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UBC Theses and Dissertations

The development of the Lielmezs-Howell-Campbell equation of state for pure fluids Herrick, Troy Alexander 1986

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T H E D E V E L O P M E N T OF T H E L I E L M E Z S - H O W E L L - C A M P B E L L E Q U A T I O N OF S T A T E FOR P U R E F L U I D S b y TROY A L E X A N D E R H E R R I C K A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E i n T H E F A C U L T Y OF G R A D U A T E S T U D I E S D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A A u g u s t 1986 © TROY A L E X A N D E R H E R R I C K , 1 9 8 6 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e h e a d o f my d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ^ £ yv\ / c V r T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main M a l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e D E - 6 f3/81 ,» ABSTRACT The Lielmezs-Howell-Campbell (LHC) equation of state, o r i g i n a l l y proposed for saturated states, was further developed to study P-V-T and thermodynamic properties of 14 select pure * f l u i d s . Using the dimensionless T coordinates: T*=( ^ -1 )/( | ^ -1 ) two new temperature dependent functions, a{T*)~ and a(T ) c , were proposed. Test comparisons were made between the LHC, Soave 1972, Redlich-Kwong, Lee-Kesler, Benedict-Webb-Rubin, and Boublik-Alder-Chen-Kreglewski equations for 14 organic, inorganic and quantal f l u i d s over P-V-T ranges where available data permitted. The LHC equation performed as consistently and accurately as the 5 already established equations in c a l c u l a t i n g P-V-T properties, Joule-Thomson c o e f f i c i e n t s , inversion curve coordinates, fugacities, the two heat capacity, Cp and Cv, values as well as the thermodynamic departure functions of: enthalpy, entropy, internal energy, Gibbs and Helmholtz free energies. A l l i i results were compared on the bases of root mean square percent, RMS-%, error and the central processing unit (CPU) time. F i n a l l y , thermodynamic departure plots were produced using the LHC equation. Table of Contents ABSTRACT i i TABLE OF CONTENTS iv LIST OF FIGURES v i i LIST OF TABLES x ACKNOWLEDGEMENTS xix A. INTRODUCTION 1 B. PREVIOUS WORK 12 1 . THE IDEAL GAS 12 2. THE VAN DER WAALS EQUATION 13 3. THE REDLICH-KWONG EQUATION ...19 4. MORE CORRESPONDING STATES 24 5. THE LIELMEZS-HOWELL-CAMPBELL EQUATION 30 6. NON-CUBIC EQUATIONS OF STATE 33 7. PURPOSE OF THIS THESIS 41 iv C. RESEARCH PROCEDURE 43 D. RESULTS AND DISCUSSION 60 1. P-V-T CALCULATIONS 68 2. P-V-T CALCULATIONS BY VIRIAL EXPANSION 73 3. THERMODYNAMIC PROPERTIES 80 4. JOULE-THOMSON COEFFICIENTS AND THE INVERSION CURVE 95 5. SPECIAL STUDY OF NORMAL AND PARA-HYDROGEN 99 6. EVALUATING THE CPU TIME 100 7. SUMMARY 102 E. CONCLUSIONS AND RECOMMENDATIONS 106 1. CONCLUSIONS 106 2. RECOMMENDATIONS 112 NOMENCLATURE 114 REFERENCES 123 v A P P E N D I X A -VAN DER WAALS E Q U A T I O N D E R I V A T I O N S 133 A P P E N D I X B - G E N E R A L I Z E D TWO C O N S T A N T E Q U A T I O N D E R I V A T I O N S 137 A P P E N D I X C - G E N E R A L I Z E D TWO C O N S T A N T E Q U A T I O N V I R I A L E X P A N S I O N S 142 A P P E N D I X D - G E N E R A L I Z E D TWO C O N S T A N T E Q U A T I O N THERMODYNAMIC D E R I V A T I O N S 147 A P P E N D I X E - B E N E D I C T - W E B B - R U B I N E Q U A T I O N THERMODYNAMIC D E R I V A T I O N S 160 A P P E N D I X F - BACK E Q U A T I O N THERMODYNAMIC D E R I V A T I O N S 169 A P P E N D I X G - F I G U R E S 196 A P P E N D I X H - T A B L E S 2 4 3 v i L i s t of F i g u r e s F i g u r e 1: Isotherms i n the L i q u i d and Vapour Phases ...15 F i g u r e 2: I n v e r t e d S q u a r e - w e l l P o t e n t i a l 39 F i g u r e 3: Joule-Thomson E x p a n s i o n P r o c e s s 57 * F i g u r e 4: a vs T f o r Propane 61 F i g u r e 5: Pressure-Volume Diagram f o r Methane 196 F i g u r e 6: Pressure-Volume Diagram f o r Ethane 197 F i g u r e 7: Pressure-Volume Diagram f o r Propane 198 F i g u r e 8: Pressure-Volume Diagram f o r n-Butane 199 F i g u r e 9: Pressure-Volume Diagram f o r n-Pentane 200 F i g u r e 10: Pressure-Volume Diagram f o r n-Hexane 201 F i g u r e 11: Pressure-Volume Diagram f o r Methanol 202 F i g u r e 12: Pressure-Volume Diagram f o r t - B u t a n o l 203 F i g u r e 13: Pressure-Volume Diagram f o r Water 204 F i g u r e 14: Pressure-Volume Diagram f o r Hydrogen S u l f i d e 205 F i g u r e 15: Pressure-Volume Diagram f o r Normal Hydrogen ..206 F i g u r e 16: Pressure-Volume Diagram f o r Para Hydrogen 207 F i g u r e 17: Pressure-Volume Diagram f o r Neon 208 F i g u r e 18: Pressure-Volume Diagram f o r Argon 209 F i g u r e 1 9 : E n t h a l p y D e p a r t u r e f o r Methane 210 F i g u r e 20: E n t h a l p y D e p a r t u r e f o r Ethane 211 F i g u r e 21: E n t h a l p y D e p a r t u r e f o r Propane 212 F i g u r e 22: E n t h a l p y D e p a r t u r e f o r n-Butane 213 v i i Figure 23: Enthalpy Departure for n-Pentane 214 Figure 24: Enthalpy Departure for n-Hexane 215 Figure 25: Enthalpy Departure for Methanol 216 Figure 26: Enthalpy Departure for t-Butanol 217 Figure 27: Enthalpy Departure for Water .....218 Figure 28: Enthalpy Departure for Hydrogen Sulfide 219 Figure 29: Enthalpy Departure for Normal Hydrogen 220 Figure 30: Enthalpy Departure for Para Hydrogen 221 Figure 31: Enthalpy Departure for Neon 222 Figure 32: Enthalpy Departure for Argon 223 Figure 33: Entropy Departure for Methane 224 Figure 34: Entropy Departure for Ethane 225 Figure 35: Entropy Departure for Propane 226 Figure 36: Entropy Departure for n-Butane 227 Figure 37: Entropy Departure for n-Pentane 228 Figure 38: Entropy Departure for n-Hexane ..229 Figure 39: Entropy Departure for Methanol 230 Figure 40: Entropy Departure for t-Butanol 231 Figure 41: Entropy Departure for Water 232 Figure 42: Entropy Departure for Hydrogen Sulfide 233 Figure 43: Entropy Departure for Normal Hydrogen 234 Figure 44: Entropy Departure for Para Hydrogen 235 Figure 45: Entropy Departure for Neon 236 vi i i Figure 46: Entropy Departure for Argon 237 Figure 47: Inversion Curve for Methane 238 Figure 48: Inversion Curve for Propane 239 Figure 49: Inversion Curve for n-Butane 240 Figure 50: Inversion Curve for Para Hydrogen .....241 Figure 51: Inversion Curve for Argon 242 ix L i s t of Tables Table 1: Overall Volumetric RMS-% Error ..69 Table 2: Overall Pressure RMS-% Error 71 Table 3: Overall Temperature RMS-% Error 72 Table 4: Volumetric V i r i a l Expansion Volume- Calculation Error 74 Table 5: Volumetric V i r i a l Expansion- Pressure Calculation RMS-% Error 75 Table 6: Volumetric V i r i a l Expansion- Temperature Calculation RMS-% Error 76 Table 7: Pressure V i r i a l Expansion- Volume Calculation RMS-% Error 77 Table 8: Pressure V i r i a l Expansion- Pressure Calculation RMS-% Error 78 Table 9: Pressure V i r i a l Expansion- Temperature Calculation RMS-% Error . .79 Table 10: Enthalpy Departure RMS-% Error 82 Table 11: Entropy Departure RMS-% Error 85 Table 12: Helmholtz Free Energy RMS-% Error 86 Table 13: Gibbs Free Energy Departure RMS-% Error 88 Table 15: Internal Energy Departure RMS-% Error 89 Table 15: Fugacity RMS-% Error 91 Table 16: Isobaric Heat Capacity Departure RMS-% Error 92 x Table 17: Isometric Heat Capacity Departure RMS-% Error 93 Table 18: Joule-Thomson Coe f f i c i e n t RMS-% Error 98 Table 19: Inversion Curve RMS-% Error 99 Table 20: Volume RMS-% Error/CPU Time for Region I 243 Table 21: Volume RMS-% Error/CPU Time for Region II .....243 Table 22: Volume RMS-% Error/CPU Time for Region III 244 Table 23: Volume RMS-% Error/CPU Time for Region IV 245 Table 24: Volume RMS-% Error/CPU Time for Region V 246 Table 25: Pressure RMS-% Error/CPU Time for Region II 247 Table 26: Pressure RMS-% Error/CPU Time for Region IV 248 Table 27: Pressure RMS-% Error/CPU Time for Region V 249 Table 28: Temperature RMS-% Error/CPU Time for Region I 250 Table 29: Temperature RMS-% Error/CPU Time for Region II 250 Table 30: Temperature RMS-% Error/CPU Time for Region III ....251 Table 31: Temperature RMS-% Error/CPU Time for Region IV 252 Table 32: Temperature RMS-% Error/CPU Time for Region V ......253 Table 33: V i r i a l Expansion in Volume to Solve for V in Region I I - RMS-% Error/CPU time 253 Table 34: V i r i a l Expansion in Volume to Solve for V in Region IV- RMS-% Error/CPU time 254 Table 35: V i r i a l Expansion in Volume to Solve for V in Region V- RMS-% Error/CPU time 255 x i Table 36: V i r i a l Expansion in Volume to Solve for P in Region I I - RMS-% Error/CPU time 255 Table 37: V i r i a l Expansion in Volume to Solve for P in Region IV- RMS-% Error/CPU time 256 Table 38: V i r i a l Expansion in Volume to Solve for P in Region V- RMS-% Error/CPU time 257 Table 39: V i r i a l Expansion in Volume to Solve for T in Region I I - RMS-% Error/CPU time 257 Table 40: V i r i a l Expansion in Volume to Solve for T in Region IV- RMS-% Error/CPU time 258 Table 41: V i r i a l Expansion in Volume to Solve for T in Region V- RMS-% Error/CPU time 259 Table 42: V i r i a l Expansion in Pressure to Solve for V in Region I I- RMS-% Error/CPU time 259 Table 43: V i r i a l Expansion in Pressure to Solve for V in Region IV- RMS-% Error/CPU time 260 Table 44: V i r i a l Expansion in Pressure to Solve for V in Region V- RMS-% Error/CPU time 261 Table 45: V i r i a l Expansion in Pressure to Solve for P in Region I I - RMS-% Error/CPU time 261 Table 46: V i r i a l Expansion in Pressure to Solve for P in Region IV- RMS-% Error/CPU time 262 Table 47: V i r i a l Expansion in Pressure to Solve for P in Region V- RMS-% Error/CPU time 262 Table 48: V i r i a l Expansion in Pressure to Solve for T in Region I I - RMS-% Error/CPU time 263 Table 49: V i r i a l Expansion in Pressure to Solve for T in Region IV- RMS-% Error/CPU time 263 Table 50: V i r i a l Expansion in Pressure to Solve for T in Region V- RMS-% Error/CPU time 264 Table 51: Enthalpy for Region I- RMS-% Error/CPU time 265 Table 52: Enthalpy for Region I I - RMS-% Error/CPU time 265 Table 53: Enthalpy for Region I I I - RMS-% Error/CPU time 266 Table 54: Enthalpy for Region IV- RMS-% Error/CPU time 266 Table 55: Enthalpy for Region V- RMS-% Error/CPU time 267 Table 56: Entropy for Region I- RMS-% Error/CPU time 267 Table 57: Entropy for Region I I - RMS-% Error/CPU time 268 Table 58: Entropy for Region I I I - RMS-% Error/CPU time ..268 Table 59: Entropy for Region IV- RMS-% Error/CPU time 269 Table 60: Entropy for Region V- RMS-% Error/CPU time 269 Table 61: Helmholtz Free Energy for Region I- RMS-% Error/CPU time 270 Table 62: Helmholtz Free Energy for Region I I - RMS-% Error/CPU time 270 Table 63: Helmholtz Free Energy for Region I I I - RMS-% Error/CPU time 271 Table 64: Helmholtz Free Energy for Region IV- RMS-% Error/CPU time 271 Table 65: Helmholtz Free Energy for Region V- RMS-% Error/CPU time 272 Table 66: Gibbs Free Energy for Region I- RMS-% Error/CPU time 272 Table 67: Gibbs Free Energy for Region I I - RMS-% Error/CPU time 273 Table 68: Gibbs Free Energy for Region I I I - RMS-% Error/CPU time 273 Table 69: Gibbs Free Energy for Region IV- RMS-% Error/CPU time 274 Table 70: Gibbs Free Energy for Region V- RMS-% Error/CPU time 274 Table 71: Internal Energy for Region I- RMS-% Error/CPU time .275 Table 72: Internal Energy for Region I I - RMS-% Error/CPU time 275 Table 73: Internal Energy for Region I I I - RMS-% Error/CPU time 276 Table 74: Internal Energy for Region IV- RMS-% Error/CPU time 276 Table 75: Internal Energy for Region V- RMS-% Error/CPU time .277 x i v T a b l e 7 6 : F u g a c i t y f o r R e g i o n I - RMS-% E r r o r / C P U t i m e 2 7 7 T a b l e 7 7 : F u g a c i t y f o r R e g i o n I I - RMS-% E r r o r / C P U t i m e 2 7 8 T a b l e 7 8 : F u g a c i t y f o r R e g i o n I I I - RMS-% E r r o r / C P U t i m e 278 T a b l e 7 9 : F u g a c i t y f o r R e g i o n I V - RMS-% E r r o r / C P U t i m e 2 7 9 T a b l e 8 0 : F u g a c i t y f o r R e g i o n V - RMS-% E r r o r / C P U t i m e 2 7 9 T a b l e 8 1 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n I - RMS-% E r r o r / C P U t i m e 2 8 0 T a b l e 8 2 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n I I - RMS-% E r r o r / C P U t i m e 2 8 0 T a b l e 8 3 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n I I I - RMS-% E r r o r / C P U t i m e 281 T a b l e 8 4 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n I V - RMS-% E r r o r / C P U t i m e 281 T a b l e 8 5 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n V - RMS-% E r r o r / C P U t i m e 2 8 2 T a b l e 8 6 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I - RMS-% E r r o r / C P U t i m e 2 8 2 T a b l e 8 7 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I I - RMS-% E r r o r / C P U t i m e 2 8 3 T a b l e 8 8 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I I I - RMS-% E r r o r / C P U t i m e 2 8 3 T a b l e 8 9 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I V - RMS-% E r r o r / C P U t i m e 2 8 3 x v T a b l e 9 0 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n V - RMS-% E r r o r / C P U t i m e 2 8 4 T a b l e 9 1 : P r o p a n e - T h e r m o d y n a m i c P r o p e r t i e s A l o n g t h e C r i t i c a l I s o t h e r m - RMS-% E r r o r / C P U t i m e 2 8 4 T a b l e 9 2 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I - RMS-% E r r o r / C P U t i m e 2 8 4 T a b l e 9 3 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I I - RMS-% E r r o r / C P U t i m e 2 8 5 T a b l e 9 4 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I I I - RMS-% E r r o r / C P U t i m e 2 8 5 T a b l e 9 5 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I V - RMS-% E r r o r / C P U t i m e 2 8 6 T a b l e 9 6 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n V - RMS-% E r r o r / C P U t i m e 2 8 6 T a b l e 9 7 : I n v e r s i o n C u r v e - RMS-% E r r o r / C P U t i m e 2 8 7 T a b l e 9 8 : L H C e q ( 3 . 3 ) / G r a b o s k i E q u a t i o n C o m p a r i s o n f o r n - H 2 -RMS-% E r r o r / C P U t i m e 288 T a b l e 9 9 : L H C e q ( 3 . 3 ) f o r p - H 2 - RMS-% E r r o r / C P U t i m e 288 T a b l e 1 0 0 : L i e l m e z s - H o w e l l - C a m p b e l l E q u a t i o n C o e f f i c i e n t s . . . . 2 8 9 T a b l e 1 0 1 : T e m p e r a t u r e R a n g e s f o r t h e A p p l i c a t i o n o f E q u a t i o n ( 3 . 3 ) o n H r , S r , A r , G r , U r a n d f / P 2 9 0 T a b l e 1 0 2 : T e m p e r a t u r e R a n g e s f o r t h e A p p l i c a t i o n o f E q u a t i o n ( 3 . 3 ) o n C p a n d C v 2 9 0 x v i Table 103: Temperature Ranges for the Application of Equation (3.3) on Joule-Thomson C o e f f i c i e n t s and the Inversion Curve 291 Table 104: P-V-T Input Data for Region I 292 Table 105: P-V-T Input Data for Region II ;....292 Table 106: P-V-T Input Data for Region III 293 Table 107: P-v-T Input Data for Region IV 294 Table 108: P-V-T Input Data for Region V 295 Table 109: Thermodynamic Property Input Data for Region I ....296 Table 110: Thermodynamic Property Input Data for Region II ...297 Table 111: Thermodynamic Property Input Data for Region III ..298 Table 112: Thermodynamic Property Input Data for Region IV ...299 Table 113: Thermodynamic Property Input Data for Region V ....300 Table 114: Enthalpy and Entropy Departure Function Data-Region I 301 Table 115: Enthalpy and Entropy Departure Function Data-Region II 302 Table 116: Enthalpy and Entropy Departure Function Data-Region III 303 Table 117: Enthalpy and Entropy Departure Function Data-Region IV 304 Table 118: Enthalpy and Entropy Departure Function Data-Region V 305 xvi i Table 119: Cp and Cv Property Input Data for Region I 306 Table 120: Cp and Cv Property Input Data for Region II 306 Table 121: Cp and Cv Property Input Data for Region III 307 Table 122: Cp and Cv Property Input Data for Region IV 307 Table 123: Cp and Cv Property Input Data for Region V ........308 Table 124: Joule-Thomson Property Input Data for Regions I and II 309 Table 125: Joule-Thomson Property Input Data for Region III ..309 Table 126: Joule-Thomson Property Input Data for Region IV ...310 Table 127: Joule-Thomson Property Input Data for Region V ....311 Table 128: Inversion Curve Input Data 312 xvi i i ACKNOWLEDGEMENTS I wish to thank Professor Janis Lielmezs for his leadership, i n s p i r a t i o n , guidance, and support during t h i s project. I am also indebted to Dr. P h i l i p H i l l , of the U.B.C. Mechanical Engineering Department for supplying me with and dire c t i n g me to various references. I would l i k e to thank Mrs. Hana Aleman for many useful ideas and suggestions. A very special thankyou is extended to Ms. Kim Burton and Ms. Natasha Aruliah for the i r patience during my extended, d a i l y computer sessions. F i n a l l y , the f i n a n c i a l assistance of the Natural Sciences and Engineering Research Council of Canada is gr a t e f u l l y acknowledged. x ix A. I N T R O D U C T I O N T h e L i e l m e z s - H o w e l l - C a m p b e l l ( L H C ) e q u a t i o n o f s t a t e a t p r e s e n t i s o n l y a p p l i c a b l e f o r d e t e r m i n i n g t h e P - V - T p r o p e r t i e s a l o n g t h e s a t u r a t e d v a p o u r - l i q u i d e q u i l i b r i u m c u r v e . T h e p u r p o s e o f t h i s t h e s i s i s t o d e v e l o p t h e L H C e q u a t i o n , f o r p u r e c o m p o u n d s , a s a v i a b l e a l t e r n a t i v e t o some m o r e c o m m o n l y u s e d e q u a t i o n s o f s t a t e i . e . t h e R e d l i c h - K w o n g , S o a v e 1 9 7 2 , B e n e d i c t - W e b b - R u b i n (BWR), L e e - K e s l e r a n d B o u b l i k - A l d e r - C h e n - K r e g l e w s k i ( B A C K ) e q u a t i o n s . T h e L H C e q u a t i o n w i l l b e c a p a b l e o f c a l c u l a t i n g b o t h P - V - T a n d t h e r m o d y n a m i c p r o p e r t i e s i n b o t h t h e s u b c r i t i c a l a n d s u p e r c r i t i c a l r e g i o n s . A t t h e s a m e t i m e , c a l c u l a t i o n s w i l l b e c o m p a r e d t o t h o s e o f t h e e q u a t i o n s o f s t a t e m e n t i o n e d a b o v e , o n t h e b a s e s o f a c c u r a c y ( RMS-% e r r o r ) a n d s p e e d o f c a l c u l a t i o n ( C P U t i m e ) . T h e n e e d f o r a c c u r a t e 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 ( P - V - T ) a n d t h e r m o d y n a m i c d a t a i s b e c o m i n g m o r e a n d m o r e p r e s s i n g i n t h e c h e m i c a l p r o c e s s i n d u s t r i e s . P r e s e n t e m p h a s i s i s o n o p t i m i z a t i o n , c o s t r e d u c t i o n , a n d e f f i c i e n c y . A c c u r a c y i s p a r t i c u l a r l y i m p o r t a n t i n t h e c r i t i c a l r e g i o n , w h e r e a n e r r o r i n t h e t h i r d s i g n i f i c a n t f i g u r e f o r i n s t a n c e c a n r e s u l t i n l e g a l d i s p u t e s p e r t a i n i n g t o m i l l i o n s o f d o l l a r s i n l o s t r e v e n u e , p l a n t s t o r a g e p r o b l e m s , o r p r o b l e m s w i t h t h e s i m u l t a n e o u s u s e o f a p i p e l i n e b y s e v e r a l f i r m s ( 5 4 ) . T h e e q u a t i o n o f s t a t e i s t h e m o s t c o m m o n l y 2 used method of generating P-V-T and thermodynamic properties. Industrial use of an equation of state has documented some c a p a b i l i t i e s and complications (20) such as the need to add more pure component parameters in order to increase the f l e x i b i l i t y of the equation. These new parameters would only be used in special cases l i k e for polar and quantal f l u i d s , or in the c r i t i c a l region. Some equations of state l i k e the Soave 1972, Lee-Kesler, BWR and BACK equations have c o e f f i c i e n t s that are based on empirical data i . e . acentric factors or reference f l u i d s . However, the accuracy of the c o e f f i c i e n t s is dependent on the accuracy of the empirical data upon which they are based. Therefore the r e l i a b i l i t y of the c o e f f i c i e n t s declines proportionally to the a v a i l a b i l i t y and accuracy of the input data, bringing their a p p l i c a b i l i t y more into question when regions l i k e the c r i t i c a l or compressed l i q u i d regions are poorly described by P-V-T data. There are regions where data are predicted quite accurately (vapours) and regions where the exact opposite i s true (compressed l i q u i d s and c r i t i c a l f l u i d s ) . The main problem i s that these areas are not well documented, even for normal f l u i d s due to a lack of experimental data upon which to compare the accuracy of calculated to experimental values. The underlying reasons for f a i l u r e are an attempt to f i t an equation over too wide a data range without having the necessary empirical 3 data available upon which to base or check the f i t , or an i n a b i l i t y to c a r e f u l l y examine the subsequent P-V-T and thermodynamic property curves generated from the new equation (20,21). A l l equations of state are intended to summarize large masses of P-V-T data, while at the same time to minimize computer storage requirements and costly search procedures (37,51). Furthermore, when the appropriate integrations and d i f f e r e n t i a t i o n s are c a r r i e d out, the researcher has access to any sought after thermodynamic property (38). Interpolation and extrapolation in this manner are obviously faster and more accurate than those by graphical means. A l l state equations may accurately predict P-V-T properties but this is no assurance that the thermodynamic properties, which cannot be measured d i r e c t l y but rather are determined from derivatives of the equation of state, w i l l likewise be accurately predicted. For a pure substance, the likehood that thermodynamic properties w i l l be accurately predicted i s increased by requiring that an equation of state provide i d e n t i c a l fugacities for both coexisting phases a l l along the saturated vapour-liquid equilibrium curve (19,52). Three decisions need to be made in setting up an equation of state (1): (i) the type and amount of required input data, ( i i ) the data range to be covered and ( i i i ) the accuracy of the 4 calculated r e s u l t s . Obviously, the most desirable equation would require minimal input data, for instance the c r i t i c a l coordinates of Tc,Pc, and Vc, supplemented by certain molecular properties, a l l of which would be readily obtainable. C r i t i c a l coordinates are associated with corresponding states. Two compounds are said to be in corresponding states when they have the same reduced coordinates Tr=T/Tc, Pr=P/Pc and Vr=V/Vc (6). Corresponding states provide a method for generalizing many compounds based upon the P-V-T data of one reference compound. Unfortunately t h i s only applies to moderately large non-spherical.molecules l i k e propane or n-butane (40,41) while polar f l u i d s (water, ammonia, methanol) and quantum f l u i d s (n-H 2, p-H2,neon) are poorly described. Additional properties l i k e molecular weight, dipole moment or de Boer's quantum mechanical parameter would allow for substance s p e c i f i c i t y in treating molecular anomalies l i k e p o l a r i t y and quantum e f f e c t s . The resultant equation of state would therefore have both generalized and substance s p e c i f i c c h a r a c t e r i s t i c s . At the same time the equation of state should y i e l d accurate results in such anomalous regions as the c r i t i c a l point and along the saturation curve, where the l i q u i d and vapour phases are in equilibrium at the same temperature and pressure (1,6). The c r i t i c a l point, in p a r t i c u l a r , is the region of greatest importance because the f l u i d may display bulk 5 c h a r a c t e r i s t i c s unlike those of either l i q u i d s or vapours (opacity, volumetric f l u c t u a t i o n s ) , while at the same time display l i q u i d or vapour c h a r a c t e r i s t i c s in isola t e d regions. Such regional behaviour changes make property measurement d i f f i c u l t and highly inaccurate under these conditions. Unfortunately, most state equations are therefore also inaccurate in the c r i t i c a l region due to uncertainties in the experimental data upon which they are based. F i n a l l y , the most desirable equation of state would cover as large a P-V-T data range as possible with maximum accuracy. The study of mathematical relationships between pressure, volume and temperature for a pure f l u i d can be either t h e o r e t i c a l or empirical. F i r s t , the t h e o r e t i c a l approach i s discussed. Here, kinetic theory or s t a t i s t i c a l mechanics are concerned with the treatment of intermolecular forces (1). Van der Waals' equation is of the th e o r e t i c a l variety. Molecules are perceived as being r i g i d spheres with long range a t t r a c t i v e forces drawing them together, and repulsive forces separating any that are in contact. Defining separate mathematical terms for each force, Van der Waals created an equation of state that connected a l l f l u i d phases, including the saturated liquid-vapour equilibrium curves, and ultimately proposed the "law of corresponding states" (1,43,99). The theoretical approach however provided a limited 6 range of application, useful for vapours only. Venturing beyond the scope of an equation of t h i s type can lead to absurd results (5), as was the case when the Van der Waals equation was applied to the compressed l i q u i d and c r i t i c a l regions. The results were very inaccurate (43). Where theoreti c a l equations are b e n e f i c i a l is that they provide a s t a r t i n g point from which to proceed with the second type of equation, which i s the empirical variety. One r e c a l l s that the Redlich-Kwong, Soave 1972 and Peng-Robinson equations (8,13,60) are a l l based upon the Van der Waals equation. More w i l l be said about Van der Waals' equation and i t s successors later in t h i s chapter. As mentioned above, the second avenue, and perhaps the most successful, i s the empirical or semi-empirical approach. An equation of t h i s variety provides results that are accurate over wide density ranges (1) because the equation i t s e l f i s based upon experimental P-V-T data, and hence more clos e l y r e f l e c t s the actual P-V-T surface. The emphasis here i s not upon molecular c h a r a c t e r i s t i c s or interaction, but on the predictive accuracy of the equation for design purposes. There are problems that ari s e when one i s forced to extend an empirical state equation outside i t s intended region of use (20). Again the compressed l i q u i d and c r i t i c a l regions are s t i l l inaccurately described, a l b e i t they are an improvement over the 7 theoretical approach. Also quantum and polar compounds are not well described by the corresponding states approach. Furthermore, in certain instances, mathematical solutions to the equations are simply impossible. For instance, the equations proposed by Joffe and Zudkevitch (24) or Lu and co-workers (29-32) have t h i s problem. These two equations work well along the saturation curve and in the surrounding s u b c r i t i c a l l i q u i d and vapour regions, but f a i l in the s u p e r c r i t i c a l region. The temperature dependent function is not designed to accomodate temperatures greater than Tc. Prior to Van der Waals 1 work, kinetic theory perceived f l u i d s as being mathematical points having neither volume nor molecular interaction. These were the c h a r a c t e r i s t i c s of a so-called "ideal gas" (6,32). In 1873, Van der Waals introduced two constants: a and b. The former was a cohesion parameter (32) to account for the forces of attraction between molecules (Van der Waals forces), while the l a t t e r corrected for the repulsive forces and the f i n i t e size of a molecule. By assuming the c r i t i c a l isotherm to have a double i n f l e c t i o n at the c r i t i c a l point, Van der Waals established compound dependent values for a and b based on the c r i t i c a l point coordinates. More w i l l be discussed on t h i s in chapter 2, but the double i n f l e c t i o n point c r i t e r i o n i s the primary c h a r a c t e r i s t i c of t h i s and a l l subsequent cubic equations of state (2). Along the saturation curve, the Van der Waals 8 equation has three real roots in terms of the compressibility factor, Z; the smallest and the largest roots are l i q u i d and vapour values respectively, while the intermediate root has no physical significance other than to id e n t i f y a region of metastable equilibrium associated with Maxwell's equal area rule (3). Outside the saturation curve the cubic equation yields only one root. The e a r l i e s t attempt to improve the Van der Waals equation was by Redlich and Kwong in 1949. They modified the Van der Waals equation by changing the a t t r a c t i o n term. The c o e f f i c i e n t "a" e s s e n t i a l l y became a temperature dependent function based upon fugacity equilibrium conditions along the saturated vapour-liquid equilibrium curve mentioned e a r l i e r . The c o e f f i c i e n t "b" remained constant, but was subsequently introduced into the a t t r a c t i o n term. The end result was increased accuracy in predicting compressed l i q u i d volumes. The modified at t r a c t i o n term more cl o s e l y approximates the steep isotherms associated with the compressed l i q u i d region of the P-V curve than did the Van der Waals equation. In the vapour region, the volume, V, i s generally no less than two orders of magnitude larger than b, so the l a t t e r in n e g l i g i b l e by comparison. In subsequent years, additional attempts have been made to improve the Redlich-Kwong equation by introducing a t h i r d 9 corresponding states parameter and modifying the temperature dependent f u n c t i o n a l i t y of the c o e f f i c i e n t "a" based on fugacity.. equilibrium along the liquid-vapour saturation curve, a l l with some degree of success; the work of Wilson (5), Barner (5), Chueh and Prausnitz (5), Joffe and Zudkevitch (12), Lu et a l (29-32), Soave (8,9,10), Raimondi (14), F u l l e r (15) are just some examples. Lielmezs, Howell and Campbell (LHC) also modified the * Redlich-Kwong equation by applying the newly developed T coordinates introduced by the f i r s t author (36,62-64). It i s the further development of their equation which i s the subject of this thesis, and more w i l l be said in subsequent chapters. Cubic equations of state based on the Redlich-Kwong model have certain l i m i t a t i o n s (5). F i r s t , polar and quantum f l u i d s are poorly handled since f i t t i n g the vapour pressure curve and obtaining fugacity equilibrium i s d i f f i c u l t (10). Second, inspite of a l l the improvements r e s u l t i n g from the transformation of the cohesion parameter "a" into various temperature dependent functions and the introduction of a t h i r d corresponding states parameter, the c o n t r o l l i n g factor remains the volume dependency of the equation. One has only to note that predictive accuracy for volumetric calculations in the compressed l i q u i d region was improved every time the constant value of Zc was reduced, due to a modification in the volume dependency of the a t t r a c t i o n 1 0 parameter i . e . Van der Waals (Zc=0.375), Redlich-Kwong and Soave 1972 (Zc=0.333), Peng-Robinson (Zc=0.3074). Adjusting the temperature fu n c t i o n a l i t y results in diminishing or negative returns (5). The accuracy of an empirical equation i s only as good as the data upon which i t is based. One improvement may lower the RMS-% error by 0.5% for one compound and yet raise i t by the same amount for another. In most instances the accuracy of the equation is improved by a fraction of a percentage point (0.1-0.5%) in P-V-T calculations due to a l t e r i n g the temperature dependence of the attraction term. This c r i t i c a l point i n f l e x i b i l i t y is a problem inherent in corresponding states theory. Volumetric calculations for compounds such as argon, hydrogen sulphide, methane, ethane and propane (Zc range 0.27-0.29) y i e l d errors as high as 20% at the the c r i t i c a l point (5,34). Since this weakness is inherent in cubic equations as a cl a s s , one must strik e a p r a c t i c a l balance between accuracy on the one hand and s i m p l i c i t y and g e n e r a l i z a b i l i t y on the other (5). Also available for P-V-T calculations and thermodynamic property prediction are the more complex non-cubic equations of state, a l l obeying the Van der Waals c r i t e r i o n that the c r i t i c a l isotherm have a double i n f l e c t i o n at the c r i t i c a l point. Some of the more popular non-cubic equations are the 8 constant 11 Benedict-Webb-Rubin (BWR) e q u a t i o n ( 3 7 - 3 9 ) , the Lee-Kesler equation (40-42), a generalized BWR equation with 13 constants and the Boublik-Alder-Chen-Kreglewski equation (45-48) with 24 constants. Because of their complexities, the above equations are more d i f f i c u l t to solve by hand or to program into a computer (21). Therefore working with cubic equations not only seems more at t r a c t i v e in repet i t i v e calculations (5), but also when coupled by the fact that there i s generally no advantage in predictive accuracy in using a non-cubic equation over a cubic equation for thermodynamic property prediction. Both classes of equation maintain roughly the same l e v e l of accuracy. This w i l l hopefully become apparent as comparisons are made between the predictive accuracies (RMS-% error) and c a l c u l a t i o n (CPU) times, when both cubic and non-cubic equations are used to calculate P-V-T and thermodynamic properties. 