to 450°C. Results and discussion Oscillations at sub-critical temperatures is very significant. It becomes smaller when the fluid goes to a supercritical state. The variation of flow rates has a great effect on measurements. When the flow rate is higher, the density measured is closer to the real value in the supercritical region [Table 4.5]. At a flow rate of 0.0353kg/s, it's almost the same as that at temperatures above the critical region. In the subcritical region, the density measured decreases with flow rates. The oscillation is small at higher flow rates [Figure 4.18]. The transition region from subcritical to supercritical has good agreements with data taken from N B C / N R C steam table (Haar et al., 1984). The discharge coefficient was calculated for each running condition. It has a peak near the critical point [Table 4.6, 4.7]. Since the mass flow rate is constant, increasing the temperature increases Reynolds number. The discharge coefficient was fitted to 61 an equation at the flow rate of 0.0212kg/s: Cd = 1 - 7.43i?e {- 0 5 ) (4.23) Applying this correlation to measurements with flow rates of 0.0212kg/s and 0.0227kg/s gives a good agreement. For the flow rate of 0.0169kg/s, it gives higher dis-charge coefficient (0.95 compared to 0.697 in supercritical region). While at 0.0353kg/s, it gives lower one (0.953 compared 1 in supercritical region). The fitted values of Cd and the original values measured in the pure water experiments are shown in Figure B.54 in Appendix B. The discharge coefficients for each running condition are listed in Table 4.8. Flow rate (kg/s) Experimental density (kg/m 3) Error (%) 0.0353 179.27 3 0.0227 164.08 5 0.0212 176.41 11 Table 4.5: Densities measured at different flow rates (25MPa, 395°C, Cd=l) 4.4.4 Water oxygen tests Procedure Water-oxygen tests were performed using the same procedure as the pure water test described above. The heat flux in the test section was kept constant during the experiments. Different oxygen concentrations and water flow rates were applied. Ramping was used in preheater 2 to increase the outlet temperature from 350°C to 410°C in 30 or 40 minutes. The experiments were run at 24 and 25MPa for the mixtures with 3% and 7.8% oxygen. The effects of flow rate are also examined. 62 Temperature Density (kg/m3) Density (kg/m3) Ratio of Cd (•co (From NBS/NRC table) (From experiments) densities 350 625.7 499.24 0.798 0.893 355 608.7 471.85 0.775 0.880 360 589.4 454.45 0.771 0.878 365 567.2 437.95 0.772 0.879 370 540.4 426.91 0.790 0.889 375 505.2 404.70 0.801 0.895 380 450.02 376.8 0.837 0.915 385 314.86 319.9 1.016 1.008 390 216.82 198.17 0.914 0.956 395 184.58 179.27 0.971 0.985 400 166.63 175.54 1.053 1.026 405 154.41 155.99 1.010 1.005 410 145.24 155.71 1.072 1.035 Table 4.6: Acuuracy of the pressure differential measurement for cold water (Flow rate: 0.0353kg/s, 25MPa) Results and discussion Figure 4.19 compares the densities of 3% oxygen obtained at water flow rates of 0.0169kg/s and 0.0212kg/s. The flow rate increases the densities when the discharge coefficients taken to be 1. This is the same as that for pure water. Fluctuation is still significant at temperatures below critical region. I t becomes stable in supercritical region. The discharge coefficients were the same as those for pure water according to the water flow rate [Table 4.8]. Increasing the oxygen concentration in the mixture decreases its density [Figure 4.20]. This effect is significant in the critical region and becomes smaller when the temperature is higher. The density calculated from the RKS EOS was compared with measurements. Figure 21 gives the density measured comparing to the results from the RKS EOS at the pressures of 25MPa for 8% oxygen. The estimated deviation is 17%. I t was reduced to 0.5% at a temperature of 400°C. More results are given in Appendix 63 Temperature Density (kg/m3) Density (kg/m3) Ratio of Cd (°C) (From N B S / N R C table) (From experiments) densities 355 608.7 536.88 0.882 0.939 360 589.4 500.81 0.850 0.922 380 450.02 411.56 0.915 0.956 390 216.82 232.39 1.072 1.035 395 184.58 164.08 0.889 0.943 400.1 164.19 144.93 0.883 0.940 410 145.24 128.11 > 0.882 0.939 420 131.93 115.73 0.877 0.936 425 126.82 116.28 0.917 0.958 Table 4.7: Accuracy of the pressure differential measurement for cold water (Flow rate: 0.0212kg/s, 25MPa) Flow rate (kg/s) Discharge coefficient 0.0353 1.0 0.0227 1 - 7.43itel-0-6> 0.0212 1 - 7.43fle<-°-5> 0.0169 0.697 Table 4.8: Discharge coefficients at different flow rates B. The results were closer at higher temperature except the experiments on June 5, 2001. The agreement is worse in the subcritical and critical region. The predicted values are lower. The difference increases when the temperature decreases. The excess volume was calculated from: VE = Vm - xVH2o - yVo, (4.24) with: VE being the excess volume, Vm the specific volume of the mixture, x the fraction of water, y the fraction of oxygen, VH2O the specific volume of water, and Vo 2 the specific volume of oxygen. Figure 4.22 gives the excess volume of water oxygen mixture with 3% and 7.8% 64 oxygen at the pressure of 24MPa and 25MPa. The excess volume is a maximum in critical region. Peaks vary with oxygen fraction and pressure. Increasing the pressure decreases the peak value and shifts it to the right. The temperature difference where peaks occur is around 3°C from 24MPa to 25MPa. The peak at 25MPa is 1/3 of that at 24MPa. When the oxygen concentration was increased to 7.8% from 3%, the peak increased almost 50%. 4.4.5 Evaluation of the error of measurements The main contribution to the measurement error was the error in differential pressure through the venturi. The zero offsets had been tested in different ways (Appendix A) . It makes slightly differences for each case. The error caused by the zero offset needs to be considered. The estimated error of transducer is 2.1%. Another error associated with the density measurement is the oxygen flow rate measurment(5%). Since density is related to mass flow rate and differential pressure, the total error in density measurement can be calculated from the correlation (Wheeler et al., 1996): where wp is the uncertainty of density measurement, wm is the uncertainty of mass flow rate and w&p is the uncertainty of pressure drop measurement. The estimated error of the density measurement is 10% in the range of 350 to 450°C. Since density changes with pressure, the assumption of taking the system pressure for the venturi introduces an error. The pressure drop through the venturi is very small compared to system pressure [Table 4.9]. It gives an additional 0.5% error. The heat losses and changes of the flow state in the venturi result in a change of temperature. The error caused by the isothermal assumption can affect validation of the prediction for mixtures in the critical region. The temperature error is within 65 2.5°C. Flow rate Oxygen Pressure Pressure drop D P / P (kg/s) concentration (%) (MPa) (kPa) (%) 0.0367 2.0 25.0 122.475 0.490 0.0367 5.0 25.0 123.489 0.494 0.0361 2.34 25.1 117.96 0.470 0.0183 7.82 25.3 30.431 0.120 Table 4.9: Comparison of pressure drop and absolute pressure at 450°C 66 Figure 4.1: UBC/NORAM Pilot Plant 67 Union 1 Union 2 / Heated Sections Union 3 \ Union 4 Union 5 Barrel Connectors Figure 4.2: Test section Preheater One Preheater Two Test Section Parallel • Series 240 ground rr>rr\ 240 rlvJ SCR Figure 4.3: Electrical heating schematic 68 B2 SI S10 B3 S l l S20 B4 SB! SB 20 Figure 4.4: Thermocouple distribution in test section (1) B2 SI S10 B3 Sll S20 B5 SB1 SB9 SB20 Figure 4.5: Thermocouple distribution in test section (2) DPMI H By pass valve Figure 4.6: Pressure and differential pressure measurement in test section 69 Figure 4.7: Cp of pure water at 2 6 M P a Figure 4.8: Cp at 2 6 M P a with 2% 0 2 70 80 60 * RKS EOS x Hard Sphere EOS + Experimental 550 600 650 700 Temperature (K) Figure 4.9: Cp at 24MPa with 5% 02 500 550 600 650 Temperature (K) 800 Figure 4.10: Cp at 24MPa and 26MPa with 2% and 5% 02 using R K S EOS 71 45 -40 -35 -30 • o S> 25 20 15 • 10 5 i i i i i i i i i i u i m 0 -I , , 1 , , 1 . 1 340 350 360 370 380 390 400 410 420 Temperature (C) [•Experimental A a'=1.0 »a'=1.1 -a'=1,3~ Figure 4.12: Cp with different exponent for r^. Elsewhere in this thesis, a' = 1 in the R K S model 72 600 650 700 750 Temperature (K) Figure 4.13: Adiabatic exponent for the mixture 1600 1200 800 400 I I + 24MPa 2%02 < 28MPa 2%02 + 28MPa 5%02 550 600 650 700 Temperature (K) Figure 4.14: Sound speeds from R K S EOS 73 W.1180 Figure 4.15: Venturi (dimension is in inches, 1 degree taper section is 2.0" long) 550 600 650 700 Temperature (K) Figure 4.16: Pressure drop for the venturi at the maximum flow (Cd = 1) 74 • • 2 2.5 Time (minutes) Figure 4.17: Measurement of cold water (0.0123kg/s, 25.07MPa, 20°C) 600 500 400 300 200 100 350 360 370 380 390 400 410 420 430 440 450 Temperature (C) « 0.0212kg/s <• 0.0353kg/s Figure 4.18: Comparions of density for pure water under different flow rates (Cd 75 600 500 * , 0 -I 1 1 1 — 1 1 1 1 > 360 370 380 390 400 410 420 430 440 Temperature (C) I > 0.0169kg/s . 0.0212kg/s Figure 4.19: Comparions of density of water oxygen under different flow rates (3%0 2, Cd=l, Runs 29 and 30) 600 -r 100 0 4 , , 1 . . 