1 2 B. PREVIOUS WORK An equation of state is a mathematical relationship, for gases and l i q u i d s , between the state parameters pressure, volume and temperature: (2.1) f(P,V,T)=0 Equation (2.1) is c h a r a c t e r i s t i c of a l l state equation models, whether i t may represent an ideal or real f l u i d . 1. THE IDEAL GAS We may define, at vanishing pressure (P -> 0), an equation of state for the "ideal gas" as : (2.2) PV=RT where the mechanical work (PV) and the kinetic (thermal) energy (RT) of the system (87) are related. This equation can be derived 13 from the simple kinetic theory of gases assuming that molecules are point masses and that there are no intermolecular forces between them (6) i.e. the energy states of the system are only temperature dependent. However for real gases, these assumptions are i n v a l i d . In real gases, molecules possess f i n i t e volumes, and there e x i s t s intermolecular forces of attraction and repulsion, that i s i f the pressure increases and the s p e c i f i c volume decreases, the volume occupied by the molecules becomes signi f icant. To account for these considerations, in view of Andrews' (100) experimental findings, Van der Waals, in 1873, proposed an equation of state. 2. THE VAN DER WAALS EQUATION Van der Waals s p l i t the interactive forces of a t t r a c t i o n and repulsion, Pa and Prep respectively, into two terms: (2.3) P=Prep+Pa The a t t r a c t i v e forces had their greatest influence on distant 1 4 molecules, while repulsive forces acted on contacting ones. Since kinetic theory gave the a t t r a c t i v e forces as being proportional to the square of the density (6,43,99), they could be written as Pa=a/V2, where "a" is a constant of prop o r t i o n a l i t y . Van der Waals realized that molecular overlap existed in a dense gas, meaning that the actual volume occupied by the molecules was less than the free volume of the system (26). On the other hand, in the d i l u t e region, there i s l i t t l e or no molecular overlap. The volume occupied by the molecules i s e s s e n t i a l l y the free volume of the system. Assuming that the repulsive forces only affect the kinetic or thermal pressure, Van der Waals expressed the repulsion term as Prep=RT/(V-b). The resul t i n g equation now reads: (2 4) P= R T - ^ U-v z , 4 t ; ^ v-b V 7 In the d i l u t e gas region, there is l i t t l e or no molecular overlap and b is e s s e n t i a l l y n e g l i b l e . As V approaches i n f i n i t y , equation (2.4) reduces to the ideal gas equation. In the dense gas region, the e f f e c t of b on V i s s i g n i f i c a n t , as they are of the same 1 5 o r d e r o f m a g n i t u d e . A s s i g n i n g n u m e r i c a l v a l u e s t o t h e t w o c o n s t a n t s ( A p p e n d i x A ) , a a n d b , r e q u i r e s t h a t a n a s s u m p t i o n b e m a d e a b o u t t h e c r i t i c a l i s o t h e r m ; t h e c r i t i c a l i s o t h e r m i n t e r s e c t s t h e c r i t i c a l p o i n t o n t h e P - V c u r v e a t a d o u b l e i n f l e c t i o n p o i n t ( 1 , 2 ) [ F i g u r e 1 , i s o t h e r m T c ] . F i g u r e 1 : I s o t h e r m s i n t h e L i q u i d a n d V a p o u r P h a s e s M o r e s p e c i f i c a l l y , a l o n g t h e c r i t i c a l i s o t h e r m : ( 2 . 5 ) 16 and ( 2 . 6 ) l d 2 p ) - n Applying the conditions of equations ( 2 . 5 ) and ( 2 . 6 ) to equation ( 2 . 4 ) gives: ( 2 . 7 ) a= |RTcVc and ( 2 . 8 ) Substituting equations ( 2 . 7 ) and ( 2 . 8 ) back into equation ( 2 . 4 ) presents the c r i t i c a l compressibility factor,Zc,as: ( 2 9 ) 7r= P c V c = 2 l ^ . y j RTc 8 1 7 Applying equations (2.7)-(2.9) converts equation (2.4) to reduced form: where Tr, Pr and Vr are the reduced quantities Tr=T/Tc, Pr=P/Pc and Vr=V/Vc respectively. Van der Waals defined "corresponding states" to exist when two substances have i d e n t i c a l reduced coordinates. Equation (2.10) then takes the form of a "law of corresponding states" (99). Equation (2.4) can be transformed into a cubic equation: (2.11) Z3-Z2(B+1)+ZA-AB=0 (2.10) Pr = 8 T r 3 3Vr-1 Vr 2 where: (2.12) Z = PV RT (2.13) A= aP R 2T 2 18 and (2.14) B= bP RT B e c a u s e of t h e Van d e r Waals c r i t e r i o n of d o u b l e i n f l e c t i o n a t t h e c r i t i c a l p o i n t , e q u a t i o n (2.11) has 3 r e a l r o o t s i n Z a l o n g t h e s a t u r a t i o n c u r v e [ F i g u r e 1, i s o t h e r m T 2 ] . The s m a l l e s t and l a r g e s t r o o t s a r e t h e s a t u r a t e d l i q u i d and v a p o u r v a l u e s r e s p e c t i v e l y . The i n t e r m e d i a t e r o o t has no p h y s i c a l s i g n i f i c a n c e o t h e r t h a n t o d e f i n e a r e g i o n of m e t a s t a b l e e q u i l i b r i u m under t h e s a t u r a t i o n c u r v e , a s s o c i a t e d w i t h M a x w e l l ' s e q u a l a r e a r u l e . ( M a x w e l l ' s e q u a l a r e a r u l e s t a t e s t h a t t h e v a p o u r p r e s s u r e a t c o e x i s t e n c e f o r each t e m p e r a t u r e (T<Tc) i s t h e i s o b a r t h a t i n t e r s e c t s t h e l o o p c o n n e c t i n g b o t h e x p e r i m e n t a l p a r t s o f t h e i s o t h e r m i n t h e c o e x i s t e n c e r e g i o n and g e n e r a t e s two e q u a l a r e a s [ F i g u r e 1, i s o t h e r m T 2 ] ) O u t s i d e t h e s a t u r a t i o n c u r v e [ F i g u r e 1, i s o t h e r m T , ] , e q u a t i o n (2.11) y i e l d s o n l y one r e a l r o o t ( 3 ) . The c o m p l e t e d e r i v a t i o n o f e q u a t i o n s (2.7) t o (2.14) c a n be seen i n A p p e n d i x A. Van d e r Waals t r i e d t o c o r r e c t e q u a t i o n (2.4) f o r some d i s c r e p e n c i e s i n t h e c r i t i c a l r e g i o n due t o m o l e c u l a r a s s o c i a t i o n (50) but t h i s r e m a i n e d l a r g e l y u n s u c c e s s f u l . 1 9 In recent years, several teams of researchers, Viswanath et a l (25,26), Adachi and Lu (32), and Law and Lielmezs (35) have modified the o r i g i n a l Van der Waals equation. Viswanath and co-workers rendered both constants, a and b, into functions of volume. The cohesive force was considered to be weak function of volume while b was transformed into a logarithmic function in V. The other research teams, Adachi and Lu, Law and Lielmezs, made the c o e f f i c i e n t "a" into a temperature dependent function. The former investigation used a logarithmic function with a c o e f f i c i e n t generated in a manner similar to that of the Soave 1972 equation (8). The l a t t e r workers used a function i d e n t i c a l to that of the Lielmezs-Howell-Campbell equation (7), about which more w i l l be said later in t h i s chapter. 3. THE REDLICH-KWONG EQUATION The f i r s t successful attempt to improve the Van der Waals equation was by Redlich and Kwong in 1949 (60).The two constants, a and b, introduced by Van der Waals, were expressed in terms of dummy parameters ,J2a and fib, which are universal constants for a l l compounds (24). They also modified the attraction term of equation (2.3) thereby making i t both a function of reduced temperature, and quadratic in volume. The resulting equation was: 20 (2.15) P- - T r o-4(v+b) where: (2.16) a= n a R 2 T c 2 Pc and (2.17) b= ObRIc At the c r i t i c a l point temperature,Tc, equation (2.15) may be expressed as: ( 2 . , 8 ) P . Sl| - v T_|_ T Following the derivation (Appendix B) of Epstein (17), for a generalized Redlich-Kwong type equation, we have: fta=0.4274802334. 21 J2b=0. 0866403500. . . and Zc=0.333 Equation (2.18) may be transformed into cubic form by applying equations (2.12)-(2.14) thereby y i e l d i n g : (2.19) Z3-Z2+Z(A-B-B2)-AB=0 Equation (2.19) has 3 real roots in Z along the saturation curve with the smallest and largest roots being the saturated l i q u i d and vapour values respectively. The intermediate root has no physical s i g n i f i c a n c e . Only one real root exists outside the saturation curve [Figure 1]. The Redlich-Kwong equation does not perform well for l i q u i d density c a l c u l a t i o n s of complex f l u i d s (long chain hydrocarbons, polar and quantum f l u i d s ) due to the inadequate form of the temperature function in the a t t r a c t i o n term (5,8,13). It however 22 improved on the d e s c r i p t i o n of the h i g h d e n s i t y r e g i o n of s i m p l e f l u i d s ( a r g o n , k r y p t o n , a n d xenon) i n the c r i t i c a l r e g i o n compared to the Van der Waals e q u a t i o n , but i t was s t i l l i n a c c u r a t e ( 5 ) . There have been s e v e r a l a t t e m p t s t o improve the Redlich-Kwong e q u a t i o n by e x p r e s s i n g the temperature dependence of the a t t r a c t i o n term i n a d i f f e r e n t f u n c t i o n a l form. Chueh and P r a u s n i t z d e f i n e d t emperature dependent, e f f e c t i v e c r i t i c a l c o n s t a n t s w i t h which the p r o p e r t i e s of quantum f l u i d s can be made to c o i n c i d e w i t h those of normal f l u i d s l i k e argon or n-butane, w h i l e m a i n t a i n i n g the form of e q u a t i o n ( 2 . 1 5 ) . They r e p l a c e d the u n i v e r s a l c o n s t a n t s of the Redlich-Kwong e q u a t i o n , fta and Rb, w i t h c o n s t a n t s e v a l u a t e d from the s a t u r a t e d v o l u m e t r i c p r o p e r t i e s of each pure compound. These two phase dependent s e t s of c o n s t a n t s a r e r e s t r i c t e d t o c a l c u l a t i n g molar volumes a t s a t u r a t i o n f o r Tr not l e s s than 0.56 ( 2 8 ) . J o f f e e t a l (12,20,24) s u b s t i t u t e d P-V-T d a t a , f o r s e l e c t e d pure compounds, i n t o the Redlich-Kwong e q u a t i o n , t h e r e b y s o l v i n g f o r vapour p r e s s u r e , w h i l e m a i n t a i n i n g f u g a c i t y e q u i l i b r i u m . The r e s u l t i n g J2a and flb v a l u e s produced f o r each p o i n t were then used t o g e n e r a t e two s e t s of temperature dependent f u n c t i o n s , fta(T) and ftb(T). One s e t d i d a p p l y t o s a t u r a t e d l i q u i d s and the o t h e r t o s a t u r a t e d v apours. 23 Lu et a l (18,29-32) made Ra and Rb into functions correlated in terms of reduced temperature,using the same method as Joffe and co-workers. Their r e s u l t s are s l i g h t l y deceptive however, as they f a i l to compare their r e s u l t s with those of other researchers. The question therefore remained unanswered about whether their work represented any s i g n i f i c a n t improvement over the previous methods. Enthalpy departures were determined at the normal b o i l i n g point only, rather than along the whole saturation curve or outside i t , again an incomplete comparison. This work w i l l show that the accuracy of an equation cannot be based so l e l y upon the results of one point of reference. The accuracy of a c a l c u l a t i o n varies d r a s t i c a l l y from the s u p e r c r i t i c a l region to the saturated vapour region with each individual compound. Harmens (34) also generated temperature dependent functions for Ra and Rb, but the c r i t i c a l point values remain at those given by Redlich and Kwong (17,60). The functions are incapable of generating Ra and Rb at the c r i t i c a l point, so the values must be supplied s p e c i f i c a l l y for the c r i t i c a l temperature, Tc. Pangiotopoulos and Kumar (12) made a and b into temperature dependent parameters but their method of c a l c u l a t i o n requires the use of the saturated l i q u i d volume to calculate the saturated vapour volume. The method e s s e n t i a l l y requires half the answer in advance. Furthermore, their proposed method breaks down as the 24 c r i t i c a l point i s approached along the saturation curve. There is one major drawback to a l l the temperature dependent functional values of Oa,fib,a or b, and that is that they are a l l based on saturation data. It i s dangerous to use these temperature dependencies in the s u p e r c r i t i c a l region (20), without testing them since their behaviour is unknown there, and may y i e l d large errors since they are being extrapolated into a temperature range (T>Tc) outside the one upon which they are based (T<Tc). Joffe and Zudkevitch (24) even state that fia and fib be taken as constants for T>Tc when using their equation. 4. MORE CORRESPONDING STATES The p r i n c i p l e of corresponding states does not lead to a unique r e l a t i o n for a l l substances, even though selected substances do have nearly the same reduced vapour pressure behaviour. With this problem in mind, Pitzer designed the acentric factor to take into account the non-sphericity of molecules (6,14,101). He then expanded the compressibility factor as a Taylor series in co and truncated the series a f t e r the linear term, thereby introducing co as a new corresponding states parameter. The acentric factor is defined as (33): 25 (2.20) co=( l o g 1 0 ( sr) " l o g 1 0 ( Pc' -) )/ l o g 1 0 ( Pc' -) Ps' Ps' where: Pc ' = c r i t i c a l pressure of the simple f l u i d P c = c r i t i c a l pressure of the sought-after f l u i d Ps'=saturated vapour pressure of simple f l u i d at Tr=0.7 Ps=saturated vapour pressure of sought-after f l u i d at Tr=0.7 Vapour pressure at Tr=0.7 was chosen because, for a simple f l u i d , Ps'/Pc'=0.10, hence equation (2.20) reduces to: The only problem with Pit z e r ' s d e f i n i t i o n was that data at Tr=0.7 are not usually available. The accuracy of co i s therefore dependent upon vapour pressure data taken from d i f f e r e n t sources which are then usually combined to undergo regression analysis. This introduces a great deal of uncertainty, for instance, for methane, there are fiv e d i f f e r e n t values of co in use (11). (2.21) co= l o g 1 0 ( H) -1 26 Recent work in the area of two constant equations of state has centred around expressing the Redlich-Kwong equation in a th i r d corresponding states parameter. Pitzer considered that co i s an appropriate t h i r d parameter (6). Other candidates for thi s are various orientation parameters (11), the Riedel factor (61) and the molecular shape factor (53). Most studies, however, involve the acentric factor. More frequently used equations requiring the acentric factor for P-V-T and thermodynamic property ca l c u l a t i o n s are those of Soave (8-10), Raimondi (14), Barner(5) and Peng and Robinson (13). The Soave 1972 equation (8) i s probably the simplest acentric factor dependent modification of the Redlich-Kwong equation of state. The more recent Peng-Robinson equation also enjoys considerable popularity, but the at t r a c t i o n parameter of equation (2.3) contains a more complex quadratic in volume than does the Soave equation. The Soave equation has the c o e f f i c i e n t "a" as a function of temperature and the acentric factor (8): ( 2- 2 2> ~ VTWBT where: 27 (2.23) a(T)=a«a(T) (2.24) a(T)=[1+m(1-Tr°' 5)] 2 (2.25) m=0.480+1 .574CJ-0. 1 76CJ* Equations (2.16)-(2.20) also apply to the Soave equation. Equations (2.24) and (2.25) are combined with saturated P-V-T data to solve for a(T) subject to both pressure and fugacity e q u i l i b r i a (5,8-10). The a(T) function was then expressed as equation (2.24). The Soave equation is applicable to l i n e a r , globular or s l i g h t l y , polar substances at Tr=0.7 (5,7),such as hydrocarbons and l i g h t gases (eg: N 2,C0 2,H 2S). The Soave equation i s e a s i l y applicable to vapour densities, vapour enthalpy departure functions and vapour-liquid e q u i l i b r i a c a l c u l a t i o n s . Unfortunately, calculated l i q u i d densities are larger than the experimental values, with deviations increasing for compounds with the largest acentric factors (5). F u l l e r (15) attempted to r e c t i f y t h i s problem by defining another temperature dependent function 0 to be used in conjuction with Soave's equation, thereby correcting b, in the a t t r a c t i o n term, for temperature. Chung, Hamam and Lu (16) argued that the more complicated F u l l e r 28 equation offered no real improvement for non-polar compounds. F u l l e r had based his conclusions upon ov e r a l l average deviation res u l t s that included both water and ammonia. When these two highly polar compounds were excluded from the res u l t s , his equation proved to be i n f e r i o r to the generalized 2 parameter method. Graboski and Daubert (23) modified the Soave equation by proposing a new acentric factor dependent equation to replace equation (2.25) as well as a new a-equation s p e c i f i c a l l y for normal hydrogen. The equation, an experimental one, is very similar to one that is proposed in t h i s work for the c r i t i c a l isotherm. Graboski and Daubert's a equation is accurate for reduced temperatures in excess of 2.5, which i s e s s e n t i a l l y the s u p e r c r i t i c a l region. More importantly however,their a-equation does not produce a value of 1.0 at the c r i t i c a l point, which e s s e n t i a l l y means that the c r i t i c a l isotherm does not have a double i n f l e c t i o n here. Equations (2.5) and (2.6) therefore do not apply to the Graboski-Daubert equation for normal hydrogen. The exponential equation proposed in t h i s work does have a point of double i n f l e c t i o n at the c r i t i c a l point however. Skogestad (19) offered suggestions, from an i n d u s t r i a l perspective, on how to improve the Soave equation. He suggests that the c o e f f i c i e n t "m" become a function of vapour pressure 29 rather than c o , over the temperature range of i n t e r e s t . The new m-function would be used in calculations where extreme accuracy is required. This i s not a corresponding states approach however. As a follow-up to his o r i g i n a l equation, Soave proposed a new equation for polar and quantum compounds (9,10), by introducing two new compound dependent parameters. The two parameters are to be determined from saturated vapour pressure data, so as to minimize the RMS-% deviation of the calculated vapour pressures from the experimental ones. He also presents several methods for c a l c u l a t i n g the two c o e f f i c i e n t s (9). As stated e a r l i e r , the Peng-Robinson (13) equation has a fu n c t i o n a l i t y dependent on co and is similar in form to that of the Soave 1972 equation, but has a modified a t t r a c t i o n term. The newer term was chosen to more clos e l y approximate the compressibility factors of many compounds (Zc range 0.27-0.29). Zc i s 0.3074 for the Peng-Robinson equation while the Soave 1972 equation has Zc=0.333. As such, the Peng-Robinson equation provides better density re s u l t s than the Soave equation and comparable gas density and enthalpy departure r e s u l t s , but i s harder to solve (5,6) than the Soave equation. Skogestad (19), however points out that,from a p r a c t i c a l point of view, the Peng-Robinson equation i s no real improvement over the Soave 1972 equation, as i t hardly a f f e c t s vapour-liquid c a l c u l a t i o n s , since 30 Zc remains fixed at the c r i t i c a l point. 5. THE LIELMEZS-HOWELL-CAMPBELL EQUATION The Lielmezs-Howell-Campbell (LHC) equation (7), f i r s t introduced in 1983, is the most recent of the Redlich-Kwong modifications to be introduced. Unlike the Soave or Peng-Robinson equations, there i s no dependence on the acentric factor. Rather, the c o e f f i c i e n t "a" i s a temperature dependent function of a * dimensionless parameter, T . The LHC equation reads s i m i l a r l y to that of equation (2.22) with the following differences: (2.26) a(T)=a-a(T*) (2.27) a(T*)=1+p(T*) q and (2.28) T*=( f- -1 )/( Tc -1 ) Tnb Equations (2.15)-(2 . 19) also apply to the LHC equation. The temperature dependency between a(T ) and T along the liquid-vapour coexistence curve i s established by solving the LHC 31 equation for saturated vapour pressure and fugacity e q u i l i b r i a . The resultant values are then f i t t e d to equation (2.27) to produce the c o e f f i c i e n t s p and q. * The T values, introduced and developed by Lielmezs, have shown a wide range of a p p l i c a b i l i t y , from the prediction of s e l f - d i f f u s i o n c o e f f i c i e n t s (62), latent heats of vaporization (36,63), pressure-temperature relations (64) and thermal conductivity (65), as well as the subject matter of t h i s thesis, * the development and further study of the LHC equation (7). The T coordinate system is associated with the phenomenological scaling and renormalization group theory (35,36). The scaling laws of renormalization theory are such that i f the reduced coordinate d e f i n i t i o n i s chosen appropriately, i t i s possible to create universally symmetrical shaped curves (50) about the c r i t i c a l point. Because the dimensionless temperature d e f i n i t i o n involves * both the c r i t i c a l and normal b o i l i n g temperatures, the T coordinates are force f i t at Tc and Tnb, producing values of 0.0 and 1.0 respectively. The s i m i l a r i t y to Trouton's rule i s evident, in the l a t t e r value. Trouton's rule states that molar entropies of evaporation for d i f f e r e n t compounds have roughly the same value at the normal b o i l i n g points. The advantages of the T coordinates are: (i) no need for quantum corrections for n-H2, p -H 2 and neon ( i i ) no need to 32 account for differences in intermolecular forces, and ( i i i ) the acentric factor is not required to correct for substances not * following corresponding states. One unfortunate problem of the T coordinates is their i n a b i l i t y to describe substances l i k e C0 2, where sublimation takes place at 1 atmosphere. By s t r i c t d e f i n i t i o n , for this compound there i s no normal b o i l i n g point. There may be the p o s s i b i l i t y of applying a correction, similar to the quantum correction of Chueh and Prausnitz (28), to remove * thi s problem, or to redefine the T coordinate system for C0 2 so that i t i s based upon the sublimation temperature. This question remains open however. The purpose of this thesis i s to develop the LHC equation (hence adjust and modify equation(2.27)) for use on pure compounds outside the saturation curve. At present the equation i s inapplicable to unsaturated states i e . the s u b c r i t i c a l and s u p e r c r i t i c a l regions. The LHC equation must also be developed for use in thermodynamic cal c u l a t i o n s . There is one added twist however, as discovered by Shaw and Lielmezs (48), and that i s the existence of a si n g u l a r i t y along the c r i t i c a l isotherm when derivatives of the LHC equation are used. The purpose of t h i s research thesis is to extend the LHC equation to include non-saturated states, both the s u b c r i t i c a l and s u p e r c r i t i c a l regions, and present a series of thermodynamic property calculations within these regions. 6. NON-CUBIC EQUATIONS OF STATE Up to t h i s point, only cubic equations of state have been discussed. Another class of equations, however, i s that of non-cubic equations. The Benedict-Webb-Rubin (BWR) equation, f i r s t proposed in 1940 and based on eight substance dependent constants, i s a modified Beattie-Bridgeman equation (6,41), generated by curve f i t t i n g isometrics of (P-RTp)/p 2 vs T. The BWR equation enjoys considerable popularity today (61), while i t s predecessor, the Beattie-Bridgeman equation, has slipped into obscurity. Use of the BWR equation is r e s t r i c t e d to r e l a t i v e l y few substances (38) because of the large amount of empirical data required to generate the necessary c o e f f i c i e n t s . I t s e l f , the Benedict-Webb-Rubin equation is a multi-valued function of 34 density,p: (2.29) P=RTp + (B0RT - A 0 - ^ ) p 2 + (bRT-a)p 3 + aap 6 + f4~ (1+7P 2)exp(- 7p 2) with A 0,B 0,C 0,a,b,c,a and 7 being the eight substance dependent constants. Density was chosen as the dependent variable because neither pressure nor temperature could be represented by simple functions without s i n g u l a r i t i e s resulting along the saturated liquid-vapour equilibrium curve (37). Van der Waals' c r i t e r i o n of double i n f l e c t i o n at the c r i t i c a l point, equation (2.5) and (2.6), also holds for the BWR equation after replacing p by 1/V in equation (2.29), thus providing for two roots along the vapour-liquid saturation curve. Solutions in p, for equation (2.29), must be obtained by non-linear means. A t h i r d root exists between the l i q u i d and vapour densities, which i s a metastable equilibrium value [Figure 1, isotherm T 2 ] . Studies have shown that vapour pressure c a l c u l a t i o n s for pure compounds are greatly overestimated at temperatures below 35 the normal bo i l i n g point (38) and that enthalpy departures tend to be overestimated (39) in general, for a l l f l u i d states. Using the o r i g i n a l c o e f f i c i e n t s supplied by Benedict,Webb and Rubin, vapour pressure results of the C, to C 7 n-alkanes have errors of 1-3%. Benedict suggested that C 0 be f i t to temperature so as to calculate vapour pressures, while the other seven c o e f f i c i e n t s remain constants. Orye looked at t h i s p o s s i b i l i t y (38). Edmister,Vairogs and Klekers (39) went in a d i f f e r e n t d i r e c t i o n and made a l l c o e f f i c i e n t s into functions of the acentric factor, bringing the f i r s t corresponding states parameter into the BWR equation. S t a r l i n g added three more constants,D 0,E 0 and d, while transforming C 0 and a into temperature dependent functions (41). The subsequent equation now bears his name. F i n a l l y , Yamada (51) divided data sets into two separate regions based on reduced density (pr<1.8 and pr>1.8), and proposed a d i f f e r e n t set of acentric factor dependent equations for each region to provide BWR c o e f f i c i e n t s . The former region, pr<1.8, i s described by 8 equations (24 c o e f f i c i e n t s ) , while 16 equations (44 c o e f f i c i e n t s ) are necessary to describe the region as high as pr=2.8. Yamada's c o e f f i c i e n t s are for general use and are based on several compounds: argon,methane,ethane,n-butane,n-pentane,n-heptane and nitrogen. 36 In 1975, Lee and Kesler (40) set out to modify the Benedict-Webb-Rubin equation, while at the same time to develop a equation that summarizes Pitzer's t h i r d parameter corresponding states p r i n c i p l e . The equation has the form: where: (2.30) z-( ^ ^ + ^ + D- 5 + b, b, b (2.31) B=b,- ^ " JL Tr T r T T r 3 c •> c (2.32) C=c,- ^ + T~r~3" (2.33) D=d,+ and d_2_ Tr (2.34) Z=Z0+ — (Zr-Z 0) CJjf The compressibility factor, Z, of equation (2.34) i s composed of two compressibility factor components: (1) Z 0 for a simple f l u i d and (2) Zr for a reference f l u i d . Equation (2.30) i s i n i t i a l l y 37 solved for Z 0 using the appropriate simple f l u i d constants (0, 7, b,-b« f c,-c 3, d 1 f d 2) and equations (2 . 31)-(2.33). Equation (2.30) i s then resolved for Zr using a d i f f e r e n t set of 12 constants for the reference f l u i d . F i n a l l y Z 0 and Zr are combined as Z using equation (2.34). The c o e f f i c i e n t s for the simple f l u i d (based on Ar,Kr and CH„) and the reference f l u i d (based on n-octane) were obtained by solving the Lee-Kesler equation subject to two constraints: (1) fugacity equilibrium between the two coexisting phases along the saturated vapour equilibrium curve and (2) that the c r i t i c a l isotherm intersects the c r i t i c a l point on the P-V curve at a double tangency. Each contraint applies to the simple and reference f l u i d separately. Normal octane was chosen as the reference f l u i d because i t was the heaviest hydrocarbon for which extensive P-V-T data were available for c o e f f i c i e n t determination. However, Munoz and Reich (42) l a t e r found they could improve l i q u i d volumetric results by basing the reference f l u i d on n-decane, then adjusting the constants to f i t the low temperature region and the properties of other substances (propane, n-butane, n-pentane). The Lee-Kesler equation accurately represents the volumetric and thermodynamic properties of both non-polar and s l i g h t l y polar compounds. The range of application extends from Tr=0.3-4.0 and 38 Pr=0-10.0, with greatest predictive accuracy in the subcooled and superheated regions. The accuracy of the Lee-Kesler equation does, however, diminish at the saturation curve and at the c r i t i c a l point (40). McFee,Mueller and Lielmezs (41) concluded that the Lee-Kesler equation had a high degree of r e l i a b i l i t y and was the most accurate o v e r a l l , for a l l compounds that they tested, when i t was compared to the BWR and Sta r l i n g equations. One p r a c t i c a l problem exists when using the Lee-Kesler equation to calculate thermodynamic departure functions. Both the simple and reference f l u i d volumes must be supplied by calc u l a t i o n when any thermodynamic departure function is to be used, which requires considerable computer time. The Boublik-Alder-Chen-Kreglewski (BACK) equation i s based on the "Augmented Van der Waals Theory" for f l u i d s (49), meaning that the constant "a" of the Van der Waals equation (equation (2.7)) is now a density dependent function. It uses the inverted square well potential (101) [Figure 2] which means that three assumptions are made: ( 1 ) a t t r a c t i v e forces are constant over a f i n i t e distance (2) repulsive forces are f i n i t e when molecules touch and (3) repulsive forces are zero at a f i n i t e distance. For ca l c u l a t i o n purposes, the free energy of a f l u i d of hard spheres with added att r a c t i o n potentials, which are weak compared to the kinetic energy, is expanded in a perturbation function. 3tr -UCD 39 - s 7 -U(r)=°° f o r 0<r<(a-S, ) U(r)=3U° from r=(a-S,) to r = a - i r -u F i g u r e 2: Inverted Square-well P o t e n t i a l The B A C K equation g i v e s the c o m p r e s s i b i l i t y f a c t o r made of a h an a t t r a c t i o n term (Z ) and a r e p u l s i o n term (Z ): (2.35) Z = p = Z h + Z a (2 36) Z h= 1 + ( 3 a - 2 ) $ + ( 3 a 2 - 3 a + 1 ) S 2 - a 2 $ where: r n m (2.37) Za=ZImDnm( £-) ( g l ) nm K l v (2.38) $ = 0.74048 ~ ^ V ° 1 7 r N 0 g3 V (2.39) V ° = V ° ° [ 1 - C exp( (2.40, (2.41) a=1+0.3cj (2.42) v ° 0 = 4 7 | 40 (2.43) ^ = 0.6CJTC ( 2 4 4 ) H^.