1 • 350 360 370 380 390 400 410 420 Temperature (C) | • 8% 02, 24MPa » 8% 02, 25MPa * 3% 02, 25MPa | Figure 4.20: Comparions of density of water oxygen under different conditions, (Cd from Table 4.8, 0.0169kg/3, Runs 30,33 and 34) 76 0 -I 1 1 1 1 1 1 1 1 ' 1 350 360 370 380 390 400 410 420 430 440 450 Temperature (C) |» RKS EOS A Experimental Figure 4.21: Water-oxygen density measurement compared with R K S EOS prediction (25MPa, 8% 02, Cd from Table 4.8) 0.005 0.0045 0.004 » 0.0035 n i. 0.003 v E 3 O > in 0.0025 0.002 0.0015 0.001 0.0005 350 360 370 380 390 400 Temperature (C) 410 420 • 24MPa, 3% 02 * 25MPa, 7.8% 02 » 25MPa, 3% 02 430 Figure 4.22: Excess volume of water oxygen mixtures 77 CHAPTER 5 CONCLUSIONS Based on the calculations using the R K S model and the Hard Sphere model and the experiments, the following conclusions are made: 1. For the phase boundary calculation, both of the two models can give good predictions around 25MPa . The Hard Sphere E O S is better for high pressures. Tem-perature and pressure decrease the accuracy of the predictions. 2. The experiments of heat capacity are acceptable. The R K S E O S can be used for heat capacity calculations. Because of the influence on temperature, different parameters need to be applied to get good agreements at temperatures below and above the cri t ical point. The use of parameters for higher temperatures increases the error in phase boundary calculations. 3. Densities of water oxygen mixtures were measured by a venturi. The data for the supercritical region is acceptable. Because it is a two-phase flow, the densities measured in subcritical region are not reliable. If the flow rate is higher, the density measurement is closer to the known value. The error of the measurement in super-crit ical region is 10%. The calculated excess volume shows that the peak is affected by oxygen and pressure as heat capacity. 4. The R K S E O S was used to predict densities. It has a good agreement wi th ex-perimental data in the supercritical region. Bu t the difference in the subcritical region is large. The predicted value is much lower than that obtained from experiments. 78 C H A P T E R 6 R E C O M M E N D A T I O N S Since the models cannot give good predictions for both lower and higher tempera-tures around critical region using the same parameters, a new relation for temper-ature dependent parameters is needed. The effect of oxygen concentration of the thermodynamic properties needs to be considered. This could be done by fitting the experimental data to those parameters. Another solution is to use other equations of state. The effects of pressure on the calculations are noticeable. Although for phase boundary calculations at higher pressure the Hard Sphere model is better, the error is still large. It is necessary to modify the models when applied high pressure. For density measurements, the data in the two-phase regions may show the in-fluence of two-phase flow. The influence of vapor on the pressure is important. A detail study on the phenomena needs to be performed to get better measurements. It has been proven that higher flow rates give better measurements. But the run fpr pure water with the flow rate of 0.0353kg/s, the density is slightly higher than known value. This resulted in a discharge coefficient higher than 1. Even though it is in the range of the error, it will be necessary to find the suitable flow range for density measurements. Also, to improve the density measurements, the effects of a zero offset in the pressure transducer for a pressure drop in the venturi and the orifice flow meter needs be reduced during the experiments. It is necessary to check the zero offset and redo the calibrations periodically. 79 REFERENCES Abrams, D.S. and J . M . Prausnitz, "Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems", AIChE Journal, Vol.21, N o . l , 116-128(1975). Butenhoff, T.J . , M . G . E . Goemans and S. J . Buelow, "Mass Diffusion Coefficients and Thermal Diffusivity in Concentrated Hydrothermal NaNC>3 Soltions", J . Phys. Chem., Vol.100, No.14, 5982-5992(1996). Boscovic, S, "Measurements of Constant-Pressure Heat Capacity And Convective Heat Transfer Coefficient of Water-Oxygen Mixtures at Near Critical Condition", M.A.Sc. Thesis, Mechanical Engineering Department, U B C , (2001). Cansell, F., S. Rey and P. Beslin, "Thermodynamic Aspects of Supercritical Fluids Processing: Applications to Polymers and Wastes Treatment ", Revue de L'institut Francais du Prtrole, Vol.53, NI , 71-98(1998). Carnahan, N.F. and K . E . Starling, "Equation of State for Non-attracting Rigid Spheres", J. Chem. Physics, Vol.51, No. 2 July, 635-636 (1969). Christoforakos, M . , and E . U . Franck, "An Equation of State for Binary Fluid Mixtures to High Temperatures and High Pressures", Ber Bunsenges. Phys. Chem., 90, 780-789 (1986). Clifford, T. and K . Bartle, "Chemical reactions in supercritical fluids", Chemistry & Industry, Vol.17, 449-452(1996). 80 Dahl, S. and M . L . Michelsen, "High-Pressure Vapor-liquid Equilibrium with a UNIFAC-Based Equation of State", AIChE Journal, Vol.36, No.12, 1829-1836(1990). Dahl, S., A . Fredenslund and P. Rasmussen, "The M H V 2 Model: A UNIFAC-Based Equations of State Model for Prediction of Gas Solubility and Vapor-Liquid Equilibria at Low and High Pressures", Ind. Eng. Chem. Res., Vol.30, 1936-1945(1991). Franck, E .U . , "Supercritical Water", Proc. of the 13th International Conference on the Properties of Water and Steam, P.R. Tremaine, P .G. Hi l l , D.E. Irish and P.V. Balakrishnan, N R C Research Press, 22-34(2000). Haar, L. and S.H. Shenker, "Equation of State for Dense Gases", J. Chem. Physics, Vol.55, No. 10 November, 4951-4958(1971). Haar, L . , J.S. Gallagher and G.S. Kell , NBS/NRC Steam Tables Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units, McGraw-Hill International Book Company, (1984). Heilig, M . , and E . U . Franck, "Phase Equilibria of Multicomponent Fluid Systems to High Pressures and Temperatures", Ber. Bunsenges. Phys. Chem., Vol.94, 27-35(1990). Hirschfelder, J.O., C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liq-uids, John Wiley & Sons, Inc., New York, (1964). Japas M . L . and E . U . Franck, "High Pressure Phase Equilibria and PVT-Data of the Water-Oxygen System Including Water-Air to 673K and 250MPa", Ber Bunsenges. Phys. Chem., Vol.89, 1268-1274(1985). Koo, M . , W. K . Lee and C. H. Lee, "New reactor system for supercritical water oxidation and its application on phenol destruction", Chem. Eng. Sci., Vol.52, No.7, 81 1201-1214(1997). Larsen, B. L. , P. Rasmussen and A . Fredenslund, "A Modified UNIFAC Group-Contribution Model for Prediction of Phase Equilibria and Heats of Mixing", Ind. Eng. Chem. Res., Vol.26, 2274-2286(1987). Modell, M . , "Treatment of Pulp M i l l Sludges by Supercritical Water Oxidation", Modell Development Corparation, Massachusettes(1990). Reid, R.C. , J . M . Prausnitz and B.E . Poling, The Properties of Gases and Liquids, McGraw-Hill Book Company, New York, (1987). Rogak, S., " Measurements of The Constant-Pressure Heat Capacity of Water-Oxygen Mixtures at Near-Critical Conditions ", Proc. of the 13th International Conference on the Properties of Water and Steam, P.R. Tremaine, P.G. Hi l l , D.E. Irish and P.V. Balakrishnan, N R C Research Press, 149-156(2000). Saur, A . M . , F. Behrent and E . U . Franck, " Calculation of High Pressure Counterflow Diffusion Flames up to 3000bar", Ber. Bunsenges. Phys. Chem. Vol.97, No. 7, 900-908 (1993). Sengers, J .V. , R .F . Kayser, C . J . Peters and H . J . White, Jr, "Equations of State for Fluids and Fluids Mixtures", IUPAC, (2000). Shmonov, G. , R . J . Sadus, and E . U . Franck, "High-Pressure Phase Equilibria and Su-percritical pVT Data of the Binary Water + Methane Mixture to 723K and 200MPa", J. Phys. Chem., Vol.97, 9054-9059(1993). Soave, G. , "Improvement of The Van der Waals Equation of State", Chem. Eng. Sci., Vol.39, No.2, 357-369(1984). 82 Thiele, E., "Equation of State for Hard Spheres", J. of Chem. Physics, Vol.39, No.2 July, 474-636(1963). Upadhyay, M . "On Discharge Coefficient of Venturi Meter", Power Handling &: Pro-cessing, Vol.5, No.3 Sept., (1993). Wheeler, A . J . and A.R . Ganji, Introduction To Engineering Experimentation, Pren-tice Hall, Englewood Cliffs, New Jersey (1996). White, F . M . , Fluid Mechanics, W C B McGraw-Hill, 4th Edition (1999). Wu, G. , M . Heilig, H . Lentz and E . U . Franck, "High Pressure Phase Equilibria of Water-Argon System", Ber. Bunsenges. Phys. Chem., Vol.94, No.7, 27-35(1990). van Wylen, G.J . , and R .E . Sonntag, Fundamentals of Classical Thermodynamics, John Wiley & Sons, Inc., New York, (1978). van Wylen, G.J . , R .E . Sonntag and C. Borgnakke, Fundamentals of Classical Ther-modynamics, John Wiley & Sons, Inc., New York (1993). Zerres, H . and J . M . Prausnitz, "Thermodynamics of Phase Equilibria in Aqueous-Organic Systems with Salt", AIChE Journal, Vol.40, No.4 Apri l , (1994). 