= Tc/(1+ 2 ) k i C / U 2.5kTc' The BACK equation has a double i n f l e c t i o n at the c r i t i c a l point. Therefore equations (2.5) and (2.6) apply to the c r i t i c a l isotherm. Equation (2.37) contains the 24 constants,Dnm, of the expanded perturbation function. The values of a l l 24 constants, based upon argon data, are meant to be used for a l l compounds. O r i g i n a l l y , the three substance dependent constants V 0 0 , a and U°/k, were obtained by solving the BACK equation while maintaining both vapour pressure and fugacity e q u i l i b r i a along the liquid-vapour saturation curve (46). However Chen and Kreglewski (66) have since presented equations (2.41), (2.42) and (2.44) to replace the e a r l i e r more cumbersome method of obtaining V 0 0 , a and U°/k. The BACK equation i s a t t r a c t i v e for use because of i t s accuracy in P-V-T calculations, however i t requires fiv e substance dependent constants (V 0 0,a,C,U°/k,and CJ). While the P-V-T results are accurate for globular molecules, they are not as good for long chain molecules l i k e n-octane or n-decane 41 (46,47). Chen and Kreglewski (45) compared calculated internal energy,Ur, results to observed values at the normal b o i l i n g point, but the comparison should be carr i e d out over various additional temperature and pressure ranges, for a true measure of accuracy. Furthermore, Shaw and Lielmezs (48) found that they could obtain the same accuracy at the normal b o i l i n g point as the BACK equation for Ur, by using cubic equations of state or for that matter their linear combinations. Should cubic equations continue to provide comparable results to the BACK equation for a l l data points then the l a t t e r species of equations are c l e a r l y the more a t t r a c t i v e to use due to their s i m p l i c i t y . 7. PURPOSE OF THIS THESIS The Lielmezs-Howell-Campbell equation w i l l be extended to calculate P-V-T properties in both the s u b c r i t i c a l and s u p e r c r i t i c a l regions beyond the saturation curve. Furthermore, the LHC equation w i l l be tested for i t s a b i l i t y to calculate thermodynamic properties (enthalpy, entropy, internal energy, Helmholtz and Gibbs free energies, fugacity, isobaric and isometric heat capacities, Joule-Thomson c o e f f i c i e n t s , inversion curves) for a l l data ranges. The LHC equation i s then to be compared, on the basis of the above res u l t s , to other state equations: the Redlich-Kwong, Soave 1972, BWR, BACK and the Lee-Kesler equations. 43 C. RESEARCH PROCEDURE As developed, the Lielmezs-Howell-Campbell (LHC) equation of state i s currently applicable only to P-V-T (7) and internal energy cal c u l a t i o n s (48) along the saturated liquid-vapour equilibrium curve. As discovered by Shaw and Lielmezs (48), the LHC equation contains a si n g u l a r i t y along the c r i t i c a l isotherm * due to the nature of the T derivatives.The P-V-T calculations remain unaffected along the Tc isotherm, however. In l i n e with the si n g u l a r i t y problem, the region where temperature exceeds Tc also remains undefined, due to the non-integer nature of the c o e f f i c i e n t q in equation (2.27). Since * T values (equation (2.28)) possess a negative sign for T > Tc, a(T ) remains undefined mathematically for a non-integer value of q. It i s necessary to eliminate these two problems in order to make the LHC equation universally applicable to a l l regions of thermodynamic c a l c u l a t i o n . Equation (2.27) was modified, i . e . regionalized, as follows below. Equation (2.27) is rewritten as: (3.1) a(T*)*=1+p(T*) q while proposing: for Tr < 1.0 44 ( 3 . 2 ) a ( T * ) - = 1 - p | T * i Q f o r T r > 1 . 0 a n d ( 3 . 3 ) a ( T * ) c = e x p ( d T * ) f o r T r = 1 . 0 E q u a t i o n s ( 3 . 1 ) a n d ( 3 . 2 ) u s e i d e n t i c a l v a l u e s o f p a n d q , a s p r e s e n t e d b y L i e l m e z s e t a l ( 7 ) , b a s e d o n t h e s a t u r a t i o n s t a t e d a t a o f e a c h c o m p o u n d ; t h e t w o a r e s u f f i c i e n t f o r P - V - T c a l c u l a t i o n s o v e r a l l t e m p e r a t u r e r e g i o n s . B o t h e q u a t i o n s ( 3 . 1 ) a n d ( 3 . 2 ) y i e l d a v a l u e o f 1 . 0 a t T c , w h i c h i s n e c e s s a r y i n o r d e r t o m a i n t a i n t h e d o u b l e i n f l e c t i o n r e q u i r e m e n t a t t h e c r i t i c a l p o i n t , a s s o c i a t e d w i t h e q u a t i o n s ( 2 . 5 ) a n d ( 2 . 6 ) . A l s o , t h e a ( T ) * a n d a ( T ) ' f u n c t i o n s j o i n a t T c t o f o r m a c o n t i n u o u s c u r v e , r a n g i n g f r o m t h e t r i p l e p o i n t p o i n t t e m p e r a t u r e u p i n t o t h e s u p e r c r i t i c a l r e g i o n , e v e n t h o u g h t h e i r r e s p e c t i v e d e r i v a t i v e s a r e d i s c o n t i n u o u s a t T r = 1 . 0 . I t w a s t h i s l a t t e r p r o b l e m t h a t p r o m p t e d t h e d e v e l o p m e n t o f e q u a t i o n ( 3 . 3 ) . I n a c t u a l i t y , i t i s m a n d a t o r y t h a t e q u a t i o n ( 3 . 3 ) h a v e f i n i t e d e r i v a t i v e v a l u e s a l o n g t h e c r i t i c a l i s o t h e r m ; t h e d e r i v a t i v e s o f e q u a t i o n s ( 3 . 1 ) a n d ( 3 . 2 ) w o u l d c o v e r a l l t e m p e r a t u r e s e x c e p t T c . A s a n a l t e r n a t i v e , e q u a t i o n ( 3 . 2 ) w a s e x p r e s s e d a s h a v i n g b e e n : 45 (3.4) a(T*)-=1+p|T*|q but c a l c u l a t i o n results proved to be of i n f e r i o r accuracy to those of equation (3.2). To proceed with the study of the LHC equation of state, a l l data sets for P-V-T and thermodynamic properties were divided up into f i v e regions: (I) saturated l i q u i d , (II) saturated vapour, (III) s u b c r i t i c a l l i q u i d (Tr<1.0 and Pr<1.0), (IV) s u b c r i t i c a l vapour (Tr<1.0 and Pr<1.0) and (V) s u p e r c r i t i c a l f l u i d (either or both Tr>1.0,Pr>1.0). P-V-T and thermodynamic property c a l c u l a t i o n studies were conducted separately in each region. Regions ( I I I ) and ( I V ) , in addition, included P-V-T and thermodynamic property calculations based on the correction factor proposed by Tannar, Lielmezs and Herrick (68). These correction factors (equation (3.5) ) allow for P-V-T prediction in regions (III) and (IV) based on li n e a r departures from the saturation curve. P-Psat s Psat ; (3.5) Cf=1+m( Tsat f c ) + n ( 46 Tsat i s the saturation temperature found along isobar P, and Psat is the saturation temperature found along isotherm T. I n i t i a l l y the pressures, volumes and temperatures were calculated using the LHC equation in the form: (3.6) P= R T - a ( T ) V-b V(V+b) where a(T) i s the appropriate temperature dependent function of (3.7) a(T)=aa(T*) +,aa(T*)",aa(T*)c,aCfa(T*) + * The correction factor, Cf, i s only associated with a(T ) + . Volumetric results were compared to those of the Soave 1972, Redlich-Kwong and the Lee-Kesler equations. A l l equations previously mentioned were used for temperature and pressure ca l c u l a t i o n s except that the Benedict-Webb-Rubin (BWR) equation replaced the Lee-Kesler equation. This was necessary since there was no available method for determining both the simple and 47 reference f l u i d c o m p r e s s i b i l i t i e s , Z 0 and Zr, from equation (2.34) without accessing an external n-octane P-V-T data source to supply Zr. Since both Z 0 and Zr values are unavailable, pressure and temperature calculations cannot be c a r r i e d out. Additionally, the LHC equation was expanded in v i r i a l s for V and P and solved as above; the results being compared to those of the Soave 1972 and Redlich-Kwong equations, also v i r i a l l y expanded in the same fashion. Only equations in the Redlich-Kwong form (equation (3.6)) were included as v i r i a l expansions. Generalized V and P v i r i a l expansions are as follows: (3.8) Z=1+ *1T) + C1T) + B I T ) + and (3.9) Z=1+B'(T)-P+C(T)-P 2+D'(T)-P 3 + where: (3.10) B(T)=b-(3.11) C(T)=b 2+ (3.12) D(T)=b 3- b 2 g j , T ) 48 Abbreviating B'=B(T), C'=C(T) and D'=D(T), we have: (3.13) (3.14) C (T) = C-B2 (RT) 2 and (3.15) D'(T)= D-3BC+2B3 (RT) 3 A derivation of equations (3.8) and (3.9) is presented in Appendix C. When equations (3.10)-(3.15) are used in conjunction with the correction factor, Cf (equation (3.5)), an apparent pressure dependency i s introduced. It should be noted that the correction factor, Cf, is a function of dimensionless temperature and pressure in a manner analogous to the acentric factor, to (equation (2.20)). The acentric factor, to, i s based upon the saturation pressures of a simple f l u i d and a sought-after f l u i d a r b i t r a r i l y at Tr=0.7. On the other hand, in p r i n c i p l e , i t could be made to vary with any reduced temperature along the saturation curve. Therefore to that extent, the acentric factor i s an apparent function of pressure. 49 Indeed, the correction factor, Cf (equation (3.5)), i s defined such that i t reduces to unity along the saturation curve. When t h i s happens, the v i r i a l c o e f f i c i e n t s of equations (3.8) and (3.9) become functions of temperature only, and both equations (3.8) and (3.9) reduce to the ideal gas state equation as the appropriate l i m i t s are set (V -> °° for equation (3.8) and P -> 0 for equation (3.9) re s p e c t i v e l y ) . The LHC equation correction factor method contains one added complication in P and T c a l c u l a t i o n s . Tsat and Psat values are functions of pressure and temperature respectively along the saturation curve, so i f P or T are the sought-after values, there is an associated unknown with each variable. To a l l e v i a t e t h i s problem, another state equation, including the LHC method of equation (3.1), must be used to generate values of Tsat and Psat when needed. Methods considered in this work were the Harlacher-Braun equation (61), Aitken's method of interpolation (extrapolation), splines under tension, and Chebychev polynomials. Although a t t r a c t i v e because of i t s s i m p l i c i t y , Antoine's equation can not be considered because i t i s not recommended for use above a pressure of 2 atmospheres (61). The LHC equation without the correction factor could be used to calculate Tsat and Psat, but the methods chosen were well established and believed to provide the best accuracy. It should 50 be noted here that although the LHC correction factor method was designed to y i e l d a value of Cf=1 along the saturation curve, th i s when calculated, was not obtained. The calculated Cf was always found to be very close to unity (within 2.0%) but never quite so, because of the approximation errors inherent in the multiple linear regression method of f i t t i n g an equation. A bisection procedure is presented in this work (explained below) to solve for P and T values. When P was the sought-after variable, one of equations (3.6),(3.8) or (3.9) must generate the same value of Tsat as does the chosen method of saturation coordinate c a l c u l a t i o n . This also applies to sought after values of T; i d e n t i c a l Psat values must be generated. Convergence in not always guaranteed, especially in region I I I . The procedure l i s t e d i s used for c a l c u l a t i n g a single unknown P/T ("/" is to be read as "or") given known values of V and T/P. 1. Set upper and lower l i m i t s of in t e r v a l to be searched, Pu/Tu and PL/TL, respectively. The upper l i m i t can never exceed Pc/Tc. 2. Set P/T half way between PL/TL and Pu/Tu. 3. Calculate Tsat/Psat by combining equations (3.1) and (3.5)-(3.7) for each of the three values P/T, PL/TL and Pu/Tu. The v i r i a l expansions, equations (3.8) and (3.9) can 51 also be used in place of equation (3.6). 4. Calculate Tsat/Psat using the Harlacher-Braun equation, Aitken's method, splines under tension or Chebychev polynomials (always set Psat as a function of T). Use of the Harlacher-Braun equation i s strongly recommended for l i q u i d s . 5. determine the signs (+ve or -ve) res u l t i n g from the differences between the corresponding Tsat/Psat values of equations (3) [ a l l 3 solutions] and (4). 6. Take the P/T interval over which a sign change is evident, upper of lower half of the i n t e r v a l , and repeat steps 1-6 u n t i l an interval of 10~8 remains. Although time consuming, the algorithm presented above was the only way available with which to compare two vastly d i f f e r e n t methods of saturation property generation as the Harlacher-Braun equation and Aitken's method. The former can be used by dir e c t algebraic substitution into equation (3.6), (3.8) or (3.9); not so with the l a t t e r however. The method of splines under tension has one more additional complication. There is no allowance for extrapolation, which i s necessary in the low temperature region, especially for l i q u i d s . As such, the spline method f a i l e d frequently. Thermodynamic departure calculations provided the next basis for comparison between the LHC equation and the other equations of state. The departure functions considered are enthalpy,Hr, entropy,Sr, internal energy,Ur, Helmholtz and Gibbs free energies,Ar and Gr, fugacity c o e f f i c i e n t s , f / P , as well as the isobaric and isometric heat ca p a c i t i e s , Cp and Cv. Equation (3. can be converted to: M 1 ^ D - R T _ QabRTF u , l b ; V-b fibV(V+b) where F for the LHC equation i s : (3.17) F= ^a(T*) The F function (48,61) allows one to represent a two constant equation in generalized form (equation (3.17)) and yet i s o l a t e each 2 constant equation of state's temperature dependent f u n c t i o n a l i t y as F, in equation (3.17). The LHC equation has F functions for each region as follows: 53 (3.18) (3.19) (3.20) F= l^i+pT*^) F= ^exp(d-T*) F= l _ ( i - p | T * | q ) (Tr < 1.0) (Tr = 1.0) (Tr > 1.0) and with the correction factor: (3.21) F= ! F(1+pCfT* q) Thermodynamic departure functions arise from equation (3.16) and are: (3.22) (3.23) (3.24) (3.25) Ar. RT' " l n ( 1 - v > - § i F - l n ( 1 + " l n ( h] fib V S r = 1 RT T ln(1" + i ( F + T If) ln(l+ + ln< H Ur = Ra 3 F l n ( l + b) RT fib 3 T ± n V ' V ; Hr = Ra i ^ i n ( 1 + b) + Z - i RT fib 3 T i n U V ; L 1 54 (3.26) (3.27) (3.28) (3.29) where: (3.30) and (3.31) w i t h (3.32) (3.33) (3.34) H-Z-1- l n ( 1 - *> §|F l n d + § ) - l n ( ^ ) ln( |) = Z-1-ln(z)- l n d - - §|F ln(l+ £) Cpr=( g|R.ln(l+ g ) ) ( 2 T f| +T2 0)-R Cv=Cp+T V 3T 3T J R fibV(V+b)v 1 3T ; V-b fiabRTF f 1 1 x _ RT 2 ObV(V+b)v V V+b ' (V-b) 2 F+T ||= " 1 3T Tr • pq/( Tc Tnb -1 ) ,*q-1 F+T —= 1 3T z d / ( Tc_ _ • T r 7 v Tnb ' F+T ||= - 1 3T fF* pq/( Tc Tnb -1 ) ,* ,q-1 (Tr < 1.0) (Tr=1.0) (Tr > 1.0) and with the correction factor: ( 3 3 5 ) F+T m P T c T * q - ( P q C f T * q - 1 } / ( Tc_ _ » * 1 3T Tsat 1 { T r ^ T Tnb 1 ' Furthermore, for Tr < 1 . 0 , T r = 1 . 0 , and Tr > 1 . 0 , respecti ( 3 . 3 6 ) ( 3 . 3 7 ) ( 3 . 3 8 ) 3F i m 2 3 2 F 2T ^ +T P Q / ( 1 Tc -1 ) Tr T Tc *q-1 3F , m 2 3 2F. 2T ^ +T 3 T ' dF,, Tc Tr' Tnb Tnb p q ( q - D / ( ^ b ~ 1 ) -1 ) 2 + Tc Tr' v Tnb -1 ) 2 T H + T 2 * l 3T 1 3 T 7 2 TF2" pq/( Tc Tnb -1 ) ,* i q - 1 Tr p q ( q - D / ( | f b " 1 ) 2 ] l T * ,q 56 and with the correction factor: (3.39) 2T 21 ' 3T 3 2F df7' 2pq *q-1 T r 2 Cf T m Tsat / ( Tc Tnb -1 ) pq(q-1)Cf *q-2 / ( Tc_ _ ) T r 3 T ' v Tnb ' Derivations of equations (3.16)-(3.39) may be found in Appendix D. In addition, this Appendix also contains the F functions and their derivatives, as well as the Soave 1972 and the Redlich-Kwong equations of state. Appendices E and F contain the departure function derivations of the BWR and BACK equations respect i v e l y . Equations (3.12)-(3.31) were solved by substituting P-V-T data into the appropriate equations, and the LHC equation results were compared to those of the Redlich-Kwong,Soave 1972,BWR and BACK equations. The f i n a l bases of testing the LHC equation and comparing the results to i t s contemporary equations are the c a l c u l a t i o n of the isenthalpic Joule-Thomson c o e f f i c i e n t s and the inversion curve locus. These two calculations are the severest tests that 57 an equation of state can undergo (56). The inversion curve [Figure 3] i s the set of points where a l l Joule-Thomson c o e f f i c i e n t s are zero. Gas behaviour along t h i s locus i s that of i d e a l i t y i.e. no intermolecular force interactions. Both inside and outside the inversion curve, the gas exhibits real gas behaviour, meaning that intermolecular forces e x i s t . The region inside the inversion curve contains po s i t i v e values of the Joule-Thomson c o e f f i c i e n t . Upon isenthalpic expansion, the gas w i l l cool, i n d i c a t i n g that the a t t r a c t i v e forces between molecules are dominant. Outside the inversion curve, isenthalpic expansion of a gas y i e l d s negative Joule-Thomson c o e f f i c i e n t s . Heating occurs and repulsive forces are dominant (55). F i g u r e 3: J o u l e - T h o m s o n E x p a n s i o n P r o c e s s 58 The isenthalpic Joule-Thomson c o e f f i c i e n t i s defined as: (3.40) M=(T £ 1 - V)/Cp A 2 where, A 2 i s given previously and: R V - b F i n a l l y , the inversion curve i s : (3.42) 1 -X 1 - 1-X fiaX Qb(1+X) F T 9F 1+X 3T = 0 with: (3.43) X= £ 5 9 The F functions and their derivatives, for the LHC equation, have been previously defined as equation (3.18)-(3.21) and (3 . 32)-(3.39) respectively. Derivations of equations (3 . 40)-(3.43) can be found in Appendix D, as well as those corresponding to the Redlich-Kwong and Soave 1972 equations. In addition, Appendices E and F contain the Joule-Thomson and inversion curve derivations for the BWR and BACK equations. Equation (3.40) can be solved e x p l i c i t l y in u while an approximate solution in X is obtained from equation (3.42) by non-linear means. Dilay and Heidemann (56) determined the P and T values from a known volume using the Soave 1972 and Peng-Robinson equations. An equation of the same nature as (3.42) yielded T, while P was provided by equation (3.16) using the values of V and T. The order of cal c u l a t i o n in t h i s work however was modified. A value of temperature was selected and substituted into equation (3.42) to solve for volume. The V and T values were then substituted into equation (3.16) to determine P. J u s t i f i c a t i o n for the order of cal c u l a t i o n in this work is simply that T i s more l i k e l y to be the known variable than V. The results of a l l tests on P-V-T, departure function, Joule-Thomson and inversion curve calculations are presented and discussed in the next chapter. 60 D. RESULTS AND DISCUSSION The introduction of the temperature dependent a-functions of equations (3.1)—(3.3) enables one to use the LHC equation in a l l of regions I-V without any d i r e c t reference to the magnitude or type of intermolecular forces involved, or to the description of the molecular structure of the f l u i d (7). Instead, the method proposed requires knowledge of the three empirical substance-dependent parameters p,q and d. The former two parameters are based on saturated vapour-liquid equilibrium data. Knowing these two, allows one to determine the t h i r d c o e f f i c i e n t , d, based on curve f i t t i n g equations (3.1) and (3.2). This new function, equation (3.3), was only designed to treat a very narrow (generally not more than ± 20K in most cases) temperature range around the c r i t i c a l isotherm. Figure 4 shows equation (3.3) in r e l a t i o n to equations (3.1) and (3.2) for propane. The temperature range around the c r i t i c a l point was adjusted u n t i l the RMS-% error associated with each set of thermodynamic property c a l c u l a t i o n results was minimized. The general result was that RMS-% errors either declined or remained constant. In very few instances did they increase, and when t h i s did occur, the increase was generally no more than 0.5%. The temperature ranges of a p p l i c a b i l i t y for equation (3.3) can be found in Tables 101-103 for a l l compounds tested i e . where data were available. F i g u r e 4: a v s T f o r P r o p a n e o a l+P*TSTAR**0 •i t- l-P*ITSTRRI* a :Q o o EXP(0*TSTflR) <£> _ l a" "I I I i 1 I I I | 1 I 1 1 1 I I I I I 1 I -0.8 Q.Q 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 TSTRR 62 The study of equations (3.1 )-(3.3) and their respective derivatives was carried out in three stages over regions I-V, as defined in chapter 3, using the c o e f f i c i e n t s l i s t e d in Table 100. Stage 1 applied equation (3.1), labelled LHC*, to regions I-IV. Stage 2 dealt with region V, where both equations (3.2) and (3.4) were compared appropriately. Equation (3.4) was proposed as an alt e r n a t i v e to equation (3.2) and was tested in the same manner. LHCI i s the notation based on equation (3.4) and LHCII refers to equation (3.2). Since some data in region V had values of T<Tc and P>Pc, equation (3.1) was used in conjunction with both equation (3.2) and (3.4) where meaningful. In stage 3, equation (3.3) was tested for a l l data used in the two previous stages, in an attempt to improve the results of prior c a l c u l a t i o n . At the end of stage 2, i t had been established that LHCII was superior to LHCI, and would, along with equation (3.1), be used to form the basis for establishing equation (3.3). The data tables presented in this work compile the results for a l l three stages i nvolved. Fourteen compounds were used in t h i s study. They break down as follows: Normal Alkanes: methane to n-hexane H bonded substances: methanol, t-butanol, H 20 Quantum f l u i d s : n-H 2 I p-H2 63 S i m p l e f l u i d s : a r g o n , n e o n S u l p h u r c o m p o u n d s : H 2 S T h e r a w d a t a w e r e t a k e n f r o m p u b l i s h e d s o u r c e s ( T a b l e s 1 0 4 - 1 2 8 ) a n d c o n v e r t e d t o t h e a p p r o p r i a t e u n i t s f o r s t u d y . W h e r e c e r t a i n v a l u e s w e r e u n a v a i l a b l e , t h e r m o d y n a m i c i d e n t i t i e s w e r e u s e d i n c o n j u n c t i o n w i t h a v a i l a b l e d a t a , t o f i l l t h e g a p . W h e r e n e c e s s a r y , t w o d i m e n s i o n a l i n t e r p o l a t i o n was a l s o c a r r i e d o u t , when o n e d a t a s o u r c e d i d n o t c o r r e s p o n d t o a n o t h e r . T h i s w a s g e n e r a l l y t h e c a s e w h e n v o l u m e s w e r e r e q u i r e d f o r t h e r m o d y n a m i c d a t a . T h e r e w e r e o t h e r m e t h o d s o f c a l c u l a t i o n t h a t h a d t o b e u s e d , f o r i n s t a n c e t o d e t e r m i n e a l l J o u l e - T h o m s o n c o e f f i c i e n t s o f n e o n b y f i n i t e d i f f e r e n c e d i f f e r e n t i a t i o n , s i n c e t h e o n l y a v a i l a b l e d a t a was T - P d a t a a l o n g i s e n t h a l p s . F u r t h e r m o r e , when t h e i s o m e t r i c h e a t c a p a c i t y v a l u e s o f p a r a - h y d r o g e n w e r e t o b e o c a l c u l a t e d i n r e g i o n V ( C p was c o n s t a n t i n r e g i o n s I - I V ) i t was n e c e s s a r y t o d e v e l o p a n i d e a l g a s h e a t c a p a c i t y e q u a t i o n b y c u r v e f i t t i n g e x p e r i m e n t a l d a t a s i n c e t h e r e was n o n e a v a i l a b l e . 6 4 When a l l data sets were gathered, the P-V-T and thermodynamic properties were calculated using the various equations mentioned l a t e r in this chapter. A l l calculated values were compared point by point with their experimental counterparts for accuracy, as measured by the percent error: ( 4 . 1 ) %-ERROR=100^ Xexp-Xcalc Xexp where Xexp is the experimental value of a parameter, and Xcalc i s the calculated value of a parameter. A l l percent error values were then used to calculate the RMS-% error which i s a measure of the o v e r a l l accuracy of an equation of state for a p a r t i c u l a r compound. The RMS-% error is defined as: ( 4 . 2 ) RMS-% ERROR= ^ I(%-ERROR)2 0 • 5 where N i s the t o t a l number of data points. The RMS-% errors and 6 5 central processing unit (CPU) times for each c a l c u l a t i o n are presented on a compound by compound basis, over the same data ranges in the appropriate tables. The mean RMS-% errors are determined using an i d e n t i c a l data base for each equation of state. Furthermore, the BWR and BACK equations are limited to use only where the necessary c o e f f i c i e n t s are available. Taking this into account, Tables 1 to 19 were compiled on the basis of information only for compounds that a l l equations could handle, thus allowing for a di r e c t comparison of the accuracy of each equation. The equations are ranked from f i r s t to la s t ( i e . most accurate to least accurate) on an RMS-% error basis and l i s t e d in the tables presented in thi s chapter. There were, however, one or two cases where an equation was not included in the f i n a l results due to i n s u f f i c i e n t data upon which to make a comparison. Several comments can be made at the outset. F i r s t , no equation i s consistently superior to the others in a l l regions of c a l c u l a t i o n . This applies to any par t i c u l a r form of the LHC equation with a correction factor. No individual correction factor method i s superior in a l l cases to the others, although the Harlacher-Braun (LHCHB) and the Aitken (LHCA) methods in pa r t i c u l a r distinquish themselves from the other correction factor methods (splines under tension (LCHSP) and Chebychev 66 polynomials (LHCCH)). The LHCHB method probably allows for the greatest accuracy of any P-V-T property ca l c u l a t i o n s , but i t s weakness l i e s in the fact that i t i s limited to certain compounds, where the available c o e f f i c i e n t s permit i t s use. Reid, Prausnitz and Sherwood (61) advise caution in that the Harlacher-Braun method i s sometimes inaccurate in the c r i t i c a l region of some compounds, and should therefore be checked prior to i t s use. Of a l l compounds studied in this work, only water had this problem; the calculated c r i t i c a l pressure was 4.4 atmospheres greater than the established value. Water was therefore not included in the study of the Harlacher-Braun method. In addition t e r t i a r y butanol and para-hydrogen had no Harlacher-Braun c o e f f i c i e n t s available for use. The Aitken method proved to be the most v e r s a t i l e , and i t provided results that in many instances r i v a l e d or improved upon those of the Harlacher-Braun method. Hence Aitken's method was chosen for use in the thermodynamic c a l c u l a t i o n s . As discovered l a t e r , the correction factor method i s actually a l i a b i l i t y in thermodynamic property ca l c u l a t i o n s , accounting for some results that are inaccurate by orders of magnitude above the other equations. The correction factor method also has a limited region of usage (T<Tc and P<Pc), since saturation data i s always required. 67 One f i n a l comment remains concerning the RMS-% error values. Because the RMS-% error is proportional to the sum of the squares of the percentage errors at a l l points, a very large error in one point can have a profound impact on the ov e r a l l value. As such, very large RMS-% errors are somewhat deceiving. Since percent errors are not homogeneously d i s t r i b u t e d , one should also check the percent error breakdown for regions of large error, after viewing the overall RMS-% values. A l l equations tested in t h i s work generally had f a i l u r e s occuring in the same regions of thermodynamic properties. In summary, the equations studied in th i s work are denoted as follows: LHC* equation (3.1) LHCI equations (3.1) and (3.4) LHCII equations (3.1) and (3.2) LHCIII equations (3.l)-(3.3) LHCA LHC correction factor equation- Aitken's method LHCHB LHC correction factor equation- Harlacher-Braun equation LHCSP LHC correction factor equation- splines LHCCH LHC correction factor equation- Chebychev polynomials S72 Soave 1972 equation RK Redlich-Kwong equation 68 BWR Benedict-Webb-Rubin equation LK Lee-Kesler equation BACK Boublik-Alder-Chen-Kreglewski equation 1. P-V-T CALCULATIONS The LHC equation in i t s cubic polynomial form and in the form of equation (3.6) was compared to the Soave 1972 and the Redlich-Kwong equations in their similar forms. Volumetric results from the two constant equations were then compared to those of the Lee-Kesler equation, while the Benedict-Webb-Rubin equation provided the comparison for the temperature and pressure r e s u l t s . Replacing the Lee-Kesler equation by the BWR equation was necessary since there was no available method for determining both the simple and reference f l u i d c o m p r e s s i b i l i t i e s , Z° and Zr, from equation (2.34) without accessing external n-octane P-V-T data to supply Zr. This was not done however. Since both Z° and Zr values are unavailable, pressure and temperature calculations cannot be carried out. For volumetric results, one or more forms of the LHC equation proved to be superior to the Soave or Redlich-Kwong equations in a l l regions except V, where the trend was reversed. The re s u l t s are in Table 1, however be aware that the accuracy between the d i f f e r e n t regions is not meant to be compared but 69 rather only those results within each individual region, which explains why there are no columns in the table. Table 1: Overall Volumetric RMS-% Error Region I LK(10.45), LHC*(14.44), S72(16.65), RK(21.39) II LHC*(6.04), S72(7.01), LK(7.67), RK(7.81) III LK(4.16), LHCHB(4.86), LHCA(5.17), LHC*(10.79), S72O5.10), LHCCH(30.70) IV LHCA(1.06), LHCHB(1.09), LHC*(1 . 1 7 ) , LK(1.53), RK(1.86), S72(1.91), LHCCH(10.97) V LK(5.21), RK(14.43), S72(14.58), LHCII(14.59), LHCI(16.15) Also noteworthy i s the comparison of the CPU times of the Lee-Kesler and the two constant equations, as given by Tables 20-24. Inspite of the complexity of the Lee-Kesler equation, i t s CPU times for volumetric calculations were smaller than those of a l l the cubic equations. The algorithm, ZERO2 (104), provides a simple volumetric root to the Lee-Kesler equation using non-linear means (combinations of linear interpolation, r a t i o n a l interpolation and bisection- Bus and Dekker method). The algorithm, RPOLY2 (105), for solving cubic polynomials uses 70 Euclid's method and determines a l l roots, both real and imaginary. Also of note is that the routine ZER02 provides a numerical value that is accurate to within 10" 1 0 of the actual solution, while RPOLY2-generated solutions are only accurate to within 10"6 of the actual numerical value. The researcher i s able to set the error tolerance of ZER02, while he has no control over the error tolerance of RPOLY2. For pressure calculations, the BWR equation was the most accurate in a l l regions considered, with LHC* appearing f i r s t in region II and fourth in region IV, ahead of both the Soave and Redlich-Kwong equations. The Harlacher-Braun and Aitken correction factor equations finished as the second and t h i r d most accurate respectively in region IV. It should be noted that a l l equations f a i l e d to provide acceptable pressure values in regions I and III and'because of their large RMS-% errors were not included in Table 2. The large errors in region V were high due to the presence of compressed l i q u i d data. The results are presented in Table 2 as follows: 71 Table 2: Overall Pressure RMS-% Error Region II LHC*(1.11), S720.27), BWR(1.51), RK(1.79) IV BWR(0.28), LHCA(0.63), LHCCH(0.64),LHC*(0.68), S72(0.78), LHCHB(1.00), LHCSP(1.03), RK(1.13) V BWR(43.43), LHCII (94.72) , S72(95.19), LHCK97.98), RK(112.38) The CPU times (Tables 25-27) of the four correction factor equations were much larger than those of the other equations, mainly due to the bisection method employed; some differences between the CPU times of the four d i f f e r e n t methods are obviously due to the method of saturation property determination i t s e l f . When temperatures are calculated, LHC* and LHCII appear to be the best LHC combination over a l l f i v e regions, i n d i v i d u a l l y appearing as either the f i r s t or second most accurate equation. The accuracy of LHC* does drop however in region I I I . The BWR equation generally r i v a l s LHC* in regions I-IV. The Soave equation i s superior to LHCII in region V, improving gradually in each region, climaxing in region V. The RMS-% errors at times are somewhat deceptive since a 5% error in 300°K for example i s 15°K. The results are summarized-in Table 3: 72 Table 3: Overall Temperature RMS-% Error Region I BWR(2.74), LHC*(9.72), S 7 2 O 0 . 1 6 ) , RK(12.76) II LHC*(0.72), BWR(0.76), S72(0.77), RK(1.06) III LHCHB(3.14), BWR(3.59), RK(7.95), LHC*(9.73), S72(9.99) IV LHC*(0.32), LHCA(0.33), LHCHB(0.33), S72(0.34), RK(0.39), BWR(0.46), LHCCH(5.54) V S72(5.47), LHCIK5.82), BWR(6.09), RK(6.58), LHCI(10.93) Again the bisection method of the correction factor accounted for the large CPU times indicated. Temperature results for individual compounds are summarized in Tables 28-32. Summarizing these P-V-T calculations, the LHC* equation in tandem with LHCII appears to be the most consistent combination in terms of s i m p l i c i t y and accuracy. These two equations comprise the LHC equations (3.1) and (3.2), and were used to construct the P-V-T diagrams of Figures 5-18. The reader should note that the hyperbolic shape c h a r a c t e r i s t i c of the vapour curves i s not apparent in logarithmic coordinates but would become so i f linear coordinates were used. 73 2. P-V-T CALCULATIONS BY VIRIAL EXPANSION The LHC, Soave 1972 and Redlich-Kwong equations are a l l v i r i a l l y expanded in the form of equations (3.8) and (3.9), and then used to solve for P-V-T values for vapours only. It was found that a l l P-V-T calculations f a i l e d for l i q u i d s , hence only regions II, IV and V are treated here. Walas (101) states that the isotherms of the v i r i a l expansions do not contain Maxwell's loops (Figure 1), so even with a large number of terms they cannot represent the l i q u i d state and coexistence of l i q u i d and vapour phases. Volumetric v i r i a l expansion results were not compared to those of the pressure expansion because i t was not always possible to solve each expansion over the same data ranges. Walas (101) points out that at high densities, both i n f i n i t e series in pressure and volume diverge. He also states that for a given truncation (four v i r i a l terms for both equations (3.8) and (3.9) in t h i s work) the v i r i a l expansion in volume is usually more accurate than the v i r i a l expansion in pressure. The volumetric solutions of equation (3.8) showed that LHC* is superior in the s u b c r i t i c a l regions, while LHCII proved to be i n f e r i o r to a l l but LHCI. The results are presented as Table 4: 74 Table 4; Volumetric V i r i a l Expansion Volume Calculation RMS-% Error Region II LHC*(2.50), RK(4.34), S72(4.40) IV L H C * ( 1 . 2 1 ) , LHCHB(1.23), LHCA(1.24), LHCSP ( 1 . 2 4 ) , LHCCB(1.24), S72(1.37), RK(1.86) V R K ( 2 . 3 4 ) , S72(2.52), LHCIl(3.36), LHCl(8.94) The results for individual compounds are found in Tables 33-35. Pressure calculation results from equation (3.8) are d i f f i c u l t to draw conclusions from. LHC* i s second in regions II and IV, giving results that are very close behind the most accurate equations, S72 and LHCA respectively. The Soave and Redlich-Kwong equations respectively provide results that are the worst in region IV but are comparable and even better than LHCII in regions II and III. The correcton factor equations do not perform well in region IV, the exception being LHCA, which finished f i r s t . The LHCI equation yields the poorest values in region V. The results are presented as Table 5: 75 Table 5: Volumetric V i r i a l Expansion Pressure Calculation RMS-% Error Region II S72U.97), LHC*(4.00), RK(4.42) IV LHCA(0.72), LHC*(0.74), S72(0.90) f RK(1.16), LHCCH(2.28), LHCHB(2.40) LHCSP(2.88) V RK(712.13), S72(837.02), LHCII(969.36), LCHI(972.47) The high errors in region V are due to the presence of compressed l i q u i d values in the data set. The results on a compound by compound basis can be viewed in Table 36-38. The CPU times (Table 37) are also higher for the correction factor equation due to the bisection procedure employed. Temperature results from the volumetric expansion again show that the LHC* equation is very accurate in regions II and IV. In the l a t t e r case, LHC* i s fourth but only 0.02% behind the leader, the Soave equation, with LHCA and LCHHB both 0.01% behind the Soave equation. There i s no real advantage to using any one of the four when the results are so close. In region V, the Redlich-Kwong equation is c l e a r l y superior, with LHCII and LHCI th i r d and fourth place respectively. Tables 39-41 present the results on a compound basis; Table 6 shows the o v e r a l l RMS-% 76 e r r o r s . T a b l e 6: V o l u m e t r i c V i r i a l E x p a n s i o n T e m p e r a t u r e C a l c u l a t i o n RMS-% E r r o r R e g i o n I I L H C * ( 1 . 0 8 ) , S 7 2 ( 1 . 0 9 ) , R K ( 1 . 6 4 ) I V S 7 2 ( 0 . 3 7 ) , L H C A ( 0 . 3 8 ) , L H C H B ( 0 . 3 8 ) , L H C * ( 0 . 3 9 ) , R K ( 0 . 5 0 ) r L H C C H ( 2 4 . 4 8 ) V R K ( 0 . 9 5 ) , S 7 2 ( 1 . 3 2 ) , L H C I I ( 1 . 5 7 ) , L H C I ( 2 . 2 2 ) T h e C P U t i m e s f o r t h e c o r r e c t i o n f a c t o r m e t h o d ( T a b l e 4 0 ) a r e a g a i n h i g h d u e t o t h e b i s e c t i o n p r o c e d u r e s i n v o l v e d . T h e b i s e c t i o n a l g o r i t h m ( C h a p t e r 3) i s a n i t e r a t i v e p r o c e d u r e , i n a d d i t i o n t o w h i c h e q u a t i o n s ( 3 . 6 ) , ( 3 . 8 ) a n d ( 3 . 9 ) a r e s o l v e d b y n o n - l i n e a r m e t h o d s , h e n c e C P U t i m e a c c u m u l a t e s . T h e p r e s s u r e e x p a n s i o n s o l v e d f o r v o l u m e s h o w s t h a t t h e L H C * e q u a t i o n i s c l e a r l y s u p e r i o r t o t h e S o a v e o r t h e R e d l i c h - K w o n g e q u a t i o n s i n r e g i o n s I I a n d I V , b u t i n t h e l a t t e r r e g i o n , a l l c o r r e c t i o n f a c t o r m e t h o d s a r e s u p e r i o r t o i t . T h e o v e r a l l w i n n e r a m o n g t h e m i s L H C H B , f o l l o w e d b y t h e o t h e r c o r r e c t i o n f a c t o r m e t h o d s , a l l w i t h i d e n t i c a l a c c u r a c i e s . I n r e g i o n V , b o t h t h e S o a v e a n d R e d l i c h - K w o n g e q u a t i o n s r e s p e c t i v e l y , a r e c l e a r l y s u p e r i o r t o L H C I I , w i t h L H C I f o l l o w i n g t h a t . T h e g e n e r a l l y h i g h 77 RMS-% errors in region V are due to the presence of compressed l i q u i d data. The results for individual compounds are compiled in Tables 42-44, with Table 7, appearing below, showing the o v e r a l l r e s u l t s . Table 7: Pressure V i r i a l Expansion Volume Calculation RMS-% Error Region II LHC*(24.01), RK(24.53), S72(24.56) IV LHCHB(2.73), LHCA(2.74), LHCCH(2.74), LHCSP(2.74), LHC*(2.82), S72(2.74), RK(3.17) V RK(20.57), S72(35.95), LHCI I ( 44 . 05 ) , LHCK51.37) In c a l c u l a t i n g P from pressure expansions, i t was found that a l l correction factor equations f a i l e d to converge. As such, LHC* was the only LHC variation in regions II and IV, where i t performed best in the former and worst in the l a t t e r . In region V, LHCI and LHCII performed best in that order. The individual compound results are presented in Table 45-47, while Table 8 l i s t s the o v e r a l l performance of each equation. Table 8: Pressure V i r i a l Expansion Pressure Calculation RMS-% Error Region II LHC*(8.31), S72(8.53), RK(8.81) IV S72(1.75), RK(1.91), LHC*(2.20) V LHCK90.94), LHCII ( 1 20. 1 8) , RK(396.08), S72(588.97) The correction factor methods also f a i l e d when temperatures were to be determined from pressure expansions. LHC* was more accurate in region II, followed closely by the Soave equation, but the order was reversed in region IV by a somewhat wider margin over that of region I I . In both cases, the Redlich-Kwong equation was badly outdistanced. In region V, the Soave equation was c l e a r l y f i r s t followed by LHCII. Both outdistanced the LHCI and Redlich-Kwong equations. Tables 48-50 contain individual compound results, while Table 9 contains the compilation of overall equation results. 79 Table 9: Pressure V i r i a l Expansion Temperature Calculation RMS-% Error Region II LHC*(2.99), S72(3.15), RK(l9.22) .IV S72(1.01), LHC*(3.30), RK(10.25) V S72O.30), LHCIK1.61), LHCK2.07), RK(3.07) In summary, the LHC*-LHCII tandem i e . equations (3.1) and (3.2) i s c l e a r l y the most successful LHC combination for v i r i a l expansions. Both function in a l l cases, a l b e i t , not always to the same degree of accuracy. One or more of the correction factor methods provides highly accurate results in some cases (Tables 4-7), the most consistent being the Harlacher-Braun and Aitken methods. The volume and pressure v i r i a l expansions based on the correction factor methods are unorthodox because the v i r i a l c o e f f i c i e n t s defined by equations (3 . 10)-(3.15) are not only functions of temperature, but pressure as well. This makes the two v i r i a l equations more d i f f i c u l t to solve due to their added complexity, which may in fact be responsible for the pressure expansion's f a i l u r e to calculate P and T by any correction factor method. Equations (3.1) and (3.2) appear to be the best possible 80 a equation combination in a l l regions studied. Also a t t r a c t i v e is their s i m p l i c i t y , when compared to any combination involving a correction factor. 3. THERMODYNAMIC PROPERTIES The thermodynamic departure functions calculated here are enthalpy, entropy, Helmholtz and Gibbs free energies, internal energy, fugacity and the isobaric and isometric heat cap a c i t i e s . The LHC equation correction factor method chosen was Aitken's method. This was simply because this method was proven to be highly accurate in c a l c u l a t i n g P-V-T properties, and i t is applicable to a l l compounds studied in this work. The LHC equation (LHC*) w i l l be studied in the s u b c r i t i c a l region while LHCI and LHCII w i l l treat the s u p e r c r i t i c a l region. Adding a new aspect to the studies in this section i s the introduction of equation (3.3), to treat the c r i t i c a l isotherm. Equations (3.1)—(3.3), as LHCIII, w i l l eliminate the s i n g u l a r i t y at Tc. The Redlich-Kwong, Soave 1972, BWR, and BACK equations w i l l also be studied, for comparative purposes. Enthalpy results indicate that for a l l regions studied, the LHCIII equation finishes t h i r d or fourth in a l l cases. The behaviour of the Soave and Redlich-Kwong equations i s 81 inconsistent. Generally they perform best in regions III and IV, where they equal or exceed the performance of LHCIII. LHCI y i e l d s the best results of a l l two constant equations in region V, f i n i s h i n g second. LHCA produces very inaccurate results in regions I, III and IV. It was however the second most accurate equation, behind the BWR equation in region II. The LHCI equation performed best with vapours, f i n i s h i n g second and t h i r d in regions IV and II respectively. The BWR equation was the most accurate equation of state in region II and V, while only LHCA was more inaccurate in the remaining three regions. There is a wide margin of performance with this complex equation. The BACK equation reached i t s peak performance being the most accurate in region I, while i t finished at highs of f i f t h to seventh in the other regions. This behaviour compares with that of Chen et a l (45) where they studied the saturated l i q u i d curves of compounds (Helmholtz free energy and internal energy). Unfortunately, they f a i l e d to include results outside region I, even though they stated to have conducted studies outsided the saturated l i q u i d curve. The enthalpy residual results are presented in Table 10, while the individual compound results are found in Tables 51-55. Table 10: Enthalpy Departure RMS-% Error 82 Region I BACK(3.66), S72(6.35), LHCIII(7.03), RK(7.99), LHC*(10.58), BWR(13.61), LHCA(70.49) II BWR(14.96), LHCA(22.49), LHC*(25.25), LHCIII(25.36), S72(25.98), RK(27.32), BACK(9945.06) III S72(418.57), RK(420.56), LHCIII(422.29), LHC*(422.33), BACK(429.51), BWR(430.44), LHCA(496.77) IV RK(154.63), LHC*(169.52), LHCIII(169.62), S72071.72), BACK( 188.73) , BWR(209.30), LHCA(488.64) V BWR(2.18E15), LHCI(2.84E15), LHCII(4.62E15), LHCIII(4.62E15), RK(5.20E15), BACK(5.57E15), S72(7.39E15) Of note are the high magnitude errors associated with region V. The size of the errors themselves seems to be caused by two problems. F i r s t , sign changes in the calculated departure function values cause large errors when compared to the experimental values. The calculated results seem to approach the ideal gas region before the experimental values do. Second, there are instances where the experimental data approach the ideal gas enthalpy y i e l d i n g a value of 10" 1 6 for example. This means that 83 even when a calculated departure value of unity is produced, the percent error (equation (4.1)) between the two numbers i s of astronomical magnitude ( 1 0 1 8 % ) . Carrying out the computer calculations in single precision programming would mean that the experimental value would be zero, hence the RMS-% error would be undefined at that point. The large errors calculated above es s e n t i a l l y overshadow the others. It is necessary to remember that RMS-% errors are not homogeneously d i s t r i b u t e d but rather increase gradually along an isobar, with some sign changes occurring, as the ideal gas state i s approached. If one studies the CPU times in Tables 51-55, one notices that a l l values are generally of the same order of magnitude. This i s because a l l enthalpies are calculated d i r e c t l y using equations (3.25), so no CPU time i s spent employing non-linear root finding methods. Calculated entropy results show that no equation i s capable of providing accurate results in a l l f i v e regions. The Soave equation provides the most accurate entropy results of any equation studied in this work. It i s the most accurate in regions I, II and I I I , but i t s accuracy declines u n t i l i t i s the most inaccurate equation in region V. LHCIII finishes second to fourth in a l l regions studied, just infront of LHC* in regions I-IV and LHCII in region V. LHCIII i s an improvement over both LHCI and LHCII in the fiv e regions. LHCI does however behave better than 8 4 LHCIII in region V. The Soave equation is the most accurate equation outside region V, with the Redlich-Kwong equation close behind. In region IV, the order is reversed however. LHCIII i s i n f e r i o r to both in regions II-IV. The accuracy of the BWR equation ( f i f t h out of seven equations) i s superior only to the BACK and LHCA equations in regions I and III, while the BACK equation rises from sixth and seventh in regions I — 111 up to f i r s t in regions IV and V. The results are summarized below in Table 11, while in d i v i d u a l compound results appear in Tables 56-60. The very high errors in region V are generally due to sign changes in the calculated values. 85 Table 11: Entropy Departure RMS-% Error Region I S72(38.95), LHCIII(39.53), LHC*(41.12), RK(42.86), BWR(49.99), BACK(57.20), LHCA(99.00) II S72(350.54), RK(351.65), LHCA(353.63) , LHCIII(354.25), LHC*(355.94), BWR(362.48), BACK(1222.62) III S72(40.36), RK(41.89), LHC*(45.87), LHCIII(45.88), BWR(51.38), BACK(95.09), LHCA(109.24) IV BACK(77.34), RK(523.71), S72(525.70), LHCIII(527.05), LHC*(529.02), LHCA(553.45), BWR(531.64) V BACK(136.86), LHCI(5543.11), LHCIII(5545.67), LHCII(5546.83), RK(5552.67), BWR(5558.58), S72(5562.89) Helmholtz free energy is the most inaccurately calculated property by a l l equations. The RMS-% error increases from l i q u i d to vapour to s u p e r c r i t i c a l region for a l l equations. The LHCIII and LHC* equations produced the most accurate r e s u l t s of any two constant equations in region I I I . The LHCIII equation finished second to the Soave equation in region I. In regions II and IV, the more complex equations, BACK, BWR and LHCA appear to be the most accurate. In region V, the BACK equation i s the most accurate followed by the LHCI equation. The 8 6 individual compound results are found in Tables 61-65, while Table 12 contains the overall RMS-% errors for each region. Table 12: Helmholtz Free Energy  Departure RMS-% Error Region I S72(38.95), LHCIII(39.53), RK(39.98), LHC*(41.12), BWR(49.99), BACK(57.20), LHCA(99.00) II BACK(6036.87), BWR(4.17E4), LHCA(4.21E4), S72(4.21E4), LHCIII(4.21E4), LHC*(4.21E4), RK(4.21E4) III LHCIII(48.61), LHC*(48.61), S72(61.80), RK(207.96), BWR(258.28), LHCA(703.16) , BACK(1714.34) IV BACK(1.41E4), LHCA(3.15E5 ) , BWR(3.15E5), LHCIII(3.16E6), LHC*(3.16E6), S72(3.16E6), RK(3.16E6) V BACK(2.11E4), LHCI(1.54E5), RK(l.55E5), LHCIII(1.56E5), LHCII(1.55E5), BWR(1.56E5), S72(1.56E5) The Gibbs free energy cal c u l a t i o n s are generally more accurate than the respective Helmholtz energy values, with the possible exception of region I I I . The LHC equations in a l l forms decline in accuracy as one goes from regions I to V. LHCIII is the most accurate equation in region I. Its accuracy declines as 87 one goes from regions II to V and results become mixed. It generally remains as one of the two most accurate LHC equations under study here for Gibbs free energy however. The Soave equation shows up well in region IV in second place behind the BACK equation, while in other regions i t occupies either sixth or seventh spot. The Redlich-Kwong equation reaches a low of f i f t h in regions I and II while i t rises as high as t h i r d in regions IV and V. The BACK equation is the most accurate equation in regions IV and V. Tables 66-70 summarize the individual compound results, while Table 13 compares the overal l r e s u l t s . 88 Table 13: Gibbs Free Energy Departure RMS-% E r r o r Region I LHCIII(57.02), LHCA(63.43), LHC*(63.86), BWR(79.85), RK(267.22), S72(1380.88), BACK(2368.50) II BWR(512.79), LHCA(518.83), LHC*(518.35), LHCIII(513.37), RK(519.98), S72(533.77), BACK(814.68) III LHCA(718.75), LHCI11 (760.45), LHC*(760.45) , RK(823.81), BWR(841 .59), S72(131 5.44) , BACK(1872.89) IV BACK(599.78), S72(915.07), RK(1002.52), LHC*(1026.16), LHCIII(1026.18), LHCA(1057.25), BWR(1074.81) V BACK(1473.43), LHCI(8.77E4), RK(8.82E4), LHCIII(8.83E4), LHCII(8.83E4), BWR(8.85E4), S72(8.91E4) For i n t e r n a l energy r e s i d u a l s , the Soave equation f i n i s h e d as one of the three most accurate equations i n a l l f i v e r e g i o n s . I t f i n i s h e d second i n regions I and I I I , and t h i r d i n the remaining r e g i o n s . In s p i t e of the f a c t that LHCIII f i n i s h e d no higher than t h i r d to f i f t h i n a l l regions s t u d i e d , i t s accuracy s t i l l remains co m p e t i t i v e with that of any c u b i c equation t h a t beat i t . The non-cubic BWR and BACK equations g e n e r a l l y p r e d i c t i n t e r n a l energy best. 89 The BACK equation was the most accurate in region I, III and V, while i t was second and last in region IV and II r e s p e c t f u l l y . The BWR equation was f i r s t in regions II and IV. Table 14 summarizes the overal l internal energy res u l t s , while the individual compound results are in Tables 71-75. Table 14: Internal Energy Departure RMS-% Error Region I BACK(4.31), S72(7.60), LHCIII(8.51), RK(9.35), LHC*(13.38), BWR(15.53), LHCA(79.56) II BWR(18.49), LHCA(29.07), S72(31.99), LHC*(32.38), LHCIII(32.52), RK(35.23), BACK(1193.03) III BACK(30.79), S72(32.41), RK(34.48), LHCI11 (36.33), LHC*(36.37), BWR(45.59), LHCA(120.23) IV BWR(19.42), BACK(23.27), S72(27.66), LHCIII(27.99), LHC*(27.99), RK(32.72), LHCA(48.87) V BACK(113.25), BWR(137.00), S72(440.75), RK(470.68), LHCIII(573.52), LHCII(576.15), LHCI(596.35) The fugacity results presented below in Table 15 are deceptively inaccurate. This is due to the presence of outlying argon r e s u l t s . Tables 7 6 - 8 0 should be studied very c a r e f u l l y to see the ef f e c t of this one compound. Actually the inaccuracy of 90 the argon calculated fugacities can probably be contested since the fugacities which they were compared to (calculated from Helmholtz free energy values, using a thermodynamic identity) are themselves suspect. Studying Table 76-80, one notices that normal f l u i d fugacities are generally well calculated by the various equations of state, and that the pattern should most l i k e l y hold for a simple f l u i d l i k e argon which has no polar or quantum anomalies. In keeping with the pattern established in the previous tables, the Soave equation can be considered to be the most unreliable of a l l equations of state tested for fugacity. Tables 76-80 w i l l v e r i f y t h i s . In regions I-IV, the BWR equation can be considered the most r e l i a b l e while i t drops to f i f t h in region V. There i s no pattern established for the other equat ions. 91 Table 15: Fugacity RMS-% Error Region I BWR(2320.73), RK(3333.92), LHC*(3921.45), LHCIII(3921.70), BACK(3944.53), LHCA(3948.66), S72(8.60E4) II BWR(3863.38), RK(3957.01), LHC*(3963.22 ) , LHCA(3963.22), LHCI11 (3963.35), BACK(4002.67) , S72(4349.42) III BWR(8.33E13), BACK(8.33E13), LHCIII(8.35E13), LHC*(8.35E13), RK(8.35E13), S72(8.38E13 ) , LHCA(8.42E13) IV BWR(3603.74), LHCA(3625.29), BACK(3625.33 ) , RK(3627.34), LHC*(3628.67), LHCI11 (3628.84), S72(3665.74) V LHCI(2.18E4), LHCIII(2.34E4), LHCII(2.34E4), RK(2.44E4), BWR(2.45E4), BACK(2.46E4) , S72(4.44E6) Isobaric heat capacity calculations show that LHCIII finishes as the th i r d most accurate equation in a l l regions. The most accurate equation o v e r a l l i s the Redlich-Kwong equation which finishes f i r s t in regions I, III and V and second in regions II and IV. At the opposite end of the spectrum i s the BACK equation, which i s the most inaccurate equation for ca l c u l a t i n g Cp in a l l but region III, where i t fin i s h e d second to last ahead of the LHCA equation. The results are presented in 92 Table 81-85. Table 16: Isobaric Heat Capacity  Departure RMS-% Error Region I RKU2.03), S 7 2 U 6 . 0 6 ) , LHCI 11 ( 59 . 98) , LHC*(64.74), BWR(260.90), LHCA(643.92), BACK(5102.90) II BWR(43.19), RKU8.90), LHCI11(49.56) , S72(49.68), LHC*(51.47), LHCA(58.40), BACK(1.10E10) III RK(29.27), S72(41.63), LHCI11 (55.66), LHC*(57.25), BWR(279.15), BACK(1041.10), LHCA(2.78E9) IV BWR(28.27), RK(30.69) r LHCI11(31.14) , S72(31.15), LHCA(45.92), LHC*(46.87), BACK(2490.15) V RK(30.88), S72(36.92), LHCI11 (46.78) , BWR(153.53), LHCII (320. 02) , LHCK321.00) BACK ( 4867 . 24 ) For constant volume heat capacity, the most accurate equation o v e r a l l is the Soave equation, f i n i s h i n g f i r s t in a l l but region IV. Next would be the LHCIII equation which finished second in a l l but region IV, where i t was f i r s t . The BACK equation was the most inaccurate, f i n i s h i n g l a s t in a l l but region III where i t finished second to l a s t , in front of LHCA. The Redlich-Kwong equation had mixed results. Tables 86-90 contain the individual compound resu l t s , while Table 17 contains 93 the o v e r a l l r e s u l t s . Due to a lack of compound resu l t s , the BWR equation was not included in Table 17. Table 17: Isometric Heat Capacity  Departure RMS-% Error Region I S72( 1 18.74), LHCIII(154.36) , LHC* ( 183.89) , LHCA(1181 .28), RK(1869.85), BACK(8.11E6) II S72(81.10), LHCIII(86.29), LHC*(102.83), LHCA(124.73), RK(128.29), BACK(1.69E10) III S72(96.33), LHCIII(139.41), LHC*(142.98), RK(13050.15), BACK(9.00E8), LHCA(6.54E9) IV LHCIII(33.84), S72(36.46), RK(41.41), LHC*(45.93), LHCA(54.02), BACK(5.93E5) V S72(89.80), LHCIII(96.68), RK(279.99), LHCII(481.58), LHCI(491.31), BACK(5.24E7) Errors in Cv can be attributed to several sources. The major sources of error are in determining the Cv (equation (3.29)) and Cpr (equation (3.28)) values at the same points. Cp i s determined from Cpr and the ideal gas heat capacity polynomials taken from Reid, Prausnitz and Sherwood (61), then used to calculate Cv. Thus Cv contains the errors associated with equations (3.28) and o (3.29), as well as any that might be associated with Cp, although 94 there i s no way of knowing what th i s l a t t e r error might be. The superiority of LHCIII over LHCI, LHCII and LHCA due to equation (3.3) has been demonstrated in Tables 10-17, but as yet, no evidence of improved accuracy along the c r i t i c a l isotherm has been demonstrated. Table 91 presents such evidence for propane. For a l l departure functions except Helmholtz free energy, the LHC equation based on equation (3.3) is as accurate as or better than the other equations tested. Clearly equation (3.3) and i t s derivatives are a success. In concluding the section on thermodynamic properties, i t should be pointed out that with several exceptions, enthalpy, entropy, internal energy and fugacity results were considered to be f a i r l y accurately calculated for a l l equations. The Helmholtz and Gibbs free energy values corresponding to the same points were notoriously inaccurate, even though a l l values are connected by thermodynamic i d e n t i t i e s , extraneous of the equations of state themselves. In many instances, the LHCIII equation, based on equations (3.1)-(3.3), i s not the most accurate equation of state, but i t s sim p l i c i t y and accuracy r e l a t i v e to the other LHC forms (LHCI, LHCII, LHCA) are also favourable. The LHCIII equation i s used to generate thermodynamic departure plots for enthalpy (Figures 19-32) and entropy (Figures 33-46). Special note should be taken 9 5 when studying the aforementioned figures. The reader w i l l notice sudden s h i f t s in the curves near the c r i t i c a l point. These s h i f t s are the result of the tr a n s i t i o n s between equations (3.l)-(3.3), according to their temperature ranges of a p p l i c a b i l i t y (Table 101). Although the transitions between the three equations are not e s t h e t i c a l l y appealing when viewed graphically, the RMS-% error values of Tables 10-17 indicate that LHCIII i s generally more accurate than LHC*, and hence i s an improvement over i t . 4. JOULE-THOMSON COEFFICIENTS AND THE INVERSION CURVE The Joule-Thomson c o e f f i c i e n t and the inversion curve locus of the LHC equation ( a l l forms) are defined by equations (3.40) and (3.42). A l l equations used to calculate thermodynamic properties were again used in t h i s section. The one notable exception is that the BACK equation i s not included in the inversion curve study since i t s performance in determining Joule-Thomson c o e f f i c i e n t s was poor i . e . i t did not produce crossover points. A l l values of n were negative. Joule-Thomson c o e f f i c i e n t and inversion curve calculations are thought to be the severest tests that an equation of state can be subjected to. The inversion curve results presented here are special because, with the exception of methane and argon, none of the other compounds have been included as part of 9 6 inversion curve studies of t h i s type before. The LHCIII equation was probably the most consistent of a l l equations used to calculate M, f i n i s h i n g second in a l l but region II, where i t was t h i r d . There is no one equation which i s consistently superior to LHCIII. The BACK equation i s consistently l a s t in regions I and I I I . The BWR equation also has th i s problem. As far as calculating the inversion curve locus i s concerned, LHCIII is the most accurate of a l l equations tested, and was therefore the LHC-type equation chosen to be included in the inversion curve plots of Figures 47-51. One sees from the Table 19 that the LHC equation and the Redlich-Kwong equation are the two most accurate. From Figures 47-51, the LHC equation appears to be most accurate in the lower region of the curve, while the Redlich-Kwong and Soave equations appear to be more accurate in the intermediate region of the inversion curve. The accuracy of the LHC equation declines with increasing temperature, while the exact opposite occurs for both the Soave and Redlich-Kwong equations. The accuracy of the LHC equation in the lower arm region of the inversion curve appears to compensate for the inaccuracy at higher temperatures, when the RMS-% error i s calculated. The superior performance of the LHC equation at lower temperatures i s useful for design purposes because t h i s is 97 the region in which refrigerators and l i q u i f i e r s usually operate in the lower portion of their cycles (59). From Figures 47-51, one sees that there i s great v a r i a b i l i t y in predicting the Boyle temperature (the temperature at which Pr=0 along the upper arm of the inversion curve (Figure 4)) using an equation of state. This behaviour i s not unique to t h i s work alone, but i s also apparent in the work of Dilay and Heidemann (56) and J u r i s and Wenzel (57). The LHC equation consistently provides the largest value of the Boyle temperature for a l l compounds studied in this work, which i s in l i n e with the increasing inaccuracy that becomes apparent as temperature increases. The Boyle temperatures, in reduced form, for methane (Figure 47) and argon (Figure 51) are approximately 7.3 in both cases, while the work of Hendricks et a l (59) displays them graphically as being approximately 5.3 and 5.4 respectively. From Figure 47 and 51, the Redlich-Kwong equation predicts the Boyle temperature for methane and argon as approximately 5.3 in both cases, followed by the Benedict-Webb-Rubin equation at 5.6 and 6.0 respectively. The Redlich-Kwong equation i s therefore the most accurate equation for predicting the Boyle point temperature. F i n a l l y , special note should also be made of the behaviour of the LHC equation in the c r i t i c a l region (Tr=1.0, Pr=1.0) in 98 Figures 47-51, where the tra n s i t i o n s between equations (3. 1) — (3.3) appear graphically. The o v e r a l l Joule-Thomson c o e f f i c i e n t and inversion curve results are present in Tables 18 and 19 respectively, while the individual compound results are in Tables 92-96 and 97 respect i v e l y . Table 18; Joule-Thomson Coefficient RMS-% Error Region I LHCA(57.20), LHCI11 (309 . 40) , LHC*(363.94), S72(404.11), RK(783.21) II BWR(22.43), S72(34.05), LHCIII(34.16), RK(36.39), LHCA(45.07), LHC*(53.40), BACK(15206.49) III LHCA(256.88), LHCI11(712.28), LHC*(865.09), S72(946.15), RK(3577.43) IV S72(13.52), LHCIII(13.78), LHC*(14.59), RK(15.38), BWR(16.29), LHCA(30.50), BACK(2.09E5) V S72(52.62), LHCI11 (60.39), LHCII(64.95), RK(66.82), BWR(110.89), LHCI(124.99), BACK(7.22E6) 99 Table 19: Inversion Curve RMS-% Error LHCIII(24.27), RK(24.49), LHCII(25.40), LHCIU5.12), S72(50.11 ), BWR(54.59) The consistency and accuracy of the LHCIII equation in ca l c u l a t i n g values of M and the inversion curve locus makes i t the most desirable of a l l equations tested. 5. SPECIAL STUDY OF NORMAL AND PARA-HYDROGEN After establishing equation (3.3), i t was subsequently noted that i t resembled the equation proposed by Graboski and Daubert (23) for normal hydrogen at Tr>2.5, even though they were not designed for the same purpose. The major difference between equation (3.3) and the Graboski-Daubert (GD) equation was that the former yielded a value of 1.0 at the c r i t i c a l temperature, while the l a t t e r did not. In c a l c u l a t i n g the c o e f f i c i e n t d, of equation (3.3), i t was discovered that the a-function accurately f i t both n-H2 and p-H2 over reduced temperature ranges of 0.228-105.327 and 0.230-106.125 respectively, with a maximum error of 3.34% in the former and 3.12% in the l a t t e r . This special case of convergent evolution invited a comparison between the two normal hydrogen equations over a l l f i v e regions, as well as a study of the similar p-H2 version of the LHC equation. The 1 00 r e s u l t s p r e s e n t e d i n T a b l e s 98 a n d 9 9 i n d i c a t e t h a t t h e L H C e q u a t i o n i s c l e a r l y m o r e c o n s i s t e n t t h e n t h e GB e q u a t i o n , a l t h o u g h t h e r e s u l t s b e t w e e n t h e t w o a r e g e n e r a l l y m i x e d i n r e g i o n V . O f s p e c i f i c i n t e r e s t h o w e v e r i s t h a t t h e GB e q u a t i o n p r o v i d e s a v e r y s m a l l e r r o r f o r n ( T a b l e 9 8 ) i n r e g i o n V . T h i s e r r o r i s v e r y d e c e p t i v e b e c a u s e t h e GB e q u a t i o n p r o v i d e s f o r n o c r o s s o v e r p o i n t h e n c e n o p o s i t i v e v a l u e s o f t h e J o u l e - T h o m s o n c o e f f i c i e n t w e r e g e n e r a t e d . A l l J o u l e - T h o m s o n c o e f f i c i e n t s a r e n e g a t i v e o r z e r o . T h e r e f o r e t h e GB e q u a t i o n c a n b e c o n s i d e r e d a f a i l u r e f o r c a l c u l a t i n g n i n t h i s r e g i o n . T h e L H C e q u a t i o n was n o t i n t e n d e d t o b e u s e d e x c l u s i v e l y i n t h e f o r m s i m i l a r t o t h e GB e q u a t i o n , i n f a c t i t i s r e c o m m e n d e d t h a t i t b e u s e d i n t h e f o r m o f L H C I I I , f o r b e s t r e s u l t s . 