83 APPENDIX A CALIBRATION CHARTS AND TABULATED DATA A.l Differential pressure calibration A. 1.1 Test discription The calibration of the D P cell D P T 429 with a diaphragm 5-44 ( D P T 429-5-44) was porformed with air as working fluid. The diaphragm was designed for the pressure drop range of 220kPa. A hand pump (Omega, P C L - 2 H P ) was attached to the positive (+) pressure port to produce the pressure difference. The negative (-) port was open to the atmosphere. The differential pressure was read on the portable manometer (Omega, PCL-200) and the voltage output from the carrier demodulator was read from the computer display. The zero offset and the span were adjusted for zero and maximum differential pressure. Zero voltage corresponded to zero voltage. 10V output corresponded to 220kPa. A. 1.2 Raw data and correlation The calibration data are given in Table A . l . A linear correlation between pressure and voltage was obtained: 84 A P = 22.1293V Voltage (V) Differential pressure (kPa) 0.0 0.0 0.8992 20.02 1.7978 40.01 2.7014 59.7 3.6206 80.1 4.518 100.2 5.4272 120.1 6.3273 140.0 7.238 160.1 8.1579 180.0 9.069 200.0 9.9957 220.0 Table A . l : Calibration data for differential pressure transducer D P T 429-5-44 A.1.3 Zero offset calibration The system pressure had a significant effect on the zero differential pressure offset voltage. It is necessary to find the zero offset at the pressures of the experiments. Three different ways were applied here. A simple way is to open the bypass valve of the D P cell. From this way, the zero offset was found [Table A.2]. The voltage of the reading from the transducer is linear to the square of the mass flow. A linear correlation between the voltage and the square of the mass flow is used to find the zero offset: V = km2 + VQ (A.2) 85 By closing the bypass valve of DPT429, the DP cell voltages were recorded at different flow rates of cold pure water with a given pressure. The flow rate ranges from 0.01789kg/s to 0.03733kg/s. These data (flow rate of cold pure water and voltage read from the computer) were fit to the correlation at the pressure of latm, 24.24MPa, 25MPa and 26MPa. The zero offsets were obtained by setting the flow rate as zero. Pressure (MPa) Zero offset (V) 0.101 -0.0253 24.24 -0.2015 25.0 -0.168 26.0 -0.1799 Table A. 2: Zero offset for differential pressure transducer DPT 429-5-44 at different pressure Even if the bypass is open, the flow can cause the variation of the readings of the pressure differential. Another experiment was done at lower flow rate to reduce this effect. During the experiment, the flow rate was lowered as less as possible by closing the bypass valve V7 of water feeding at the pressures of 24MPa, 25MPa and 26MPa. The flow rate could not be zero to maintain the system pressure. Flow rate (1/min) Pressure (MPa) Zero offset (V) 0.441 24.18 -0.14836 24.97 -0.15225 26.03 -0.16613 0.2056 24.10 -0.14602 25.02 -0.15725 26.00 -0.17147 close to zero 23.99 -0.14046 25.02 -0.14773 25.99 -0.16495 Table A.3: Zero offset obtained with different flow rate for differential pressure trans-ducer DPT 429-5-44 From the above, we can see that there is slightly change of the zero offset with pressure and flow rate [Table A.3]. For the experiments, the zero offset at the pressure 25MPa is taken as -0.14773V and at the pressure of 24MPa, it is -0.14V. 86 A.2 Oxygen flow calibration A Foxboro E13DH-ISAM2 Transmitter is used to measure the oxygen flow rate. It contains a orifice ( 0.034in ) and pressure differential measurement across the orifice. It gives 4mA to 20mA output signal. Since the data aquisition system only accept voltage signal, a 500O resistor is used to get 2-10V output from the transmitter. A 24 volts power supply is used to provide the power for the transmitter. From Bernouli's equation, we have ^ = kAP (A.3) P Since AP linear to the output signal, it becomes ^ = k'(V - Vo) (AA) P where Vo is the zero offset. k' can be found by fitting data to the correlation. The transmitter were calibrated at three different pressures: atmospheric pressure (lOlkPa), 22MPa and 27.2MPa. At lOlkPa and 27.2MPa, a bubble meter was used to measure the outlet flow rate. In the former case, nitrogen was used as working fluid. At 27.2MPa, oxygen was used. A t 22MPa, the mass flow rate was measured by an omega mass flow meter (Range 0-5V). Nitrogen was used as working fluid. Raw data and correlation 87 The calibration data were listed in Table 4 to 6. At each pressure, the coefficient k was found. It changes from 0.0627 at latm to 0.047 at 22MPa, then to 0.048 at 27.2MPa. At the pressure of 27.2MPa, the mass flow rate can be calculated from: m = 4.16(V - V0)1/2 (A.5) where m is mass flow rate (kg/h), V is output voltages (V) and V0 is zero offset of the transmitter (V). Output of transmitter (V) Bubble flow meter reading (1/min) Flow rate (kg/h) V- V0 m2/density 1.9970 0.0 0.0 0.0 0 2.0584 0.8384 0.0780 0.0614 0.0039 2.1557 1.261 0.1173 0.1587 0.0089 2.2173 1.531 0.1424 0.2203 0.0131 2.2482 1.638 0.1523 0.2512 0.0150 2.3131 2.833 0.1705 0.3161 0.0187 2.4190 2.100 0.1953 0.4220 0.0246 2.5363 2.415 0.2246 0.5393 0.0325 2.7376 2.852 0.2652 0.7406 0.0454 2.8764 3.122 0.2903 0.8794 0.0544 3.0491 3.408 0.3169 1.0521 0.0648 3.2181 3.696 0.3437 1.2211 0.0762 3.6250 4.227 0.3931 1.6280 0.0997 3.9624 4.701 0.4372 1.9654 0.1233 4.3432 5.160 0.4799 2.3462 0.1486 4.9134 5.752 0.5349 2.9164 0.1846 5.0370 5.896 0.5483 3.0400 0.1940 Table A.4: Calibration data of oxygen flow rate at lOlkPa Operation procedure It was observed that the overload of the transmitter will change the relation of mass flow rate and the voltage reading. A ball valve was used as bypass to protect the transmitter. When the oxygen is pressurized and depressurized, the bypass valve 88 Output of Omega flow meter Flow rate transmitter (V) reading (V) (Wh) V-V0 m2 / density 2.0600 0.0 0.0 0.0 0.0 2.0756 0.203 0.5891 0.0156 0.0015 2.1277 0.310 0.8996 0.0677 0.0035 2.1827 0.409 1.1869 0.1227 0.0061 2.2660 0.512 1.4858 0.2060 0.0096 2.3122 0.602 1.7470 0.2522 0.0132 2.3977 0.701 2.0343 0.3377 0.0179 2.5503 0.805 2.3361 0.4903 0.0236 2.6796 0.912 2.6466 0.6196 0.0303 2.7766 1.010 2.9310 0.7166 0.0372 2.9467 1.117 3.2415 0.8867 0.0455 3.1009 1.215 3.5259 1.0409 0.0538 3.2585 1.297 3.7639 1.1985 0.0613 3.5547 1.454 4.2195 1.4947 0.0771 3.9368 1.589 4.6113 1.8768 0.0920 4.4958 1.795 5.2091 2.4358 0.1175 5.0186 1.972 5.7227 2.9586 0.1418 5.8211 2.214 6.4250 3.7611 0.1787 6.5829 2.417 7.0141 4.5229 0.2130 7.5668 2.695 7.8209 5.5068 0.2648 8.3414 2.823 8.1923 6.2814 0.2905 9.3874 3.038 8.8163 7.3274 0.3365 Table A.5: Calibration data of oxygen flow rate at 22MPa should be open. Close the bypass valve when oxygen is needed. The needle valve in the oxygen line is used to adjust the oxygen flow rate. The oxygen flow causes mechanic forces on the transmitter. This reduces the zero offset. Also, the zero offset changes with the pressure. It is necessary to readjust the zero offset to 2V before the oxygen flow starts to get more accurate measurement. 89 Output of transmitter (V) Bubble flow meter reading (1/min) Flow rate (kg/h) v-v0 m2/density 1.9945 0.0 0.0 0 0 2.0035 6.203 0.4943 0.0009 0.0007 2.0213 8.902 0.7093 0.0268 0.0014 2.0494 12.40 0.9880 0.0549 0.0027 2.0753 14.84 1.1825 0.0808 0.0039 2.0822 15.28 1.2175 0.0877 0.0041 2.0912 15.99 1.2741 0.0967 0.0045 2.1093 16.99 1.3538 0.1148 0.0051 2.1315 18.21 1.4510 0.1370 0.0058 2.1346 19.64 1.5649 0.1401 0.0068 2.1461 20.29 1.6167 0.1516 0.0072 2.1709 22.34 1.7801 0.1764 0.0088 2.1958 23.86 1.9012 0.2013 0.0100 2.2229 23.21 2.0087 0.2284 0.0129 2.2638 27.13 2.1617 0.2693 0.1755 Table A.6: Calibration data of oxygen flow rate at 27.2MPa 90 240 Voltage (V) Figure A . l : Calibration data and data fit for D P T 429-5-44 2.5 2 0.5 0 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Transmiter output (V) Figure A.2: Calibration data and data fit for transmitter of oxygen flow rate 91 0.3 V-VJ Figure A.3: Oxygen flow rate vs. voltage reading 92 APPENDIX B EXPERIMENTAL DATA .1 Test summaries 1. Date: Sept. 2, 1998 (Runs 1-6) Objectives water-oxygen and ethanol test. Operation Pure water was run first. At 12:44, water-oxygen mixture sampled from in-termediate for 10 minutes. Because of the problem of the pump, ethanol was stopped during 1:00 to 1:14. Observation and comments Surface temperature for the test section and reactors are around 10°C lower than bulk temperature. 2. Date: March 10, 1999 (Runs 7-10) Objectives Pure water and water-oxygen heat transfer test. Operation Different heat fluxs were applied in the test section. Observation and comments Steam leak was observed. When the oxygen system was pressurized, pressure 93 stopped increasing after 2800psi. But the booster was running 1 cycle/min. Leak check was performed. No substantial leak was detected. 3. Date: June 17, 1999 (Runs 11-16) Objectives water-oxygen heat capacity test. 4. Date: Dec. 8, 1999 (Runs 17-18) Objectives Heat capacity test for water-oxygen. Operation In the test section, heat flux was kept 300V. In Preheater 1, 300V power was applied. Ramping was used in Preheater 2. Observation and comments When the ramping stopped, the temperature at the Ph2 outlet was only 378°C not 400° C. So, the data acquisition was continued for a while until the temper-ature got to 400° C. In the second case (02TALL.txt) , the test was interrupted by oxygen fluctua-tion. It was repeated and saved as the same file. 5. Date: June 19, 2QQ0 (Runs 19-22) Objectives The test section was unheated for the heat loss tests in the test section. Operation No heat flux applied in the test section. Heat capacity and heat transfer coef-ficient data were obtained. Observation and comments At 13:03, the surface temperature tripped the alarm. Pump speed was increased 94 to cool the system. 