6. E V A L U A T I N G T H E C P U T I M E I n g e n e r a l , i t i s v e r y d i f f i c u l t t o p r o v i d e a u n i f o r m c r i t e r i o n f o r c o m p a r i n g t h e c a l c u l a t i o n t i m e b e t w e e n d i f f e r e n t s t a t e e q u a t i o n s b e c a u s e o f d i f f e r e n c e s i n f o r m ( c u b i c a n d n o n - c u b i c ) o r d i f f e r e n c e s b e t w e e n s u b r o u t i n e s . C o m p u t e r c a l c u l a t i o n t i m e r e s u l t s s h o u l d t h e r e f o r e be i n t e r p r e t e d w i t h c a u t i o n . I n t h i s w o r k , we d e f i n e t h e c e n t r a l p r o c e s s i n g u n i t ( C P U ) t i m e s t u d i e d a s t h e t o t a l t i m e s p e n t c a l c u l a t i n g o n e s p e c i f i c 101 P-V-T or thermodynamic property for each compound, over a given region (I-V). The CPU time, however, does not include the time spent in c a l c u l a t i n g the percent error between experimental and calculated data points (equation (4.1)) or the RMS-% error (equation (4.2)). CPU time i s the time spent in carrying out r e p e t i t i v e calculations for each individual data point. Using the Soave 1972 equation in cubic polynomial form (equation (2.19)) as an example, the time spent evaluating the constants "a, b and m" (equations (2.15), (2.16) and (2.25) respectively) would not be included as part of the CPU time because they are compound s p e c i f i c constants and therefore need only to be evaluated once. The times necessary to establish the c o e f f i c i e n t s of equation (2.19) (determined from equations (2.23), (2.24), (2.13) and (2.14) in that order), solve the polynomial to obtain the one sought-after root as well to convert the root to the correct form (Z to V using equation (2.12)) for use in percent and RMS-% error calculations would a l l be included in the CPU time. Since these calc u l a t i o n s are r e p e t i t i v e , they are a l l included in a "DO loop". The CPU time therefore represents the t o t a l time spent within the "DO loop" i t s e l f . Since the LHC, Soave 1972 and Redlich-Kwong equations d i f f e r only in the forms of their a-equations, changes inside the DO 102 loop are minimal. The LHCIII equation does however require a * series of "IF" statements so that the appropriate a(T ) equation (equations (3.l)-(3.3)) may be chosen depending upon temperature. In the case of the two constant equations therefore, CPU time differences r e f l e c t the differences in the a-function forms themselves, since a l l three cubic equations require an i d e n t i c a l root finding algorithm, RPOLY2 (105). Although the CPU times of the non-cubic equations (BWR, LK and BACK equations) are evaluated in the same manner as the two constant equations above, differences in CPU times between the cubic and non-cubic equations also r e f l e c t differences in the non-linear root finding methods (RPOLY2 (105) in the former and ZER02 (104) in the l a t t e r ) . These differences w i l l be discussed in greater d e t a i l later in this chapter. 7. SUMMARY Tables 1-19 show that equations (3.1)-(3.3) and their derivatives have proven to provide the most consistent and accurate results of any LHC-type equation tested. Therefore these three equations and their derivatives are recommended as being the equations upon which to base the LHC equation for general use in a l l of the five regions, as defined in t h i s work. A l l three a equations are convenient to use, and are accurate over the 1 03 temperature ranges t e s t e d . E q u a t i o n s (3.1) and (3.2) a l o n e a r e s u f f i c i e n t f o r c a l c u l a t i n g P-V-T p r o p e r t i e s , w h i l e a l l t h r e e of e q u a t i o n s ( 3 . 1 ) — ( 3 . 3 ) and t h e i r d e r i v a t i v e s a r e t o be used i n c a l c u l a t i n g thermodynamic p r o p e r t i e s . The CPU t i m e s a s s o c i a t e d w i t h each compound's r e s u l t s seem t o i n d i c a t e t h a t the two c o n s t a n t e q u a t i o n s (not i n c l u d i n g the c o r r e c t i o n f a c t o r methods) have s i m i l a r c a l c u l a t i o n speeds. In g e n e r a l , the l a r g e d i f f e r e n c e s e x i s t i n g between the CPU times of the s i m p l e s t two c o n s t a n t e q u a t i o n s and those of the L e e - K e s l e r , BWR and BACK e q u a t i o n s are due t o the e f f i c i e n c y of the r o o t f i n d i n g methods of the computing c e n t r e . T h i s comparison o n l y seems t o be n e c e s s a r y when s o l u t i o n s t o the e q u a t i o n s a r e not e x p l i c i t . The a l g o r i t h m e f f i c i e n c y was a c t u a l l y b e i n g e v a l u a t e d , not the e q u a t i o n s . The c o r r e c t i o n f a c t o r methods used f o r e s t a b l i s h i n g p r e s s u r e and temperature v a l u e s c o n s t i t u t e a s p e c i a l problem and was d i s c u s s e d i n the P-V-T r e s u l t s s e c t i o n . There are s e v e r a l sources of e r r o r i n e v a l u a t i n g CPU t i m e s . To b e g i n w i t h , the a l g o r i t h m s t h a t employed the b i s e c t i o n p r o c e d u r e f o r P and T s o l u t i o n s r e q u i r e d t h a t upper and lower l i m i t s be s e t on the r e g i o n t o be s e a r c h e d . G e n e r a l l y the upper l i m i t was s e t a t Tc or Pc but cases e x i s t e d where v a l u e s g r e a t e r than t h e s e were r e q u i r e d . The Soave 1972 and Redlich-Kwong e q u a t i o n s e a s i l y adapted t o t h i s w h i l e the c o r r e c t i o n f a c t o r 1 04 methods were not designed to handle values greater than Tc or Pc since they are based on saturation data. The result was that they remained stranded in regions III and IV, f a i l i n g to evaluate cert a i n points near Tc or Pc. When the Soave and Redlich-Kwong results were adjusted to accomodate t h i s deficiency in the correction factor method there was always error (estimated at ±0.003-0.010 seconds depending on the size of the data set) associated with the adjustment. One would think that increasing the size of the range to be searched would increase the CPU times. This is not necessarily so. Since linear interpolation, r a t i o n a l interpolation and bisection are employed, there were instances where CPU times remained unaffected i e . methanol pressures solved by the Redlich-Kwong and Soave equations, or where the increase was only by 0.001 seconds i e . the pressure of n-hexane by the same equations. The time of day in which the tests were conducted in the computing centre also affected the computer times. Obviously, the CPU times are larger for runs conducted during the day (n-hexane pressures from Aitken's method increases by 0.007 seconds in the daytime as compared to the evening, while the Harlacher-Braun method accounts for a 0.018 second increase at the same time of day). To minimize the effects of t h i s , a l l equations tested on the same compound were run at the same time of day, even though 105 not a l l compounds were run at the same time of day. Even the time of year may have had a marked impact on the CPU times. Obviously December and late March are the busiest times of the year while May and late August are probably the least busy. No attempt was made to evaluate the CPU time changes through the year however. Clearly there are many factors involved in interpreting CPU times. It should also be mentioned that the computing centre i n s t a l l e d a new compiler in late August of 1985, thus a l l CPU times of the P-V-T properties in regions I-IV were based on the speed of the older more sluggish compiler. At a l l times every attempt was made to minimize the v a r i a b i l i t y of the CPU times due to various sources of error. Identical algorithms for the two constant equations (excluding the correction factor versions) were used. Where v a r i a b i l i t y between programs was inevitable, tests were conducted to evaluate the seriousness of the errors. Any resulting errors were found to be no more serious than those noted e a r l i e r , so no special adjustments or precautions needed to be taken. 106 E. CONCLUSIONS AND RECOMMENDATIONS 1. CONCLUSIONS The Lielmezs-Howell-Campbell (LHC) equation, based on equations (3.1)—(3.3) and their derivatives, has been successfully developed for use in ca l c u l a t i n g P-V-T and thermodynamic properties at a l l temperatures, pressures and volumes considered. The calculated results from the LHC equation are competitive with those of the other two constant equations such as Soave 1972 and Redlich-Kwong equations. In some instances the LHC equation matches or even exceeds the performances of more complex equations, for instance the Benedict-Webb-Rubin or BACK equations. Tests have shown that no single equation consistently w i l l outperform the others over the complete range of P-V-T and thermodynamic data available. Each equation has i t s own areas of strength and weakness in terms of ca l c u l a t i o n accuracy. An impressive aspect of the LHC equation is that i t appears to be the most consistently r e l i a b l e of a l l equations tested. This may in i t s e l f be a b e n e f i c i a l aspect of the LHC equation, because while i t may place t h i r d , for example out, of seven equations, i t w i l l usually stay in t h i r d place (second or fourth in some cases), while the other equations bounce around in the standings, 1 07 anywhere from f i r s t to l a s t . The BACK equation i s one of the most notorious examples of t h i s . Thus a consistent t h i r d place and assured r e l i a b i l i t y may be better than accuracy in some cases, when the accuracy accompanied by doubt as to the r e l i a b i l i t y of the c a l c u l a t i o n . For t h i s work, massive data sets have been compiled (eg. for n-H2 we have tested 4 4 0 1 data points in region V alone), and some as yet unstudied problems have been brought to l i g h t while making comparisons between equations of state. In short RMS-% errors are regionally (regions I-V) dependent. The compressed l i q u i d and the ideal gas regions have percent errors so large at some points, that they alone are refl e c t e d in RMS-% error values. Lesser errors in points outside the ideal gas and c r i t i c a l regions become i n s i g n i f i c a n t by comparison. This alone may be one reason why extensive studies between equations of state have not been carried out, and why a l l data should be broken up into regions as was done in this work. Unlike any other two constant equation tested, the LHC equation i s based on experimental P-V-T data in c a l c u l a t i n g the c o e f f i c i e n t s p and q in equations (3.1) and (3.2) (and subsequently d in equation (3.3)). Only experimental saturated vapour-liquid P-V-T data are required to accurately describe a compound through the c o e f f i c i e n t s p,q and d. The BWR equation i s 108 also based on experimental data but i t requires much more extensive data sets (outside the saturation curve) to accurately generate c o e f f i c i e n t s . Since, in many instances the extensive data sets are unavailable, the BWR equation i s lim i t e d to a very narrow range of compounds (generally hydrocarbons and small molecules). The BWR equation has been c r i t i c i z e d in l i g h t of th i s fact for years (41,47,61). This work, the development of the LHC equation is not merely a development of an equation of state, but also a study in the application of the T coordinate system. Of s i g n i f i c a n t interest i s the fact that this is the f i r s t study to be undertaken of the * region where T>Tc as far as T coordinates are concerned, and the resultant changes that must be considered when using them. For * example, because T has a negative sign for T>Tc, equation (3.1) is not applicable, since a negative number cannot be raised to a non-integer power ( a l l values of q are non-integer numbers). This * problem was circumvented by introducing the absolute value of T in equation (3.2). In l i n e with t h i s are the s i n g u l a r i t i e s existing along the c r i t i c a l isotherm, noted by Shaw and Lielmezs (48), when the derivatives of equation (3.1) and (3.2) are used. It became necessary to develop a t h i r d equation (equation (3.3)) whose derivatives were f i n i t e at Tc. 109 The correction factor method of equation (3.5) is of limited use in i t s present form. Tannar, Lielmezs and Herrick (68) demonstrated the a p p l i c a b i l i t y of the correction, Cf, for unsaturated l i q u i d and vapour volumes (T<Tc and P<Pc), but the method remains limited to the two regions as mentioned above. This i s simply due to equation (3.5) being based on saturation data. Joffe and Zudkevitch (24) as well as Lu and coworkers (29-32) experienced the same li m i t a t i o n s in their equations when the s u p e r c r i t i c a l region was to be entered. To date, to the knowledge of thi s author, no researcher has solved t h i s problem. When further studies on the LHC correction factor method were undertaken to solve for temperature and pressure, i t was necessary to resort to a bisection procedure (chapter 3) since the LHC equation contains two unknown values i . e . P and Tsat or T and Psat. Accompanying the LHC equation must be an external means of providing Tsat and Psat, l i k e the Harlacher-Braun equation or Aitken's method of interpolation (using saturated vapour pressure data that i s supplied). Pressure and temperature were only determined when the LHC equation and the external source of saturation data arrived at i d e n t i c a l values of Tsat and Psat using the same P and T values respectively. Unfortunately, the bisection method would not always guarantee a solution, especially with temperatures or pressures near Tc or Pc 1 1 0 r e s p e c t i v e l y . The computer c a l c u l a t i o n t i m e s were f a r l a r g e r u s i n g t h e LHC c o r r e c t i o n f a c t o r method as c ompared t o t h e CPU t i m e s f r om the s i m p l e r forms o f t h e two c o n s t a n t e q u a t i o n s o f s t a t e (LHC, Soave and R e d l i c h - K w o n g ) , r e f l e c t i n g t h e i n e f f i c i e n c y o f t h e b i s e c t i o n method. However, a s s t a t e d i n C h a p t e r 3 , b i s e c t i o n was t h e o n l y u n b i a s e d p r o c e d u r e t h a t c o u l d be u s e d so as t o d i r e c t l y compare two d i s s i m i l a r . s a t u r a t e d v a p o u r p r e s s u r e s o u r c e s l i k e t h e H a r l a c h e r - B r a u n e q u a t i o n an d A i t k e n ' s method of i n t e r p o l a t i o n . CPU t i m e d i f f e r e n c e s a p a r t f r o m t h e c o r r e c t i o n f a c t o r method a r e b a s e d more upon t h e e f f i c i e n c y o f t h e c o m p u t i n g c e n t r e r o o t f i n d i n g a l g o r i t h m s t h e m s e l v e s . F o r i n s t a n c e , one a l g o r i t h m employs l i n e a r i n t e r p o l a t i o n , r a t i o n a l i n t e r p o l a t i o n and b i s e c t i o n t o f i n d a s i n g l e r o o t , w h i l e a n o t h e r u s e s E u c l i d ' s method t o p r o v i d e a l l t h e r e a l and i m a g i n a r y r o o t s o f a p o l y n o m i a l . O b v i o u s l y t h e t i m e s p e n t f i n d i n g a s i n g l e r o o t i s s h o r t e r t h a n t h a t f o r m u l t i p l e r o o t s . F o r thermodynamic p r o p e r t y c a l c u l a t i o n s , t h e c o r r e c t i o n f a c t o r method a g a i n had i t s p r o b l e m s . W i t h some o u t l y i n g p o i n t s ( n e a r t h e i d e a l gas r e g i o n ) , t h e e r r o r s were v e r y l a r g e , w h i l e t h e o t h e r , s i m p l e r , two c o n s t a n t e q u a t i o n s p r o d u c e d more a c c u r a t e v a l u e s . . In s p i t e o f t h e p r o b l e m s m e n t i o n e d a b o v e , t h e c o r r e c t i o n f a c t o r method s h o u l d n o t be a bandoned. I f a n y t h i n g , i t s h o u l d be 111 modified such that a d i f f e r e n t correction factor equation is produced for each thermodynamic property to be calculated, rather than simply having one correction factor equation, for each of regions III and IV, to be used in c a l c u l a t i n g a l l thermodynamic properties. Correction factors could be calculated for every thermodynamic property in a similar manner to that used by Tannar, Lielmezs and Herrick ( 6 8 ) for volumetric c a l c u l a t i o n s . Correction factors were established for volume using multiple linear regression techniques to determine the c o e f f i c i e n t s for equation ( 3 . 5 ) , but whether or not i t may be better to use a nonlinear correction factor equation for the LHC equation remains an open question. In summary, thi s work was the f i r s t to: (1) study a l l phase regions ( l i q u i d , vapour and s u p e r c r i t i c a l f l u i d ) ( 2 ) adapt the T coordinate system to the s u p e r c r i t i c a l region ( 3 ) conduct and document a complete study of the BACK equation ( 4 ) study the inversion curves of propane, n-butane and para-hydrogen ( 5 ) compare CPU times extensively between equations of state 1 1 2 2. RECOMMENDATIONS (1) The l i s t of compounds studied in thi s work i s by no means extensive, therefore the LHC equation should be applied, using the methods of this work, to a greater number of compounds. Accurate substance dependent property relationships could then be developed so as to predict the c o e f f i c i e n t s p, q and d. Lielmezs and coworkers (7) were able to develop an acentric factor dependent function for the c o e f f i c i e n t p, but determination of a similar function for q indicated that q tends to become constant. (2) The LHC equation, in i t s present form should be adapted to binary mixtures. Work has already been done for saturated l i q u i d s by Lielmezs and Beatson (102). However there is a sca r c i t y of data. Before binary mixture studies can be thoroughly c a r r i e d out, recommendation (1) should be completed. Having the co e f f i c i e n t s available for a larger number of pure compounds w i l l mean that any available binary mixture data can be properly accomodated. (3) Other equations l i k e the Peng-Robinson equation and the Van der Waals equation have been developed by Lielmezs and coworkers (35,70), based on equation (3.1). The methods of t h i s work can be readily adapted to these equations also. Work has already been done in adapting the Lielmezs-Law modification of the Van der Waals equation (35) to saturated binary l i q u i d mixtures by 1 1 3 Besher, Wu and Lielmezs (103). (4) A systematic method should be developed for comparing the accuracy of equations of state over a l l regions. (5) It may be possible to use equation (3.3) as a pseudo-saturation curve into the s u p e r c r i t i c a l region and therefore to set up a correction factor equation based on i t . An alternative would be to extend the Harlacher-Braun equation in a similar manner. At present, the only way to use a correction factor in the s u p e r c r i t i c a l region would be to somehow base i t on the c r i t i c a l point coordinates. This was t r i e d unsuccessfully in this work however, and abandoned. Note also that i t i s always dangerous to use an equation l i k e the Harlacher-Braun equation outside of i t s intended region of use. F i n a l l y i t i s best to avoid using Aitken's method for extrapolation into t h i s region as i t i s capable of producing very inaccurate values, and there would be no way to check the v a l i d i t y of them. How to develop the correction factor method for use in the s u p e r c r i t i c a l region remains an open question. NOMENCLATURE A dimensionless parameter in equation (2.13) A,...6 dummy variables in equations (3.30), (3.32), (3.40) and Appendix F, equations (4), (5), (6), (34), (35) and (36) A 0 Benedict-Webb-Rubin c o e f f i c i e n t Ar Helmholtz free energy departure function a (1) parameter in a two constant equation defined in equation (2.16) (2) Benedict-Webb-Rubin c o e f f i c i e n t (3) constant in generalized integral of Appendix F, equation (21) a(T) temperature dependent function of "a" for a two constant equation, given by equation (2.23) B (1) dimensionless parameter in equation (2.14) (2) Lee-Kesler c o e f f i c i e n t (3) dummy variable in Appendix F, equation (11) B 0 Benedict-Webb-Rubin c o e f f i c i e n t B,...„ dummy variables used in Appendix D, equations (39) and (40), Appendix E, equations (39), (42), (45), (53) and Appendix F, (116) and (117) B(T) temperature dependent c o e f f i c i e n t of v i r i a l 1 1 4 1 1 5 expansion in V B'(T) temperature dependent c o e f f i c i e n t s of v i r i a l expansion in P b (1) parameter in a two equation, defined by equation (2.17) (2) Benedict-Webb-Rubin c o e f f i c i e n t (3) constant in generalized integrals of Appendix F, equation (21) . b, ...a Lee-Kesler c o e f f i c i e n t s C Lee-Kesler c o e f f i c i e n t C 0 Benedict-Webb-Rubin c o e f f i c i e n t Cp isobaric heat capacity Cpr isobaric heat capacity departure function C(T) temperature c o e f f i c i e n t of v i r i a l expansion in V C'(T) temperature dependent c o e f f i c i e n t of v i r i a l expansion Cv isometric heat capacity Cf correction factor in Lielmezs-Howell-Campbell equation c (1) Benedict-Webb-Rubin c o e f f i c i e n t (2) BACK equation constant; ration of soft repulsion distance to c o l l i s i o n diameter c, ...3 Lee-Kesler c o e f f i c i e n t s 1 1 6 Lee-Kesler c o e f f i c i e n t dummy variables used in Appendix E, equations (2)-(4) temperature dependent c o e f f i c i e n t of v i r i a l expansion in V temperature dependent c o e f f i c i e n t of v i r i a l expansion in P Lielmezs-Howell-Campbell equation c o e f f i c i e n t for the c r i t i c a l isotherm a-function, given by equation (3.3) Lee-Kesler c o e f f i c i e n t BACK equation c o e f f i c i e n t , based on a simple f l u i d temperature dependent function defined by equation (3.17) function representing interaction energy dependence of BACK equation, in Appendix f, equation (68) (1) denotes a generalized function in equation (2.1) (2) fugacity Gibbs free energy departure function enthalpy departure departure function integral equations in Appendix E, equations (15), (16) and (17) and Appendix F, equations (20), (49) and (56) Boltzmann's constant 1 17 linear c o e f f i c i e n t of expansion (1) acentric factor dependent function for the Soave 1972 equation; see equation (2.37) (2) BACK equation c o e f f i c i e n t (3) c o e f f i c i e n t in the LHC correction factor equation Avogadro's number (1) c o e f f i c i e n t in the LHC correction factor equation (2) generalized exponent of integral of Appendix F, equation (21) pressure at t r a c t i v e pressure force in the Van der Waals equat ion c r i t i c a l pressure reduced pressure repulsive pressure force in the Van der Waals equation saturation pressure in LHC correction factor equation residual pressure c o e f f i c i e n t of the LHC a-functions, given by equations (3.1) and (3.2) c o e f f i c i e n t of LHC a-functions given by equations (3.1) and (3.2) gas constant 1 18 Sr entropy departure function T temperature * T reduced temperature based on both the c r i t i c a l point and normal b o i l i n g point temperatures Tc c r i t i c a l temperature Tnb normal b o i l i n g point temperature Tr reduced temperature Tsat saturation temperature in LHC correction factor equat ion U BACK equation square well potential U° BACK equation hard sphere interaction energy Ur internal energy departure function u dummy variable used in i l l u s t r a t i n g the method of integration by parts; Appendix E, equation (19) V molar volume Vc c r i t i c a l volume V° close packed molar volume of hard spheres V 0 0 close packed molar volume of hard spheres at O'R Vr reduced volume v dummy variable used in i l l u s t r a t i n g the method of integration by parts; Appendix E, equation (19) X dummy variable in equation (3.43) and Appendix A, equation (9) 1 1 9 variable in generalized integrals of Appendix F, equat ion (21) compressibility factor a t t r a c t i v e component of the compressibility factor defined by the BACK equation c r i t i c a l compressibility factor repulsive component of the compressibility factor defined by the BACK equation compressibility factor of the reference f l u i d defined by the Lee-Kesler equation compressibility factor of reference f l u i d defined by the Lee-Kesler equation Symbols (1) Benedict-Webb-Rubin c o e f f i c i e n t (2) sphericity constant for the BACK equation temperature dependent function of a for the Soave 1972 and the Lielmezs-Howell-Campbell equations; see equations (2.24) and (2.26) a-function for the LHC equation defined by equation (2.27) a-function used when Tr<1.0, in the LHC equation a-function used when Tr>1.0, in the LHC equation 1 20 c a(T ) a-function used when Tr=1.0, in the LHC equation 7 Benedict-Webb-Rubin equation c o e f f i c i e n t $ BACK equation variable in equation (2.38) 77 constant for interaction energy dependence in the BACK equation; see equation (2.40) M Joule-Thomson c o e f f i c i e n t p density Z summation sign a c o l l i s i o n diameter Ra dummy parameter in equation (2.16) Rb dummy parameter in equation (2.17) co acentric factor t o r acentric factor for reference f l u i d in the Lee-Kesler equat ion Subscr ipts a (1) denotes a t t r a c t i v e force in the Van der Waals equation (2) denotes association of Ra with the "a" given by equation (2.16) b denotes association of Rb with the "b" given by equation (2.17) c c r i t i c a l 121 m summation of m terms in the BACK equation n summation of n terms in the BACK equation nb normal bo i l i n g point o (1) denotes Benedict-Webb-Rubin c o e f f i c i e n t in equation (2.29) (2) denotes simple f l u i d in the Lee-Kesler equation r (1) denotes repulsive force in the Van der Waals equation (2) reduced coordinate (3) denotes reference f l u i d in the Lee-Kesler equat ion (4) departure function s denotes simple f l u i d in equation (2.20) Superscript a a t t r a c t i v e force in the BACK equation c denotes Tr=1.0 in LHC equation h repulsive force in the BACK equation m exponent in the BACK equation n exponent in the BACK equation 0 denotes hard cores in the BACK equation 0 0 denotes hard cores at 0°K in the BACK equation + denotes Tr<1.0 in the LHC equation denotes Tr>>0.0 in the LHC equation 122 (1) reduced coordinate notation for temperature given by the Lielmezs-Howell-Campbell equation (2) denote residual pressure in the BACK equation denotes simple f l u i d in equation (2.20) denotes f i r s t derivative with respective to temperature in Appendix F, equation (39) denotes f i r s t derivative with respective to volume in Appendix F, equation (109) REFERENCES 1. 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At t h e c r i t i c a l t e m p e r a t u r e , e q u a t i o n (1) i s : (2) P = R T c - i * ^ v-b W D i f f e r e n t i a t i n g e q u a t i o n ( 2 ) : 3P -RTc ^ 2a (3) a n d (4) a v ( v - b ) 2 v 3 2 P _ 2RTc _ 6a 3 V 7 ( V - b ) 3 V 7 At t h e c r i t i c a l p o i n t : ,r) 3P _ 3 2 P _ n { 5 ) 3V " 3 V 7 - 0 t h e r e f o r e e q u a t i o n s (3) and (4) become: 133 1 34 ,r\ RTc _ 2a ( 6 ) (Vc-b) J = W and ( 7 ) 2a = 2RTcVc Vc7 3(Vc-b) 3 Combining equation ( 6 ) and ( 7 ) to solve for b gives: ( 8 ) b- p Substituting ( 8 ) into ( 6 ) and solving for a; ( 9 ) a= |RTcVc At the c r i t i c a l point, equation ( 1 ) becomes: RTc a ( 1 0 ) Pc= Vc-b Vc Combining equations ( 8 ) , ( 9 ) and ( 1 0 ) yields: 3 RTc ( 1 1 ) Pc= 8 Vc Defining the c r i t i c a l compressibility factor, Zc, as: ( 1 2) Zc= P c V c U 1 ' RTc 1 3 5 e q u a t i o n ( 1 1 ) c a n b e r e a r r a n g e d t o y i e l d : ( 1 3 ) Zc= | F r o m e q u a t i o n s ( 1 2 ) a n d ( 1 3 ) , t h e g a s c o n s t a n t R c a n b e r e w r i t t e n a s : 8 P c V c ( 1 4 ) R= 3 T c U s i n g e q u a t i o n s ( 8 ) , ( 9 ) a n d ( 1 4 ) t o s u b s t i t u t e f o r b , a a n d R, r e s p e c t i v e l y , i n e q u a t i o n ( 1 ) g i v e s : ( 1 5 ) I f : ( 1 6 ) T r = ^ ( 1 7 ) P r = f ^ ( 1 8 ) V r = Y ^ n-t 8 P c V c w T ^ _ 3 P c V c 2 3 T c M V - V c / 3 ; e q u a t i o n ( 1 5 ) c a n be r e w r i t t e n a s : 3 1 36 Finally,Van der Waals equation can be transformed into cubic form. Equation (1) rearranges to become: (20) PV3-V2(bP+RT)+aV-ab=0 multiplying by P 2/(R 3T 3) and defining the compressibility factor, Z, as: (an z - P equation (20) i s : (22) Z 3-Z 2( + Z ( gflr) - ( I f f T ) ( |f)=0 Def ining: (23) A= and (24) B-equation (22) transforms to: (25) Z3-Z2(B+1)+ZA-AB=0 APPENDIX B -GENERALIZED TWO CONSTANT EQUATION DERIVATIONS The following derivations apply to the Redlich-Kwong,Soave 1972 and Lielmezs-Howell-Campbell equations. Along the c r i t i c a l isotherm, Tc, the three equations mentioned reduce to: ( 1 ) P = R*P C - a M ' v V-b vTv+bl Determine the c r i t i c a l point parameters by d i f f e r e n t i a t i n g equation (1): , 0 N 9P_ a , a _ RTc W > 3V~ V 2 (v+b) vTvTbP" (V-b) 2 and, after finding a common denominator for the "a" terms and d i f f e r e n t i a t i n g again, ,o\ 3 2P 2 2(2V+b) 2(2V+b) x (3) - a( v j ( v + b ) 2 v 3(v+b) 2 V 2(v+b) 3 ' 2RTc (V-b) 3 At the c r i t i c a l point, there i s a double i n f l e c t i o n , so: ( 4 ) I E = I I P = 0 V 4 ; 9V dV7 U Therefore equations (3) and (4) become: 1 37 and i c\ _ RTcVc 3(Vc+b) 3 ( b ) a " (Vc-b) 3(3Vc i ! + 3bVc+bi!) Combining equations (5) and (6) yi e l d s (7) Vc 3-3bVc 2-3b 2Vc-b 3=0 Let Vc=Xb, therefore: (8) (Xb) 3-3b(Xb) 2-3b 2(Xb)-b 3=0 Simplifying equation (8) gives: (9) X3-3X2-3X-1=0 Using Newton's method, X=3.847322101.. Introduce: (10) fia= j^ffpr and bPc (11) flb= RTc 1 39 At the c r i t i c a l point equation (1) becomes: (12) Pc = j g S g - V c ( 5 c + b ) Substituting equations (5) and (12) into equations (10) and (11) and simplifying y i e l d s : ( 1 3 ) n a = ( U - O M 2 X + 1 ) ) 2 ( X 2- 2 X" 1) and ( 1 4 ) n b = ( 2 x I l ) ? x - ! ) 2 and since X i s known: fia=0.4274802334... Rb=0.0866403500... Dividing equation (10) by (11) and solving for Tc y i e l d s : ( , 5 » T c - i < i s ' Furthermore, Pc is determined by d i v i d i n g equation (10) by the squared value of equation (11) and rearranging to give: 1 40 Defining the c r i t i c a l compressibility factor, Zc as: equation (15) and (16) are substituted into (17) to give: (18) Zc=fib(Vc/b)=flbX Therefore: Zc=0.333... F i n a l l y , equation (1) can be transformed into cubic form. After rearranging, i t becomes: (19) PV3-RTV2+V(a-RTb-Pb2)-ab=0 Multiplying by P 2/(R 3T 3) and defining the compressibility factor, Z, as: (20) z- {2 equation (19) i s : Def i n i n g : (22) A= and ( 2 3 ) B- | § equation (21) transforms t o : (24) Z 3-Z 2+Z(A-B-B 2)-AB=0 APPENDIX C -GENERALIZED TWO CONSTANT EQUATION VIRIAL EXPANSIONS In general, a two constant equation of state is expressed as: m P - R T - A ( T ) ( 1 ) P _ V ^ b V ( v + b ) Since (2) Z= ^ K l ) RT equation (1) can be rewritten in the form: m 7 - v - a ( T ) U ' Z " V-b (V+b)RT by dividing i t by V/RT. Equation (3) can be further rearranged to give: 1 a(T) (4) Z = 1-b/V (1+b/V)VRT Using the series expansions, as V approaches i n f i n i t y : (5) I, =1+ + K + K + ••• v ' 1 -b/V V V 7 V 7 and 142 1 43 ( 6 ) 1 , = 1 _ b + b 2 1+b/V V V V-e q u a t i o n (4) can be expanded i n V and r e w r i t t e n as: ( 7 ) 4 b b 2 b 3  1 + V + V* + + a ( T ) VRT V b 2 V2" v-t h e n r e a r r a n g e d a s : (8) Z=1 + V b-a ( T ) RT b2 + b a ( T ) RT , 3 _ b 2 a ( T ) RT A b b r e v i a t i n g e q u a t i o n (8) as: (9) Z=1 + S i l l + C ( T ) + D(T) v ' v V 2 v 3 t h e v i r i a l c o e f f i c i e n t s a r e : (10) B(T)=b- ^ 1 (11) C ( T ) = b 2 + and (12) D ( T ) = b 3 - b 2 ^ T ) The c o m p r e s s i b i l i t y f a c t o r , Z, can be e x p r e s s e d i n P a l s o . I n general: (13) Z=1+B' (T)•P+C' (T)•P 2 +D' (T)• P 3 +••• Now determine B'(T), C'(T) and D'(T). I f : (14) V=1/P equation (2) can be rewritten as: (15) P=ZpRT Given that: (16) B(T)= |^ at constant T, for p=0 dp and (17) B'(T)= || at constant T, for P=0 i t i s necessary to determine how B 1(T) and B(T) are related each other. Equation (18) below provides the basis for th i s r e l a t i o n . MR) = + dP. 9 2 v ' 9P 9P 9Z 3P From equation (15) one establishes: ( 1 9 ) l£ = _ L _ v ' 3P ZRT and ( 2 0 ) dp = 3Z - P Z 2RT which when s u b s t i t u t e d i n t o equation (18) y i e l d s , a f t e r r e a r r a n g i n g : (21 ) 3 £ 3P 1 ZRT 1 -P _3_Z_' Z 3P A l s o , s i n c e : ( 2 2 ) 3Z = 3Z 3p 3P dp' 3P equation ( 2 1 ) becomes: (23) l 2 = °± 3Z dp _ J _ ( 1 _ £ 1Z) ZRT v Z 3 ? ' Upon r e a r r a n g i n g , equation ( 2 3 ) i s ; ( 2 4 ) l 2 = ( -9-Z-) / 3P v 3 p ; / I ( If)' Since P=p=0 and Z=1, equation ( 2 4 ) reduces t o : ( 2 5 ) ^ = 3Z = j _ , 32iv 3P RT 3 p ; By a b b r e v i a t i n g B(T) as B and B'(T) as B', one gets ( 2 6 ) B' = B RT O b t a i n e d i n a s i m i l a r m a n n e r a r e ( 2 7 ) C -a n d A P P E N D I X D - G E N E R A L I Z E D TWO C O N S T A N T E Q U A T I O N T H E R M O D Y N A M I C D E R I V A T I O N S T h e g e n e r a l i z e d t w o c o n s t a n t e q u a t i o n o f s t a t e i s : ( 1 ) P ^ R T - a ( T ) V - b vTv+bT I f ( 2 ) a ( T ) = a - a ( T ) w h e r e a ( T ) i s a g e n e r a l i z e d t e m p e r a t u r e d e p e n d e n t f u n c t i o n f o r a n y e q u a t i o n o f s t a t e , e q u a t i o n ( 1 ) b e c o m e s : / , x D _ R T a - a ( T ) ( 3 ' p _ V = b v(v+b) D e f i n i n g : ( 4 ) a = n * ^ £ ± P c a n d / c \ K fibRTc ( 5 ) b = — P c ~ e q u a t i o n ( 3 ) c a n b e r e w r i t t e n a s : ( 6 ) P= R T V - b S 2 a R 2 T c 2 P c a ( T ) vTv+BT E q u a t i o n ( 5 ) , w h e n r e w r i t t e n s o a s t o b e e x p l i c i t i n P c , i s t h e n 147 148 substituted into equation (6) to give: (7) p= EL_ - ®* bRTca(T) K ' V-b fib V(V+b) Equation (7) i s rewritten as: (R\ P = 1Z_ _ fia bRTca(T) T V 0 ; V-b fib V(V+b) T and again as: Def ining: (10) F= ^ a ( T ) equation (9) f i n a l l y becomes: f i n P= R T - flabRTF u ; V-b flbV(V+b) For the LHC equation: (12) F= lp(1+pT* q) (Tr < 1.0) (13) F= ^exp(d-T*) (Tr = 1.0) and (14) F= 1-p|T*|q) (Tr > 1.0) 149 and with the correction factor, for unsaturated l i q u i d s and vapours (Tr < 1 . 0 and P r < 1 . 0 ) , the F function i s : (15) F= ^ 0+pCfT* q) For the Redlich-Kwong equation: ( 1 6 ) F = l / T r 1 ' 5 and for the Soave 1 9 7 2 equation: ( 1 7 ) F= ^ [ 1 + m O - T r 0 - 5 ) ] 2 with ( 1 8 ) m=0.480+1 . 5 7 4 t o - 0 . 1 76co 2 Determine the equations of the thermodynamic departure functions using equation ( 1 1 ) . In general, the Helmholtz free energy i s given by: ( 1 9 ) Ar = -Sl(P- f ^ d V - RT-ln( ^ ) Substituting equation ( 1 1 ) into ( 1 9 ) gives: , o n> fv / RT fiabRTF RTv „„, , , V x ( 2 0 ) Ar = - S J ^ - flbv(v+b) " v-)dV - RT-ln( ^ ) From Weast ( 6 9 ) , the generalized integral is for the middle term 1 50 i s : (21 ) J x(a+bx) - In a a+bx Therefore after integration, equation (20) becomes: (22) Ar=RT ; l n ( 1 " v> " § § F - l n ( 1 + - l n ( v^>] and upon rearrangement: (23) tr "ln(1-1 } - i F ' l n ( 1 + - l n ( h] The entropy departure function i s defined as: (24) Sr = 3(Ar) 3T D i f f e r e n t i a t i n g equation (22) y i e l d s : (25) Sr=R.ln(l- + §|RF-ln(l+ + R-ln( ^ ) fiaR„ 3F, , b» flbRT 3 T l n U v' and upon rearrangement (26) Sr_ i r i n ( 1 _ ^(F+T H RT- T v' + fib(F T 3T ( F + T ln ( 1 + h + l n ( gc- ) V V 1 Using the thermodynamic i d e n t i t y : (27) Ur=Ar+TSr 151 Internal energy can be determined from equations ( 2 3 ) and ( 2 6 ) and i s : < 2 8 » Using another thermodynamic i d e n t i t y : ( 2 9 ) Hr=Ur+RT(Z-1) the enthalpy departure function i s : The Gibbs free energy departure id e n t i t y i s : (31) Gr=Hr-TSr and therefore: ( 3 2 ) ||=Z-1- I n d - - § § F l n d + £) - 1„( V , ) The fugacity c o e f f i c i e n t i s defined as: ( 3 3 ) ln( |)=Z-1-ln(Z)+ | | + ln( ^ ) and by substituting equation ( 2 3 ) into ( 3 3 ) , one gets: ( 3 4 ) ln( |)=Z-1-ln(z)- l n ( l - - § | F l n d + £) The isobaric heat capacity departure function i s : 1 52 (35) Cpr = 3Hr 3T Therefore, d i f f e r e n t i a t i n g equation (30) gives (36) fla 3F C p r = ( i s R ' l n ( 1 + £ ) ) ( 2 T ^ + T V 3T 3 2F dT7 )-R The isometric heat capacity can also be determined from: (37) Cp-Cv=T( |f) ( ||) v p D i f f e r e n t i a t i n g equation (11) gives (38) ( § ) - | where: (39) _ flabR , T 3F,_ R B l _ fibV(V+b)(F T 3T ) V^b and (AO) R - fiabRTF , 1 1 RT l 4 0 ) B z _ fibv(V+b)( V V+b } ( V - b ) 2 While: (41 ) Therefore: 153 (42) Cv=Cp+T f-1-2-The isenthalpic Joule-Thomson c o e f f i c i e n t s are defined as: (43) T( 1 v 9 T ; -V /Cp Therefore: (44) M=(T £-L - V)/Cp F i n a l l y , the inversion curve defined as: (45) T ( IE) + v( —) =0 Upon d i f f e r e n t i a t i n g equation (11) one gets: ( 4 6 ) Substituting equation (41) and (46) into equation (45) and rearranging gives: (47) 1 V-b 1 -V V-b fia b fib V+b V V 1 9T ;J U V+b Defining: (48) X=b/V equation (47) rearranges to become: (49) 1-X 1 - 1-X flaX Qb(1+X) 1 +X T & 1 9T = 0 The derivatives of the F functions required for use in the departure functions, Joule-Thomson c o e f f i c i e n t and inversion curve equations are derived below, star t i n g with the LHC equat ion. * To begin with, the f i r s t and second derivatives of T are (50, §- - £ /( Sfc -, , and (51 ) d2T n Tc // Tc dT' Tnb -1 ) D i f f e r e n t i a t i n g equation (12) gives: (52) F +T I^TcpqT*^ 1 % 9T ,*q-1 dT dT Substituting equation (50) into (52) and rearranging gives: (53) F+T 9F -1 9T Tr Tc P q / ( Tnb " 1 > ,*q-1 D i f f e r e n t i a t i n g equation (52) y i e l d s : (54) 2 T 91 + T 2 9 l F = T c D Q T * q - 1 dlT Z l 9T 1 ^ P y 1 dT 7 Tcpq(q-1)T* q 2 dT dT 1 55 Substituting equation (50) and (51) into (54) and rearranging gives: (55) 2T +T2 ^1= Z 1 9T 1 I T 7 T r 7 W« Its -' )]**q-' Tr 3 p q ( q - D / ( f^b -1 ) 2 ] T * q " 2 Now d i f f e r e n t i a t i n g equation (13) y i e l d s : * (56) F+T ||=Tcd.exp(dT*)• | f substituting equation (50) into (56) and rearranging results in: (57) F+T 9F 9T~ -d / ( Tc T r 2 / v Tnb -1 ) exp(dT*) and rearranging: (58) F+T 9F 9T~ Tc_ Tr 7 v Tnb 1 ) D i f f e r e n t i a t i n g equation (56) produces ( 5 9 ) 2T | f +T2 f^=Tcd.exp(dT*). | ^ + Ted 2-exp(dT*) 9 2F d 2T dT dT Now substituting equations (50) and (51) into (59) gives: (60) 9F i m 2 92F. 2T +T 9T 2d /( Tc_ _ . T F ™ Tnb 1 ' ex :p(dT*) d 2 „ Tc Tr 3 / v Tnb -1 ) exp(dT ) 1 56 Rearranging y i e l d s : (61 ) 2 T i f + T 2 I f F * l 3T 1 ST7 dF// Tc Tr/( Tnb -1 ) 2+ — / ( _ 1 ) Tr 7 v Tnb 7 D i f f e r e n t i a t i n g equation (14) produces: (62) F+T f f T c p q | T * r 1 ^ dT One should see the note at the end of the appendix for d i f f e r e n t i a t i n g absolute values. Substituting equation (50) into (62) and rearranging gives: (63) F+T 3F -1 3T T r 2 pq/( Tc Tnb 1 ) Di f f e r e n t i a t i n g equation (62) produces: * (64) 2T« + T . 0.Tcpq|T*|5-' Tcpq(q-1)|T * ,q-2 dT dT Substituting equations (50) and (51) into (64) and rearranging y i e l d s : (65) 2 T £E +T2 - K Z l 3T 1 3T 7 T r 7 pq/( Tc Tnb 1 ) * ,q-1 Tr Tc p q ( q - D / ( "I ) ,* ,q-2 1 5 7 c f = 1 + m ( T : ? f a t ) + n ( p : p f a t ) Tsat Psat Now d i f f e r e n t i a t i n g equation ( 1 5 ) gives: ( 6 6 ) F+T ff=Tc PT* q f§£ + TcpqCfT*^ 1 § The correction factor i s defined as: ( 6 7 ) Therefore ( 6 8 ) and ( 6 9 ) 3Cf 3T " m Tsat 3 2Cf_„ 72 - 0 3T' Therefore substituting equation ( 5 0 ) and ( 6 8 ) into ( 6 6 ) gives: ( 7 0 ) F + T m P T c T * q - ( P q C f T * q ~ 1 } / ( Tc_ , w u ; 3T Tsat 1 V TF2^1 ) / k Tnb 1 ; D i f f e r e n t i a t i n g equation ( 6 6 ) y i e l d s : ( 7 1 ) 2T | f +T2 0 = TcpT* q 3 2Cf 3 T 1 - + 2TcpqT*q 1 1 3Cf" * dT 3T dT + Tcpq(q - 1)CfT* q 2 + TcpqCfT* q _ 1 dT dT * d 2T dT2" Substituting equations ( 5 0 ) , ( 5 1 ) , ( 6 8 ) and ( 6 9 ) into ( 7 1 ) and 1 58 rearranging gives: (72) 2T § *T> 0= fpT*3-> Cf T m Tsat / ( Tc Tnb -1 ) + pq(q-1 )Cf T *q-2 / ( Tc_ _ ) 2 T r 3 T / V Tnb 1 ; The F function derivatives of the Redlich-Kwong and Soave 1972 equations are also developed below. Equation (16) d i f f e r e n t i a t e s and rearranges to become: (73) F+T |£= -0.5Tr" 1' 5 = -0.5F o 1 3 T D i f f e r e n t i a t i n g equation (73) and rearranging results in (74) 2T | | +T2 -0.5T ||= 0.75F Equation (17) d i f f e r e n t i a t e s to: (75) F+T ||= ' m 0 . 5 [ l+m ( l-Tr°- 5)] and again to: (76) 2T | | +T2 0= 2 ^ r 0 . b [ l + m ( l - T r ° - 5 ) ] + | i where m i s given by equation (18). Equations (75) and (76) can be rewritten as: (77) F+T 4£= -mTr°" 5F o 1 and (78) 2T | | +T2 |^= ^[F.Tr°- 5+m] NOTE: D i f f e r e n t i a t i o n of Absolute Value Terms Given the function: (79) y=m|x|3 we can rewrite i t as: (80) y=mx3 for x>0 and (81) y= -mx3 for x<0 Therefore d i f f e r e n t i a t i n g equations (80) and (81) respectively, one gets: (82) y'=3mx2 for x>0 and (83) y'= -3mx2 for x<0 APPENDIX E - BENEDICT-WEBB-RUBIN EQUATION THERMODYNAMIC DERIVATIONS The Benedict-Webb-Rubin equation i s : (1) P=RTp + (B0RT - A 0 - !=£)p2 + (bRT-a)p 3 + aap 6 + (l+7P 2)exp(- 7p 2) If : (2) D^BoRT-Ao" (3) D2=bRT-a (4) D3=aa and (5) p=1/V equation (1) can be rewritten as: (6) P= F + V±+ V^ + V* + TV ( 1 + V ^ ) G X P ( Defining the Helmholtz free energy as: (7) Ar = -;^(P" |^)dV - RT-ln( ^ ) 1 60 one can substitute equation (6) into (7) and obtain: (8) Ar= - J 1 P i Do D , ( V7 V7 V* T 2 V 3 v ' ' v 2 ' ^ " ^ ' V' -(1+ £y)exp( dV -RT-ln( V n r) Divide equation (8) up as: ( 9 ) Ar= P i ^ Ps ^ P i dV - S i T^frexpi ^)dV -Si -pvrexp( ^ ) d V - R T . l n ( Y ^ ) Rewrite equation (9) as: (10) Ar = I 1 + l 2 + i 3-RT.ln( o^-) and solve each integral separately: (11 ) I i = -x D l , D , , D , v i + v i + V^ " D , ^ D 2 D 3  d V = V 1 + 2V^ + 5V^ Next: (12) I 2= " J v c oo T 2 V 3 exp( ^ ) d V Equation (12) must be transformed into a density dependent equation. Rewrite equation (5) as: (13) V = l / p 162 one d i f f e r e n t i a t e s to obtain: (14) dV= 4^-dp P Establish new integration l i m i t s . When V=°°, p=0 and when V=V, P=1/V. Therefore after substituting equations (13) and (14) into equation (12) and changing- the l i m i t s , the result i s : (15) i2=/Q"'|#exp(-Tp2)dp and after integration y i e l d i n g : (16) Now: (17) i,= -Si - p V r e x p ( ^ ) d V Again, applying equations (13) and (14) to equation (17) and changing the integration l i m i t s gives: (18) i 3 = /^ ,£aPi exp(- Tp 2)dp Equation (18) must be integrated by parts. In general: (19) /u-dv=uv-/V'du Knowing that (20) (21) dv= - ^£exp(- 7p 2)dp (22) du=-27p«dp and (23) 2 7T 7 exp(~7p2) one integrates equation (18) and produces: (24) I 3 = -c 3 2-exp( ^ ) - ^ r T e x p ( ^ ) + 27T' 2yT' F i n a l l y , after adding up a l l the integrals and substituting for D,, D 2 and D3, one gets: (25) Ar= ^(BoRT - A 0 - §£) + ^"(bRT-a) + a a 5VJ T^7 1-(1+ 2v2")exp( ^ ) -RT-ln( ^-) Defining the entropy departure function as: (26) Sr = -a(Ar) 9T Equation (25) d i f f e r e n t i a t e s to become: (27) Sr=-(B0R+ 1 - ^ 2c T J 7 1-(1+ 2 v ^ " ) e x p ( ^2)] + R . l n ( Y ^ ) The thermodynamic identity for enthalpy departure i s : (28) Hr=Ar+TSr+RT(Z-1) Therefore aft e r substituting equations (25) and (27) into (28), and rearranging, the result i s : (29) Hr= | r ( -0.5)+ |h V^2-+RT(Z-1 ) 1-(1 + 2v^)exp( ^ r ) V The internal energy departure identity i s : (30) Ur=Ar+TSr Substituting equations (25) and (27) into (28) and rearranging y i e l d s : (31) Ur = | r ( -0.5)+ | r [ 2V^)exp( ^ r ) V V The Gibbs free energy r e l a t i o n i s : (32) Gr=Ar+RT(Z-1) Therefore substituting equation (25) into (32) yields: (33) Gr = ^(B 0RT - A 0 - §£) + ^ ( b R T - a ) + f v ^ T^7 1-(1+ ^ J e x p * v?) V -RT«ln( Vf T r)+RT(Z-1 ) The fugacity c o e f f i c i e n t is given by: (34) ln( |)=Z-1-ln(Z)+ | | + ln( ^ ) Substituting equation (25) into (34) gives: (35) ln( |)= ^ ^(B 0RT - A 0 - §$) + ^ ( b R T - a ) aa c 5V5" T^y 1-(1+ 2v^)exp( ^ ) +Z-1-ln(z) One obtains the isobaric heat capacity departure function from (36) Cpr = 9Hr 3T Therefore, d i f f e r e n t i a t i n g equation (29) y i e l d s : (37) Cpr = -6 1 - ( 1 + 2v^)exp( ^ ) •]" R The isometric heat capacity, Cv, can be determined from: (38) Cp-Cv=T( f f ) < f*> v 1 Therefore, d i f f e r e n t i a t i n g equation (1) gives: (39) ( ||) =B,=Rp+(B0RT - §^)p 2+bRp 3 v ^ § ^ ( l + 7 P 2 ) e x p ( - 7 p 2 ) and by i m p l i c i t d i f f e r e n t i a t i o n : (40) Rp+RT f£+(B 0R + f ^ ) p 2 + 2(B 0RT - A 0 - §£)p +bRp3+3(bRT-a)p2 |f+6aap 5 ||- 1+ 7p 2 )exp(- 7p 2 ) + ^ | ^ ( l + 7 P 2 ) e x p ( - 7 p 2 ) + | ^ ( 2 7 p f f ) e x p ( - T p 2 ) + |^-(1+7P 2)(-27P |^exp(- 7P 2))=0 From equation (13) one gets: 3T V 7 3T p 3T Therefore, rearranging equation (40) and applying (41) gives <«> < f > - It P where: (43) B2= |+(B0R + | Cr(l+7P 2)exp(- 7p 2)+bR and 1 67 (44) B3=RT+2(B0RT - A 0 - p+3 (bRT-a) p 2 +6aap5+ 3(l + 7 p 2 ) - 2 7 2 p ' ^T-exp(- 7p 2) Therefore equations (39) and (42) can be applied to equation (38) to give Cv, from equation (45): (45) Cp-Cv=TB, | i 0 3 The d e f i n i t i o n of the isenthalpic Joule-Thomson c o e f f i c i e n t and the inversion curve are respectively: (46) and u= T < i > -V P /Cp (47) T ( 1£) + V ( — ) =0 The one uncalculated derivative (9P/9V) w i l l now be determined: (48) ||=RT+2(B0RT - A 0 - j=£)p+3(bRT-a)p2+6aap5 3(I+7P 2)+2 7p 2 ^ ? - e x p ( - 7 p 2 ) - ^ § ^ ( l + 7 P 2 ) e x p ( - 7 P 2 ) and again from equation (13): (49) l£= - Jl IV and therefore: (50) 3 P 2 3 P av" Applying equation (50) to equation (48) and rearranging gives (51) ( ||) =B„ = (2 7 2P a-3-37P 2) ^ e x p ( - 7 P 2 ) - 6 a a p 7 -3(bRT-a)p" -2(B 0RT - A 0 - ^-)p 3-RTp 2 The f i n a l equations for the Joule-Thomson c o e f f i c i e n t and inversion curve equations can be obtained by substituting equation (42) into (46) and equations (39) and (51) into (47) give respectively: (52) T i t " V /Cp and (53) TB1+VBa=0 where B^B, have been previously defined. APPENDIX F - BACK EQUATION THERMODYNAMIC DERIVATIONS The Back equation thermodynamic properties w i l l be derived here. The F function for the Back equation was not isolated because i t was not needed, therefore begin by determining the Helmholtz free energy. (1 ) Z= | Y = Z h + Z a where: -h_ 1+(3a-2)$+(3a 2-3a+1)$ 2-g 2£ 3  U ) Z ( 1-5) 3 and n m (3) Za=EZmDnm( pp) ( g l ) nm kT v Def ining: (4) A,=3a-2 (5) A2=3a2-3a+1 and (6) A 3=a 2 equation (2) can be rewritten and substituted into equation (1), 1 69 170 along with equation (3) to produce: n m Following the procedure outlined by Wilhelm (44), for a hard sphere equation, the residual Helmholtz free energy i s defined: ( 8 ) i = sl j £ * v where: ( 9 ) P * = fv(" ( 1P}BACK _ ( 9 P ) P e r f e c t 1 { ' J ° ° L 9V ; T V 9V ; T J a v For a perfect gas: ( 1 0 ) RT = ( l - S ) 3 I f : (11) $=0.74048 ^ = | then equations (7) and (10) can respectively be transformed into: n m P V^A^V+A^B 2 -A-,B3/V,^ mDnm, U , , V° . ( 1 2 ) RT = (V-B) 3 +§§ — ( kT> ( V" > and 171 D i f f e r e n t i a t i n g e q u a t i o n s ( 1 2 ) a n d ( 1 3 ) r e s p e c t i v e l y p r o d u c e s ( 1 4 ) 1_ 3 £ = 2 V + A 1 B + A 3 B3 / V 2 RT 3V ( V - B ) 3 6 V 2 + A 1 B V + A 2 B 2 - A 3 B 3 / V ( V - B ) 4 n m v v (m+1 )mDnm, U \ , V ° , "SI V^ ( kT> ( V " } a n d ( 1 5 ) J _ 9 P = 2 V - 3 B + B 3 / V 2 RT 3V ( V - B ) 3 - 3 V 2 - 3 B V + 3 B 2 - B 3 / V ( V - B ) * N o w , a p p l y i n g e q u a t i o n ( 9 ) t o e q u a t i o n s ( 1 4 ) a n d ( 1 5 ) t h e n r e a r r a n g i n g g i v e s : ( 1 6 ) ^ L_ 1P*.= ( A 1 + 3 ) B + ( A , - 1 ) B 3 / V 2 RT 3V ( V - B ) 3 ( A 1 + 3 ) B V - B 2 ( 3 - A ? ) - ( A 3 - 1 ) B 3 / V * M ( V - B ) " ; n m v ~ (m+1 )mDnm, U , , V ° > ~nl V 2 ( k T ) ( V ~ ) A p p l y i n g e q u a t i o n ( 1 6 ) t o e q u a t i o n ( 9 ) y i e l d s : 1 72 (17) P*_= v RT J °° (A,+3)B+(A,-1)B3/V2 ( v ^ P (A 1+3)BV-B 2(3-A 2)-(A 3~1)B 3/V , 3 i ( V - B ) " ; n m y y (m+1)mDnm, U w V° * nm V2" ( kT } ( V - > dV After integration, equation (17) becomes tia\ P * (A,+3)BV-B2(3-A,) -(A 3-1)B 3/V (18) ^f= W=BT n m Now applying equation (18) to equation (8) gives: (19) Ar_ _ ,v RT J » (A1+3)BV _ B 2(3-A,) _ (A 3-1)B 3 (V-B) 3 V(V-B) 3 (V-B) 3 n m y y mDnm, U » , V° > fiS V 1 kT> 1 V ' dV Rewrite equation (19) as: (20) ^=I,+I 2+I 3+I 4 173 Now solve each integral independently sta r t i n g with I , . From Weast (69), a generalized integral of the form of I , i s : (21 ) J x - dx (a+bx) for n?M ,2 n b' -1 (n-2)(a+bx) n (n-1)(a+bx) Given that: T _ _ r v (A 1+3)BV J T T (22) I r - / „ ( v - B ) 3 d V equation (22) integrates to: (23) I 1=(A 1+3)B 1 B V-B 2 ( V - B ) 2 Now for 1 2: ( 2 4 ) T - rv B 2 ( 3 - A , ) j T r  l 2 ~ '~ ( V - B ) 3 d V Equation (24) readily integrates to: OS) T - B 2 ( A 2 - 3 ) (25) I 2 - 2 < v - B > 2 Now, for I 3 : (26) T - r v (A 3-1 ) B 3 J T , l 3 " X . V ? V - B ) 3 d V 174 A g a i n , f r o m W e a s t ( 6 9 ) a g e n e r a l i z e d i n t e g r a l o f t h e f o r m o f I 3 i s : ( 2 7 ) T h u s : ( 2 8 ) f d x j_ x ( a + b x ) 3 ~ a 2a+bx ' a + b x + l n a + b x I 3 = ( A , - 1 ) In - 0 . 5 V - 2 B ' V - B + 0 . 5 N o w , d e t e r m i n e I 4 : n m r v ^ m D n m , U x , V ° ( 2 9 ) I , = -Si LL ^ ( d V nm T h e r e f o r e : n m U x , V ° ( 3 0 ) I . = I Z D n m ( £ = ) ( £ - ) nm IcT' v V N o w , a f t e r s u b s t i t u t i n g e q u a t i o n s ( 2 2 ) , ( 2 5 ) , ( 2 8 ) a n d ( 3 0 ) i n t o e q u a t i o n ( 2 0 ) , t h e n a p p l y i n g e q u a t i o n s ( 4 ) - ( 6 ) a n d ( 1 1 ) , b e f o r e r e a r r a n g i n g g i v e s : n m ( 3 . ) £ § - ( . » - , ) l n ( . - $ , + " ' f ' l f o 3 ' * ' • g D n » < fe) ( f ) N o w , f r o m W i l h e l m ( 4 4 ) : ( 3 2 ) U r = / e 9 P * 3 T - P * J d V W h e n e q u a t i o n s ( 4 ) - ( 6 ) a n d ( 1 1 ) a r e a p p l i e d t o e q u a t i o n ( 1 8 ) , t h e 175 result, after rearrangement i s : /„» VP*_ (3a+1)S +(3a2-3a-2) 5 2 - (a 2-1 ) 5 3 V J J ' RT " ( l - S ) 3 n m +IImDnm( ^=) ( ^  ) nm KT V Def ining: (34) A„=3a+1 (35) A5=3a2-3a-2 and (36) A6=a2-1 equation (33) becomes: n m (37) ^ 3 +ZgmDnm( ^ ) < — ) Now d i f f e r e n t i a t e equation (37) for temperature remembering that, 5, U and V° are a l l functions of T. The result i s : 176 (38) V 3P* RT 3T RT 7" ( 1 - 5 ) 3 6 A a 5 + A . 52 - A f i 5 ( i - S ) 4 n m -ZZmnDnm( ( ^ 1 ) 1 n m n m o.vv 2 n / U » , V ° > 1 3 V ° +ZZmDnm( 1-=) ( 77- ) =70-« nm kT' v V ' V° 3T where: (39) $'= 3T Rearranging equation (38) gives: (40) V 3P* V P * RT 3T RT' ( A a + 2 A S S - 3 A 6 $ 2 ) ( l - 5 ) + 3 ( A , 5 + A B 5 2 - A f i 5 3 ) i i 3T n m +ZZmnDnm( JJL) ( gl ) nm kT V 1 _ l l U* 3T Tj n m ^ v v 2 _ , U x , V ° . 1 3V° +ZZm2Dnm( ^ ) ( ^  ) g=r-D i f f e r e n t i a t i n g equation (11) with respect to temperature gives: (A n M = 0-74048 3V * ' 3T V 3T 0 Now di v i d i n g equation ( 4 1 ) by equation ( 1 1 ) yields (42 ) 1 a v ^ = ]_ a_$ V ° " 9T $ 9T Also: ( 43 ) U = U ° ( 1 + J _ ) Therefore: (44 ) ^ = K ' 9T U 0 ' kT-Divide equation ( 44 ) by ( 4 3 ) and rearrange to get (45 ) u' 9U 9T~ kT+r? Now subtract 1/T from ( 4 5 ) and rearrange to get ( 46 ) U" 9U 9T _1_ T= 1 + 2i?/kT 1 +7?/kT 1 78 S u b s t i t u t e equations (42) and (46) i n t o equation (40) to produce: (47) V 3P* _ VP* RT 3T RT"2"' A a + 2 ( A s + A , ) S + ( A s - 3 A f i ) < : ( 1 - S ) " i i 3T n m -IZmnDnm( 2_ ) ( Y l , 1 1 +2T?/kT" 1 +Tj/kT n m S u b s t i t u t i n g equation (47) i n t o (32) and d i v i d i n g both s i d e s by V g i v e s : (48) Ur = v RT 2 *" 0 0 A a + 2 ( A . ; + A u ) S + ( A , - 3 A f i ) < : i i dv 3T V n m -;:zimnDnm( ^ ) ( ^  ) 1 °nm 1 +27?/kT] dV 1+7?/kT J V n m TO J r a — kT V $ 3T V "nm 1 79 Rewrite the equation as: (49) ^ r = I 5 + I 6 + I 7 Now solve each integral separately, beginning with I 5 (50) I 5 = / A„+2(As+A A )S+(A 5-3A 6)$: i i a v 9T V Change the integration so that we integrate in terms of $. Using equation ( 1 1 ) , when V=°°, $ = 0. When V=V, From Wilhelm's paper ( 4 4 ) : ( 5 1 ) ( § | ) =31$ v From equation ( 1 1 ) : / c o x 91 Q . 7 4 Q 4 8 V 0 ( 5 2 ) 9 V =  Rearranging equation (52) gives: ( 5 3 ) 3 $ = - 0 - 7 4 0 4 8 V ° 9 V Divide equation (53) by ( 1 1 ) to give: (54) 180 Therefore substituting equations (51) and (54) into (50) then changing the l i m i t s gives, after rearranging: (55) I 5 = -31 ; Now rewriting equation (55) as: (56) I s = - 3 1 ( I 8 + I g + I 1 0 ) 2 ( A 5 + A „ ) 5 ( A c - 3 A p ) 5 2 d$ Solve for the three integrals beginning with I 8 . This integral can be determined d i r e c t l y , and rearranged to give: (57) I 8 = -1 3 ( 1 - S ) 3 The generalized integral for I 9 i s presented as equation (21) Therefore, upon integration, I 9 becomes: (58) I 9 = 2 ( A „ + A c ) 2 - 3 ( 1 - S )+ ( 1 - 5 ) 3 6 ( i - 5 ) 3 Equation (58) rearranges to become: (59) (A,+A 5) 3 S 2 - ^  3 From Weast a generalized integral for I 1 0 is 181 (60) x 2 »dx (a+bx) -1 2a (n-3)(a+bx) n 3 (n-2)(a+bx) n 2 (n-1)(a+bx) n- 1 for 1 , 2 Thus, I 1 0 integrates to: (61) I i o = ( A 5 ~3A6) which rearranges to: T^I " TTTP" 3d-s) 3 T _ (A 5-3A f i) $ (62) I 1 0 - 3 <1 -1> 3Now solve for I 6 and I 7 . I 6 integrates d i r e c t l y to: n m (63) I6=ZInDnm( §_) ( Y i ) 1 nm 1+27?/kT 1 +rj/kT Now do I 7 : n m (64) I , . ^H - n n m C fe) < ^  > j f | ^ °nm 182 S u b s t i t u t i n g equation ( 5 1 ) i n t o ( 6 4 ) and s i m p l y i n g g i v e s : ( 6 5 ) n m I , = 3! / W D n . f fe) < £ ) f which r e a d i l y i n t e g r a t e s t o : ( 6 6 ) I ,= - 3 1 n o m -ZZmDnm( p= ) ( ^ - ) nm KT V Now s u b s t i t u t e the equations f o r I 5 - I 1 0 i n t o equation ( 4 9 ) , and rearrange using equation ( 5 1 ) to g i v e : n m_ ( f l l) Ur _ _ 1 31 ( 6 7 ) — j - - j 3A,,$ + 3 A 5 $ 2 - 3 A 6 < 3 v v „ , U x , V ° x 3(Y-$)» ( V - > n m •F<„>HnD™.< fe) ( £ ) i where: ( 6 8 ) F ( T ? ) = 1 +2n/kf 1+T?/kT S u b s t i t u t e equations ( 3 4 ) - ( 3 6 ) i n t o ( 6 7 ) and m u l t i p l y by T to produce: (69) Ur RT" T 9_S_ S 9T (3a+1)$+(3a 2-3a-2)$ 2-(a 2-1)j; 3 ( 1 - 5 ) 3 n m. +ZZmDnm( £-) ( ^  ) nm kT V +F(7?)ILnDnm( nm kT Rewrite equation (69) as: (70) —= - - i i v RT $ 3T (3a-2) S+(3a 2-3a+1 ) S 2 - a 2 j; 3 . $ 3-3$ 2+3$-1 (1-S) 3 nm n f^+ZZmDnmt ( ^ n m +F(rj)ZZnDnm( g=) ( YJL ) nm KT V Simplifying, equation (70) becomes ( 7 1 ) U r - T ( 1 - s ) 3 - 7 ' ( 7 1 ) RT- 1 3T[ rrfp z n +F(T7) ZZnDnm( £-) ( ^ nm kT V 1 84 and f i n a l l y : n m ( 7 2 ) P7F= J ||(l-Z) +F(,)ZZnDnm( ( £l ) Find the Gibbs free energy residual knowing the thermodynamic ident i t y : (73, §E+Z-, Therefore, substituting equation (31) into (73) and rearranging gives: ii m ,74, n- <..-,.!»<,-$>• " , ; ^ ; ; 3 ° i i +KD»»< fe, ( £ , + Z-1 Now find the enthalpy residual function from the thermodynamic ide n t i t y : (75) 185 Therefore substituting equation (72) into (75) and rearranging gives: Hr (76) RT = (Z-1 ) $ 3T n m U x , V ° +F(TJ) ZZnDnm( £=) ( ) nm kT v Now find the entropy residual function from the identity: (77) Sr Hr Grl RT RT RT /T Substituting equations (74) and (75) into (77) and rearranging gives: (78) S r = 1 RT T T 9$,. „ v , 2 1 N 1 ys (g2+3a) g-3a<;: j -^( 1-Z)-(a 2-1 )ln( 1-$) ( l - $ ) 2 n m n m. +F(,)ZZnDnm( JL.) ( Y l ) _£ZDnm( ^ ) ( ) nm Now find the fugacity c o e f f i c i e n t given: (79) ln( |)= ||+Z-1-ln(Z) Substituting equation (31) into (79) gives: (80) ln( |)=<a»-1)ln(1-5) + ( a 2 + | ^ ^ f a^ +ggDnm( fe nm kT +Z-1-ln(Z) Knowing that: (81) Cpr= 9Hr 9T equation (76) can be rearranged as: (82) Hr=(PV-RT) 1- 1 M S 3T n + R T - F ( T J ) IEnDnm( fe) ( g l nm kT v and then d i f f e r e n t i a t e d to give: 1 87 ( 8 3 ) C p r = R 1 i i $ 3T R T ( Z - 1 ) i i ( 1 -D-T afj 3T v $ 1 ' A 3T' F ( T J)+T 3 F ( T ? ) 3T n m RZZnDnm( g=) ( g l ) nm KT v n m +RT-F(77)ZZnDnm( §=) ( g- ) nm K1 V n 1 in U* 3T +m 1 3V 0' V ° 3T U s i n g e q u a t i o n ( 6 8 ) , F ( T J ) c a n b e d i f f e r e n t i a t e d a n d r e a r r a n g e d t o g i v e : ( 8 4 ) 3 F ( T ? ) _ - 7 ? / k T 2 3T ( 1 + 7?/kT) 2 K n o w i n g t h a t : ( 8 5 ) V ° = V 0 0 [ 1 - C e x p ( L3U1 ) ] 3 kT c a n b e d i f f e r e n t i a t e d t o y i e l d : ( 8 6 ) 3V C 3T 9 V0 0 U ° C kT1 1 - C - e x p ( -: / 3U°, e x p ( - -^p - ) D i f f e r e n t i a t i n g e q u a t i o n ( 1 1 ) g i v e s : ( 8 7 ) 3£ = 0 . 7 4 0 4 8 3 V ° 3T V 3T C o m b i n i n g e q u a t i o n s ( 8 6 ) a n d ( 8 7 ) t o g i v e : ( 8 8 ) b£ 0 . 7 4 0 4 8 3T V 9 V0 0 U ° C kT' 1 ^ / 3 U ° v 1 - C - e x p ( - - j ^ - ) 2 / 3U°x e x p ( - - j ^ - ) 188 and d i f f e r e n t i a t i n g again gives: (89) 32$ = 0.74048 3T 2 V 9V 0 0U°C, 3U° kT' kT' 1-C-exp( 3U° kT 2 / 3U°v exp(- pp-) 18V 0 0U°C kT 1 9V 0 0U°C kT' 1-Cexp(-1-Oexp(- g , - ) 2 / 3U°. exp(- ir^-) exp( -kT 3U° kT » , n n 3U° , 3 U ° M )(-2C p r-rexp(- rsr~) ) kT Equation (89) rearranges to become: (90) i i i = UL 3T* 3T 3 U ° FT2" 2 T 6CU° kT 2 1-C.exp(- j[2L) 1 / 3u°; exp(- pp-) Next, equations (46) and (68) combine to produce (91) (91 ) 1 9U _ i U* 3T T F U ) Now substituting equation (42) and (91) into (83) gives: (92) Cpr=R 1 i i . , ' 5 3T RT(Z-1) 5 i i ( 1 } 315 3TV $ 1 ' 1 3T 7 F(T?)+T 3 F ( T ? ) 3T RIInDnm( ^ ) ( ¥i ) m nm +RT.F(T?) ZInDnm( ?=;) ( ) nm K 1 v m si i i _ n F(„) ' 5 3T T r 189 Equation (12) can be r e w r i t t e n as: n m (93) j f i * 1 ( V - B P +gg«"Dnm( ^ ) ( ^ ) Now f i n d 3V/3T by d i f f e r e n t i a t i n g equation (93) knowing that B, U and V° are f u n c t i o n s of T. The r e s u l t i s : ( 9 4 ) RT PV' _ PV RT -3 T=3 V3+A 1By 2+A ?B zV-A 3B i v • n 2 7v=§T 3B 3T V 3+A,BV 2+A 2B 2V-A 3B 31 3V (V-B5» '1 "J 3T 3V 2 V R +2A 1 BW , +A 1 V 2 B'+A 2 B 2 V , +2A ? VBB R -3A 3 B 2 B R (V-B) 3 n m U x / V° X 1 n m U X , V° , 1 3U -ZZmnDnm( — ) ( — ) -+ZZmnDnm( — ) ( — ) ^ ^ n m U x , V° x 1 3V° n m U x , V° x 13V V° 3T "^ 2Dnm( ^ ) ( ^  ) ^ 9 T where: ( 9 5 ) B ' = I B l i V 3 3 ; ° 3T 3T and (96) V' = | f 1 90 Now a f t e r r e a r r a n g i n g , e q u a t i o n (94) becomes: P V ' (97) RT - 3 V PV a v R T 2 9T V^ 3 / V + 2 A , B / V 2 + A 2 B 2 / V 3 ( V - B ) 3 1 / V + A 1 B / V 2 + A 2 B 2 / V 3 - A 3 B 3 / V a ( V - B ) " + 9B 9T + 3 V " v -A 1 / V + 2 A 2 B / V 2 - 3 A 3 B 2 / V 3 ( V - B ) 3 1 / V + A 1 B / V 2 + A 2 B 2 / V 3 - A 3 B 3 / V ' t ( V - B ) u n m +LLmDnm( " ) ( ) nm KT V 1 9U _ 1 U" 9T T +m 1 9 V ° _ I 9V' V 7 3 " 9T V 9T Now s u b s t i t u t e e q u a t i o n s ( 1 1 ) , ( 4 2 ) , ( 91) and (95) i n t o (97) and c o l l e c t t h e 9V /9T and 9B /9T terms t o g i v e : (98) 9V 9T p 1 RT V 3 + 2 A , $ + A , $ 2 ( 1 - I) 3 2>a-A a S : n m _ , v v 2 r , / U < , V ° v 1 PV +nlm D n m ( kf] ( V" ) V = R T 7 9T A 1 + 2 A 2 $ - 3 A 3 $ 2 ' + 3 1 + A i ^ A , $ 2 - A 3 $ 3 n m +LLmDnm( £ - ) ( ^ ) nm k T V E i i _ H F ( „ ) $ 9T T r v T ? ; 191 M u l t i p l y b o t h s i d e s b y V a n d s u b s t i t u t e e q u a t i o n s ( 4 ) - ( 6 ) , t h e n ( 2 ) t o g i v e : , h ( 9 9 ) — 7 X 3 - Zu 3 + 2 ( 3 a - 2 ) S + ( 3 a 2 - 3 a + 1 ) $ : n m + Z Z m 2 D n m ( ^ ) ( — ) = — +V 3T 3 » Z n . ( 3 a - 2 ) + 2 ( 3 a 2 - 3 a + 1 ) $ ~ 3 a 2 $ T l + T F T P n m +VZZmDnm( fe) ( g l ) nm KT V El M - H F ( „ ) 5 3T T K V I T h e r e f o r e e q u a t i o n ( 9 9 ) r e a r r a n g e s t o b e c o m e : ( 1 0 0 ) w h e r e : ( 1 0 1 ) 3 V = B_L 9T B 2 B - V ^ B l _ V 9T 3 - z " ( 3 a - 2 ) + 2 ( 3 a 2 - 3 a + 1 ) < ; - 3 a 2 $ 2 "Ps rrrn n m +VZZmDnm( fe) ( ^ - ) nm K l v $ 3T T r ZV a n d 1 92 (102) B: Z + 3 • Z _ 3+2(3a~2)$+(3a2-3a+1)$ 2 1-5 n m +LLm2Dnm( fe) ( g l ) nm K i v Now c a l c u l a t e 9P/9T from equation (93): ( 1 0 3 ) RT 9T " R T ^ (V-l)" + 3 1+A 1$+A 2 g 2-A 3 S 3 (1"S)« n m n m n m where £' i s given by equation (39). Rearrange equation (103) i n t o : (104) V_ 9P _ Z = 9_£ RT 9T T 9T A i + 2 A 2 $ - 3 A 3 $ 2 (1 - 5 ) 3 l+A_4+A j4 2 ZA iil (1-1)* n m +EZmDnm( fe) ( g l ) nm K T v 1 - 1 U* 9T T +m 1 9yJ V75" 9T Now s u b s t i t u t e equation ( 4 ) - ( 6 ) , (42) and (91) i n t o equation (104) t o g i v e : (3a-2)+2(3a2-3a+1 )$-3a 2 $ 2 + 3 1 + (3a-2) <; + (3a2-3a+1 )S 2 - a 2 < ; 3 ' ( 1 - 5 ) * +LLmDnm( ° ) ( Hi ) nm K T V m n i i - nF(„) { 3T T K V > Substituting equation (2) into (105) and rearranging gives .h (106) Y _ iP_ i i RT 9T 9T 3«Z" (3a-2)+2(3a 2-3a+1)g-3a 2$ : T 7! TTTP n m + |+ZLmDnm( g - ) ( Yl ) J nm K T V m 9_$ 5 9T ; F ( T ? ) F i n a l l y : (107) 9P_ RT 9_5_ 9T * 3 V 9T 3-Zn. (3a-2)+2(3a2-3a+1 ) S-3a 2 S 2 l T ^ I + r r i h — * - J n m + f E ^ m D n m ( fe> < v r > m 91 S 9T ^ ( T J ) ZR V Now calculate 9P/9V from equation (93) ( 1 0 8 ) RT 9V RT = ( 1 - S ) 3 3 > > T 1+A1$+A2$2-A3<;  +3| n m with ( 1 0 9 ) $ - i i 3V After substituting equations ( 4 ) - ( 6 ) and rearranging, equation ( 1 0 5 ) becomes: ( 1 1 0 ) y_ BP p _ = _aj_ RT 3V RT 3V ( 3 a - 2 ) + 2 ( 3 a 2 - 3 a + 1 ) $ - 3 a 2 $ 2 + 3 1 + ( 3 a - 2 ) $ + ( 3 a2 - 3 a + 1 ) $ 2 - a 2 $ 3 ' n m -ZIm2Dnm( %=) ( ^  ) nm KT v Now substitute equation ( 2 ) into ( 1 0 7 ) to give: .h (111) Y_ i S + P_= i i v ' RT 3V RT 3V 3 - Z ( 3 a - 2 ) + 2 ( 3 a 2 - 3 a + 1 ) S ~ 3 a 2 $ 2 1-S ( 1 - S ) 3 n m -ZLm2Dnm( ( ) 1 nm kT v v Multiply both sides by V and rearrange to: .h U > Z ) 3 V _ B < , _ V 3V 3 « Z " ( 3 a - 2 ) + 2 ( 3 a 2 - 3 a + 1 ) <;-3a 2 $ 2 T^I TTTP RT, n m U x , V ° x ZRT 1 95 The isometric heat capacity, Cv, the Joule-Thomson c o e f f i c i e n t , M , and the Inversion curve are respectively obtained from: (113) Cp-Cv=T( | f ) ( | f ) v p (114) (115) T( —) -v 3 T p /Cp T ( IE) + V ( =0 When equations (101), (102), (107) and (112) are appropriately substituted, one obtains: (116) (117) B Cp-Cv=TB3 g B •-v]/cp and (118) TB3+VBa=0 FIGURE 5: PRESSURE-VOLUME DIRGRRM FOR METHANE T r VOLUME(LITRES/MOLE) FIGURE 6: PRESSURE-VOLUME DIflGRRM FOR ETHANE FIGURE 7: PRESSURE-VOLUME DIAGRAM FOR PROPANE FIGURE 8: PRESSURE-VOLUME DIflGRRM FOR N-BUTANE o VOLUME(L1TRES/MOLE) FIGURE 9: PRESSURE-VOLUME DIAGRAM FOR N-PENTANE FIGURE 10: PRESSURE-VOLUME DIflGRRM FOR N-HEXRNE FIGURE 11: PRESSURE-VOLUME DIAGRAM FOR METHANOL VOLUME(LITRES/MOLE) FIGURE 12: PRESSURE-VOLUME DIflGRRM FOR T-BUTflNOL FIGURE 13: PRESSURE-VOLUME DIRGRflM FOR HATER 2*10"2 IO"1 10° 1Q1 io 2 VOLUME(L1TRES/MOLE) FIGURE 14: PRESSURE-VOLUME DIAGRAM FOR HYDROGEN SULFIDE FIGURE 15: PRESSURE-VOLUME DIflGRRM FOR NORMAL HYDROGEN VOLUME(LITRES/MOLE) FIGURE 16: PRESSURE-VOLUME DIRGRRM FOR PRRfl HYDROGEN VOLUME(L1TRES/MOLE) FIGURE 17: PRESSURE-VOLUME DIAGRAM FOR NEON FIGURE 18: PRESSURE-VOLUME DIflGRRM FOR RRGON FIGURE 19: ENTHALPY DEPARTURE FOR METHANE FIGURE 20: ENTHALPY DEPARTURE FOR ETHANE FIGURE 21: ENTHALPY DEPARTURE FOR PROPANE FIGURE 22: ENTHALPY DEPARTURE FOR N-BUTANE FIGURE 23: ENTHALPY DEPARTURE FOR N-PENTANE FIGURE 24: ENTHALPY DEPARTURE FOR N-HEXANE FIGURE 25: ENTHALPY DEPARTURE FOR METHANOL FIGURE 27: ENTHALPY DEPARTURE FOR NATER FIGURE 28: ENTHALPY DEPARTURE FOR HYDROGEN SULFIDE FIGURE 29: ENTHALPY DEPARTURE FOR NORMAL HYDROGEN FIGURE 30: ENTHALPY DEPARTURE FOR PARA HYDROGEN FIGURE 31: ENTHALPY DEPARTURE FOR NEON IN Q FIGURE 32: ENTHALPY DEPARTURE FOR ARGON FIGURE 33: ENTROPY DEPARTURE FOR METHANE FIGURE 34: ENTROPY DEPARTURE FOR ETHANE FIGURE 35: ENTROPY DEPARTURE FOR PROPANE FIGURE 36: ENTROPY DEPARTURE FOR N-BUTANE FIGURE 37: ENTROPY DEPARTURE FOR N-PENTflNE FIGURE 38: ENTROPY DEPRRTURE FOR N-HEXANE FIGURE 39: ENTROPY DEPARTURE FOR METHANOL FIGURE 40: ENTROPY DEPARTURE FOR T-BUTANOL FIGURE 41: ENTROPY DEPRRTURE FOR NRTER FIGURE 42: ENTROPY DEPARTURE FOR HYDROGEN SULFIDE FIGURE 43: ENTROPY DEPARTURE FOR NORMAL HYDROGEN FIGURE 4 4 : ENTROPY DEPARTURE FOR PARA HYDROGEN FIGURE 45: ENTROPY DEPARTURE FOR NEON FIGURE 46: ENTROPY DEPARTURE FOR ARGON FIGURE 4 7 : INVERSION CURVE FOR METHANE FIGURE 48: INVERSION CURVE FOR PROPRNE FIGURE 49: INVERSION CURVE FOR N-BUTRNE FIGURE 50: INVERSION CURVE FOR PRRR-HYDROGEN FIGURE 51: INVERSION CURVE FOR ARGON APPENDIX H -TABLES T a b l e 20: V o l u m e RMS-% E r r o r / C P U T i m e f o r R e g i o n I ( % ) / s e c COMPOUND LHC* S72 RK LK Me t h a n e 6 . 2 9 / 0 . 309 10 . 7 7 / 0 .3 19 10 . 5 3 / 0 . 3 2 0 18 .38/0 .053 P r o p a n e 1 1 .89/0 . 193 13 . 4 6 / 0 . 192 17 .30/0 . 203 1 .67/0 .030 n - B u t a n e 14 . 3 5 / 0 . 185 16 .03/0 . 185 20 . 4 7 / 0 . 195 1 1 .01/0 .033 n - P e n t a n e 15 .7 1/0 . 169 17 . 15/0 . 169 24 .61/0 . 183 7 .49/0 .029 n - H e x a n e 16 . 2 2 / 0 . 269 18 .39/0 . 2 6 2 25 .85/0 .273 2 .85/0 . 044 M e t h a n o I 38 . 4 2 / 0 . 155 4 1 . 7 0 / 0 . 154 59 .OO/O . 18 1 1 1 .82/0 .026 H 1 0 36 .41/1 . 247 39 .03/1 .2 15 46 . 7 8 / 1 . 27 1 33 . 3 8 / 0 . 203 H , S 7 . 17/0 . 329 8 . 9 9 / 0 . 3 3 0 10 .58/0 330 2 .97/0 .052 n-Hz 3 .03/0 260 10 -02/0 2 4 2 13 .43/0 250 8 .05/0 .04 1 p - H i 7 .87/0 24 1 9 .82/0 2 3 3 13 .31/0 243 7 .73/0 04 1 N e o n 5 . 4 4 / 0 223 7 .06/0 2 14 7 .43/0 214 5 .08/0 0 3 6 A r g o n 5 5 3 / 0 695 7 4 1/0 704 7 4 0 / 0 705 13 9 9 / 0 128 T a b l e 2 1 : V o l u m e RMS-% E r r o r / C P U T i m e f o r R e g i o n I I [ % ) / s e c COMPOUND LHC* S72 RK LK M e t h a n e 3 G2/0 309 3 9 I/O 324 3 3 0 / 0 3 19 . 1 2 2 / 0 0 4 6 E t h a n e 5 9 0 / 0 187 7 0 1 / 0 187 6 5 6 / 0 188 1 9 2 / 0 0 2 5 P r o p a n e 6 2 2 / 0 . 192 6 6 7 / 0 194 7 3 2 / 0 . 203 4 0 1 / 0 . 0 2 7 n - B u t a n e 4 9 3 / 0 . 351 a 2 4 / 0 . 