6. Date: May 24, 2001 (Run 23) Objectives This was the first experiments using venturi to measure density. The test was to measure the density of pure water at the pressure of 25MPa to calibrate the venturi. The venturi was planned to be heated up to 450°C. Operation The flow rate was initially set at the pump speed of 1600rpra. Power supply in Preheater 2 was broken. Because there was not enough heat flux to get the desired temperature, flow was reduced to 1200rpm at 12 : 53. At 1:02, flow was reduced to lOOOrpm. The experiment was done at 2:49. The inlet pressure of the feeding was set to 4000psi (P/102) by partly closing V9. Full power was used in Preheater 1, the test section and reactors. Observation and comments The offset of the transducer for pressure drop across the venturi changed from low pressure to high pressure. The fluctuation of the flow effected the pres-sure differential reading. To reduce the fluctuation, valve V9 was closed a little to increase the pressure of the feeding to 4000psi. The power on Preheater 2 was broken. The outlet temperature in the test section is lower than desired (450°C). The flow rate is decreased to 1200 rpm then to 1000 rpm. 7. Date: June 5, 2001 (Runs 24-27) Objectives The test was to get the density and heat capacity of pure water and water-oxygen mixtures. Operation The pump speed was set as 800rpm and 1600rpm. For water-oxygen tests, 3% 95 and 5% oxygen were applied. The power in test section was 300V. Ramping was used in preheater 2. Observation and comments The bypass valve of DPT429 had leakage. A thermocouple was attached at the surface of the pressure tap at the inlet of the venturi. It was observed that the temperature increased very slowly (maximum 31°C) and the leakage was very little. The oxygen flow rates from gas flow meter were also recorded. The zero offset of the transmitter for oxygen flow rate changed from 2V at the start to 1.9033V when oxygen flow stopped. 8. Date: June 6, 2001 Objectives Cold water tests for zero offset of DPT429. Operation The experiments were run at the pressures of 0.1, 24, 25 and 26MPa with flow rates from 800rpm to 1600rpm. Data were taken with the bypass valve open. 9. Date: June 20, 2001 (Run 28-35) Objectives The test was to get the density and heat capacity of pure water and water-oxygen mixtures at different pressure, flow rate and oxygen concentration. Operation In the test section, heat flux were kept constant for each running. Different heat (370V, 300, 200V) flux was applied to check the heat transfer effect on heat capacity for water-oxygen test with 8% oxygen. To get the steady change of temperature, ramping was applied. The range of temperature is between 340°C and 400°C. The ramping time was 30 minutes or 40 minutes. Observation and comments Water feeding valve V9 had leakage. This gave higher oxygen percentages than 96 expected. The zero offset of the transmitter for oxygen flow rate changed from 2V at the start to 1.81V when oxygen flow stopped. The oxygen flow rates from gas flow meter were also recorded. During the running of 370V, 1200rpm, 3600psi, 5%02, the oxygen flow was not stable. It is observed that the temper-ature at the outlet of preheater 2 was always around 10°C"s difference (lower when it was heating, higher when it was cooling) with the set point when ramping stopped. When the ramping was set from high temperature to low temperature, the heat flux in preheater 2 would increase first, this resulted the increase of the temperature. The pressure gauge near V15 was reading 200psi higher than the real pressure. Calibration of system pressure was done after the experiments. The bulk thermocouple in the middle of the test section was broken. Oxygen flow rate discussion The running time of the oxygen flow may have an effect on the zero offset shifting. During the calibration, it has around 0.03V change in a few minutes. On June 5 experiments, the change was 0.1V in 2 hours. On June 21, the change was 0.2V in 6 houres. This might be caused by mechanical reasons. Typically, when the voltage reading is about 2.6V for the oxygen transducer, the error caused by the zero shift is about 3.8%. 97 B.2 Data files for the experimental runs 9 8 k .s 2 0. x -0. -Ell 3 OI B.CU EQ 1 a. 10-, E S = a. C u Q « f S 10 5 g K S-0-C4 ( B-S: E 8 ^ 5 £ l a ; as to 1 cn cn - oi 2 ' f cn S O .So. a. i N £ CC h •5 B c -s E K5-k .s •5 E h o O CM X 0. o CO 3 a. i i 99 CM CO s O 43 > o o CO oT CO EE I 5 CO to rH a o 5.-• + -45 eo co <D CO 1 os a> ' N M n ( i o o ' o o a. i cn -a -4-3 c3 G c3 ii a> c3 >H CO a CU CO CN m • C4 eo r !Efo * J5. 3 e +3 cj 3 cn O 00 CO CO CO CO CM CO T—1 B o 43 > CO Ci CO CO CM CO 43 CU O cu is o CJ S3 c tS 43 u •v a> T3 ja a o s C3 43 s o Tj< CO 3: CO CJ rH tS O > 4 3 o CO „ CO CO a o CO rH CO s ~ o ° > «r? O CO O rH " E CO o « cfcl s > 4 3 CM o c i rH <Q •° rH 43 S3 to a 3 cj B O T3 a o CJ bO B "S B 3 (H CJ 43 3 ta o CJ 43 T3 CJ bO a O o rH U 6 c3 CM CJ CJ a S CJ CU 43 60 S O CJ cj cS "O CJ T3 cS o B cj lH cS in CM T3 B tS -tf CM B .o 43 a e s (-1 o OH CM CM CJ 43 (S > CO Oi CM 100 Running conditions Run File (.txt) Pressure (MPa) Water flow rate (rpm) 1 8-W-26 26 800 2 8-W-25 25 800 3 8-W-24 24 800 4 8-w-l 0.101 800 5 9-W-26 26 900 6 9-W-25 25 900 7 9-W-24 24 900 8 9-w-l 0.101 900 9 lO-w-26 26 1000 10 lO-w-25 25 1000 11 lO-w-24 24 1000 12 10-w-l 0.101 1000 13 ll-w-26 26 1100 14 ll-w-25 25 1100 15 ll-w-24 24 1100 16 11-w-l 0.101 1100 17 12-W-26 26 1200 18 12-W-25 25 1200 19 12-W-24 24 1200 20 12-w-l 0.101 1200 21 13-W-26 26 1300 22 13-W-25 25 1300 23 13-W-24 24 1300 24 13-w-l 0.101 1300 25 14-W-26 26 1400 26 14-W-25 25 1400 27 14-W-24 24 1400 28 14-w-l 0.101 1400 29 15-W-26 26 1500 30 15-W-25 25 1500 31 15-W-24 24 1500 32 15-w-l 0.101 1500 22 16-W-26 26 1600 24 16-W-25 25 1600 35 16-W-24 24 1600 36 16-w-l 0.101 1600 Table B.3: Cold water tests (June 6, 2001) 101 B.3 Results of the experiments 102 0 10 20 30 40 50 Time (minutes) » Oxygen flow rate » Pressure T I - — — T 2 T3 | Figure B . l : Temperature, pressure, pressure differential and oxygen flow rate (Run 12) 80.00 -| 0.00 -I . 1 —' — ' 1 — ' 320.00 340.00 360.00 380.00 400.00 420.00 440.00 Temperature (C) | • Test section 1 » Test section 2 * RKS EOS | Figure B . 2 : Cp measurement compared with RKS EOS prediction (Run 12) 103 30 Jf 1 5 O B L g. 10 lit in £ 5 Q L 10 15 20 Time (minutes) 25 Oxygen flow rate Pressure T1 30 T2 T 500 i- 450 - 400 -- 350 -• 300 ture (C) - 250 perai - 200 Tem - 150 - 100 -• 50 35 Figure B.3: Temperature, pressure, pressure differential and oxygen flow rate (Run 13) 70.00 60.00 50.00 %, 40.00 2 a. 30.00 o 20.00 10.00 0.00 -3-ilk * ' " " 0 = 1 . 1 1 , ; 320.00 340.00 360.00 380.00 400.00 Temperature (C) 420.00 440.00 • Test Section 1 * Test Section 2 - RKS EOS Figure B.4: Cp measurement compared with R K S EOS prediction (Run 13) 104 0 10 20 30 40 50 Time (minutes) • Oxygen flow rate — Pressure * T1 • T2 Figure B.5: Temperature, pressure, pressure differential and oxygen flow rate (Run 14) • • i# s * * f _ , , 1 — 1 1 r— 320 340 360 380 400 420 440 Temperture (C) I • Section 1 • Section 2 » RKS EOS I Figure B.6: Cp measurement compared with R K S EOS prediction (Run 14) 105 30 a 25 • Oxygen flow rate—Pressure * T1 T2 500 450 10 20 30 40 50 60 70 80 Time (minutes) Figure B.7: Temperature, pressure, pressure differential and oxygen flow rate (Run 15) 300 320 340 360 380 400 Temperature (C) 460 • Section 1 « Section 2 * RKS EOS Figure B.8: Cp measurement compared with R K S EOS prediction (Run 15) 106 Time (minutes) • Pressure » T1 —T2 Figure B.9: Temperature, pressure, pressure differential and oxygen flow rate (Run 16) 0-1 1 , , r- , , 330 350 370 390 410 430 450 Temperature (C) |—RKS EOS » NBSflJRC Steam Tables * Experimental Figure B.10: Cp measurement compared with R K S EOS prediction (Run 16) 107 0 5 10 15 20 25 30 35 40 Time (minutes) Pressure » Oxygen flow rate » T1 T2 Figure B . l l : Temperature, pressure, pressure differential and oxygen flow rate (Run 17) (0 . 1 ,— 1 1 1 ' 280 300 320 340 360 380 400 420 Temperature (C) • sectionl • section2 » RKS EOS Figure B.12: Cp measurement compared with R K S EOS prediction (Run 17) 108 30 o> 25 t 20 o c c a> s? 1 5 O t? Q. S 10 3 (A £ 5 a -• -• - --w « 0 2 4 6 8 10 Time (minutes) 12 14 16 18 — Pressure * Oxygen flow rate » T1,C » T2, C 450 400 300 _ o 200 a E 150 *" 100 50 0 Figure B.13: Temperature, pressure, pressure differential and oxygen flow rate (Run 18) 300 320 340 360 Temperature (C) 380 400 420 • sectionl ° section2 » RKS EOS Figure B.14: Cp measurement compared with R K S EOS prediction (Run 18) 109 500 Time (minutes) • Pressure - Pressure differential Tout Tin Tmid Figure B.15: History of temperature, pressure and pressure differential (Run 23) 800 700 600 « 500 5 400 </> c S 300 200 100 0 370 380 390 400 410 420 430 440 450 460 Temperature (C) • Experimental <* NBS/NRC Steam table Figure B.16: Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 23, Cd=l- 7.43i2e ( _ 0- 5 )) 110 60 * • 370 380 390 400 410 420 430 440 450 460 Temperature (C) | » Experimental « RKS EOS Figure B.17: DP measurement compared with R K S EOS model (Run 23, Cd = 1) 0 10 20 30 40 50 60 70 80 Time (minutes) » Pressure * Pressure differential « Tout Tin Tmid Figure B.18: Temperature, pressure, pressure differential and oxygen flow rate (Run 26) 111 700 600 500 o | 400 g 300 200 100 350 360 370 380 390 Temperature (C) 400 410 420 • Experimental » NBS/NRC Steam table Figure B.19: Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 26, Cd = 1) 100 90 80 70 60 to a. & 50 OL ° 40 30 20 10 0 4> jc « « X 350 360 370 380 390 Temperature (C) 400 Experimental » RKS EOS 0^ 410 420 Figure B.20: DP measurement compared with R K S EOS model (Run 26, Cd = 1) 112 10 15 20 Time (minutes) 25 » Pressure Oxygen flow rate • Pressure differential Tout » Tin Tmid Figure B.21: Temperature, pressure, pressure differential and oxygen flow rate (Run 27) 800 700 600 | 500 400 200 100 350 *4 H^" • • • " " X . o u , , , , , , , , , , , 360 370 380 390 400 Temperature (C) 410 420 430 Experimental • RKS EOS Figure B.22: Water-oxygen density measurement compared with R K S EOS prediction (Run 27, Cd = 1) 113 120 350 360 370 380 390 400 Temperature (C) 410 420 430 • Experimental » DP (kPa) Figure B.24: History of temperature, pressure, pressure differential and oxygen flow rate (Run 28) 114 100 350 360 370 380 390 400 410 Temperature (C) 420 430 440 450 Experimental » NBS/NRC Steam table Figure B.25: Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 28,Cd = 1 - 7A3Re^0^) 50 45 40 35 _ 30 n a. £ 25 o. a 20 15 10 5 0 300 320 340 360 380 400 Temperature (C) • Experimental » RKS EOS 420 440 Figure B.26: DP measurement compared with R K S EOS model (Run 28, Cd = 1) 115 0 -I 1 -r- 1 i 1 1- o 0 10 20 30 40 50 60 Time (minutes) Pressure Oxygen flow rate • Experimental Tout Tin Figure B.27: Temperature, pressure, pressure differential and oxygen flow rate (Run 29) 700 _ — 600 0 -I 1 1 1 1 r— 350 370 390 410 430 450 Temperature (C) [* Experimental » RKS EQS~ Figure B.28: Water-oxygen density measurement compared with R K S EOS prediction (Run 29, Cd = 1 - 7.43fle(-°-5)) 116 60 50 ... 350 370 390 410 Temperature (C) 430 Experinnerrtal » RKS EOS 450 470 Figure B.29: DP measurement compared with RKS EOS prediction (Run 29,Cd 50 45 40 35 O 30 ^ 2 5 O 20 15 10 300 JU. 320 340 360 380 400 Temperature (C) Experimental * RKS EOS 420 440 460 Figure B.30: Cp measurement compared with RKS EOS prediction (Run 29) 117 40 35 _ 30 n — 4 « i . ra «" - 25 I f ! | g | 20 jj 5 c £ S S £5 °" 10 „ *TJ —• - -- --1 1 10 20 30 40 Time (minutes) 50 500 450 400 350 300 250 a 200 | 150 100 50 0 o 1 60 « Pressure - Oxygen flow rate » Pressure differential Tout Tin Figure B.31: Temperature, pressure, pressure differential and oxygen flow rate (Run 30) 600 500 i 4 *li m 350 360 370 380 390 400 410 420 430 440 450 Temperature (C) •> RKS EOS A Experimental Figure B.32: Water-oxygen density measurement compared with R K S EOS prediction (Run 30, Cd = 0.697) 118 350 360 370 380 390 400 410 Temperature (C) 450 Experimental « RKS EOS Figure B.33: DP measurement compared with R K S EOS prediction (Run 30, Cd = 1) 300 320 340 360 380 Temperature (C) 400 420 • Experimental * RKS EOS 440 Figure B.34: Cp measurement compared with R K S EOS prediction (Run 30) 119 500 30 Time (minutes) • SB1 » S10 S11 •: S20 Figure B.35: Surface temperature in test section (Run 30) 10 20 30 40 Time (minutes) 50 Pressure Oxygen flow rate + Experimental Tout •Tin Figure B.36: Temperature, pressure, pressure differential and oxygen flow rate (Run 31) 120 500 450 400 4 350 o g 300 | 250 '«> § 200 Q 150 100 50 i » * t 1 * *4 ' 4 ^ A 1 4» • " " • • • m m . ) 350 360 370 380 390 400 410 420 430 Temperature (C) > RKS EOS » Experimental Figure B.37: Water-oxygen density measurement compared with R K S EOS prediction (Run 31, Cd = 0.697) 40 • 35 30 25 I? o. «- 20 o. Q 15 10 5 0 350 360 370 380 390 Temperature (C) 400 Experimental « RKS EOS 410 420 Figure B.38: DP measurement compared with R K S EOS prediction (Run 31, Cd = 1) 121 300 320 340 360 380 Temperature (C) 400 Experimental » RKS EOS 420 440 Figure B.39: Cp measurement compared with R K S EOS prediction (Run 31) 15 20 Time (minutes) -SB1 • S10 S11 S20 Figure B.40: Surface temperature in test section (Run 31) 122 500 S | 3 3 E o w -a c £ ^ » °- s B £ o o. 10 15 20 Time (minutes) 25 30 Pressure Oxygen flow rate — Pressure differential » Temperature 3 18 a. £ Figure B.41: Temperature, pressure, pressure differential and oxygen flow rate (Run 33) 700 600 500 400 £ 300 Q 200 100 • H • * 350 360 370 380 390 400 Temperature (C) 410 420 430 440 Experimental « RKS EOS Figure B.42: Water-oxygen density measurement compared with R K S EOS prediction (Run 33, Cd = 0.697) 123 350 360 370 380 390 400 Temperature (C) 410 420 430 • Experimental » RKS EOS Figure B.43: DP measurement compared with R K S EOS prediction (Run 33, Cd 45 40 35 30 % 25 o 15 10 5 -J..it 1 ^ 340 350 360 370 380 390 Temperature (C) 400 410 • Experimental « RKS EOS 420 Figure B.44: Cp measurement compared with R K S EOS prediction (Run 124 10 15 20 Time (minutes) 25 -SB1 S10 -S11 S20 30 Figure B.45: Surface temperature in test section (Run 33) 20 30 40 Time (minutes) Pressure » Oxygen flow rate • Pressure differential Tout Tin | Figure B.46: Temperature, pressure, pressure differential and oxygen flow rate 34) 125 500 450 400 350 n E 300 | 250 to a 150 100 50 0 ~i—i • 1. A " ' "*fl"*"1"1 •I"'L.(M.!M»**W»., 350 360 370 380 390 400 410 420 430 Temperature (C) » RKS EOS » Experimental Figure B.47: Water-oxygen density measurement compared with R K S EOS prediction (Run 34, Cd = 0.697) 45 40 35 30 ? 25 S 20 15 10 350 360 370 380 390 Temperature (C) 400 Experimental » RKS EOS 410 420 Figure B.48: DP measurement compared with R K S EOS prediction (Run 34, Cd = 1) 126 70.00 60.00 0.00 A 1 1 1 1 ' ' 300 320 340 360 380 400 420 Temperature (C) • Experimental » RKS EOS Figure B.49: Cp measurement compared with R K S EOS prediction (Run 34) 0 10 20 30 40 50 60 Time (minutes) |-~~— Pressure » Experimental Tout • • • • Tin Figure B.50: History of temperature, pressure, pressure differential and oxygen rate (Run 35) 127 700 600 500 co 1,400 g 300 200 100 • t» t . • . • • * J • * • - . V * . .*.«!> 350 360 370 380 390 Temperature (C) 400 410 . Experimental » NBS/NRC Steam table Figure B.51: Pure water density measurement compared with the densities taken from N B S / N R C steam table (Run 35, Cd = 0.697) 320 330 340 350 360 370 380 Temperature (C) Experimental » RKS EOS | 390 400 410 Figure B.52: D P measurement compared with R K S E O S model (Run 35, Cd = 1) 128 Figure B.53: Comparions of Cp under different conditions 1.2 0.8 •o o 0.4 A « ^ » : .** A r- • X 0 -\ 1 0.00E+O0 5.00E+03 1 00E+04 1 — 1 1.50E+04 2.00E+04 1 1 2.50E+04 3.00E+O4 Re • 0.0353kg/s » 0.0212kg/s A 0.0227kg/s * 0.0169kg/s Cd=1-7.43ReA(-0.5) - - Cd=0.697 Cd=1 Figure B.54: Discharge coefficients 129 A P P E N D I X C P R O G R A M 130 The main program * * + * * * + * * + + * + * + + * + + + + + + + Tir + + + * + + * + * * + * + * + + * * * + + * ^ In t h i s program the RKS EOS with HMV2 mixing rule i s applied. This i s a more general copy including the calc u l a t i o n s of: Phase boundary enthalpy, Cp(Numerical methods and the equation of the derivatives),Cv, adiabatic exponent(=Cp/Cv), volume for the mixture, l i q u i d and vapour, the compressibility, the predictions of the pressure drop for the venturi, Sound speed, and the Mac number at the throat area. OXPCT: the weight f r a c t i o n of oxygen XLV: water mole f r a c t i o n AVEMW: the average molecute of the mixture p h i _ ( ) : the fugacity c o e f f i c e n t for l i q u i d and gas of water and oxygen **++*+*******+********+************+***********^  PROGRAM EOS DIMENSION HMIX(300) ,Cp(300)LFRACT (30.0) ,CV(300) DIMENSION DDV(300),DVL(300),DVV(300) DOUBLE PRECISION OXPCT,X,Y,TOL,K,P,Tl,TMAX,DT DOUBLE PRECISION VCW,R,XLV,AVEMW,VMAX,VL,VV, T DOUBLE PRECISION VROOT,I1L,I2L,I1V,I2V,zl, zv, Z DOUBLE PRECISION phi_wl,phi_ol,phi_wv,phi_ov DOUBLE PRECISION kw,ko,kwr,kor,ynew,xnew,INTEG DOUBLE PRECISION V,DH,HMIX,HWIDEAL,HOIDEAL,LFRACT DOUBLE PRECISION DHL,DHV,HL,HV,Cp,CPWIDEAL,CPOIDEAL DOUBLE PRECISION DPDT,DPDV,DPDX,CPL,CPV,CPIL,CPIV,DLDT DOUBLE PRECISION VSOUND,VSOUNDL,VSOUNDV,Athroat,A,Ma DOUBLE PRECISION massflow,Velocityl, Velocity2, V2, DP DOUBLE PRECISION DPDL, DPDV2, CV, DDV, DVL, DW, GAMA, DVDT DOUBLE PRECISION DLDP,DPDDV INTEGER I EXTERNAL INTEG,HWIDEAL,HOIDEAL CHARACTER*40 FNAME COMMON /TOLERANCE/TOL WRITE(*,*) 'Enter f i l e name:' READ(*,5) FNAME 5 FORMAT(A40) OPEN (UNIT=5,FILE=FNAME,STATUS='NEW') OPEN (UNIT=1, FILE='INPUT',STATUS='OLD') READ (1,15) OXPCT,X,Y,TOL,K,P,T1,TMAX,DT,massflow 15 FORMAT (F3. 1, 2X, 2 (F5. 3, 2X) , F6. 4, 2X, F5 . 1, 2X, E9. 2, 2X, 2 (F5 .1, 2X) C ,F3.1,2x,F6.4) PRINT*, OXPCT,X,Y,TOL,K,P,Tl,TMAX,DT,massflow XLV=(OXPCT-100.)*32./((OXPCT-100.)*32.-OXPCT*18.) AVEMW=18.*XLV+32.