343 9 6 0 / 0 . 379 9 5 9 / 0 . 0 5 6 n - P e n t a n e 6 . 7 9 / 0 . 324 9 . 5 9 / 0 . 3 0 7 1 1 19/0. 363 9 0 9 / 0 . 0 4 7 n - H e x a n e 5 . 8 0 / 0 . 438 7. 16/0. 4 4 8 a. 2 5 / 0 . 462 2. 3 1 / 0 . 0 5 9 M e t h a n o I 1 1 . 17/0. 465 12. 16/0. 4 5 5 14 . 7 6 / 0 . 466 IO. 7 7 / 0 . 064 t - b u t a n o 1 9 . 8 1/0. 134 10. 4 2 / 0 . 134 13 . 0 9 / 0 . 163 10. 5 1/0. 0 1 9 H.O 6 . 95/1 . 252 8 . 8 6 / 1 . 2 10 10. 19/ 1 . 279 4 . 6 I/O. 162 H i S 5 . 14/0. 333 5 . 8 0 / 0 . 3 3 0 5 . 7 8 / 0 . 3 3 0 4 . 4 7 / 0 . 0 4 5 n-H. 3 . 7 1/0. 259 3 . 5 4 / 0 . 242 4 . 3 6 / 0 . 25 1 19 . 99/0. 0 3 7 p - H , 3 . 4 7 / 0 . 241 3 . 5 7 / 0 . 233 4 . 6 6 / 0 . 243 20. 7 1/0. 0 3 6 N e o n 3 . 9 3 / 0 . 224 3 . 5 9 / 0 . 2 1 4 2 . 7 9 / 0 . 2 1 3 1 . 8 3 / 0 . 0 2 8 A r g o n 7 . 12/0. 696 7 . 5 7 / 0 . 7 0 2 7. 4 8 / 0 . 706 6. 3 2 / 0 . 101 2 4 3 T a b l e 22: V o l u m e RMS-% E r r o r / C P U T i m e f o r R e g i o n III C/.)/sec COMPOUND LHC ' LHCHB LHCA i LHCCH S72 * RK LK * Me t h a n e 2 . 4 4 / 0 905 2 32/2 102 2 32/2 103 4 35/2 086 2 . 89/2 083 2 84/2 095 1 2 9 / 0 260 P r o p a n e 7 . 85/0 684 4 51/1 6 3 0 4 55 / 1 638 42 94/2 155 1 1 . 6 3 / 1 547 10 4 1/1 552 1 8 2 / 0 203 n - B u t a n e 12 . 16/0 104 7 6 7 / 0 242 7 7 1/0 242 44 0 5 / 0 165 17 . 8 3 / 0 233 16 8 1/0 232 2 0 6 / 0 028 n - P e n t a n e 10 . 6 1/0 153 3 6 3 / 0 364 3 58/0 366 43 6 8 / 0 246 16 6 1/0 350 15 28/0 353 1 7 8 / 0 044 n - H e x a n e 13 46/0 149 5 93 / 0 358 5 93 / 0 359 39 5 6 / 0 238 2 1 . 18/0 342 20 0 6 / 0 339 1 8 6 / 0 045 Me t hano1 37 . 14/0 283 1 1 4 2 / 0 738 14 55 / 0 720 1 1 17/0 503 52 . 9 3 / 0 623 5 1 16/0 627 12 7 4 / 0 066 H.O 34 . 82/ 1 385 - 53 48/3 012 27 84/3 294 42 59/3 093 40 39/3 104 19 2 0 / 0 354 H i S 6 . 4 1/0 101 3 80 / 0 246 3 79 / 0 245 87 0 5 / 0 3 0 0 8 2 3 / 0 228 8 62 / 0 230 1 6 6 / 0 032 n - H i 9 84 / 0 280 4 27 / 0 647 4 20 / 0 644 2 1 8 0 / 0 704 10 3 2 / 0 674 15 9 3 / 0 684 8 4 9 / 0 075 p - H i 9 45/0 29 1 - 4 0 6 / 0 67 1 20 3 8 / 0 708 14 3 8 / 0 698 15 5 8 / 0 7 12 8 19/0 087 N e o n 4 44 / 0 155 2 0 7 / 0 3 5 0 2 0 9 / 0 35 1 9 0 5 / 0 374 5 2 5 / 0 359 6 49 / 0 36 1 6 4 8 / 0 047 A r g o n 3 53/0 657 2 .97/1 5 14 2 .97/1 532 3 . 3 6 / 1 5 5 0 4 13/ 1 . 538 4 .67/ 1 553 3 .46/0 184 $ RMS-% e r r o r v a l u e s w e r e p r e s e n t e d In r e f e r e n c e ( 6 8 ) N o t e : T h e C o r r e c t i o n f a c t o r m e t h o d u s i n g s p l i n e s u n d e r t e n s i o n f a i l e d f o r a l l c o m p o u n d s . T a b l e 23: V o l u m e RMS-'/. E r r o r / C P U T i m e f o r R e g i o n IV ('/.)/sec COMPOUND LHC • LHCHB LHCA $ LHCCH S72 i RK LK * M e t h a n e 0. 3 4 / 0 . 735 0. 4 2 / 1 . 977 0. 42/1 .985 4 . 52/2 . 362 0. 19/ 1 . 705 0. 18/ 1 . 7 13 0. 3 0 / 0 . 165 E t h a n e 0. 8 6 / 0 . 566 0 6 1/1 228 0. 61/1 .242 1 . 59/1 . 352 1 . 2 8 / 1 . 337 1 . 18/ 1 . 337 0. 5 0 / 0 . 133 P r o p a n e 0. 7 7 / 0 7 15 0 94/1 599 0. 9 4 / 1 .623 5 . 0 9 / 1 405 1 . 3 5 / 1 667 1 27/ 1 678 0 4 0 / 0 179 n - B u t ane 1. 4 3 / 0 345 1 2 7 / 0 794 1 . 2 7 / 0 . 8 0 5 5 . 6 8 / 0 863 2 . 4 9 / 0 828 2 42/0 829 0 7 3 / 0 08 1 n - P e n t a n e 1. 4 6 / 0 180 1 10/0.425 1 . 1 1 / 0 . 4 3 0 4 . 0 4 / 0 442 3 0 1 / 0 466 2 9 1/0 467 0 20/ 0 052 n - H e x a n a 2 . 2 7 / 0 886 2 16/1 7 19 2 . 15/1.803 12 . 28/2 405 3 . 52/2 275 3 45/2 247 1 4 3 / 0 227 Me t h a n o 1 1 9 5 / 0 29 1 2 2 0 / 0 8 7 0 2 2 2 / 0 . 8 6 9 70 2 4 / 0 879 4 6 1/0 761 4 52/0 770 1 8 3 / 0 073 t - b u t a n o l 3 48/0 432 - 4 5 9 / 1 . 0 0 6 26 4 9 / 0 901 4 32/1 0 9 5 4 28 / 1 1 14 2 5 2 / 0 103 H!0 0 63/2 189 - 0 5 0 / 4 . 7 8 6 0 88/4 676 1 14/5 384 1 1 1/5 322 0 3 3 / 0 502 H. S 1 14/0 205 1 18/0 422 1 18/0.42 1 1 3 4 / 0 . 5 1 6 1 14/0 495 1 0 8 / 0 479 1 0 3 / 0 0 5 0 n-H. 0 8 6 / 0 109 0 8 5 / 0 246 0 8 5 / 0 . 2 4 9 9 4 7 / 0 336 1 5 1/0 237 1 75/0 239 8 8 8 / 0 028 p-H. 1 4 4 / 0 1 16 - 1 6 4 / 0 . 2 5 9 16 7 6 / 0 37 1 1 7 2 / 0 280 1 9 1/0 292 9 0 2 / 0 0 3 0 N e o n 1 12/0 .093 0 .72/0 . 188 0 7 3 / 0 . 1 9 0 3 4 0 / 0 .257 1 0 7 / 0 . 208 0 .95/0 2 0 0 0 9 1/0 023 A r g o n 0 65/ 0 . 570 0 . 5 9 / 1 .324 0 5 9 / 1 . 334 2 9 8 / 1 .519 0 84/1 347 0 77/ 1 3 12 0 .59/0 138 $ RMS-'/, e r r o r v a l u e s w e r e p r e s e n t e d In r e f e r e n c e ( 6 8 ) N o t e : The c o r r e c t i o n f a c t o r m e t h o d u s i n g s p l i n e s u n d e r t e n s i o n f a i l e d f o r a l l c o m p o u n d s . T a b l e 24: V o l u m e RMS-% E r r o r / C P U T i m e f o r R e g i o n V ('/.)/seo COMPOUND LHC 1 LHCI I S72 RK LK MQ t h a n e 14 3 1/7 087 14 72/7 . 286 13 . 74/7 . 3 1 3 15 75/7 . 398 0 7 2 / 0 376 E t h a n e 18 48/5 72 1 13 57/5 . 836 17 . 95/5 827 13 9 9 / 5 8 5 0 1 3 9 / 0 34 1 P r o p a n e 15 50/3 089 1 1 39/3 . 147 12 . 0 7 / 3 150 1 1 3 6 /3 159 1 5 0 / 0 189 n - B u t a n e 9 94/1 943 7 9 2 / 1 . 948 8 . 24/ 1 954 a 1 1/1 962 1 3 3 / 0 145 n - P e n t a n e 10 56/ 1 3 19 8 0 0 / 1 33 1 9 6 5 / 1 33 1 1 1 2 8 / 1 329 2 1 1/0 107 n - H e x a n e 13 34/2 803 10 86/2 866 10. 47/2 85 1 10 96/2 8 6 0 3 8 0 / 0 2 12 Me t h a n o l 36 80/0 836 36 8 0 / 0 835 37 5 2 / 0 838 42 7 3 / 0 842 18 8 1/0 054 t - b u t a n o l 2 4 8 / 0 198 3 4 0 / 0 205 3 3 1 / 0 203 2 8 7 / 0 202 2 4 5 / 0 012 H. 0 27 15/5 054 26 44/5 109 26 74/5 183 27 37/5 297 3 1 5 4 / 0 378 Hi S 10 91/1 086 a 4 1/1 100 a 3 7 / 1 0 9 9 6 9 8 / 1 0 96 1 14/0 072 n - H i 20 16/5 746 19 1 1/5 807 17 0 6 / 5 622 14 78/5 509 3 0 6 / 0 298 p - H . 19 62/5 722 19 0 1 / 5 782 15 60/5 601 13 14/5 507 2 8 4 / 0 297 N e o n 1 1 78/4 096 8 14/4 204 9 12/4 1 16 7 42/4 306 1 26/0 204 A r g o n 15 0 4 / 1 0 . 4 5 1 16 5 2 / 1 0 . 6 6 7 14 2 8 / 1 0 . 4 7 2 15 2 8 / 1 0 . 8 6 6 1 0 0 / 0 58 1 T a b l e 2 5 : P r e s s u r e RMS-'/. E r r o r / C P U T i m e f o r R e g i o n I I ('/.)/sec COMPOUND L H C * S72 RK BWR M e t h a n e 0. 5 8 / 0 . 002 0. 3 0 / 0 . 001 0. 2 2 / 0 . 001 1 . 15/0. 001 E t h a n e 1 5 4 / 0 . 0 0 1 1 . 6 3 / 0 001 1 . 8 8 / 0 001 1 . 2 1 / 0 001 P r o p a n e 1 0 0 / 0 . 0 0 1 1 . 0 9 / 0 001 1 6 9 / 0 001 0. 7 4 / 0 001 n - B u t a n e 0 5 7 / 0 002 0. 9 2 / 0 002 1 7 0 / 0 001 1. 2 5 / 0 001 n - P e n t a n e 1 3 6 / 0 001 1 . 5 9 / 0 001 2 6 3 / 0 001 1. 0 9 / 0 001 n - H e x a n e 1 8 4 / 0 002 2 . 19/0 002 3 4 5 / 0 002 1 2 7 / 0 002 M e t h a n o 1 5 2 7 / 0 003 5 . 7 5 / 0 002 7 5 8 / 0 002 -t - b u t a n o 1 3 8 9 / 0 001 4 . 0 2 / 0 001 6 31/ 0 001 -H i O 2 0 6 / 0 006 2 4 8 / 0 004 3 5 3 / 0 004 -H i S 1 0 7 / 0 002 1 2 8 / 0 001 1 6 1 / 0 . 0 0 1 -n-H. 0 8 5 / 0 002 0 6 1/0 001 1 8 4 / 0 001 -p - H i 0 72/0 002 0 6 7 / 0 001 1 8 2 / 0 001 -N e o n 0 .95/0 001 0 8 1/0 .001 0 . 6 0 / 0 .001 -A r g o n 0 . 9 0 / 0 004 1 18/0 .003 0 . 9 7 / 0 .002 3 8 4 / 0 .003 ro T a b l e 2 6 : P r e s s u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n IV ( % ) / s e c COMPOUND L H C * LHCHB LHCA LHCCH LHCSP S72 RK BWR Me t h a n e 0 . 2 4 / 0 . 0 02 1 . 0 2 / 3 . 0 49 0 . 5 5 / 0 . 528 0 . 57/1 . 58 1 - 0 . 1 6 / 0 . 004 0 . 1 5 / 0 . 004 0 . 3 4 / 0 . 0 0 6 E t h a n e 0 . 6 6 / 0 0 0 2 1 2 8 / 2 365 0 4 4 / 0 397 0 4 5 / 1 458 0 4 4 / 0 . 2 16 0 . 7 2 / 0 0 0 3 0 8 7 / 0 0 0 3 0 . 2 3 / 0 004 P r o p a n e 0 . 5 2 / 0 002 0 6 7 / 3 4 15 0 6 8 / 0 516 0 6 8 / 1 807 0 6 8 / 0 . 189 0 . 5 7 / 0 004 0 8 2 / 0 004 0 . 3 8 / 0 0 0 6 n-Bu t a n e 0 . 6 2 / 0 001 1 4 2 / 1 178 0 7 7 / 0 2 13 0 7 7 / 0 83 1 2 0 7 / 0 07 1 0 . 7 5 / 0 002 1 0 9 / 0 002 0 . 2 1/0 0 0 2 n - P e n t a n e 0 . 9 1/0 001 0 6 4 / 0 727 0 6 4 / 0 127 0 6 5 / 0 374 0 9 2 / 0 0 4 3 1 0 7 / 0 001 1 7 2 / 0 001 0 . 2 8 / 0 001 n - H e x a n e 1 . 6 4 / 0 0 0 3 14 2 8 / 3 5 1 1 1 7 7 / 0 677 14 3 3 / 1 858 - 1 8 7 / 0 0 0 5 2 4 1/0 004 1 . 46/0 0 0 6 M e t h a n o 1 1 5 5 / 0 001 13 6 3 / 1 342 1 6 7 / 0 245 13 5 2 / 0 753 14 3 7 / 0 14 1 1 8 3 / 0 0 0 2 2 6 6 / 0 0 0 2 -t - b u t ano1 3 0 2 / 0 001 - 16 2 7 / 0 309 4 06/1 132 1 1 7 6 / 0 101 3 0 8 / 0 002 3 7 6 / 0 0 0 2 -H . O 0 5 2 / 0 007 - 0 43/1 696 0 4 6 / 9 195 - 0 6 1/0 0 1 3 0 9 0 / 0 01 1 -H i S 0 9 0 / 0 001 0 8 9 / 0 903 0 8 9 / 0 151 0 8 9 / 0 454 0 8 9 / 0 09 1 0 8 6 / 0 001 0 9 0 / 0 001 -n - H i 0 5 0 / 0 0 0 0 t 1 0 2 / 0 348 0 4 9 / 0 077 0 9 8 / 0 246 2 9 3 / 0 0 3 0 0 2 7 / 0 001 1 0 7 / 0 001 -p - H i 1 2 3 / 0 0 0 0 * - 1 2 8 / 0 08 1 5 9 7 / 0 255 8 2 4 / 0 03 2 1 16/0 001 1 3 3 / 0 001 -N e o n 0 9 6 / 0 0 0 0 t 0 6 4 / 0 429 0 6 4 / 0 073 0 . 6 3 / 0 236 - 0 9 1 / 0 001 0 8 1/0 001 -j A r g o n 0 3 5 / 0 002 0 28/2 .428 0 2 B / 0 430 0 . 2 8 / 1 6 0 3 0 .3 1/0 2 12 0 4 5 / 0 0 0 3 0 3 6 / 0 0 0 3 1 . 6 7 / 0 .004 4 The c o m p u t e r r o u n d e d t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d 1SJ co T a b l e 27: P r e s s u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n V ( % ) / s e c COMPOUND LHCI LHCI I S72 RK BWR M e t h a n e 16 . 40/0 017 12 23/0 017 13 0 4 / 0 0 1 0 28 . 6 9 / 0 . 009 30 4 8 / 0 013 E t h a n e 1 7 7O/0 0 14 5 8 4 / 0 014 6 2 9 / 0 008 4 8 6 / 0 008 2 4 2 / 0 01 1 P r o p a n e 1 4 5 3 / 0 007 9 6 7 / 0 008 10 0 5 / 0 004 8 0 8 / 0 004 5 7 0 / 0 006 n-Bu t a n e 90 9 1/0 005 9 1 0 1 / 0 005 9 1 77 / 0 003 1 10 3 5 / 0 003 53 8 3 / 0 004 n - P e n t a n a 3 19 5 3 / 0 004 3 18 50/0 004 3 19 9 5 / 0 003 376 6 0 / 0 002 80 6 4 / 0 003 n - H e x a n e 206 10/0 007 208 8 5 / 0 007 208 9 0 / 0 004 235 17/0 004 96 8 2 / 0 005 Me t hano1 424 19 2 0 / 0 002 424 19 2 0 / 0 002 4 2 4 2 0 6 4 / 0 001 4 2 3 9 0 2 1/0 001 -t - b u t ano1 2 1 1/0 001 2 9 9 / 0 . 0 0 1 2 9 2 / 0 0 0 0 * 2 5 5 / 0 0 0 0 t -H.O 3216 2 4 / 0 015 32 16 2 3 / 0 . 0 1 5 2 9 9 0 6 9 / 0 0 0 9 2842 7 3 / 0 008 -H.S 43 13/0 003 43 2 9 / 0 003 44 0 7 / 0 . 0 0 2 51 3 4 / 0 002 -n-H. 59 5 6 / 0 013 59 4 5 / 0 . 0 1 4 63 1 5 / 0 . 0 0 8 120 5 5 / 0 007 -p - H . 55 35/0 013 55 2 5 / 0 014 59 4 4 / 0 008 1 15 6 1/0 007 -N e o n 17 4 0 / 0 009 14 .89/0 0 1 0 17 2 8 / 0 .005 2 1 .11/0 005 -A r g o n 20 6 7 / 0 .024 1 6 .94/0.025 16 .33/0.014 22 .90/0 .013 34 . 10/0 0 1 9 $ T h e c o m p u t e r r o u n d e d t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d T a b l e 2 8 : T e m p e r a t u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n I ( % ) / s e c COMPOUND L H C * S72 RK BWR M e t h a n e 2 . 3 1 / 0 . 018 2 . 5 3 / 0 . 0 1 2 3 .3 1/0 . 012 2 . 2 9 / 0 .011 P r o p a n e 8 . 9 1 / 0 . 013 9 . 5 0 / 0 . o o a 11 . 2 5 / 0 . 008 2 .7 1/0 . 0 0 7 n - B u t a n e 9 . 3 2 / 0 . 012 9 . 6 9 / 0 0 0 7 12 6 5 / 0 0 0 8 1 10/0 0 0 8 n - P e n t a n e 1 1 6 4 / 0 014 1 1 9 1/0 0 0 9 16 16/0 0 0 9 1 4 5 / 0 0 0 8 n - H e x a n e 22 3 2 / 0 018 23 8 2 / 0 01 1 28 16/0 0 ( 2 2 4 7 / 0 0 1 0 Me t h a n o 1 - - - -H i O - - - -H,S 6 4 6 / 0 02 1 6 5 2 / 0 0 1 3 6 9 1/0 0 1 3 -n - H i 18 7 7 / 0 0 1 2 19 2 2 / 0 0O9 27 9 2 / 0 0 0 9 -p - H i 18 8 1/0 0 1 2 19 4 6 / 0 . o o a 28 1 5 / 0 . 0 0 9 -N e o n 5 . 4 7 / 0 . 0 1 2 6 5 5 / 0 . o o a 7 . 8 7 / 0 . 0 0 9 -A r g o n | 3 . 7 9 / 0 . 04 2 3 . a a / o . 0 2 7 5 . 5 1/0. 0 29 6 . 4 1/0. 0 2 5 T a b l e 2 9 : T e m p e r a t u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n I I i n ( % ) / s e c COMPOUND L H C * S72 RK BWR M e t h a n e 0 2 7 / 0 . 0 1 7 O . 19/0 .011 o . 0 8 / 0 .011 0 4 8 / 0 01 1 E t h a n e 1 17/0 0 1 0 1 18/0 0O7 1 3 2 / 0 0 0 7 1 0 1 / O o o a P r o p a n e 0 7 7 / 0 01 1 0 7 9 / 0 0 0 7 1 15/0 0O8 0 5 4 / 0 0 0 7 n - 3 u t a n e 0 2 9 / 0 0 2 0 0 3 8 / 0 01 1 0 7 7 / 0 0 1 2 0 5 2 / 0 0 1 2 n - P e n t a n e 0 8 9 / 0 018 0 9 3 / 0 0 10 1 5 7 / 0 0 1 0 0 2 5 / 0 01 1 n - H e x a n e 1 13/0 0 2 6 1 2 4 / 0 0 1 5 2 10/0 017 0 5 5 / 0 0 16 M e t h a n o 1 3 4 0 / 0 026 3 5 5 / 0 0 1 6 4 9 2 / 0 016 -t - d u t ano1 2 4 8 / 0 008 2 5 1/0 0 0 5 4 19/0 0 0 6 -H.O 1 2 6 / 0 06 1 1 3 9 / 0 0 3 8 2 13/0 04 1 -H, S 0 7 6 / 0 017 0 8 0 / 0 0 1 1 i 0 0 / 0 0 1 2 -n - H , 0 4 6 / 0 01 1 0 4 2/0 o o a 1 1 9 / 0 0 0 9 -p - H , O. J 8 / 0 0 1 1 0 4 5 / 0 o o a i 1 9 / 0 008 -N e o n O. 6 3 / 0 011 O 5 1/0 o o a 0 3 7 / 0 0 0 8 -A r g o n 0 . 5 5 / 0 038 0 . 6 5 / 0 . 02 4 O . -14/0. 0 2 7 1 . 9 6 / 0 . 0 2 6 T a b l e 30: T e m p e r a t u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n III ( % ) / s e c COMPOUND LHC LHCHB LHCA LHCCH S72 RK BWR Me t h a n e 2 . 6 0 / 0 . 0 3 3 2 9 1/5 573 - - 2 . 7 1/0 06 3 4 7 8 / 0 067 3 . 2 2 / 0 . 058 P r o p a n e 9 . 6 6 / 0 . 0 2 6 2 60/4 237 - - 9 . 9 3 / 0 0 4 8 12 2 4 / 0 05 1 4 . 16/0 0 4 6 n-Bu t ane 10 . 0 7 / 0 . 0 0 3 3 4 2 / 0 524 - - 10. 3 1/0 007 13 3 2 / 0 008 2 . 10/0 007 n - P e n t a n e 15 . 2 3 / 0 . 0 0 5 2 59/0 840 - 12 . 13/0 067 15 . 8 4 / 0 0 10 20 6 6 / 0 01 1 2 . 0 1 / 0 0 1 0 n - H o x a n a 16 12/0.005 4 9 8 / 0 77 1 - 18 . 8 9 / 0 066 16 . 3 9 / 0 0 1 0 2 1 9 3 / 0 0 1 0 2 . 8 7 / 0 0 0 9 Me t h a n o 1 - - - - - - -H.O - - - - - - -H.S 5 3 1 / 0 . 0 0 4 6 8 9 / 0 57 1 - - 5 2 9 / 0 007 6 4 0 / 0 007 -n-H. 23 4 1 / 0 . 0 0 8 18 16/1 704 18.00/0 258 18 .65/0 108 23 7 1/0 017 33 4 0 / 0 018 -p-H. 23 8 4 / 0 . 0 0 8 - 16.67/0 270 16 .60/0 1 14 24 5 1/0 017 34 8 0 / 0 01B -N e o n 9 2 8 / 0 . 0 0 5 4 84/ 1 02 1 4 .88/0 148 5.99/0 062 10 8 5 / 0 0 1 0 12 8 5 / 0 01 1 -A r g o n 4 6 9 / 0 . 0 2 1 2 . 34/3 478 - - 4 7 4 / 0 04 1 6 5 6 / 0 . 0 4 2 7 17/0 .039 N o t e : The c o r r e c t i o n f a c t o r m e t h o d u s i n g s p l i n e s u n d e r t e n s i o n f a i l e d f o r a l l c o m p o u n d s . T a b l e 3 1 : T e m p e r a t u r e RMS-°/. E r r o r / C P U T i m e f o r R e g i o n IV ('/.)/sec COMPOUND LHC • LHCHB LHCA LHCCH LHCSP S72 RK BWR Me t h a n e 0 . 17/0. 022 0.31/4 503 0. 3 1/0. 74 1 0 6 0 / 0 . 309 - 0. 13/0. 043 0. 13/0 . 044 0. 28 / 0 . 05 1 E t h a n e 0. 5 1/0 0 1 8 0.35/2 990 0 3 5 / 0 . 556 9 53 / 0 269 1 2 . 1 7 / 0 . 687 0. 5 3 / 0 . 035 0. 65 / 0 036 0. 18 /0 0 4 0 P r o p a n e 0. 37 / 0 023 0.49/4 662 0 49 / 0 707 2 6 6 / 0 298 - 0. 39 / 0 0 4 5 0 58 / 0 046 0. 3 1 /0 0 5 2 n - B u t a n e 0. 35 / 0 0 1 0 0 . 28/ 1 677 0 28/0 289 - - 0. 4 0 / 0 018 0 6 4 / 0 019 0. 0 9 2 / 0 0 1 9 n - P e n t a n e 0 . 5 9 / 0 006 0 . 4 9 / 0 987 0 49 / 0 17 1 - - 0. 6 5 / 0 012 1 10/0 012 0. 12 /O 0 1 3 n - H e x a n e 1 . 10/0 024 1 1 . 88/4 8 13 - - - 1 . 72 / 0 0 5 0 1 6 4 / 0 052 0. 95 /O 0 5 7 M e t h a n o 1 1 . 0 8 / 0 009 1.14/1. 584 1 14/0 288 9 70 / 0 14 1 - 1 . 2 2 / 0 022 1 8 5 / 0 325 -t - b u t ano1 2 19/0 015 - 3 0 5 / 0 382 3 67 / 0 165 - 2 22 / 0 027 2 9 4 / 0 027 -H.O 0 40 / 0 0 7 0 - - - - 0 4 4 / 0 128 0 6 9 / 0 130 -H i S 0 S 8 / 0 006 0.65/1 188 0 65 / 0 200 9 .40/0 0 9 9 - 0 6 5 / 0 013 0 7 1/0 014 -n-H. 0 35 / 0 003 0. 3 2 / 0 . 557 0 32 / 0 103 9 . 17/0 049 - 0 20 / 0 006 0 7 4 / 0 007 -p-H, 1 0 4 / 0 003 - 1 0 5 / 0 1 10 a .49/0 052 1 .05/0 139 0 9 9 / 0 007 0 9 6 / 0 007 -N e o n 0 78 / 0 003 0 . 6 0 / 0 . 550 0 6 0 / 0 096 - - 0 73 / 0 006 0 6 4 / 0 006 -A r g o n 0 24/0 017 0. 17/3 . 1 16 0 18/0 570 9 . 3 S / 0 27 1 - 0 29 / 0 034 0 2 0 / 0 035 1 05 /O .045 ho T a b l e 3 2 : T e m p e r a t u r e RMS-% E r r o r / C P U T i m e f o r R e g i o n V ( % ) / s e c COMPOUND LHCI L H C I I S72 RK BWR M e t h a n e 9 -05/0 . 2 14 3 . 30/0 . 201 1 .45/0 . 121 2 .30/0 . 138 1 .79/0 . 140 E t h a n e 1 1 . 13/0 . 183 1 . 72/0 . 178 1 .93/0 . 104 t .59/0 1 16 6 . 3 3 / 0 1 15 P r o p a n e 9 . 14/0 . 101 1 . 89/0 098 2 . 19/0 .056 1 .75/0 0 6 2 1 .4 3/0 059 n - B u t a n e 9 -98/0 0 7 6 7 . 73/0 0 7 0 7 . 8 9 / 0 0 4 0 9 49/ 0 0 4 5 2 1 . 7 3 / 0 037 n - P e n t a n e 16 3 0 / 0 0 4 5 13 20/0 045 13 8 4 / 0 0 2 6 17 2 3 / 0 0 3 0 3 .97/0 024 n - H e x a n e 10 7 3 / 0 10 1 8 74/0 09S 8 8 6 / 0 0 5 6 IO 8 8 / 0 06 2 5 9 4 / 0 0 5 5 M e t h a n e I - - - - -t - b u t a n o I 2 0 9 / 0 0O7 2 10/0 006 2 0 8 / 0 004 2 0 3 / 0 004 -H.O - - - - -H.S 5 6 1 / 0 . 0 3 9 3 3 7 / 0 . 036 3 4 2 / 0 0 2 0 3 8 1 / 0 . 0 2 3 -n-H. 1 1 3 8 / 0 . 127 10. 9 3 / 0 . 128 10 7 0 / 0 . 0 8 9 12 6 1/0. 108 -p - H , 10. 2 7 / 0 . 125 9 . 9 2 / 0 . 128 9. 8 0 / 0 . 0 8 8 1 1 . 9 1/0. 108 -N e o n 7 . 11/0. 109 3. 17/0. 105 2 . 3 0 / 0 . 0 6 6 2. 4 6 / 0 . 0 7 3 -A r g o n 10. 18/0. 3 0 0 4 . 17/0. 287 2 . 15/0. 175 2 . 8 2 / 0 . 197 1 . 4 7 / 0 . 21 1 T a b l e 33: V i r i a l E x p a n s i o n i n V o l u m e t o S o l v e f o r V i n R e g i o n I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND L H C S72 RK M e t h a n e 2 5 0 / 0 . 0 0 9 0 50/0 008 O 5 8 / 0 .009 E t h a n e 1 9 2 / 0 005 2 79/0 OOS 2 5 7 / 0 0 0 5 P r o p a n e 1 2 7 / 0 OOG 2 9 2 / 0 007 2 6 5 / 0 0 0 7 n - S u t a n e 1 5 1/0 0 0 9 4 35/0 008 4 10/0 0 0 9 n - P e n t a n e 2 3 8 / 0 0 0 7 5 TO/O 007 5 4 5 / 0 0O7 n - H e x a n e 2 8 7 / 0 015 6 OG/0 015 5 8 8 / 0 0 1 6 M e t h a n o 1 6 0 7 / 0 0 1 4 10 28/0 013 10 13/0 0 1 3 t - b u t a n o 1 5 6 4 / 0 004 9 63/ 0 004 9 4 8 / 0 0 0 4 H.O 2 8 0 / 0 034 5 76/0 03 1 5 6 3 / 0 0 3 3 H.S 1 52 / 0 0 1 0 3. 12/0 009 2 8 7 / 0 0 1 0 n-H. 1 4 6 / 0 007 3. 28/0 008 3 9 5 / 0 0 0 8 p - H , I 5 9 / 0 007 3 . 41/0 COS 4 12/0 0 0 8 N e o n 1 9 2 / 0 007 1 38/0 GO • 1 14/0 0 0 8 A r g o n 1 52/0 02-1 ' 2 • •15/0 024 2 2 2 / 0 0 2 5 T a b l e 34: V i r i a l E x p a n s i o n i n V o l u m e t o S o l v e f o r V i n R e g i o n IV RMS-*/. E r r o r / C P U t i m e t n ( % ) / s e c COMPOUND LHC • LHCHB LHCA LHCCH LHCSP S72 RK M e t h a n e 0. 3 7 / 0 . 0 1 5 0. 4 5 / 0 . 0 4 6 0. 4 4 / 0 . 0 4 6 0. 4 4 / 0 . 046 0. 4 4 / 0 . 154 0. 2 2 / 0 . 0 4 3 0. 2 1/0. 0 4 5 E t h a n e 0. 8 4 / 0 . 012 0. 5 7 / 0 . 0 3 6 0. 5 7 / 0 . 0 3 7 0. 5 8 / 0 . 0 3 6 0. 5 7 / 0 . 123 0. 9 5 / 0 . 034 1 . 15/0. 0 3 5 P r o p a n e 0 . 7 0 / 0 . 0 1 5 0. 8 8 / 0 . 0 4 7 0. 8 9 / 0 . 0 4 7 0. 8 9 / 0 . 0 4 7 0. 8 9 / 0 . 174 0. 8 6 / 0 . 04 3 1 . 2 1/0. 045 n-Bu t a n a 2 2 2 / 0 . 008 2 0 3 / 0 . 0 2 3 2 . 0 3 / 0 . 0 2 3 2 . 0 2 / 0 . 0 2 3 2 . 0 3 / 0 . 068 3 . 0 7 / 0 022 3 7 2 / 0 022 n - P e n t ane 1 3 0 / 0 004 0 9 2 / 0 . 0 1 2 0 9 2 / 0 . 0 1 3 0 9 3 / 0 012 0 9 2 / 0 04 4 1 7 2 / 0 012 2 78 / 0 012 n - H e x a n e 2 20 / 0 018 3 2 3 / 0 . 0 5 7 3 2 5 / 0 . 0 5 7 3 2 4 / 0 057 3 2 5 / 0 199 2 6 1/0 05 1 3 39 / 0 053 M e t h a n o l 1 9 2 / 0 007 2 1 9 / 0 . 0 2 0 2 2 2 / 0 . 0 2 0 2 2 0 / 0 0 2 0 2 2 2 / 0 073 2 4 2 / 0 018 3 4 4 / 0 0 1 9 t - b u t a n o l 3 4 8 / 0 0 0 9 - 4 5 9 / 0 . 0 2 8 4 5 6 / 0 028 4 5 5 / 0 100 3 55 / 0 025 4 28/ 0 026 H i O 0 6 2 / 0 042 - 0 50 / 0 . 1 3 2 0 5 2 / 0 133 0 5 0 / 0 626 0 7 6 / 0 1 18 1 1 1/0 124 H i S 1 14/0 004 1 18/0.013 1 18/0. 132 1 18/0 014 1 18/0 045 1 0 7 / 0 012 1 0 9 / 0 012 n-H. 0 9 4 / 0 0O3 0 9 4 / 0 . 0 0 8 0 9 4 / 0 . 0 0 7 0 9 4 / 0 008 0 9 4 / 0 0 2 0 0 37/ 0 .007 1 8 3 / 0 0O7 p - H i 1 .48/0 003 - 1 7 0 / 0 . 0 0 8 1 7 0 / 0 008 1 7 0 / 0 022 1 2 9 / 0 .007 1 9 6 / 0 .008 N e o n 1 . 12/0 .002 0 .72/0.006 0 7 3 / 0 . 0 0 6 0 7 3 / 0 006 0 .73/0 02 1 1 0 7 / 0 .006 0 .95/0 .006 A r g o n 0 . 52/0 .012 0 .45/0.037 0 .44/0.037 0 .44/0 .037 0 .44/0 12 1 0 73/ 0 .033 0 .64/0 .035 T a b l e 3 5 : V i r i a l E x p a n s i o n In V o l u m e t o S o l v e f o r V 2 5 5 In R e g i o n V RMS-% E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND LHCI LHCI I S72 RK Me t h a n e to .94/0 .07 3 4 .07/0 .086 1 . 5 7 / 0 .067 2 . 4 5/0 .07 1 E t h a n e 12 .9 1/0 .055 1 .73/0 .062 1 . 7 9 / 0 .049 1 . 5 8 / 0 .052 P r o p a n e 9 . 6 3 / 0 .03 1 2 . 13/0 036 2 . 3 4 / 0 .028 2 . 16/0 0 3 0 n - B u t a n e 13 8 2 / 0 001 0 50/0 002 O . 9 4 / 0 001 1 .3 1/0 0O1 n - P e n t a n e 8 6 2 / 0 0 0 2 1 54/0 002 1 5 7 / 0 001 1 .66/0 002 n - H e x a n e 9 5 9 / 0 0 2 2 3 18/0 0 2 5 2 7 9 / 0 0 2 0 2 .07/0 021 Me t h a n o 1 - - - -t - b u t a n o 1 2 4 9 / 0 0 0 3 3 39/0 0O3 3 3 1/0 00 3 2 8 7 / 0 0 0 3 H,0 7 15/0 054 6 63/0 0 6 0 5 5 6 / 0 0 5 0 4 9 6 / 0 0 5 3 H, S 6 2 7 / 0 0 0 6 1 3 1/0 007 1 2 5 / 0 0 0 5 1 3 4 / 0 . 0O6 n - H i 7 5 6 / 0 . 0 5 6 6 10/0. 067 4 8 3 / 0 . 053 3 4 0 / 0 . 0 5 7 p - H j S 9 9 / 0 . 0 5 9 5 . 6 1/0. 068 4 . 3 0 / 0 . 053 3 0 O / 0 . 0 5 7 N e o n a. 9 7 / 0 . 0 4 4 3 . 12/0. 053 O. 8 5 / 0 . 042 1 . OO/O. 0 4 4 A r g o n 11. 3 3 / 0 . 1 14 4 . 4 3/0. 136 1 . 7 2 / 0 . 107 2 . 5 8 / 0 . 113 T a b l e 36: V i r i a l E x p a n s i o n i n V o l u m e t o S o l v e f o r P i n R e g i o n I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND L H C « S72 RK M e t h a n e 2 . 8 0 / 0 .002 2 . 66/0 .001 2 .65/0 .010 E t h a n e 4 4 7 / 0 001 4 48/0 001 4 . 5 7 / 0 -OOI P r o p a n e 4 4 0 / 0 OOI 4 40/0 001 4 . 5 6 / 0 .OOI n - 8 u t a n e 3 79/ 0 002 3 71/0 002 3 8 6 / 0 002 n - P e n t a n e 3 S 2 / 0 0 0 3 3 77/0 002 5 3 3 / 0 0 0 2 n - H e x a n e 5 0 6 / 0 002 5 00/0 001 4 5 9 / 0 003 Me t h a n o 1 7 9 6 / 0 003 8 15/0 0O2 9 4 9 / 0 003 t - b u t a n o l 5 9 5 / 0 0 0 1 5 93/0 001 7 5 6 / 0 001 H,0 4 5 8 / 0 ooa 4 60/0 006 5 19/0 007 H, 5 3 6 6 / 0 002 3 64/0 002 3 7 6 / 0 0O2 n-H, 2 7 2 / 0 0 0 2 2 42/0 001 3 0 7 / 0 002 p - H , 2 •34/O. 002 2 5 1/0 001 3 . 1 1/0 0O1 N e o n 2 8 2 / 0 . OO 1 2 . 75/0. OO 1 2 . 7 0 / 0 . OO 1 A r g o n ! . 4 9 / 0 . 005 ! . 62/0. 003 1 4 8 / 0 . 004 T a D l e 3 7: V i r i a l E x p a n s i o n In V o l u m e t o S o l v e f o r P In R e g i o n IV RMS-"/. E r r o r / C P U t i m e In (*/.)/sec C O M P O U N D L H C • LHCHB LHCA LHCCH LHCSP S72 RK M e t h a n e 0. 25/ 0 . 003 1 . 01/2 . 945 0. 5 5 / 0 . 51 1 0. 5 8 / 1 . 52 1 - 0. 16/0. 006 0. 16/0. 0 0 8 E t h a n e 0. 6 5 / 0 . 003 1 . 2 6 / 2 . 283 0. 4 3 / 0 . 383 0. 44/ 1 . 398 0. 4 3 / 0 . 209 0. 7 1/0. 005 0. 8 6 / 0 . 0 0 6 P r o p a n e 0. 50/0 0 0 3 0. 67/3 . 3 10 0. 6 7 / 0 . 497 0 6 7 / 1 735 0. 6 7 / 0 . 183 0. 5 6 / 0 . 006 0. 8 1/0 0 0 8 n - B u t a n e 0. 56/0 001 0 76/ 1 128 0. 7 5 / 0 206 0 7 5 / 0 799 2 2 2 / 0 . 068 0. 6 4 / 0 003 0 9 8 / 0 0 0 3 n - P e n t a n e 0 88/0 001 0 6 0 / 0 704 0 . 6 0 / 0 , 1 2 2 0 6 1/0 359 0 9 9 / 0 04 1 1 0 3 / 0 002 1 6 8 / 0 002 n - H e x a n e 1 6 2 / 0 004 14 33/3 4 14 1 77/0 644 14 37/1 782 - 1 8 5 / 0 007 2 3 9 / 0 0 0 9 M e t hano1 1 56/0 001 13 7 5 / 1 29 1 1 6 7 / 0 236 13 6 3 / 0 725 14 5 0 / 0 135 1 7 9 / 0 003 2 6 1/0 003 t - b u t ano1 3 0 2 / 0 002 - 16 2 7 / 0 298 4 06/1 0 8 5 1 1 7 6 / 0 09 6 3 0 7 / 0 004 3 7 5 / 0 0 0 5 H J O 0 5 2 / 0 0 1 0 - 0 43/1 638 0 45/8 836 0 43/1 84 1 0 6 1 / 0 019 0 9 0 / 0 0 2 5 H i S 0 9 0 / 0 001 0 8 9 / 0 879 0 9 0 / 0 142 0 8 9 / 0 436 0 8 9 / 0 087 0 8 6 / 0 002 0 9 0 / 0 0 0 2 n - H ! 0 53/0 001 1 0 5 / 0 338 0 5 2 / 0 07 5 1 0 1 / 0 237 2 9 8 / 0 0 2 9 0 2 6 / 0 001 1 10/0 001 p - H i 1 24/0 .001 - 1 2 9 / 0 07 8 6 0 2 / 0 244 8 2 9 / 0 03 1 1 16/0 .001 1 3 4 / 0 .001 N e o n 0 9 6 / 0 .001 0 . 6 4 / 0 .4 15 0 6 4 / 0 .07 1 0 .63/0 . 226 - 0 9 0 / 0 .001 0 . 8 0 / 0 .001 A r g o n 0 3 1/0 .003 0 . 24/2 . 324 , 0 24/0 .4 16 0 .24/1 .539 0 . 3 2 / 0 205 0 4 1/0 .005 0 . 32/0 .006 Ch T a b l e 3 8 : V i r i a l E x p a n s i o n I n V o l u m e t o S o l v e f o r P In R e g i o n V RMS-% E r r o r / C P U t i m e t n ( % ) / s e c COMPOUND LHCI LHCI I S72 RK Me t h a n e 34 1 . 3 7 / 0 . 02 2 34 1 . 0 6 / 0 . 0 3 1 3 42 .60/0 .014 372 . 4 6 / 0 .018 E t h a n e 37 . 3 2 / 0 .018 23 . 7 1/0 .028 23 .5 1/0 .011 24 . 7 2 / 0 .016 P r o p a n e 4 1 .06/0 .010 26 .82/0 .015 26 .66/0 .006 28 . 5 4 / 0 .009 n - B u t a n e 304 .32/0 .007 301 .2 1/0 0 0 9 3 0 0 .27/0 .004 273 3 5 / 0 0 0 6 n - P e n t a n e 637 . 4 5 / 0 0 0 4 683 4 8 / 0 0 0 5 68 1 3 1/0 003 589 9 3 / 0 004 n - H e x a n e 285 . 39/0 0 1 0 280 0 9 / 0 0 1 3 279 9 4 / 0 006 273 7 5 / 0 0 0 8 Me t h a n o I 2017 . 4 1/0 0 0 3 2017 4 1/0 0 0 3 2 0 3 0 4 0 / 0 002 1442 5 I/O 0O3 t - b u t a n o 1 2 1 1/0 0O1 2 9 9 / 0 001 2 9 1/0 OOO * 2 5 4 / 0 001 H.O 899 ! 4 8 / 0 02 1 8 9 9 1 4 7 / 0 0 2 3 7 115 6 0 / 0 013 592 1 7 2 / 0 0 1 8 H.S 278 3 3 / 0 004 276 6 0 / 0 0 0 5 275 8 4 / 0 002 265 2 4 / 0 . 0 0 3 n-H. 188 7 9 / 0 0 1 6 188 . 4 7 / 0 . 0 2 3 193. 0 7 / 0 . 0 1 0 2 7 2 . 3 2 / 0 . 014 p - H . 186 4 3 / 0 . 0 1 6 186 . 14/0. 0 2 3 19 1 . 4 3 / 0 . 0 1 0 269 . 2 9 / 0 . 014 N e o n 104 5 2 / 0 . 0 1 3 103 . 6 3 / 0 . O 19 107. 10/0. 008 112. 2 6 / 0 . 01 1 A r g o n 148 5 6 / 0 . 031 147 . 9 9 / 0 . 0 4 5 147. 6 4 / 0 . 019 157 . 1 7 / 0 . 0 2 6 t T h e c o m p u t e r r o u n d s CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d T a b l e 3 9 : V i r i a l E x p a n s i o n i n V o l u m e t o S o l v e f o r T i n R e g i o n I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC* S 7 2 RK M e t h a n e 0 35/ 0 .019 O 20/ 0 0 1 2 0 1 I/O .023 E t h a n e 1 20/0 0 1 2 1 2 1/0 008 1 3 4 / 0 .016 P r o p a n e 0 8Q/0 014 0 8 1/0 0C9 1 17/0 0 1 7 n - B u t a n e 0 33/0 0 2 0 0 37/ 0 013 0 7 7 / 0 0 2 2 n - P e n t a n e 0 97/ 0 0 1 8 0 9 9 / 0 01 1 1 6 7 / 0 02 1 n - H e x a n e 1 14/0 0 2 8 1 2 4 / 0 018 2 1 1/0 0 3 6 Me t h a n o 1 3 4 a / 0 0 2 8 3 6 2 / 0 018 5 0 2 / 0 0 3 7 t - b u t a n o 1 2 58/0 0 1 0 2 6 I/O 007 4 3 6 / 0 0 1 3 H.O 1 29/0 0 6 2 1 4 1/0 0 4 0 2 16/0 0 9 3 H, S 0 78/0 0 2 0 O 8 2 / 0 013 1 0 1 / 0 0 2 6 n-H, 0 52/0 0 1 3 o 4 1/0 0 1 0 1 2 6 / 0 O 19 p - H . 0 55/0 0 1 3 O 4 4 / 0 0 1 0 1 2 4 / 0 0 1 9 N e o n 0 65/0 0 1 3 0 52/0 0C9 0 4 0 / 0 O (8 A r g o n 0 5 4 / 0 0 4 0 0 6 2 / 0 027 0 4 1/0 054 T a b l e -10; V i r i a l E x p a n s i o n In V o l u m e t o S o l v e f o r T In R e g i o n IV RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC + LHCHB LHCA L H C C H S72 RK Me t h a n e 0 . 17/0. 024 0. 3 1/4 359 0. 3 1/0. 742 0 6 0 / 0 . 328 0. 13/0. 047 0. 13/0. 109 E t h a n e 0 . 5 1 / 0 0 2 0 0. 35/2 958 0 3 5 / 0 . 589 39 3 6 / 0 . 336 0. 5 3 / 0 . 038 0. 6 5 / 0 . 0 85 P r o p a n e 0 . 3 7 / 0 025 0 49/4 464 0 4 9 / 0 7 14 2 6 6 / 0 3 17 0. 3 9 / 0 048 0 5 8 / 0 1 12 n - B u t a n e 0. 35/0 01 1 0 27/1 639 0 2 7 / 0 312 - 0. 3 8 / 0 02 1 0 6 2 / 0 044 n - P e n t a n e 0 5 8 / 0 007 0 4 8 / 0 964 0 4 8 / 0 18 1 - 0 . 6 4 / 0 012 1 0 8 / 0 027 n - H e x a n e 1 10/0 027 1 1 88/4 699 - - 1 . 2 2 / 0 057 1 6 4 / 0 125 M e t h a n o l 1 0 8 / 0 0 10 1 14/1 547 1 14/0 303 - 1 16/0 0 2 0 1 8 4 / 0 045 t - D u t a n o 1 2 19/0 016 - 3 0 5 / 0 385 3 6 7 / 0 175 2 2 2 / 0 029 2 9 4 / 0 065 H i O 0 40/0 076 - - 38 34/ 1 367 0 4 4 / 0 135 0 6 9 / 0 359 H i S 0 6 8 / 0 007 0 6 5 / 1 154 0 6 5 / 0 2 1 1 30 5 7 / 0 120 0 6 6 / 0 014 0 7 1/0 032 n - H i 0 35/0 004 0 3 3 / 0 545 0 3 3 / 0 109 44 0 5 / 0 063 0 2 0 / 0 007 0 74/0 015 p - H , 1 0 4 / 0 004 - 1 0 6 / 0 1 16 37 9 6 / 0 064 0 9 9 / 0 007 0 9 7 / 0 016 N e o n 0 78/0 003 0 6 0 / 0 537 0 6 0 / 0 101 - 0 7 3 / 0 006 0 6 4 / 0 .015 A r g o n 0 24/0 0 1 9 0 17/3 .045 0 . 17/0 606 29 .6 1/0 328 0 2 8 / 0 036 0 18/0 085 N o t e : The c o r r e c t i o n f a c t o r m e t h o d u s i n g s p l i n e s u n d e r t e n s i o n f a i l e d f o r a l l c o m p o u n d s . to cn co T a b l e 4 1 : V i r i a l E x p a n s i o n In V o l u m e t o S o l v e f o r T In R e g i o n V RMS-% E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND LHCI L H C I I S72 RK M e t h a n e 2 . 15/0 .113 1 .06/0 .15 1 0 .76/0 .063 O . 5 7 / 0 . 178 E t h a n e 2 .87/0 .060 2 . 17/0 .082 2 . 15/0 .033 1 . 5 3 / 0 084 P r o p a n e 1 .92/0 0 5 2 O . 79/0 07 I 0 8 0 / 0 0 2 6 0 5 8 / 0 1 22 n - B u t a n e 1 5S/0 001 0 16/0 OOI O 14/0 OOO * 0 I 7 / 0 OOI n - P e n t a n e 1 3 1/0 002 0 17/0 C02 0 17/0 001 0 18/0 0 0 2 n - H e x a n e 0 8 1/0 0 2 2 1 2 1/0 029 1 18/0 01 1 0 7 3 / 0 0 2 8 M e t h a n o l - - - -t - b u t a n o l 1 80/0 0 0 4 1 96/0 0O5 1 9 5 / 0 0 0 2 1 3 7 / 0 OOS H.O 6 16/0 125 5 8 1/0 147 5 4 3 / 0 0 6 4 3 6 5 / 0 234 H.S 1 38/0 0 1 0 1 0 5 / 0 013 1 . 0 5 / 0 0 0 5 0 7 5 / 0 0 1 3 n-H. 2 76/0 0 4 9 2 65/0 0 7 0 2. 4 4 / 0 0 3 4 1 6 9 / 0 0 9 9 p - H . 1 17/0 0 4 8 0 94/ 0 . 0 7 0 0. 3 3 / 0 . 0 3 4 0. 2 4 / 0 . 0 9 9 N e o n 2. 6 0 / 0 . 0G3 1 . 41/0. 033 0. 4 7 / 0 . 0 3 7 0. 4 8 / 0 . 102 A r g o n 2 . 3 6 / 0 . 163 1 . 0 6 / 0 . 214 0. 3 S / 0 . 0 9 1 0. 3 8 / 0 . 2 5 7 £ T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d T a b l e 42: V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r V i n R e g i o n II RMS-% E r r o r / C P U t i m e i n ( 7 , ) / s e c COMPOUND LHC- S72 RK M e t h a n e 20 7 1/0 0 0 3 2 1 0 7 / 0 .002 21 0 3 / 0 .003 E t h a n e 25 2 1/0 001 25 4 1/0 0O2 25 3 8 / 0 0 0 2 P r o p a n e 26 23/0 0 0 2 26 53/0 0O2 26 4 9 / 0 0 0 2 n - B u t a n e 31 9 7 / 0 0 0 3 32 70/0 002 32 6 6 / 0 0 0 3 n - P e n t a n e 35 7 6 / 0 0 0 3 36 50/0 0O2 36 4 6 / 0 0 0 3 n - H e x a n e 19 6 3 / 0 0 1 6 20 42/0 003 20 3 6 / 0 0 0 4 M e t h a n o 1 27 12/0 004 28 80/0 003 28 7 5 / 0 0 0 4 t - b u t a n o 1 29 5 3 / 0 002 30 76/0 OO 1 30 7 2 / 0 0 0 2 H.O 22 10/0 0 3 5 23 02/0 007 22 9 8 / 0 0 0 9 H, S 23 73/0 0O3 24 0 7 / 0 003 24 0 4 / 0 0 0 3 n-H. 20 0 3 / 0 0 0 3 20 2 1 /0 0O2 20 2 I/O 0 0 2 p-H, 19 5 1/0 0 0 3 19 52/0 002 19 . 5 1 /O 0 0 2 N e o n 19 98/O 002 20 0 7 / 0 0O1 20. 0 3 / 0 0 0 2 A r g o n i 60/0 006 •. 4 3 ' "0 00-> 1 4 7 6 / 0 0 0 6 T a b l e 4 3 : V i r i a l E x p a n s i o n In P r e s s u r e t o S o l v e f o r V In R e g i o n IV RMS-'/. E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC * LHCHB LHCA LHCCH LHCSP S72 RK M e t h a n e 0. 6 4 / 0 . 004 0 7 8 / 0 . 0 1 2 0. 7 8 / 0 . 0 1 2 0 7 7 / 0 . 0 1 2 0. 7 8 / 0 . 0 07 0. 7 4 / 0 007 0 72 / 0 . 0 0 9 E t h a n e 1 . 6 7 / 0 003 1 5 1 / 0 . 0 0 9 1 . 5 2 / 0 . 0 0 9 1 5 2 / 0 . 0 0 9 1 . 5 2 / 0 0 0 6 1 . 77/ 0 006 1 9 1/0 008 P r o p a n e 1 . 9 4 / 0 004 2 0 3 / 0 . 0 1 2 2 0 3 / 0 . 0 1 1 2 0 3 / 0 . 0 1 2 2 0 3 / 0 007 2 . 0 3 / 0 008 2 23 / 0 0 1 0 n - B u t a n e 7 . 6 4 / 0 001 7 5 6 / 0 . 0 0 5 7 5 6 / 0 . 0 0 5 7 5 6 / 0 . 0 0 6 7 5 6 / 0 0 0 3 7 . 8 1/0 003 8 0 2 / 0 004 n - P e n t a n e 4 . 3 4 / 0 001 4 17/0.004 4 18/0.003 4 17/0.004 4 18/0 002 4 . 5 8 / 0 002 5 15/0 0 0 3 n - H e x a n e 3 . 8 0 / 0 004 3 3 2 / 0 . 0 1 3 3 3 6 / 0 . 0 1 3 3 3 3 / 0 . 0 1 3 3 3 6 / 0 0 0 9 4 . 10/0 009 4 65/ 0 01 1 M e t h a n o 1 4 9 2 / 0 002 5 0 7 / 0 . 0 0 5 5 10/0.006 5 0 9 / 0 . 0 0 6 5 10/0 004 5 . 5 8 / 0 004 6 29/0 004 t - b u t a n o l 3 5 6 / 0 002 - 4 6 4 / 0 . 0 0 7 4 6 2 / 0 . 0 0 7 4 6 1 / 0 0 0 5 3 62 / 0 004 4 34/0 0 0 6 H.O 0 7 5 / 0 01 1 - 0 6 3 / 0 . 0 3 5 0 6 4 / 0 . 0 3 6 0 6 3 / 0 023 0 8 8 / 0 023 1 . 19/0 0 3 0 H. S 1 0 2 / 0 002 1 0 2 / 0 . 0 0 4 1 0 2 / 0 . 0 0 4 1 0 1 / 0 . 0 0 3 1 0 2 / 0 0 0 3 0 9 8 / 0 003 1 .01/0 004 n-H. 1 6 6 / 0 001 1 .64/0.002 1 6 4 / 0 . 0 0 3 1 6 4 / 0 . 0 0 2 1 6 4 / 0 002 2 13/0 002 1 .58/0 0 0 2 p - H . 2 0 7 / 0 0 0 0 * - 1 9 4 / 0 . 0 0 2 1 9 4 / 0 . 0 0 2 1 9 4 / 0 001 2 39 / 0 002 1 .80/0 0 0 2 N e o n 1 2 2 / 0 001 0 .82/0.002 0 8 3 / 0 . 0 0 3 0 8 3 / 0 . 0 0 2 0 8 3 / 0 .002 1 17/0 001 1 .06/0 0 0 2 A r g o n 2 18/0 .003 2 . 1 5 / 0 . 0 0 9 2 . 14/0.009 2 . 1 4 / 0 . 0 0 9 2 . 14/0 .006 2 30 / 0 .007 2 .26/0 .008 $ T h e c o m p u t e r r o u n d s CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d Ch o T a b l e 4 4: V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r V i n R e g i o n V RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHCI LHCI I S72 RK M e t h a n e 20 .34/0 .019 24 .75/0 .028 17 . 5 5 / 0 .014 20 .9 1/0 .018 E t h a n e 12 . 5 7 / 0 .013 1 .99/0 . 0 2 0 1 .8 7/0 .009 1 .82/0 .0 12 P r o p a n e 9 . 10/0 .008 1 . 8 9 / 0 .012 2 .07/0 .OOG 1 .92/0 . 0 0 7 n - B u t a n e 10 . 12/0 .001 0 . 79/0 OOO t 1 .01/0 OO 1 1 3 3 / 0 OO 1 n - P e n t a n e 8 5 7 / 0 0 0 0 t 4 40/0 001 4 39/ 0 0 0 1 4 4 0 / 0 OOO * n - H e x a n e 6 25/ 0 0 0 5 4 37/0 0 0 7 4 0 8 / 0 0O3 3 8 4 / 0 02 1 Me t h a n o 1 - - - -t - b u t a n o 1 2 3 9 / 0 001 3 45/0 0 0 2 3 3 7 / 0 0 0 2 2 9 2 / 0 001 H i O 6 5 7 / 0 0 1 5 6 0 8 / 0 0 1 6 5 3 2 / 0 01 1 4 8 2 / 0 0 1 4 H i S 4 80/0 0 0 2 2 3 0 / 0 . 0 0 2 2 18/0 0O2 2 2 1/0 001 n-H, 304 5 1/0. 0 1 5 273 0 2 / 0 . 0 2 2 225 . 7 7 / 0 . 01 1 111. 8 6 / 0 . 0 1 3 p - H . 250. 4 9 / 0 . 0 1 4 224 . 9 4 / 0 . 02 1 182 . 3 1/0. 0 1 0 9 1 . 7 5 / 0 . 0 1 4 N e o n 13 . 4 6 / 0 . 0 1 2 2 . 9 1/0. 0 1 8 1 . 8 8 / 0 . 0 0 8 1 . 43/O. 01 1 A r g o n 18 . 6 8 / 0 . 0 2 9 2 1 . 7 0 / 0 . 0 4 5 15. 5 9 / 0 . 02 1 18 . 16/0. 0 2 7 4> T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d 1 T a b l e 4 5 : V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r P i n R e g i o n I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND L H C S72 RK Me t h a n e 7 . 39/0 .010 7 . 6 3 / 0 .010 7 .6 t/O .010 E t h a n e 8 1 1/0 00 6 8 18/0 006 8 2 8 / 0 0 0 6 P r o p a n e 8 49/0 007 8 57/0 006 8 7 5 / 0 0 0 7 n - B u t a n e 1 1 66/0 01 1 1 1 8 6 / 0 0 1 1 12 14/0 01 1 n - P e n t a n e 6 82/ 0 O 1 4 7 0 4 / 0 013 13 0 5 / 0 0 IO n - H e x a n e 12 5 t/O 0 10 12 70/0 0 1 0 7 6 4 / 0 0 1 4 M e t h a n o 1 9 0 7 / 0 0 1 4 9 6 8 / 0 0 1 4 10 9 8 / 0 0 14 t - b u t a n o 1 t o 5 4 / 0 0 0 5 IO 6 8 / 0 0O4 1 I 9 t/O OC5 H i O 7 19/0 034 7 5 6 / 0 03 ! a OS/O 034 H.S 7 95 / 0 01 1 8 10/0 0O9 8 2 1/0 0 IO n-H. 7 13/0 008 7 5 3 / 0 o c a 7 2 4 / 0 008 p - H . 6 90/0 007 7 22/C 007 6 9 0 / 0 008 N e o n 5 97/0 008 7 . 0 2 / 0 007 S 9 7 / 0 007 A r g o n 5 54,'C 022 5 7C/C ^2 2 5 G 4 / O 0 2 3 T a b l e 4 6 : V i r i a l E x p a n s i o n I n P r e s s u r e t o S o l v e f o r P In R e g i o n IV RMS-% E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND LHC * S72 RK M e t h a n e 0 . 