*(1-XLV) WRITE(5,* WRITE(5,* WRITE(5,* WRITE(5,* WRITE(5,* WRITE(5,* 'The running condition:' 'Oxygen weight f r a c t i o n : ', OXPCT, ' '• 'Pressure:',P,'Pa' 'Water, mole f r a c t i o n : ',XLV 'Temperature difference:',DT, 'K' 'Massflow:', massflow,'kg/s' WRITE(5,1) 'T(K) ', 'P(Pa) ', 'X', 'Y', 'Hmix(kJ/kmol) ', 'Cp(kj/kg/k) ' , 131 C 'Cv(kJ/kg/K) * , 'GAMA* , ' VL (m.3/mol) ', ' VV (m3/mol) 1 , 'V(m3/mol) ' , C 'ZL',1ZV',' Z' ,'Vsound(m/s)','DP(Pa)' ,'Ma' 1 FORMAT (A4, 5X, A5, 6X, Al, 8X, Al, lOx, A13, 5x, Al 1, 4x, A l l , 3x, A4 , 4x, C A10,4x,A10,4x,A9,5x,A2,7x,A2,7x,Al,7x,A12,8x,A6,1lx,A2) A=3.14159*(6.2E-3)**2 ./4 . Athroat = 3.14159*(3.E-3)**2./4 . 1=1 VCW=57.1E-6 R=8.314 VMAX=K*VCW VL=VCW/2. VV=VL T=T1 20 VL=VROOT(P,T,VL,X) VV=VROOT(p,T,VV,Y) I1L=INTEG(VL,VMAX,X,T,1) I2L=INTEG(VL,VMAX,X,T,2) I1V=INTEG(VV,VMAX,Y,T,1) I2V=INTEG(VV,VMAX,Y,T,2) zl = p*vl/R/T zv = p*vv/R/T phi_wl = exp(ill+i21*(1-x))/zl phi_ol = exp(ill-i21*x)/zl phi_wv = exp(ilv+i2v*(1-Y))/zv phi_ov = exp(ilv-i2v*Y)/zv kw = phi_wl/phi_wv ko = phi_ol/phi_ov * kwr = Y/X * kor = (1-Y)/(1-X) yNEW = kw*X xNEW = 1-(1-Y)/KO IF((ABS(X-XNEW).GT.TOL).OR.(ABS(Y-YNEW).GT.TOL)) THEN X=(X+XNEW)12. Y=(Y+YNEW)/2. * PRINT*, x,y,xnew,ynew * STOP GOTO 20 END IF WRITE(*,*) 'CALCULATING ENTHALPY OF MIXTURE' * SINGLE PHASE REGION IF(XLV.GT.X .OR. XLV.LT.Y) THEN WRITE(*,*) 'IN SINGLE PHASE REGION' V=VROOT(P,T,VL,XLV) DH=INTEG(V,VMAX,XLV,T,3) WRITE(*,*) 'DH(kj/kmol)=',DH HMIX(I)=DH+XLV*HWIDEAL(T)+(l.-XLV)*HOIDEAL(T) IF(XLV.GT.X) THEN LFRACT(I)=1. ELSE LFRACT(I)=0. ENDIF Cp(I)=(HMIX(I)-HMIX(1-1))/DT 132 Calculating the Cp using the equation of the derivatives. Cp(I)=XLV*CPWIDEAL(T)+(1.-XLV)*CPOIDEAL(T)-R-C T*INTEG(V,VMAX,XLV,T,4)-(T*(DPDT(T,V,XLV))**2 . )/(DPDV(T,V,XLV)) —Gama calculation DDV(I)=V CV(I)=CP(I)*1000.*AVEMW-V*DPDT(T,V,XLV)-P*(DDV(I)-DDV(1-1))/DT CV(I)=CP(I)+T*DPDV(T,V,XLV)*((DDV(I)-C DDV(I-l))/DT)**2. DVDT=(DDV(I)-DDV(I-l))/DT CV(I)=CP(I)-T*DVDT*DPDT(T,V,XLV) CP(I)=CP(I)/AVEMW CV(I)=CV(I)/AVEMW PRINT*, DPDV(T,V,XLV),((DDV(I)-DDV(1-1))/DT)**2. GAMA=CP(I)/CV(I) Z=P*V/(R*T) Z1=P*VL/(R*T) ZV=P*VV/(R*T) SOUND speed and pressure drop. Assuming the density is constant, and the throat diameter is 3mm. VSOUND=(V)*(-GAMA*(DPDV(T,V,XLV))* 1000./AVEMW)**0.5 VSOUND=V*((GAMA*(DPDT(T,V,XLV)*1000./(AVEMW*DVDT)))**0.5) Velocityl=massflow*V*1000./(AVEMW*A) Athroat=(massflow*V*1000./AVEMW)*((1/(R*l.4*T*1000./AVEMW))* C ((1. 4 + 1)/2)**(2.4/0.4))**0. 5 Velocity2=massflow*V*1000./(AVEMW*(3.141592 6* (3. 0E-3)**2 . / 4) ) DP=(0.5/(V*1000/AVEMW))*(Velocity2**2.-Velocityl**2.) Ma=Velocity2/Vsound WRITE(5, 90) T,P,XNEW,YNEW,HMIX(I),Cp(I) , Cv(I) , GAMA,VL,VV,V, C ZL, ZV, Z,VSOUND,DP,Ma WRITE (*, 90) T, P, XNEW, YNEW, HMIX (I) , Cp (I) ,Cv(I) , GAMA, VL, VV, V, C ZL, ZV, Z, VSOUND,DP,Ma 90 FORMAT(F 6.1,G12.3,2F10.6,F16.7 , 2F15.7,F10.6, 3F14.10, 3F9. 5, F15.8 C ,F20.8,F10.5) TWO PHASE REGION ELSE LFRACT(I)=(XLV-Y)/(X-Y) WRITE(*,*) 'IN TWO PHASE REGION, L=',LFRACT(I) V=VL*LFRACT(I)+VV*(1-LFRACT(I)) DHL=INTEG(VL,VMAX,X,T,3) DHV=INTEG(VV,VMAX> Y,T,3) HL=DHL+X*HWIDEAL(T)+(l.-X)*HOIDEAL(T) HV=DHV+Y*HWIDEAL(T)+(1.-Y)*HOIDEAL(T) HMIX(I)=LFRACT(I)*HL+(1.-LFRACT(I))*HV Cp(I)=(HMIX(I)-HMIX(I-l))/DT 133 DLDT=(LFRACT(I)-LFRACT(I-1))/DT DLDP=-( (XLV-Y)/((X-Y)**2.))/DPDX(T,VL,X) + (XLV/( (X-Y) )**2. -C X/((X-Y)**2.))/DPDX(T,VV,Y) PRINT*,DLDP . DPDDV=(LFRACT(I)/DPDV(T,VL,X)+(1-LFRACT(I))/DPDV(T,VV,Y) C +(VL-VV)/DLDP)**(-1.) PRINT*,LFRACT(I)*DPDV(T,VL,X),(1-LFRACT(I))/DPDV(T,VV,Y) PRINT*, (VL-W)/DLDP -- Calculating the Cp using the equation of the derivatives. DCPL=T*INTEG(VL,VMAX,X,T,4) DCPV=T*INTEG (W, VMAX, Y, T, 4 ) CPL=CPWIDEAL(T)*X+(1.-X)*CPOIDEAL(T)-R-DCPL C -T*(DPDT(T,VL,X))**2./(DPDV(T,VL,X)) CPV=CPWIDEAL(T)*Y+(1.-Y)*CPOIDEAL(T)-R-DCPV C -T* (DPDT (T, VV, Y) ) **2 . / (DPDV (T, VV, Y) ) CP(I)=(LFRACT(I)*CPL+(1.-LFRACT(I))*CPV+(HL-HV)*DLDT)/AVEMW --Gama calculation DVL(I)=VL DW (I) =VV DDV(I)=V CV(I)=CP(I)* 1000.*AVEMW-LFRACT(I)*(VL*DPDT(T,VL,X)+P*(DVL (I) -C DVL(I-l))/DT) C - (1-LFRACT (I) ) * (W*DPDT (T, W, Y) +P* (DW (I) -DW (1-1) ) /DT) DVDT=LFRACT(I)*(DVL(I)-DVL(1-1))/DT+(1-LFRACT(I))*((DVV(I)-C DW (1-1) ) / DT) +DLDT* (VL-VV) DVDT=(DDV(I)-DDV(I-1))/DT CV(I)=CP(I)-T*DVDT*DPDT(T,V,XLV) PRINT*,DVDT, DPDT(T,V,XLV),V,DPDT(T,V,XLV)*1000./(DVDT*AVEMW) STOP CP(I)=CP(I)/AVEMW CV(I)=CV(I)/AVEMW GAMA=CP(I)/CV(I) ZL=P*VL/(R*T) ZV=P*VV/(R*T) Z=P*V/(R*T) SOUND speed and pressure drop. DPDL=-( (XLV-Y)/(X-Y)**2.)*DPDX(T,VL,X) + (XLV/(X-Y)**2 . -C X/(X-Y)**2.)*DPDX(T,VV,Y) DPDV2=DPDL*(VL-VV)+LFRACT(I)*DPDV(T,VL, X) + C DPDV(T,VV,Y)*(1-LFRACT(I) ) VSOUND=V*((GAMA*(DPDT(T,V,XLV)*1000./(AVEMW*DVDT)))**0.5) Velocityl=massflow*(LFRACT(I)*VL+(1-LFRACT(I))*VV)* 1000.0/ C (AVEMW*A) Athroat=(massflow*VV*1000./(y*18. + (1-y)*32.))*((1/(R*1000.*1.4 * C T/(y*18. + (1-y)*32.)))*((!.4 + l)/2)**(2. 4/0. 4))**0. 5 134 Velocity2=massflow*(LFRACT(I)*VL+(1-LFRACT(I))*VV)*1000./ C (AVEMW*(3.E-3)**2.*3.141592 6/4.) DP=(0.5/((LFRACT(I)*VL+(1-LFRACT(I))*VV)* 1000./AVEMW))* C (Velocity2**2.-Velocityl**2.) Ma=Velocity2/Vsound WRITE(5,100) T,P,XNEW,YNEW,HMIX(I) , Cp(I),Cv(I) , GAMA, VL,VV,V, C ZL,ZV,Z,VSOUND,DP,Ma WRITE(*,100) T,P,XNEW,YNEW,HMIX(I) ,Cp(I) , Cv(I),GAMA,VL,VV,V, C ZL,ZV,Z,VSOUND,DP,Ma 100 FORMAT(F5.1,G12.3,2F10.6,F16.7,2F15.7,F10.6,3F14.10,3F9.5,F15.8 C , F20.8,F10.5) END IF T=T+DT 1=1 + 1 IF(T.LT.TMAX) GOTO 20 CLOSE(5) STOP END * Integrate the enthalpy from absolute zero to T for water *************************************************************** FUNCTION HWIDEAL(T) DOUBLE PRECISION T,TH,HWIDEAL TH=T/100. C ARBITRARILY DEFINE ZERO ENTHALPY AT ABS. ZERO HWIDEAL=(14 3.05*TH-183.54*TH**1.25/1.25+82.75*TH**1.5/1.5 C -3.6989*TH**2./2.)*100 RETURN END *************************************************************** * Integrate the enthalpy from absolute zero to T for oxygen *************************************************************** FUNCTION HOIDEAL(T) DOUBLE PRECISION T,TH,HOIDEAL TH=T/100. C ARBITRARILY DEFINE ZERO ENTHALPY AT ABS. ZERO HOIDEAL=(37.4 32*TH+.020102*TH**2.5/2.5+17 8.57*TH**(-0.5)/0.5 C -236.88*TH**(-1.))*100 RETURN END * Cp(T) for oxygen and water *************************************************************** FUNCTION CPWIDEAL(T) DOUBLE PRECISION CPWIDEAL,T,TH TH=T/100. 135 CPWIDEAL=143.05-183.54*TH**0.25+82.751*TH**0.5-3.6989*TH RETURN END FUNCTION CPOIDEAL(T) DOUBLE PRECISION CPOIDEAL,T,TH TH=T/100. CPOIDEAL=37.432 + 0.020102*TH** 1.5-178.57*TH**(-1.5)+236.88*TH**(-2) RETURN END *************************************** * This function is to define the equations that is need to * be integrated. *************************************************************** FUNCTION F(V,X,T,NTYPE) DOUBLE PRECISION F,V,X,T,R,DPDV,DPDX,DPDT,SECDPDT INTEGER NTYPE EXTERNAL DPDV,DPDX,DPDT R=8.314 IF (NTYPE.EQ.l) THEN F=-V/(R*T)*DPDV(T,V,X)-1./V ELSE IF(NTYPE.EQ.2) THEN F=DPDX(T,V,X)/(R*T) ELSE IF(NTYPE.EQ.3) THEN F=-T*DPDT(T,V,X)-V*DPDV(T,V,X) ELSE F=SECDPDT(T,V,X) END IF END IF END IF RETURN END i t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * The following three functions are to get the derivative * of the equation of state. FUNCTION DPDV(T,V,X) DOUBLE PRECISION DPDV,T,V,X,PRKS,TOL DOUBLE PRECISION V1,V2,DV EXTERNAL PRKS COMMON /TOLERANCE/TOL DV=V*TOL V1=V-DV V2=V+DV DPDV=(PRKS(T,V2,X)-PRKS(T, VI, X))/(2 . *DV) RETURN END FUNCTION DPDX(T,V,X) 136 DOUBLE PRECISION DPDX,T,V,X,PRKS,TOL DOUBLE PRECISION XI,X2 EXTERNAL PRKS COMMON /TOLERANCE/TOL X2=X+TOL IF(X2.GT.l.) X2 = l . Xl=X-TOL IF(X1.LT.0.) X1=0. DPDX=(PRKS(T,V,X2)-PRKS(T, V, XI) )/(X2-X1) RETURN END FUNCTION DPDT(T,V,X) DOUBLE PRECISION DPDT,T,V,X,PRKS DOUBLE PRECISION Tl,T2,DT,TOL EXTERNAL PRKS COMMON /TOLERANCE/TOL DT=T*TOL T1=T-DT T2=T+DT DPDT=(PRKS(T2,V,X)-PRKS(TI, V, X) ) / (2 . *DT) RETURN END ************************************************ * Secondary dirivative of pressure respect to temperature at constant * volume. *********************************************************************** FUNCTION SECDPDT(T,V,X) DOUBLE PRECISION SECDPDT,DPDT,T,V,X,DT DOUBLE PRECISION TOL,Tl,T2 EXTERNAL DPDT COMMON /TOLERANCE/TOL DT=T*TOL T1=T-DT T2=T+DT SECDPDT=(DPDT(T2,V,X)-DPDT(T1,V,X))/(2.*DT) RETURN END *********************************************************************** * Finding the roots of the EOS. Given pressure,temperature, x, * find the volume starting with i n i t i a l guess V. *********************************************************************** FUNCTION VROOT(P,T,V,X) DOUBLE PRECISION VROOT,P,T,V,X,PRKS, TOL, slope, DV DOUBLE PRECISION vnew,DPDV COMMON /TOLERANCE/TOL 1 = 1 100 SLOPE= DPDV(T,V,X) DV= (P-PRKS(T,V,X))/SLOPE IF(ABS(DV).GT.V/20.) DV=V/20.*DV/ABS(DV) VNEW=DV+V 1=1 + 1 IF(I.GT.200) THEN WRITE(*,*) 'NO ROOT V IN 200 ITERATIONS' RETURN 137 END IF IF(ABS(VNEW-V).GT.V*TOL) THEN V=VNEW GOTO 100 END IF VROOT=VNEW RETURN END ************************************************* * Put in the physical properties used in the Equation of State * Units are K, Pa,m*3/mol **************************************************************** * Calculate parameters used in the EOS equation *++++++++++++++++++++++++++++++++++++++++++++++++ FUNCTION PRKS(T,V,X) DOUBLE PRECISION PRKS, T, V, X,P,TCW,TCO,PCW,PCO,VCW,VCO,R DOUBLE PRECISION ACCW, ACCO,MW,MO,FW,FO,AW,AO,KOW,BW,BO DOUBLE PRECISION WW,WO,GCE,Awo,Two,THETAw,THETAo, THETAwo DOUBLE PRECISION THETAow,GRE,GE,ALFAw,ALFAo,A2,B2,C2,A,B DOUBLE PRECISION ALFAmix,Trw,Tro,Yw, Yo TCW=64 7.3 TCO=154.6 PCW=22.12E6 PCO=5.04E6 VCW=57.1E-6 VCO=7 3.4e-6 R=8.314 ACCW=.34 4 ACCO=.025 KOW=0. C Calculate parameters used in the RKS equation Trw=T/TCW Tro=T/TCO Yw=l-Trw**0. 5 Yo=l-Tro**0.5 IF (Trw .GT. 1) THEN Fw=(l+1.0873*Yw)**2 ELSE FW=(1+1.087 3*Yw-0.6377*Yw**2.0+0.6345*Yw**3.0)**2.0 ENDIF IF (Tro .GT. 1) THEN FO=(l+0.8252*Yo)**2 ELSE FO=(1+0.8252*Yo+0.2512*Yo**2.0-1.7039*Y0**3.0)**2.0 ENDIF AW=0.42748*R**2.*TCW**2./PCW *FW AO=0.42748*R**2.*TCO**2./PCO *FO BW=0.08664 *R*TCW/PCW BO=0.08 664*R*TCO/PCO Minxing Rule B=X*BW+(l.-X)*BO 138 Excess Gibbs Energy Combinatorial term WW=x*0.9**(2.0/3.0)/(x*0.9**(2./3.) C +(l.-x)*1.764**(2.0/3.0)) WO=(l.-x)* 1.764**(2.0/3.0)/(x*0.9**(2.0/3.0) C +(l.-x)*1.764**(2.0/3.0)) GCE=(x*log(WW/x)+(1.-x)*log(WO/(1.-X)))*R*T Residual term Awo=688.7-0.9018*(T-298.15) Two=EXP(-Awo/T) Two=EXP(-Awo/T**l.14) THETAw=l.4*x/(x*l.4+(l.-x)*1.91) THETAo=l.91*(1.-x)/(x*l.4+(1.-x)*1.91) THETAwo=THETAw*Two/(THETAw*Two+THETAo) THETAow=THETAo*Two/(THETAw+THETAo*Two) GRE=R*T*(x*1.4*log((l.-THETAow)/THETAw) C +(1.-x)*1.91*LOG((1.-THETAwo)/THETAo)) GE=GRE+GCE ALFAw=AW/(BW*R*T) ALFAo=AO/(BO*R*T) A2=-0.0047 B2=-0.478 C2=0.47 8*(x*ALFAw+(l.-x)*ALFAo)+0.004 7*(x*ALFAw**2.+ C (l.-x)*ALFAo**2.)-GE/(R*T)-(x*log(B/BW)+(1.-x)*log(B/BO)) IF ((B2**2.-4.*A2*C2) .LT. 0.0) THEN PRINT*, (ALFAmix has no root) ELSE ALFAmix=(-B2-SQRT(B2**2.-4.*A2*C2))/(2.*A2) A=ALFAmix*B*R*T ENDIF IF(V.LE.B) THEN WRITE(*,*) 'ERROR v<B IN PRKS',V,B ENDIF PRKS=R*T/(V-B) - A/(V**2.+B*V) RETURN END **************************************************************** This function is for the integration ************************************* FUNCTION INTEG(A,B,C1,C2,NTYPE) DOUBLE PRECISION INTEG,a,b,x,total, st, integral, TOL, Cl, C2 INTEGER NTYPE COMMON /TOLERANCE/TOL total=0 st=A/50. X=A i f (b-x.lt.st) st=b-x c a l l step(st,x,integral,TOL,Cl,C2,NTYPE) total=total+integral i f (x.lt.b)•goto 10 139 INTEG=TOTAL RETURN end This routine calculates the integral from x to x+st, returns with the updated x, the integral, and a new estimate for an appropriate step size. subroutine step(st,x,integral,TOL,CI, C2, NTYPE) DOUBLE PRECISION dx(30),i(30),st,x,integral, xj ,sum,F,TOL,SMALL DOUBLE PRECISION C1,C2 integer k ,neval,j,NTYPE EXTERNAL F SMALL=l.E-8 k=l dx(k)=st/2.**(k-1) xj=x+dx(k) I(k)=(F(x,Cl,C2,NTYPE)+F(xj,Cl,C2,NTYPE))*dx(k)12. continue k=k+l neval=2.**(k-1)-2.**(k-2) dx(k)=dx(k-l)12. xj=x+dx(k) sum=0 do 5 j=l,neval sum=sum+F(xj,CI,C2,NTYPE) xj=xj+dx(k-1) continue I(k)=Sum*dx(k)+1(k-1)12 . IF(DX(K).GT.SMALL) THEN if(abs(I(k)-I(k-1)).gt.tol) GOTO 1 ELSE write(*,*) 'step size too small in integral' END IF integral=I(k) write(*,*) x,st,k,integral,I(k-1) X=X+ST ST=DX(K)*10. return end 140 T o r i T OO *o r- r- r- r-Tr "»f *r *r rr o o o o o o o o o o o o o o o o o o O f N » f s o oo O i OO 00 00 OO OO OS ; t t ^ Tf TT Tf ' n I s" r j \ r > ^ O ( N ON ON ON — — — f N T rr T i i n i/-) V> o o o o o o o o o o o o o o O O O O O O O O O O O O O O O ' r- m r- oo TT TT O r O o ro ro v i rN oo s o v> ^ ( N O OO ^ 00 ^ ON O P- rj-O fN in o o o sO s o r o os o ON s O — T f — OO O r o r o ON ON o oo o o — r - v i oo «n v i r o o P - ro O i T f Os s O 1 oo vs ON < IT l T f — O t o o O r~~ CN oo p- o sO ON ON r-^  rr i n O ' r i »n oo — - o m r -O T f T f p- <N — 1 ^ SO ON p- — oo <N o r- o CN ro OO — i Vi ob r—* v i t f l O - M r > r - m C N 1 f N Tf — f N 1 r o s o r o < r - ON T f v i i m — T f r- i T f m o NO i ON oo — r- i — r o sb ob • f N so o T f i T f T f m m 1 f N f N r N f N f N r N f N f N f N f N f N f N f N f N f N f N f N f N f N i „ f p- o ro NO ON fN «n 00 fN >n oo fN «n ro ro ro T f T f T f T m »n v > sO •O sO r- r-v . 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'. fN ro T f i n sb ob ON d — p-i CO T f v i sb p^  o o o o o o o o fN fN fN fN fN fN fN fN fN fN rO ro ro ro ro ro ro ro i n V l Vi VI VI i n i n VI VI Vi »n V) Vi Vi Vi Vt Vi i n V, Vl Vj Vi Vi m »n Vl • n Vt Vi Vi v. Vi V, • n V) i n 141 r i ^ 7< r- oo oo oo «n v , <n <n o o o o o o o o b o b © oo - i > n p» o ON cn o oo <n oo oo n IN n oo v i r- m TT I —. m «n o — <N T f r-- < OS T f Ov T f m r t Tt i n M fN (N rN fN fN (N CN m r- o ON ON © V v> \ 0 O O O O © O © © © r» oo — -f ON S O — — v i f O ao O O o o o © r N s O O N r n sO s O s O s O © © © © o o o o r - * r N N O © T f c c m p - r N s O — o o o o o o o o 3 . O Vt O NO NO r-fN fN fN fN fN fN 0 0 — fN Vi fN O O NO oo v » SO NO r f OO OO fN fN OO NO —« p» oo fN fN fN fN oo r- m ON fN oo fN o ov NO fN — m ON i n r- NO m O O N C O t n in m O m cn T f — 0 0 N O r- m oo TT 0O O N O N O fN fN fN rn fN fN fN fN in r-O N fN ON rn 3 •o sd so O O O o o © bob O N r- t f r— ON OO fN NO 0 O - i / 1 fN r- fN oo TT © oo m C7i rr m © v i Tf' i n NO fN OO T f fN fN rn fN fN fN •n NO , vO NO o o ' © © b o b © C N r-oo «n in p-fN ON m o <n in T f ON P - SO od — o r-T f T f m m fN fN r- >n rn rn T f N O so m ON f N m oo T f O N v i O N m O N i n m m m fN fN so r-so so o o o o b © fN T f O 0 sO m oo m o r- oo T f O fN v i fN rn v i so rn so r-m m fN fN m o v i O r- oo oo ON s O NO s O NO 0 0 0 0 < _ ) ' — ) < — J o © © o o © © o p © © p © p p ^ O -O fN M 'n - M <r i n \ o i— i — O O O N O N O i — r — - r — - r - p - r — r - ^ r ~ o o O © © 0 © 0 © 0 0 © © © o o o p p p o b o o o o o o o m oo O fN sO rn O NO so — O N T f so ON N O — m oo T f oo T f m oo P -— rn fN — m — oo T f O 0 SO T f T f ON N O m © OO O N © rn rn m T f fN fN fN fN © O TT © m oo © OO rn m O N sO fN fN — fN NO NO ON m P » T f NO 0 0 r- T T — © — fN r- ON T f fN <n — oo r-© oo © r-ON T f T f in p- p- NO T f O m T f oo T f C N m O N — i n ON rn rN NO — od ON NO T f — rN m T f i n T f T f T f T f T f  T f fN fN fN fN fN fN fN m >n r- © r- o m ON O 0 O 0 O O N 0 0 N O fN O © m — fN — i O — >n T f oo r- — ON m m m in T f Tf" i n ON P » v i m T i ^ i — oo T f T f T f T f fN fN fN fN ON ON NO i n rN f i «n m oo m CN oo oo vo <n NO — O fN rn so T f m in in — od oo r- v> O — fN m m in I fN fN fN — v i rN fN OO — P - ON fN — rn fN p- oo oo © © oo rn NO © r— rn P -•n m ^ o NO —I — m T f p- p- od — TT m rs fN m T f i n NO i n m in i n fN fN fN fN — 0 0 «n fN — — fN m 0 0 oo oo oo o © o o o © © o © b o b oo oo fN r-NO TT —. m so oo T f r» © © r-» m rn — v i rn rn »n ON v i ON 0 O OO V . oo oo T f p-. in — ON od — — © © P - 0 0 Ov © «n v i v> so fN fN fN fN —« © oo m r» fN T f ON SO ON fN v, oo — ON — sd — oo ON P -NO ON T f v i oo © OO T f — fN oo NO © fN so i i n oo — i fN rn i n • vi O N rn" I so m in • i n m v i i OO ON so r-T f fN © so 0 0 sO © © 0 0 SO OO O N ON SO O O fN fN r- oo so v i m fN r» r- oo r-fN so ON m O N © — rn »n b T f od m m fN — m in v i i n oo oo fN v i — i m T f — NO ON fN OO r- © T f sq fN sd — o m in p» —' m — Vt s O r- oo b T f © O N f N ON m oo T f ON m in so «n T f f N © m © — ON rn oo oo OO O NO — 0 0 ON m — ov m © in so oo fN rn r- — r- p-V- l T f T f T f T f T f m so O fN m so oo oo Ov fN so m © fN m so v i Ov so m T f T f — m so — — O N —• m — m m (— r- oo rN ON © ON m v. v-i r- oo ON m — •n r f T f fN T f —. v i © Ov m —• so so o — sd b m m T f T f T f T f T f — fN T f © fN O i n — © fN m fN in T f (N m T f oo fN — T f T f so fN ON T f fN ON SO fN ON m r— © ON fN Ov m fN fN fN o r» T f i n v-» m T f © oo • t— r- rs r-fN so Ov oo m oo o oo fN © oo — i m oo ON T f T f r- — m i i n — m Ov so so ON T f m T f T f -O , »n © T f m —« © r- fN ON fN v> oo ^- i rn T f tn T f fN < fN —OO ON — fN — — r- TT in r*- r** oo oo oo oo T f oo rs so © T f T f T f T f 0 0 fN m in m m OO OO I sd © sO 0 O m m T f O N . o m r— so ; fN sd I fN — OO fN © © v-i Ov «n v i in so <n in T f m © T f od — o Ov m m CN so T f so m OO O fN fN rs cn r- so ON sO O N r-i n oo - in — Ov © so m oo m m fN © ON r-; fN sd O N rn ON oo r- r-fN fN fN fN rNOf!>oor - .r -«r--.ooo\-- m v i o o f N N O — so — ^ l t « 0 ^ ^ ^ < 5 N O ^ o o o n ^ • - ^ - ^ ^ N O O O - n « ^ O ^ T r o ^ ^ n f N - f N T f v ^ v S v ^ f n v S v S s O s O s O s O s O s O s O s O O O O C M O O O O O O O O M C f l O O O O M M O O O O O O O O C f l O O M O O O O O O O O O O O O M b b b b b b b b b b b b o o o o o o o o o o o o o o b b b b o o b o o b b b oo o T f fN SO s o ON ON O © — ON m >n m m oo oo oo m o m T f m O 0 0 so so v i i n O N Ov Ov boo oo r- N O NO r- oo v. m vt OO 0 0 oo _ © so — O N so T f — CN m m T f ON O N O N bob i n so so ov o — m so so oo oo oo O N ON m o r- m T f T f Ov Ov P - O N r» m fN o T f T f O N O N r— v i m O o o o o o o o o m ON — Vi oo in fN fN Ov Ov © © ON — < fN T f I SO S O • OO OO i i OO fN ' r» O N > SO so i 0 0 OO r— r- r- r-0 0 oo oo OO m r- T f O T f oo m © P -fN fN — Ov ON ON bob r- T f — so OO O r- r- oo oo oo oo oo r- T f — T f r-v. fN ON — — o Ov Ov ON Ov — S O T f © O Ov Ov © O © © © ON r- so — m m oo oo oo oo oo oo so so f- ON 0 0 oo oo oo r f r-. r-~ — ©r--Novi — — r--©p--ooviTfONrnoor ,--©ONOov,NO m T f i n N o ^ o v i T f m r N O r * - i n — oOTfoomr~-fN»noo — f N m T f T f — o o v i f N O N s o m o i ^ T f © r ^ T f © r ^ c n O s o c o O N v , r N M T f O s o o o s C A i > M M o o o o r * r ^ ^ ^ ^ s O w i ^ ^ ^ ^ n n n N r N | f N — O N O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O b b b b b b b b b b b b b b b b b b b b b b b b b b r - o o o m p * - — v i — r - T f f N © © © — r n v i O N m O N V i m — —• —* cn — c n s o o o © r n v ^ o o © r n ^ O N f N v , o o — Trr-- — T f o o f N v o O T f o o O N O N O N O N O O O © — — — — f N f N f N r n c n r n T f T f T f i n i n s p s p s p O O O O O O O O O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O V O N O N O N O N O N o o c i o o o o o o o d o o o o o o o o d c i o d o o o o d o l l i l l l l l l l i l i i i i l i i l l i i l i l i l l l i l i l i l l i l i l i l l l i i i i i lg i i i i i i i i l i i i i i i l i i i i l i l i i l i i i l l i l l i l i l i i i i l l l i l l i i i l o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d §§§!§ !§§ i§ i i i i i ! § i i §§ i§ i l iP§§§§ i§§§ o o o o o o o o o o o o o o d o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d 1 I 1 I I I | | | | 1 § 1 1 I I 1 I I § § 8 S S S S 8 S S S 8 8 S 8 S 8 S 8 S S S S S S 8 S 3 S S S 8 S PPPPPpopNppp P P P P O O © 0 © 0 0 © © © 0 0 © 0 © 0 © 0 © © © 0 © © 0 0 © © 0 © 0 0 © 0 0 © © o o o o o o o o o o o o o o o b b b b b b b b b o b b b b b b b b o b b o b b b b b b b b b o b o o b b -0 <M- T*- 1^ 3^ 5^ ?^ ?5 *Ci ^ » ®>* """* ^ N NO P*» ^ 5 f*- so ON ^G> i n O N S O SO • ^ 5 rn rn ^ 3 fN OO O 0 0 » * - . i * f t r ^ ^ n r t ^ N _ _ /^jT+-iy-^r».o>©fNTfsor-ON—' r n v i r - O N — T f s O O O O m P ^ O O O O O O O O O O O O O N O N O N O N O N O © © © — — r - o o O N O — r n T f v i s o o o o N O f N c n v i s o o o o v — C N T T V I I v i s 6 r * - o b o N © — f N m T f s o . - — • - — — — — . . . . . . _ - -n M n n n ^ t t t ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ N O ^ s O ^ s c f N f N f N f N f N f S f N f N f N f N f N f N f N f ^ OOsOONTf — O N O O O O © O N © f N — T f T f V i N O s O r n f N o O O N V ^ O f N O N V i r ^ r ^ r ^ v ^ — f M r n o o r n r ^ r n O r n r ^ i n s O N O M n ^ P f ? f N ? P o f N ^ M - ^ r T f l T f o O f N N O O v r n t ^ ° ^ « ^ ^ C I ^ n S S ^ 2 ! 0 2 ^ S S ^ N S S S 3 S N O T f f N © o s r - V ! T f r N © o o r ^ i n r n f N © o o r ^ i n f n f N © o o r ^ v i T ^ S O s O s O s O V , V i V i v ^ v i ^ T f T f T f T f N f s i f N f S ( N f N ( N f N f N M f N N s O o O f N T f T f s D T f T T f r n r ^ — s O T f f N f N V i T f s O i n r ^ c O f s r - r ^ ^5 ON T f —* fN m ^ > OO v> <© © t**" v i fN fN m O N P** NO ON T f \0 ON V I T f m sO •— 1 fN ON ^ 3 rn Ov OO r*- T f sO OO rn rn ON "™ T f rn ^ 3 fN OO r^ - T f ^3 r n ^ o r ^ f N v ^ c w — o o v i r n v i — r n © T f i n T f r n f N r n P ^ T f v > r n o o r N s ^ o o o m m — o o o r - p - s o s o p - p ^ o o © — T f s o o v f N ^ o v i O v i — c o T f f s ! o o o ^ ^ ^ M O ^ S ' f l » n ^ n o ^ ^ J ^ ^ ^ • o ^ ^ o — r n t n r - O N O r N T f v o o o o f N T r r - O N — m i n o o o m v - i o o © m v , o o — r r ^ ^ f N « n o o N ^ M - ^ M N ^ a - n r ^ - ^ ^ ^ M r N — — — — — — f N r N f N r N r N r n r n r n r n r n T f T f T f T ^ ^ v i v i v i i n i n i n ' v i v i v i v S v i v i i n i n v i v ^ — ON r-m — oo so cn v i m Ov fN cn © oo — T f m O P*- — rn — — ON m ON NO CN ON O — fN fN ON OO P^- so ON — fN ON — Ov — T f fN S O fN r-Tf — oo r- oo — 0 0 P^ fN N O — m — r-. rn fN — O O N so so ON V) I— m vt sO rn r*~ sd — sd © T f v i v i so rn fN — © P- ON so m m V* — O m — m so r- m m oo © r-» • m Tf m oo • en so v i • oo ON © rn < m Tf oo so i — OO SO Vl 1 SO T f Ov © I T f OO — T f sd ON so so p** P -r- CN i oo i n • so oo i © r-© fN 0 0 oo cn rN © © fN m oo oo — o V T f r- r» ON 0 0 fN cn m © m ON m' fN oo oo ON OO ON ON ON Ov cn N O oo r-Vl © 0 0 Ov fN O N O sO fN 0 0 fN sO rn ON SO sO m © T f ON sq od sd r— r- r-. T f ON fN 0 0 © fN fN VI © r-o so ON SO rn © T f fN V, T f Tf m fN © O N O N i sO m ON ON SO sd fN sO so — o O N ON sO s o fN ON O © fN OO — r-rn v i OO T f s d b © V i T f V l ON sq r- CN v i m TT TT ON OO P-» SO oo oo oo oo r- oo ON r-© C C fN ON — T f 0 0 ON — fN V I O N T f oo 0 0 fN fN — fN T f i n © ON fN m s d P*- ON m C N — © m r f m fN 0 0 oo oo oo 0 0 fN — T f m oo ON © T f C N © ON ON ON r-© ON OO OO P— P-T f ON ON m ON oo T f i n © r- fN v i — 0 0 SO cc — r-fN T f oo od s d rn SO Vl T f p^ SO Vl r- r~~ sO 0 0 V ) S O P» rn s O © — ON v i m fN fN fN Ov — s O V I O T f ON fN — sO sO O s O b s d fN -o m — o oo T f m C N o r-- r- [— r-oo v i m rn fN O N r*- oo T f — fN ON so OO fN fN Vi © P- v i rn oo fN v i T f T f r-. — b m vi r-^  r-- i n m — ON OO P*- SO sO NO SO SO — Vi ON sO 0 0 T f fN SO — — Vi © m rn T f cn m cn rn m m o © © © O © — ON so T f ON Vl oo so Vi v i in v i 0 0 0 0 0 0 ON ON ON bob 0 0 OO OO o o o + + + UJ LU UJ o o o SO sO sO fN fN fN bob — m f N — ONfNOO — sOf N f N s O — T f m T f m © r * * T f f N O O v o o o o r n r - - f N P - f N s o — vi\ O N o r - - r - o o o o O N m r n m m r n m m m m v i s o r - - v i T f T f — r - f N P ^ f N P - f N P - - f N O v O O s O V i r n c N O O v T f T f T f T f T f T f T f m rn r- © — SO fN T f fN — i n i n v i 0 0 oo 0 0 Ov O N ON oo oo oo o © © + + + UJ UJ UJ o © © NO vO sO fN fN fN bob o o o o o o o o o o o o o o o o o © © o © o © © + + + + + + + + U J U J L U U J U J U J U J U J o o © o © o © o s O s O s O s O s o s O s O s O fN fN fN fN fN fN fN fN r- — vi oo v i T t p^ -r*- v i NO — so so — p- p- P - 0 0 0 0 ON — sO — sO — sO — P"-Ov © © — — fN fN m T f T f T f T f T f T f b © b b © © © P** fN so Ov ON 0 0 ON sO — V i ON rn P - — P-* NO T f fN — Ov OO m m rn m m fN rN 0 0 0 0 OO 0 0 0 0 0 0 0 0 O N ON ON ON ON ON ON b © © b © © b OO OO 0 0 OO OO 0 0 OO © O © © © O © + + + + + + + UJ UJ UJ UJ UJ UJ UJ o o © © © o © •O sO sO NO SO SO NO fN fN fN fN fN fN CN © © © © b © O O T f T f o o v i — c n o o m © © T f o o r « - f N f N V i m f N N O T f m — oo — i n s o m — T f O v s o O N V - i f N v , m v i o o — v i o o r n r ^ f N o o T f © s o m O o o s O T f f N P - r N C C m O O T f O v V i O N O f N P * - r n O N T f O s O m m T f T f i n v i s o s o p ^ o o o o O N O v © © — rNf N T f T f T f T f T f T f T f T f T r T f T f T f T r V l i n V ^ V l i n s O c n O N m O N m s o — p^oorNsop*-— OO — r - V i — sOVisOONONONfN m — fN — — — — f N m T f s o c o — f N O O T f O N O f N O O T f O s O f N O O V i m m T f i n i n s O s o r - o o o o O N O N O v i v i v i v i v i v v i v i v i v i i n v i s o i O O © © O © © O N o o s o m o p ^ - f N s o o v i O N T r s o p ^ o v O — — m O N r n r ^ — T f o o — v i c e — v o o — T f o o — T f s O T f m — © o o s o v i m — © o o s o v t r n — © c c f N f N f N f N f N — — — — — — © 0 0 0 © O O N O O O O O O O O O O O O O O O O C C C C O O O O O O O O C O O O O O P -O N O N O N O N O N O V O N O N O V O N O N O N O N O N O N O N O N O N _ b b b b b b b b b b b b b b b b b o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 © © © © 0 0 © © © © 0 0 0 © 0 © 0 + + + + + + + + + + + +• + + + + + -•-U J U J U 1 U J U J U J U 4 U J U J U J U J U J U J U J U J U J U J U J O © 0 © 0 © © © 0 0 0 0 © © 0 0 0 © s O s O s O s O s O v O s O s O s O ^ O s O ' O s O O s O s O s O s O r ^ f N f N f N f N r N f N f N f S f M f S f N f N f N f N f ^ b b b b b b b b b b b b b b b b b b o b b b o b b o b b o o f N © P - - P - P - r n r n r n T f T f O O O m p ^ o m s o o N f N v i o o — T f p - — T f • O ' n m — O N o o s O T f r n — O V O O N O O N O N O N O N O O C C O O O O C C O O P - - P » P -r - p - p * - p * * i — p - r ^ » p * - i — r - p » r - - ( — O N O N O N O N O N O N O N O N O N O N O N O N O V b © © © © © © o 1 © © © • O © © © © © © © © © 1 © O © ' © © © o o < o o © © p © © l n ^ o r • o o ( > 0 - f N ^ ^ ' n v 0 ^ o o ( > 0 - f N ^ n t ^ ^ ^ » ^ ^ - M ^ ^ ^ ^ o o o p p p CC Ov © — fN rn i n i n i n i n ^ i n i A i n i n i n ' n v, v i >n in v i v v i Vi c c o o o o c c o o o o o o o o o o c c o o o o o o © © © © O O O O O O © © © + + + + + + + +• + + + + + U J U J U J U J U J U J U J U J U J U J U 4 U J U J o © © © o o o © o © © © © • O sO *0 sO ' O "O NO ~0 SD *0 NO \C f N f N f N f N f N f N f N r N f N f N f N f N f N o o b b o o o o o o o o b © © © p o p p p p p p p © r - ^ o d o v b — r i m T f ' n ' ^ d p ^ o o o N r - r - r » o o o o o o o o o o o o o o c c o o o o m v i v i vt v i v, v v i v v, v. v i v i 142 APPENDIX D FILE LISTS 143 1. In win98, the files are in the directory of \ws\ \ws\test\: *.xls: files for the density measurement and comparison of heat capacity and densities. 8rpm: runs at 800rpm (Runs 24 and 25) lOrpm: runs at lOOOrpm (Run 23) 12rpm: runs at 1200rpm (Runs 30 to 35) 14rpm: runs at 1400rpm (Runs 28 and 29) 16rpm: runs at 1600rpm (Runs 26 and 27) June-99: runs in June 1999 (Runs 11 to 16) Dec-99: runs in December, 1999 (Runs 17 and 18) Coldwater: cold water tests (Runs 1-36) DPCalibration: calibration data for DPT429 02Calibration: calibration data for oxygen flow rate PCalibration: calibration data for system pressure Venturi.dwg: drawing for the venturi \DATA\ : txt files listed in Appendix B. 2. In Linux , the files are in the directory \wangshuo\ \THESIS\: all the files needed for the thesis. *.tex files are latex files and *.eps files are graphs. \EOS\: \CODE\ contains the code for the program. hm*.f files are for the RKS EOS and hs*.f files are for the Hard Sphere EOS. \ D A T A \ contains all the output files used in thesis, hm: the RKS EOS; hs: the Hard Sphere EOS; the digit after hm or hs means the weight percentage of oxygen and the last two digits represent pressures. For example: hm2-25 = RKS EOS, 2 wt % oxygen, 2 5 MPa 144 

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