78/0 .013 0 . 46/0 .035 0 .44/0 .038 E t h a n e 1 . 52/0 .010 1 .14/0 .027 1 . 2 7 / 0 .030 P r o p a n e 1 .53/0 .013 1 . 14/0 .035 1 .32/0 .037 n - B u t a n e 5 .33/0 0O6 3 74/0 015 3 .90/0 .017 n - P e n t a n e 3 .43/0 0 0 3 2 52/0 009 2 9 8 / 0 0 1 0 n - H e x a n e 3 14/0 0 1 5 2 6 2 / 0 04 1 3 0 8 / 0 04 3 Me t h a n o 1 3 4 2 / 0 0O6 3 0 0 / 0 015 3 6 5 / 0 0 1 7 t - b u t a n o l 3 22/0 0 0 7 3 14/0 0 2 0 3 8 1/0 0 2 2 H.O 0 71/0 0 3 6 0 69/ 0 096 0 9 6 / 0 102 H.S 0 80/0 0O4 0 8 1/0 0 1 0 0 8 6 / 0 01 1 n-H. 1 8 0 / 0 . 0 0 2 1 . 26/0 0 0 6 1 0 5 / 0 . 0 0 6 p - H . 2 . 14/0. 0 0 2 1 . 6 6 / 0 . 006 1 . 2 9 / 0 . 0 0 7 N e o n 1 . 2 0 / 0 . 0O2 0. 9 9 / 0 . 005 O. 8 9 / 0 . 0 0 6 A r g o n 1 . 8 0 / 0 . 0 1 0 1 . 2 7 / 0 . 028 1 . 2 2 / 0 . 0 3 0 N o t e : T h e c o r r e c t i o n f a c t o r m e t h o d s f a i l e d f o r a l l c o m p o u n d s . T a b l e 4 7 : V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r P In R e g i o n V RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHCI LHCI I S72 RK M e t h a n e 32 .30/0 .08 1 30 .97/0 .090 138 . 2 7 / 0 .079 125 .08/0 . 0 8 6 E t h a n e 26 77/0 . 07O 21 .78/0 .077 2 1 .94/0 .07 1 22 . 17/0 . 0 7 4 P r o p a n e 28 86/0 0 3 8 24 78/0 043 24 . 5 3 / 0 038 25 . 6 3 / 0 0 4 0 n - B u t a n e 1 12 19/0 0 3 3 160 36/0 035 74 3 8 / 0 0 3 2 74 . 5 9 / 0 0 3 3 n - P e n t a n e 590 67/0 0 2 3 9 2 3 3 7 / 0 024 144 6 2 / 0 0 2 3 7 1 3 4 / 0 0 2 3 n - H e x a n e 34 49/0 04 1 35 38/0 044 34 7 5 / 0 039 33 82/Q 0 4 2 M e t h a n o 1 8 1 0 6 / 0 0 1 4 a 1 0 6 / 0 014 83 2 3 / 0 0 1 5 8 0 . 8 5 / 0 . 0 1 5 t - b u t a n o 1 2 0 9 / 0 0 0 2 3 15/0 003 2 9 6 / 0 0 0 3 2 5 9 / 0 0 0 2 H.O 4 1 59/0 077 4 1 2 3/0 079 4 1 5 8 / 0 075 40 6 4 / 0 0 7 9 H, S 44 54/0 0 1 6 43 0 9 / 0 017 45 5 4 / 0 0 1 6 45 5 3 / 0 0 1 7 n-H. 111. 53/0. 0 6 5 1 37 . 6 5 / 0 . 07 1 3 7 9 9 6 2 / 0 064 2 4 8 0 2 0 / 0 0 6 7 p - H , 108 . 13/0. 064 134 . 4 9 / 0 . 07 1 3724 5 0 / 0 064 2 4 2 9 7 7 / 0 0 6 6 N e o n 23 . 73/0. 0 4 6 2 1 . 9 2 / 0 . 052 22 . 3 7 / 0 . 044 22 4 5 / 0 . 0 4 7 A r g o n 25 . 14/0. 112 23 . 3 1 /0 127 87 . 2 3 / 0 . 108 90 5 1/0. 1 19 T a b l e 4 8 : V i r i a l E x p a n s i o n In P r e s s u r e t o S o l v e f o r T In R e g i o n I I RMS-% E r r o r / C P U t i m e I n ( % ) / s e c COMPOUNO LHC * S72 RK Me t h a n e 2 .65/0 .030 2 .87.0 .020 22 .04/0 .056 E t h a n e 2 .83/0 .017 2 .90/0 .012 16 .72/0 .030 P r o p a n e 2 .79/0 .018 2 .86/0 .012 18 2 1/0 .033 n-Bu t a n e 3 .87/0 .03 1 4 .00/0 .02 1 26 7 2 / 0 .067 n - P e n t a n e 3 9 9 / 0 028 4 12/0 019 25 3 7 / 0 05G n - H e x a n e 2 3 9 / 0 04 2 2 5 J / 0 028 t s 7 1/0 0 7 4 Me t h a n o 1 4 O l / O 0 4 5 4 28/0 029 t 7 12/0 0 7 4 t - b u t a n o l 3 6 4 / 0 013 3 73/0 009 20 8 6 / 0 0 2 3 H,0 2 3 4 / 0 107 2 6 0 / 0 073 13 6 9 / 0 183 H,S 2 7 0 / 0 03 1 2 84/0 02 1 17 0 6 / 0 0 5 4 n-H: 2 9 7 / 0 0 2 6 3 27/0 016 19 . 6 2 / 0 0 4 3 p - H , 2 9 1 / 0 0 2 5 3 15/0 015 19 . 2 0 / 0 . 04 1 N e o n 2. 6 3 / 0 . 02 1 2. 6 4 / 0 . 014 18 . 7 2 / 0 . 0 3 7 A r g o n 2 . 1 8/0. 067 2. 3 0 / 0 . 045 17 . 0 8 / 0 . 120 T a b l e 4 9 : V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r T i n R e g i o n TV RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUNO LHC* S72 RK M e t h a n e 2 .09/0 .035 0 . 27/0 .073 9 . 0 8 / O . 180 E t h a n e 2 .58/0 .027 0 .71/0 .059 8 . 7 9 / 0 . 142 P r o p a n e 2 .31/0 .035 0 6 1/0 076 7 2 8 / 0 182 n - B u t a n e 7 4 8 / 0 0 1 5 1 53/0 03 1 19 2 7 / 0 0 3 7 n - P e n t a n e 4 9 1/0 0 0 9 t 2 1/0 02 0 15 1 1/0 0 4 9 n - H e x a n e 4 15/0 011 1 5 1/0 086 1 2 9 7 / 0 2 15 M e t h a n o 1 3 8 8 / 0 015 1 48/0 03 1 1 1 4 2 / 0 0 7 5 t - b u t a n e 1 3 1 1/0 0 2 0 2 27/0 045 8 6 4 / 0 106 H.O O 9 7 / 0 107 0 48/0 224 3 3 3 / 0 526 H, S 1 0 6 / 0 0 1 0 O 64/0 022 4 7 7 / 0 05 1 n-H, 4 27/0 0 0 5 0. 75/0 01 1 13 4 0 / 0 0 2 9 p - H , 4 4 2 / 0 006 t . 22/0 012 t3 6 I/O. 0 3 0 N e o n t . 8 6 / 0 0 0 5 0. 79/0. 01 1 5 . 6 4 / 0 . 0 2 5 A r g o n 3 . 17/0. 0 3 0 Q 66/0. 059 10. 2 4 / 0 . 1 44 N o t e : The c o r r e c t i o n f a c t o r a i e t n c d s f 3 ' ! e d f o r a 1 1 c o r a o o u n d s T a b l e 50: V i r i a l E x p a n s i o n i n P r e s s u r e t o S o l v e f o r T i n R e g i o n V RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHCI LHCI I S72 RK M e t h a n e 2 .09/0 . 140 1 .09/0 . 190 O . 7 6 / 0 .113 2 .3 1/0 . 2 5 0 E t h a n e 2 . 6 9 / 0 .076 2 17/0 103 2 . 15/0 . 055 3 . 5 2 / 0 13 1 P r o p a n e 1 .57/0 .059 0 8 2 / 0 0 8 0 0 8 0 / 0 .054 2 . 7 4 / 0 0 9 9 n - B u t a n e 1 13/0 01 1 0 52/0 001 0 14/0 001 3 6 8 / 0 0 0 2 n - P e n t a n e 1 14/0 002 0 6 3 / 0 0 0 2 O 18/0 OOI 3 9 5 / 0 0 0 3 n - H e x a n e 1 0 7 / 0 029 1 57/0 0 3 6 1 19/0 019 4 4 7 / 0 0 4 6 M e t h a n o1 - - - -t - b u t a n o l 2 OS/0 0O6 2 16/0 0 0 6 1 9 5 / 0 004 3 6 2 / 0 0 0 8 H.O 4 4 2 / 0 152 4 12/0 178 5 1 1/0 101 1 6 0 / 0 259 H. S 1 2 7 / 0 0 1 2 1 17/0 0 1 6 1 0 5 / 0 009 3 15/0 02 1 n-H. 2 8 5 / 0 . 077 2. 7 4 / 0 . 105 2 . 4 4 / 0 063 3 4 5 / 0 . t 4 4 p - H , 1 . 3 5 / 0 . 0 7 7 1 . 13/0. 105 O. 3 3 / 0 064 2 . 6 4 / 0 . 144 N e o n 2 . 7 4 / 0 . 083 1 . 5 7 / 0 . 1 12 0. 4 8 / 0 . 065 2 . 3 8 / 0 . 15 1 A r g o n 2 . 4 3 / 0 . 20 1 1 . 19/0. 27 1 0. 3 5 / 0 . 164 2 . 4 3 / 0 . 363 T a b l e 5 1 : E n t h a l p y f o r R e g i o n I RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * LHCI 1 I LHCA S72 RK BWR BACK Me t h a n e 5 70 / 0 001 3 35 / 0 001 7 42 / 0 001 3 46 / 0 0 0 0 * 1 1 OO/O 0 0 0 * 1 1 9 5 / 0 0 0 0 1 5 6 / 0 001 E t n a n e 7 0 7 / 0 0 0 1 7 0 7 / 0 002 7 07/ 0 002 5 69/0 001 3 44 / 0 001 6 97/0 0 0 0 * 7 17/0 002 P r o p a n a 7 5G/0 002 7 29/ 0 002 64 30/O 002 6 92/0 0 0 1 2 16/0 001 5 0 2 / 0 0 0 1 2 24/ 0 OO 1 n-But a n o 7 59/ 0 0 0 1 7 59/ 0 00 1 122 04 / 0 00 1 7 9 1/0 00 1 10 65 / 0 0 0 1 19 6 6 / 0 0 0 0 7 4 3 / 0 0 0 0 * n - P e n t a n e 9 79/ 0 0 0 1 9 79/0 001 90 75/0 001 8 9 1/0 0 0 0 * 7 70/0 001 * 3 37/0 0 0 0 * 1 9 8 / 0 001 n - H e x a n e 33 0 3 / 0 001 1 1 0 3 / 0 002 194 29/0 003 10 0 5 / 0 001 9 5 1/0 001 10 54/0 0 0 0 * 4 26/0 0 0 2 Me t n a n o 1 23 5 5 / 0 001 23 5 5 / 0 002 1970 99/0 003 25 53/0 00 1 17 1 1/0 001 - -H.O 90 36/ 0 005 90 36/0 006 625 32/0 006 90 36/0 003 90 39 / 0 003 - -n - H. 237 4 5 / 0 0 0 1 237 4 5 / 0 002 123 0 2 / 0 002 242 4 I/O 001 423 9 6 / 0 0 0 1 - 195 2 6 / 0 002 A r c j o n 3 3 5 / 0 003 3 1 1 /O 004 6 69/0 004 1 4 9 / 0 00 1 I 1 36/0 002 37 . 78/0 0 0 1 0 9 8 / 0 O03 I The c o m p u t e r r o u n d s t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d T a h l e 52: E n t h a l p y f o r R e g i o n II RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC* LHCI I I LHCA S72 RK BWR BACK Me t h a n e 37 8 6 / 0 0 0 0 + 37 8 6 / 0 0 0 0 * 30 63 / 0 . 0 0 1 37 37 / 0 001 35 2 8 / 0 001 29 . 6 6 / 0 0 0 0 * 32 0 3 / 0 0 0 0 * E t n a n e 2 1 7 1/0 001 2 t 7 1/0 002 19 9 7 / 0 002 2 1 0 5 / 0 0 0 0 * 2 1 B 5/0 0 0 0 * 6 . 18/0 001 12 2 1/0 001 P r o p a n e 22 12/0 001 22 75/0 001 4 1 86/0 001 22 34/0 0 0 0 * 26 18/0 0 0 0 * 5 . 4 4 / 0 001 695 18 0 /O 001 n-Bu t a n e 20 0 0 / 0 001 20 0 0 / 0 001 8 40/0 001 19 10/0 0 0 0 * 24 7 1/0 001 4 . 5 4 / 0 0 0 0 * 8 6 6 / 0 001 n - P e n t a n e 20 6 3 / 0 001 20 6 3 / 0 001 1 1 9 9 / 0 001 20 6 8 / 0 0 0 0 * 28 84 / 0 0 0 0 * 1 . 9 5 / 0 000* 10 4 4 / 0 001 n - H e x a n e 38 7 6 / 0 00 1 38 76 / 0 002 4 1 1 1/0 003 38 75/0 001 43 9 5 / 0 001 29 20/0 0 0 0 * 30 8 8 / 0 -002 M e t h a n o l 64 70 / 0 001 64 70 / 0 003 46 76/0 003 64 44/0 001 7 1 15/0 001 - -H. 0 1612 0 4 / 0 005 16 12 0 4 / 0 005 1736 7 8 / 0 006 1735 22/0 002 1736 8 9 / 0 0 0 3 - -n-H. 182 4 0 / 0 001 182 40 / 0 002 190 0 8 / 0 001 168 77/0 001 19 1 9 9 / 0 001 - 172 7 2 / 0 001 A r g o n 15 6 8 / 0 003 15 7 8 / 0 003 3 5 0 / 0 . 0 0 4 15 5 9 / 0 002 10 4 1 / 0 . 0 0 2 27 73 / 0 . 0 0 1 3 2 1/0 0 0 3 ro 1 T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . T a b l e 5 3 : E n t h a l p y f o r R e g i o n I I I RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c C O M P O U N D L H C * L H C I I I L H C A S72 RK BWR B A C K Me t h a n e 1 93/0 007 1 9 2 / 0 008 a 0 5 / 0 . 0 0 9 1 9 5 / 0 004 17 0 3 / 0 0 0 3 29 4 3 / 0 003 0 73/ 0 0 0 6 P r o p a n e 1 3 76/0 005 13 6 4 / 0 0 0 6 126 3 2 / 0 . 0 0 7 12 9 4 / 0 003 6 2 4 / 0 0 0 3 15 3 1/0 002 8 12/0 005 n - B u t a n e 26 33/0 024 26 3 3 / 0 0 2 9 205 4 3 / 0 . 0 1 9 12 10/0 008 1 1 6 7 / 0 008 29 8 1/0 006 1 1 2 2 / 0 0 1 3 Mo t h a n o 1 1 195 4 1/0 002 1 195 4 1/0 002 1 185 3 6 / 0 . 0 0 2 1 195 38/ 0 001 1 195 5 5 / 0 001 - ' -H i O 26 66/0 01 1 26 6 6 / 0 012 1 19 15/0.002 26 4 0 / 0 001 1 3 9 / 0 001 - -n - H . 284 0 1 / 0 009 284 0 1 / 0 01 1 38 2 9 / 0 . 0 1 2 3 12 13/0 0 0 6 596 5 6 / 0 006 - 2 10 5 4 / 0 0 0 9 A r g o n 1647 28/0 0 0 5 1647 2 8 / 0 006 1647 2 9 / 0 . 0 0 7 1647 2 8 / 0 0 0 3 1647 2 7 / 0 003 1647 19/0 002 1697 9 8 / 0 0 0 5 T a b l e 54: E n t h a l p y f o r R e g i o n I V RMS-'/. E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND L H C » LHC11 I LHCA S72 RK BWR BACK M e t h a n e 24 64/0 004 23 9 4 / 0 004 4 1 5 9 / 0 . 0 0 5 22 54/0 002 21 7 1 / 0 002 19 5 0 / 0 001 2 1 0 2 / 0 004 £ t h a n e 14 48/0 004 14 4 8 / 0 005 10 9 7 / 0 006 15 4 5 / 0 002 18 7 8 / 0 002 6 4 8 / 0 002 12 7 6 / 0 004 P r o p a n e 36 45/0 003 36 4 5 / 0 005 46 9 8 / 0 005 36 3 4 / 0 002 39 2 3 / 0 002 3 1 0 1/0 002 35 0 3 / 0 004 n-Bu t a n e 1 2 59/0 003 12 5 9 / 0 003 23 0 0 / 0 004 12 4 6 / 0 002 20 1 8 / 0 002 3 18/0 001 7 68/ 0 002 n - P e n t a n e 15 45/0 002 15 45/0 002 12 74/0 002 15 6 1/0 001 25 3 7 / 0 001 1 6 3 / 0 001 10 9 7 / 0 001 n - H e x a n e 1054 8 1/0 006 1054 8 1/0 007 3259 17/0 009 107 1 0 6 / 0 004 93 1 3 4 / 0 004 1376 15/0 003 1209 9 8 / 0 006 Me t hano1 34 8 27/0 002 348 27/0 002 335 18/0 003 347 9 4 / 0 001 345 5 6 / 0 001 - -H.O 249 93/0 016 249 9 3 / 0 0 2 0 3.83E8 /O 023 172 99/0 009 154 7 3 / 0 008 - -n-H, 1 1 7 07/ 0 004 1 17 .07/0 .006 1 18 .88/0 007 1 17 .09/0 .003 12 1 5 4 / 0 003' - 1 18 1 1/0 004 A r g o n 28 .2 1/0 0 1 0 29 . 6 0 / 0 .012 26 .06/0 014 28 . 57/0 006 25 7 7 / 0 006 26 . 33/0 005 23 6 7 / 0 0 1 0 M Ch Ch T a b l e 5 5 : E n t h a l p y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI LHC 1 I L H C I I I S72 RK BWR BACK My t h a n e 26 19 . 7-1/0 0 9 5 2 7 7 2 . 0 5 / 0 122 277 1 . 83/0 107 2 8 9 4 . 2 0 / 0 0 5 3 2 8 3 5 .77/0 0 5 1 274 1 .41/0 038 3 0 8 0 . 1 5 / 0 087 E t h a n e 1 .23E 16 /0 09 1 6.4 IE 15 /O 120 6.4 1E 15 /O 102 1 . 17E 16 /O 0 5 0 1 .23E 16 /O 09 1 2 . 56E 15 /O 036 1 .86E 16 10 0 8 3 P r o p a n o 7.58E 15 /0 109 2 . 59E 16 /O 146 2 . 59E 16 /O 125 4.OOE16 /O 06 1 2.41E 16 /O 058 1 . 27E 16 /O 045 2,04E 16 /O 102 n Bu t a n o 604 .83/0 0 4 6 598 . 8 6 / 0 04 7 598 . 6 7 / 0 052 4 4 5 . 0 0 / 0 0 2 5 4 7 8 . 2 0 / 0 024 122 . 4 6 / 0 017 2 4 1 . 9 4 / 0 0 3 8 n - P e n t a n e 9 3 . 9 3 / 0 003 3 0 . 1 0 / 0 003 1 5 . 6 5/0 003 7 . 7 7 / 0 002 15 . 6 8 / 0 001 2 . 7 8 / 0 0 0 1 8 . 3 7 / 0 002 n - H e x a n e 72 . 5 8 / 0 0 1 0 5 4 . 4 0 / 0 014 56 . 6 7/0 01 1 6 0 . 4 6 / 0 0 0 5 5 4 . 5 0 / 0 0 0 6 5 8 . 8 0 / 0 004 5 6 . 5 8 / 0 0 0 9 Me t hano1 19864.3 /0 005 19864.3 /0 005 19864 . 3 /O 0 0 5 19864.3 /O 0 0 3 19864 . 5 /O 0 0 3 - -H,0 5 3 0 . 4 9 / 0 036 762 .74/0 045 762 . 74/0 043 937 . 19/0 0 2 0 7 7 0 . 9 6 / 0 0 19 - -n - H , 3 3 8 8 5 0 . 0 / 0 108 3 3 8 7 6 2 . 0 / 0 24 3 338834 .0/0 180 3 3 8 4 4 2 . 0 / 0 106 338 193.0/0 104 - 3 7 4 3 2 6 . 0 / 0 175 A r g o n 2 7 5 3 . 6 7 / 0 1 16 84 1 . 42/0 190 84 1 43/0 137 1229 . 53/0 065 1400.18/0 105 6 2 4 . 3 2 / 0 0 4 6 9 3 5 . 9 1/0 106 l o o l e 56: E n t r o p y f o r R e g i o n I RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC ' L H C I I I LHCA S72 RK BWR BACK Mst n a n e 3 3G/0 002 2 95 / 0 003 3 16/0 003 1 22/0 002 5 34 / 0 002 7 73 / 0 001 70 .94/0 0 0 3 E t h a n e 5 0 3 / 0 002 5 0 3 / 0 002 5 0 3 / 0 002 4 G6/0 00 1 4 0 8 / 0 0 0 1 5 9 1/0 001 64 . 5 4 / 0 002 P r o p a n e 5 4 3 / 0 002 5 43 / 0 002 56 25/0 003 5 10/0 002 1 6 0 / 0 002 4 48/0 0 0 1 66 . 6 4 / 0 003 n - B u t a n e 5 5 5 / 0 00 1 5 55/0 0 0 0 } 109 9 2 / 0 00 1 5 44/0 00 1 5 77 / 0 0 0 1 13 7 7 / 0 001 67 . 18/0.001 n- P e n t a n e 8 2 1/0 00 1 8 2 1/0 001 83 0 2 / 0 001 7 68 / 0 00 1 3 1 1/0 001 3 13/0 0 0 1 62 . 8 6 / 0 001 n - H e x a n e 19 9 8 / 0 003 1 1 2 1/0 003 208 72/0 003 10 0 9 / 0 002 3 53 / 0 002 1 1 23/0 002 55 . 15/0 0 0 3 t nano1 7 44/0 002 7 4 4/0 002 6B0 78/0 002 1 1 16/0 002 20 17/0 0 0 1 - -M , 0 4 1 3 3/0 006 4 1 33/0 007 290 4 1/0 009 4 1 3 3 / 0 005 4 1 3 5/0 004 - -n-H, 706 10/0 002 706 10/0 002 534 0 5 / 0 002 726 28/0 0 0 1 888 8 0 / 0 0 0 1 - 2 10 .04/0 0 0 2 A r g o n 2 4 0 3 1/0 004 238 . 36/0 004 226 . 93/0 005 238 44/0 004 256 4 0 / 0 003 309 .03/0 002 13 .09/0 0 0 5 t Tne c o m p u t e r r o u n d s t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d rO Ch T a b l e 5 7 : E n t r o p y f o r R e g i o n II RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * L H C I I I LHCA S72 RK BWR BACK Me t h a n e 4 17 4 7 / 0 001 4 17 4 7 / 0 001 437 25/0 001 4 16 93/0 0 0 0 * 4 2 0 0 1 / 0 0 0 0 * 430 9 4 / 0 0 0 1 60 9 4 / 0 001 E t h a n e 37 1 6 2 / 0 001 37 1 6 2 / 0 001 373 7 4 / 0 001 370 80/0 001 368 12/0 001 383 6 2 / 0 0 0 0 * 56 9 7 / 0 001 Propane 352 6 8 / 0 001 35 1 7 5 / 0 001 330 10/0 001 352 4 3 / 0 001 347 3 7 / 0 001 370 3 3 / 0 001 7985 5 6 / 0 001 n - B u t a n e 6 5 0 / 0 00 1 6 5 0 / 0 0 0 0 * 4 0 0 / 0 001 5 50/0 0 0 1 6 9 7 / 0 001 1 4 6 / 0 0 0 1 87 6 3 / 0 001 n - P e n t a n e 10 4 0 / 0 0 0 1 10 4 0 / 0 00 1 5 19/0 001 10 79/0 0 0 1 13 0 5 / 0 OO 1 0 8 2 / 0 0 0 1 86 2 2 / 0 001 n - H e x a n e 25 3 1/0 003 25 3 1/0 003 9 0 4 / 0 003 25 3 1/0 002 27 4 8 / 0 0 0 2 15 0 2 / 0 002 92 4 5 / 0 003 M e t h a n o l 52 1 9 3 / 0 003 52 1 9 3 / 0 003 5 17 84/ 0 003 52 1 9 0 / 0 002 522 3 6 / 0 002 - -H i 0 23 10/0 006 23 1 1/0 008 24 0 1 / 0 . 0 0 8 24 0 1 / 0 0 0 6 24 0 1 / 0 004 - -n - H i 167 2 5 / 0 002 167 2 5 / 0 002 169 7 4 / 0 003 163 9 7 / 0 001 170 7 7 / 0 001 - 105 6 4 / 0 002 A r g o n 1307 .60/0 004 1304 .23/0 .005 1 3 1 6 . 0 7 / 0 . 0 0 5 1272 .02/0 .003 1278 .53/0 .003 1335 . 19/0 .002 188 .59/0 .005 * T h e c o m p u t e r r o u n d s t h e CPU t l m B t o t h e n e a r e s t m i l l i s e c o n d . T a b l e 58: E n t r o p y f o r R e g i o n I I I RMS-*/. E r r o r / C P U t i m e 1n ( % ) / s e c C O M P O U N D L H C * L H C I I I LHCA S72 RK BWR BACK Met h a n e 3 5 5 / 0 008 3 5 5 / 0 0 1 0 6 7 1/0.011 2 0 6 / 0 007 12 6 3 / 0 007 26 6 4 / 0 004 64 3 1/0 01 1 P r o p a n e 6 5 4 / 0 007 6 5 5 / 0 008 93 7 4 / 0 . 0 1 0 5 3 0 / 0 0 0 6 3 2 5 / 0 005 8 3 9 / 0 004 6 1 2 1/0 009 n-Bu t a n e 24 9 9 / 0 029 24 9 9 / 0 034 188 0 6 / 0 . 0 2 4 5 6 7 / 0 014 3 2 9 / 0 0 1 3 22 3 8 / 0 008 56 4 2 / 0 023 M e t h a n o l 103 3 7 / 0 002 103 3 7 / 0 002 99 7 4 / 0 . 0 0 3 103 36/0 0 0 2 103 4 1/0 002 - -H i O 24 4 1/0 002 24 4 1/0 0 0 3 54 5 8 / 0 . 0 0 2 30 3 0 / 0 001 15 8 7 / 0 001 - -n-H i 589 6 8 / 0 0 1 2 589 6 8 / 0 014 378 3 7 / 0 . 0 1 6 6 15 0 3 / 0 0 0 9 82 1 9 8 / 0 008 - 168 6 0 / 0 01S A r g o n 140 4 1/0 007 148 4 2 / 0 008 148 4 5 / 0 . 0 0 9 148 4 2 / 0 0 0 6 148 3 8 / 0 005 148 12/0 004 198 4 3 / 0 0 0 9 to cr. co T a b l o 5 9 : E n t r o p y f o r R e g i o n IV RMS-'/. E r r o r / C P U t i m e I n (*/.)/sec COMPOUND LHC * L H C I I I LHCA S72 RK BWR BACK Me t hane 825 5 8 / 0 005 8 17 18/0 005 804 10/0.006 8 16 68/0 004 8 17 78 / 0 004 824 15/0 003 54 6 6 / 0 006 E t hane 762 5 4 / 0 007 762 5 4 / 0 008 777 7 7 / 0 008 759 2 1 / 0 005 754 7 2 / 0 0 0 5 766 4 0 / 0 003 56 5 0 / 0 008 Propane 186 1 0 6 / 0 006 186 1 0 6 / 0 006 1874 7 2 / 0 008 1851 9 8 / 0 005 1839 4 5 / 0 004 1864 7 3 / 0 002 64 23/0 007 n-Butane 10 29/0 003 10 2 9 / 0 003 1 1 0 0 / 0 004 10 43/0 002 14 0 2 / 0 002 2 35/ 0 002 85 8 5 / 0 004 n -Pentane 13 5 3 / 0 002 13 5 3 / 0 002 8 7 9 / 0 003 13 5 9 / 0 002 19 0 1 / 0 002 0 92/0 001 83 17/0 0 0 3 n-Hexane 48 6 7 / 0 008 48 6 7 / 0 0 1 0 2 14 90 / 0 0 1 0 49 4 1/0 006 4 1 9 7 / 0 0 0 6 79 8 4 / 0 004 93 5 7 / 0 01 1 Me thano1 105 9 6 / 0 004 105 9 6 / 0 004 108 0 5 / 0 005 106 0 8 / 0 003 106 5 1 / 0 003 - -hi i 0 66 45 / 0 023 66 4 5 / 0 026 5.42E7 /0 030 8 1 10/0 015 5 1 . 3 2 / 0 014 - -n-H, 1 17 . 7 3 / 0 006 1 17 .73/0 007 1 18 .49/0 008 1 17 . 80/0 005 1 19 .44/0 .005 - 101 4 5 / 0 008 A rgon 18 1 . 4 8 / 0 014 176 .06/0 .017 182 .87/0 019 178 . 5 8 / 0 01 1 179 .03/0 .010 183 .09/0 .006 103 . 37/0 .018 T a b l e 60: E n t r o p y for Reg ion V RMS-% E r r o r / C P U time In (%)/sec COMPOUND LHCI LHCI I LHCII I S72 RK BWR BACK Me t hane 18644 49/0 129 18658 64/ 0 158 18648 52/0 142 18658 51/ 0 09 1 18655 0 8 / 0 086 18670 5 4 / 0 055 9 1 9 8 / 0 152 E thane 9 175 99/0 123 9 190 67/0 154 9 189 34/0 136 9206 33/0 086 9175 9 9 / 0 123 9204 77/ 0 05 1 76 6 1/0 145 Propane 6 130 29/0 149 6 127 85/0 186 6 127 5 1/0 165 6145 9 9 / 0 103 6 145 99 / 0 103 6 14 1 8 1/0 062 283 9 1/0 167 n-Butane 22 96/0 055 19 9 6 / 0 056 19 8 8 / 0 063 8 0 3 / 0 04 1 10 5 0 / 0 037 26 8 6 / 0 023 55 4 1/0 065 n- f e n tane 4 7 9 1/0 003 14 75/0 004 7 69/ 0 004 4 5 1/0 002 7 9 1/0 002 0 94/0 002 77 8 8 / 0 004 n-Hexane 2 9 14/0 0 14 14 49/0 017 15 6 4 / 0 015 14 34/0 0 1 0 15 9 4 / 0 009 13 9 0 / 0 006 90 2 6 / 0 016 Methanol 102 44/0 006 102 44/0 006 102 4 4 / 0 007 102 43/0 004 102 4 8 / 0 004 - -h,0 1 2 96/0 044 10 9 1/0 055 10 9 1/0 05 1 10 99/0 032 8 6 5 / 0 029 - -n-H, 1 358 1 6 /0 259 13590 8 /0 3 14 1 3540 3 /0 250 14 114 8 /0 182 17428 5 /0 172 - 4044 6 6 / 0 304 Argon 475 1 0 2 / 0 1 58 48 11 42/0 190 48 1 1 0 8 / 0 180 4902 49/ 0 1 10 4857 3 0 / 0 105 485 1 23/0 066 282 0 0 / 0 185 Ch 'I at) l a G l : He 1 mho 1 12 F r e e E n e r g y f o r R e g i o n I RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC • LHCI I I LHCA S72 RK BWR BACK Me i h a n e 9 •13/0 001 9 43/0 0 0 1 9 43/0 001 10 0 1 / 0 001 16 9 2 / 0 001 15 29/0 001 227 1 7 / 0 001 t t h a n e 4 5 1/0 001 4 5 1/0 001 4 5 1/0 001 4 33/0 001 5 30/0 0 0 1 7 8 0 / 0 001 236 6 1 / 0 00 1 P r o p a n e 1 92/0 001 1 9 1/0 002 1 74/0 001 2 25/0 001 10 5 7 / 0 001 1 59/0 00 1 234 9 4 / 0 00 1 iv- l i u t a n e \ 57/0 0 0 0 t 1 57 / 0 001 1 5 9 / 0 0 0 0 * 2 22 / 0 0 0 0 * 13 7 2 / 0 0 0 0 * 17 3 0 / 0 0 0 0 * 277 10/0 0 0 0 * n- P e n t a n e 3 28/0 000 * 3 28/0 001 3 25/0 0 0 0 * 3 70/0 0 0 0 * 25 6 7 / 0 0 0 0 * 1 50/0 0 0 0 * 27 1 4 2 / 0 0 0 0 * n - He x a n e 35 42/0 002 35 83/0 002 35 8 8 / 0 002 38 5 1/0 002 2 13 6 0 / 0 002 32 8 7 / 0 0 0 1 1 127 4 2 / 0 002 Me t h a n o 1 24 2 2 / 0 001 24 22/0 001 24 24/0 001 22 8 9 / 0 001 22 01/0 001 - -M .- 0 G337 8 6 / 0 005 6337 8 6 / 0 006 6337 9 5 / 0 005 6337 55/0 004 6339 7 5 / 0 0 0 5 - -n - i l i 256 55/0 00 1 256 55/0 00 1 256 5 6 / 0 00 1 257 37/0 00 1 244 18/0 0 0 1 - 59 4 3 / 0 001 A rgoi'i 2a-t 45/0 003 284 46/0 003 284 45/0 003 285 1 3/0 002 283 17/0 0 0 3 274 66/0 002 1 4 2 4 / 0 003 1 T h e c o m p u t e r r o u n d s t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d T a b l e 62: H e l m h o l t z F r e e E n e r g y f o r R e g i o n I I RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * L H C I I I LHCA S72 RK BWR BACK Me t n a n e 2 4 8 4 7 8 /0 001 24847 8 /0 001 24845 9 /0 001 24854 1 / o 001 24827 0 /O 001 2473 1 7 /O 007 1909 9 0 / 0 001 E t n a n e 2 . 4 8 E 5 /0 001 2.48E5 /0 001 2.48E5 /0 001 2 . 48E5 /O 001 2 . 49E5 /O 00 1 2 . 4 6 E 5 /O 001 15773 0 /O 001 P r o p a n e 2 119 1 0 /O 001 2 119 1 0 /O 0 0 0 * 2 119 1 7 /O 001 2 1 190 1 / o 001 2 1252 1 /O 00 1 2 1007 4 /O 0 0 0 * 15662 5 /O 001 n-Bu t a n e 1 4 0 / 0 0 0 0 + 1 4 0 / 0 001 1 4 0 / 0 0 0 0 * 1 44/0 0 0 0 * 1 8 8 / 0 0 0 0 * 0 26/0 0 0 0 * 1 1 4 3 4 / 0 0 0 0 * n - P e n t a n e 12 9 2 / 0 OOO* 12 92/0 0 0 0 * 12 9 4 / 0 0 0 0 * 13 0 4 / 0 0 0 0 * 17 3 1/0 0 0 0 * 0 1 1/0 0 0 0 * 132 8 4 / 0 0 0 0 * n - H e x a n e 48 8 3 / 0 002 48 8 3 / 0 002 48 8 3 / 0 002 48 8 6 / 0 002 47 6 3 / 0 002 5 1 84 / 0 0 0 1 106 6 9 / 0 002 Me t h a n o l 464 0 2 / 0 002 464 0 2 / 0 002 464 0 2 / 0 . 0 0 2 464 0 3 / 0 002 463 7 5 / 0 0 0 2 - -H,0 1000 0 9 / 0 005 1000 0 9 / 0 006 1077 8 8 / 0 004 1077 8 8 / 0 003 1077 8 7 / 0 004 - -n - H< 183 8 5 / 0 001 183 85/0 001 183 8 5 / 0 001 184 15/0 001 183 18/0 001 - 86 7 4 / 0 .001 A r c j o n 2 14 9 5 / 0 003 2 14 96/0 003 2 14 95/0 003 2 15 0 6 / 0 002 2 14 9 9 / 0 0 0 3 2 13 7 1/0 002 89 8 7 / 0 .003 4 T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . T a b l e 6 3 : H e l m h o l t z F r e e E n e r g y f o r R e g i o n I I I RMS-'/. E r r o r / C P U t i m e i n (*/.)/sec COMPOUND LHC* L H C I I I LHCA S72 RK BWR BACK Me t h a n e 48 6 7 / 0 007 48 6 7 / 0 008 136 6 8 / 0 . 0 0 7 7 6 . 4 2 / 0 . 0 0 5 5 6 3 . 0 7 / 0 0 0 6 672 6 3 / 0 004 5265 6 5 / 0 007 P r o p a n e 2 1 15/0 005 2 1 15/0 006 69 0 5 / 0 . 0 0 5 20.7 9/0.004 1 1 .49/0 0 0 5 24 3 4 / 0 003 263 7 0 / 0 0 0 5 n - B u t a n e 27 7 0 / 0 022 27 7 0 / 0 027 25 10 0 0 / 0 . 0 1 1 5 3 . 0 7 / 0 . 0 1 1 1 6 0 . 3 7 / 0 013 239 2 5 / 0 009 1228 0 2 / 0 0 1 3 Me t h a n o 1 107 6 7/0 001 107 6 7 / 0 002 109 3 1/0.001 107.67/0.001 107.6 1/0 001 - -HiO 32 4 2 / 0 002 32 4 2 / 0 002 74 7 5 / 0 . 0 0 1 3 5 . 9 9 / 0 . 0 0 1 4 5 . 5 0 / 0 001 - -n - H . 154 15/0 009 154 15/0 01 1 188 10/0.010 152 .01/0.008 1 3 0 . 5 6 / 0 0 0 9 - 83 6 2 / 0 009 A r g o n 96 9 1/0 005 96 9 1 / 0 006 96 .91/0.006 96 .91/0.004 96 . 9 1/0 0 0 5 96 9 0 / 0 003 99 .98/0 0 0 5 T a b l e 64: H e l m h o l t z F r e e E n e r g y f o r R e g i o n IV RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC" L H C I I I LHCA S72 RK BWR BACK Me t h a n e 1 .79E6 /0 003 1 . 79E6 /O 004 1 .79E6 /O 004 1.79E6 /O 003 1 . 79E6 /O 003 1.79E6 /O 002 6 1655 9 /O 003 E t h a n e 1 .95E5 /O 004 1.95E5 /O 005 1.95E5 /O 0 0 5 1 .95E5 /O 004 1 .95E5 /O 004 1 .95E5 /O 0 0 3 9 137 2 9 / 0 004 P r o p a n e 2 . 2 2 E 5 /O 004 2 . 2 2 E 5 /O 005 2.22E5 /O 004 2.22E5 /O 003 2.22E5 /O 004 2 . 22E5 /O 0 0 3 10355 7 /O 004 n-Bu t a n e 382 4 3 / 0 002 382 4 3 / 0 003 2 7 5 . 4 7 / 0 002 401 40/0 002 589 4 3 / 0 002 85 .77/0 001 9 104 4 4 / 0 002 n - P e n t a n e 576 3 4 / 0 001 576 3 4 / 0 002 2 7 2 . 8 5 / 0 001 606 0 0 / 0 001 901 12/0 001 6 1.11/0 0 0 1 8042 3 0 / 0 001 n- He xa ne 4 1 16/0 006 4 1 16/0 007 3 1 . 54/0 007 4 1 37/0 005 42 4 1/0 006 4 0 . 2 4 / 0 004 104 2 1/0 006 Me t hano1 244 6 7 / 0 002 244 6 7 / 0 003 3 4 4 . 5 0 / 0 002 244 67/0 002 244 6 4 / 0 002 - -H.O 247 6 8 / 0 017 247 6 8 / 0 02 1 4 . 49E6 /O 018 268 22/0 012 256 5 9 / 0 014 - -n-H. 129 4 3 / 0 005 1 29 43/ 0 006 129.44/0 005 129 53/0 004 129 2 1/0 004 - 95 8 4 / 0 005 A r g o n 13 1 16/0 01 1 13 1 17/0 012 13 1. 13/0 01 1 131 . 18/0 009 13 1 16/0 0 1 0 130.89/0 007 98 25/0 0 1 0 I ' a D l e 6 5 : H e l m h o l t z F r e e E n e r g y f o r R e g i o n V RMS-°/ E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC I LHCI I L H C I I I S72 RK BWR BACK Met h a n e 1.38E5 /0 09 0 1 . 4 IE5 /0 1 19 1 . 4 1E5 /O 101 1 .4 1E5 /O 074 1 4 1E5 /O 084 1 . 4 IE5 /O 059 3 1 136 0 1 / 0 0 9 0 E i h a n e 3 . 7 4 E 5 /0 087 3.80E5 /0 1 17 3.78E5 /O 095 3 . 8 IE 5 /O 07 1 3 74E5 /O 087 3.79E5 /O 056 5 1249 7 /O 086 P r o p a n e 5 . 3 8 E 5 /0 105 5.55E5 /0 1 40 5.55E5 /O 1 16 5.58E5 /O 086 5 54E5 /O 097 5.56E5 /O 067 37448 7 8 / 0 108 n - Bu t a n e 46 1 2 6 / 0 038 128 4 1 /O 04 1 128 . 27/0 044 114. 5 9 / 0 033 91 0 2 / 0 036 107.26/0 025 102 4 29/ 0 038 n - P e n t a n e 16074 9 /0 002 1148 9 6 / 0 003 1148. 12/0 002 749 . BO/0 002 3532 9 2 / 0 002 129 . 5 3 / 0 001 265 15 5 /O 002 n - H e x a n e 28 0 2 / 0 0 1 0 4 6 79/0 013 46 . 83/0 01 1 45 . 3 8 / 0 008 4 1 2 3 / 0 009 4 6 . 0 0 / 0 006 173 5 5 / 0 0 1 0 Me t nano1 99 9 2 / 0 004 99 9 2 / 0 004 99 . 9 2/0 005 99 . 9 2 / 0 004 99 9 2 / 0 004 - -H,0 10 35/0 03 1 10 3 8 / 0 04 1 10. 38/0 037 10. 5 7 / 0 026 1 1 3 6 / 0 029 - -n-H i 4 5 8 3 6 2 / 0 182 46 19 84 / 0 238 4598 . 95/ 0 160 4 6 3 0 . 4 1/0 152 51 19 9 3 / 0 17 1 - 4 167 .49/0 182 A r g o n 1 1639 1 1/0 1 1 1 12 125 9 /O 143 12 125. 9 /O 130 12355. 9 /O 092 12273 6 /O 104 12267.5 /O 072 323 3 5 / 0 1 1 1 T a b l e 6 6 : G i b b s F r e e E n e r g y f o r R e g i o n I RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC* L H C I I I LHCA S72 RK BWR BACK M e t h a n e 27 1 1/0 001 27 . 10/0.001 26 8 0 / 0 . 0 0 2 469 0 4 / 0 . 0 0 1 77 7 1 / 0 002 72 4 5 / 0 001 932 5 1 / 0 002 E t h a n e 29 2 6 / 0 002 29 . 2 6 / 0 . 0 0 2 29 2 6 / 0 . 0 0 1 637 17/0.001 15 9 8 / 0 001 64 2 7 / 0 001 1 183 7 1/0 001 P r o p a n e 12 3 7 / 0 002 12 . 3 7 / 0 . 0 0 2 1 1 2 0 / 0 . 0 0 2 858 .22/0.001 86 7 7 / 0 001 14 5 0 / 0 001 1466 3 3 / 0 001 n - B u t a n e 5 17/0 001 5 . 1 7 / 0 . 0 0 0 * 5 34/0.001 437 . 6 8 / 0 . 0 0 0 * 63 0 6 / 0 0 0 0 * 83 6 5 / 0 001 835 9 4 / 0 0 0 0 * n - P e n t a n e 126 5 7 / 0 001 126 . 5 7 / 0 . 0 0 0 * 124 1 7 / 0 . 0 0 0 * 6 0 8 0 .8 2 / 0 . 0 0 0 * 12 16 8 5 / 0 0 0 0 * 83 8 5 / 0 001 10498 6 /O 001 n - H e x a n e 64 45/0 002 65 . 2 0 / 0 . 0 0 3 64 2 0 / 0 . 0 0 2 974 . 7 9 / 0.002 229 9 1/0 002 65 9 9 / 0 002 1653 16/0 002 M e t h a n o l 28 8 7 / 0 001 2 8 . 8 7 / 0 . 0 0 2 28 9 0 / 0 . 0 0 2 36 .48/0.001 26 6 2 / 0 001 - -H i O 1224 1 9 /O 005 1224 1 9 / 0 . 0 0 5 1224 1 9 / 0 . 0 0 5 12559 .0 /0.004 12245 4 /O 005 - -n - H i 148 0 7 / 0 002 148 0 7 / 0 . 0 0 2 148 .08/0.001 180 .57/0.001 140 0 8 / 0 001 - 38 3 8 / 0 001 A r g o n 182 0 7 / 0 003 182 0 8 / 0 . 0 0 4 1 8 2 . 0 7 / 0 . 0 0 3 208 . 4 5 / 0 . 0 0 3 180 7 6 / 0 .003 174 . 24/0 002 9 2 6 / 0 0 0 3 * T h e c o m p u t e r r o u n d s t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d fO T i i D l e 6 7 . G t b b s F r e e E n e r g y f o r R e g i o n 11 RMS-'/. E r r o r / C P U t i m e In C / . ) / s e c COMPOUND LHC* LHCII 1 LHCA S72 RK BWR BACK Me t hane 1 140 59/ 0 001 1 140 5 9 / 0 001 1 1 4 0 . 5 4 / 0 . 0 0 0 * 1 163 27/ 0 001 1 140 4 0 / 0 001 1 137 32/0 OOO* 150 7 4 / 0 001 E t nane 1 140 0 5 / 0 001 1 140 0 5 / 0 001 1 139 9 3 / 0 001 116 1 76/0 001 114 1 7 1/0 00 1 1 133 75/0 001 150 8 1/0 001 Propane 1 107 28/ 0 001 1 107 3 6 / 0 001 1 107 30/0 001 1 132 67/ 0 0 0 0 * 1 1 10 4 1/0 001 1098 6 4 / 0 001 3 7 3 0 4 2 / 0 001 n - B u t ane 1 6 9 / 0 00 1 1 6 9 / 0 0 0 0 * 1 6 9 / 0 001 5 8 6 / 0 0 0 0 * 2 2 6 / 0 0 0 0 * 0 3 1/0 00 1 132 0 1 / 0 0 0 0 * n-Pen tane 19 6 7/0 00 1 19 6 9 / 0 001 19 7 1/0 0 0 0 * 55 9 0 / 0 0 0 0 * 26 3 3 / 0 0 0 0 * 0 12/0 001 17 1 2 4 / 0 001 n-Hexane 37 0 0 / 0 002 37 0 0 / 0 002 37 0 0 / 0 003 35 18/0 002 36 5 7 / 0 002 38 2 4 / 0 002 1 18 7 7 / 0 002 Me thano1 488 0 9 / 0 002 488 0 9 / 0 002 488 0 9 / 0 002 489 23/0 002 487 7 9 / 0 002 - -M, 0 22 0 9 / 0 005 22 0 9 / 0 006 23 17/0 004 23 17/0 004 23 17/0 004 - -n-H. 148 . 3 9 / 0 002 148 . 39/0 002 148 39/0 001 150 6 9 / 0 001 147 7 2 / 0 001 - 79 3 4 / 0 001 A r g o n 182 . 17/0 003 182 . 18/0 . 003 182 . 17/0 003 183 77/0 0 0 3 182 . 18/0 003 18 1 . 13/0 .002 84 . 3 7 / 0 .003 4 The computer rounds the CPU time to the n e a r e s t m i l l i s e c o n d . l a c l e 6 8 : G lbbs F ree Energy f o r Reg ion III RMS-% E r r o r / C P U time in (%) / sec COMPOUND LHC * LHCII I LHCA S72 RK BWR BACK Me thane 92 3 1/0 007 92 3 1/0 008 130 83/0 007 1356 9 6 / 0 006 262 0 7 / 0 007 257 2 7 / 0 005 2698 5 6 / 0 0O7 Propane 48 5 1/0 005 48 5 1/0 006 168 6 9 / 0 005 237 83/0 004 23 7 1/0 005 57 10/0 004 589 7 5 / 0 0 0 5 n-Bu tane 25 9 9 / 0 023 25 9 9 / 0 027 1070 64/0 015 79 1 97/0 01 1 134 4 6 / 0 0 1 3 177 0 2 / 0 009 1298 0 9 / 0 014 Me thano1 2 18 22/ 0 002 2 18 22/ 0 002 2 18 0 4 / 0 00 1 2 18 23/0 001 2 18 23/0 001 - -H.O 42 19/0 002 42 19/0 003 95 9 4 / 0 001 74 32/0 00 1 59 10/0 001 - -n-H. 97 14/0 009 97 14/0 01 1 12 1 76/0 0 1 0 160 42/0 008 80 7 9 / 0 0 0 9 - 57 0 5 / 0 0 1 0 A rgon 2874 9 8 / 0 006 2874 9 9 / 0 006 2874 9 9 / 0 006 2875 0 1 / 0 005 2874 9 8 / 0 0 0 5 2874 9 6 / 0 004 2905 14/0 0 0 6 M 0 0 T a b l e 6 9 : G i b b s F r e e E n e r g y f o r R e g i o n IV RMS-'/. E r r o r / C P U t i m e i n ('/.)/sec COMPOUND LHC • L H C I I 1 L H C A S72 RK BWR B A C K Me t h a n e 1844 0 9 / 0 004 1844 19/0 004 1846 5 7 / 0 004 1851 3 1 / 0 003 1844 8 0 / 0 . 004 1844 7 0 / 0 002 135 9 0 / 0 004 E t h a n e 1928 0 3 / 0 005 1928 0 3 / 0 006 1926 4 4 / 0 005 1935 0 5 / 0 004 1929 0 5 / 0 004 1925 7 7 / 0 003 1 18 77/0 0 0 5 P r o p a n e 1826 43/0 005 1826 4 3 / 0 005 1826 8 7 / 0 004 1833 7 2 / 0 004 1827 7 7 / 0 004 1823 6 7 / 0 002 139 85/0 004 n-Bu l a n e 4 36/0 002 4 36/ 0 002 2 58 / 0 002 16 4 3 / 0 002 6 0 5 / 0 003 0 8 4 / 0 002 1 15 2 1/0 002 n - P e n t a n e 6 6 8 / 0 002 6 6 8 / 0 002 4 13/0 002 2 1 9 7 / 0 001 9 7 1/0 001 0 52/ 0 001 1 1 1 29/0 001 n - H e x a n e 1448 5 3 / 0 007 1448 5 3 / 0 007 1669 14/0 007 62 1 6 7 / 0 005 1275 2 4 / 0 006 1803 4 0 / 0 004 3 4 8 0 49/0 007 M e t h a n o l 15 1 22/0 002 15 1 2 2 / 0 004 150 8 4 / 0 003 151 0 7 / 0 002 15 1 15/0 002 - -H i O 178 43/0 018 178 4 3 / 0 0 2 0 4.76E6 /0 019 159 8 5 / 0 013 185 3 7 / 0 014 - -n-H, 1 18 . 9 5 / 0 005 1 18 .95/0 006 1 18 .97/0 005 1 19 .53/0 004 1 18 . 7 4 / 0 005 - 92 . 6 2 / 0 .005 A r g o n 125 .02/0 .011 125 .03/0 .013 124 .99/0 .012 125 . 3 5 / 0 .009 125 .02/0 .010 124 .78/0 .007 96 . 9 8 / 0 .011 T a b l e 70: G i b b s F r e e E n e r g y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( y . ) / s e c COMPOUND LHCI LHCI I L H C I I I S72 RK BWR BACK M e t h a n e 401 16 61/0 093 4 0 6 3 0 5 4 / 0 123 4 0 6 3 0 . 4 5 / 0 . 103 4 1943 9 7 / 0 074 40792 0 1 / 0 087 4 0 8 0 5 . 1 4 / 0 06 1 1248 4 8 / 0 096 E t h a n e 2 . 9 1 E5 /0 088 2 . 9 3 E 5 /0 1 18 2.93E5 / 0 . 0 9 8 2.94E5 / O 07 1 2 . 9 1 E 5 / O 088 2 . 9 3 E 5 / O 057 6 107 9 9 / 0 09 1 P r o p a n e 2 . 7 7 E 5 /0 107 2 . 8 0 E 5 / O 143 2 . 8 0 E 5 / O 131 2.82E5 / O 087 2 . 8 0 E 5 / O 09 3 2 . 8 0 E 5 / O 069 2396 4 4 / 0 109 n - B u t a n e 93 65/0 0 4 0 92 7 8 / 0 042 92 . 7 6 / 0 0 5 0 85 8 3 / 0 03 1 9 1 3 3 / 0 037 89 . 2 3 / 0 026 1 15 10/0 0 4 0 n - P e n t a n e 10 9 1/0 002 1 7 8 / 0 003 1 . 74 / 0 003 17 7 9 / 0 002 2 5 0 / 0 002 0.7 1/0 001 125 9 6 / 0 003 n - H e x a n e 36 0 2 / 0 0 1 0 34. 7 7 / 0 014 35 2 6 / 0 012 31 7 1/0 008 35 1 1/0 0 1 0 34 . 8 6 / 0 007 129 5 6 / 0 0 1 0 M e t h a n o l 4 8 4 9 48/0 004 4849 4 8 / 0 005 4849 4 8 / 0 005 4849 5 1 / 0 004 4 8 4 9 5 0 / 0 004 - -H.O 1 1 65/0 032 1 1 6 7 / 0 042 1 1 6 7 / 0 038 27 2 4 / 0 025 12 9 0 / 0 0 3 0 - -n-H. 369 5 24/0 186 3765 4 9 / 0 242 3743 8B/0 164 6384 3 6 / 0 152 3896 . 8 6 / 0 177 - 3 0 5 3 9 5 / 0 190 A r g o n 4697 65/0 1 14 4 8 5 0 0 3 / 0 146 4849 8 8 / 0 135 5427 8 7 / 0 09 2 4899 1 1/0 108 489 1 .37/0 073 190 4 5 / 0 1 16 T a b l e 7 1 : I n t e r n a l E n e r g y f o r R e g i o n I RMS-'/. E r r o r / C P U t i m e In (*/.)/aec COMPOUND LHC • LHCI1 I LHCA S72 RK BWR BACK Me t h a n e 7 5 8 / 0 002 4 3 6 / 0 002 9 4 3 / 0 002 4 33/0 001 12 9 8 / 0 001 13 7 0 / 0 001 1 9 9 / 0 001 E t h a n e 8 7 7 / 0 001 8 7 7 / 0 00 1 B 7 7 / 0 002 6 9 4 / 0 001 3 9 0 / 0 001 7 8 6 / 0 001 8 4 5 / 0 001 P r o p a n e 9 9 3 / 0 001 8 8 6 / 0 001 73 8 8 / 0 002 8 37/0 001 2 6 8 / 0 001 5 68 / 0 0 0 0 * 2 6 1/0 002 n - B u t a n e 9 2 5 / 0 000 * 9 25/0 001 139 6 6 / 0 001 9 57/0 001 12 8 0 / 0 0 0 0 * 22 6 9 / 0 001 8 9 7 / 0 001 n - P u n t a n e 1 1 5 9 / 0 OOO * 1 1 5 9 / 0 001 102 87 / 0 00 1 10 48 / 0 001 9 0 5 / 0 001 3 77 / 0 0 0 0 * 2 2 8 / 0 OOO * n - H e x a n e 42 18/0 002 12 74 / 0 003 2 14 37/0 003 1 1 6 5 / 0 001 10 8 8 / 0 001 1 1 4 8 / 0 00 1 4 70/0 002 Me t hano1 26 13/0 002 26 13/0 002 2 133 68/ 0 002 27 88 / 0 001 19 2 8 / 0 002 - -H.O 99 8 1/0 005 99 8 1/0 006 801 0 7 / 0 007 99 8 1/0 003 99 8 5 / 0 003 - -n-H . 606 5 2 / 0 00 1 606 52/0 00 1 238 6 9 / 0 002 648 23/0 00 1 1 169 3 0 / 0 0 0 1 - 497 18/0 001 j A r g o n 4 3 5 / 0 003 4 0 3 / 0 003 7 9 6 / 0 004 1 8 5 / 0 002 13 17/0 001 43 5 1/0 002 1 2 0 / 0 0 0 3 • The c o m p u t e r r o u n d s t h e CPU t i m e s t o t h e n e a r e s t m i l l i s e c o n d T a b l e 72: I n t e r n a l E n e r g y f o r R e g i o n II RMS-% E r r o r / C P U t i m e In (°/.)/sec COMPOUND L H C » L H C I I I LHCA S72 RK BWR BACK M e t h a n e 40 8 3 / 0 001 40 83 / 0 00 1 3 1 19/0 001 39 9 2 / 0 0 0 0 * 37 0 0 / 0 0 0 0 * 29 . 6 3 / 0 001 32 5 7 / 0 001 E t h a n e 30 0 6 / 0 001 30 0 6 / 0 001 27 59/0 001 29 0 6 / 0 001 30 27 / 0 001 8 . 4 9 / 0 0 0 0 * 16 6 2 / 0 001 P r o p a n e 30 14/0 00 1 30 98 / 0 001 57 80 / 0 001 30 44 / 0 001 35 7 7 / 0 001 7 . 6 7 / 0 0 0 0 * 8 2 3 6 5 6 / 0 001 n - B u t a n e 27 5 9 / 0 0 0 0 * 27 59/0 001 10 98 / 0 001 26 6 0 / 0 001 34 3 6 / 0 0 0 0 * 6 . 2 7 / 0 0 0 0 * 12 0 6 / 0 001 n - P e n t a n e 28 6 5 / 0 0 0 0 * 28 65 / 0 0 0 0 * 16 6 5 / 0 00 1 28 72/0 001 39 9 6 / 0 001 2 . 6 8 / 0 0 0 0 * 14 4 9 / 0 0 0 0 * n - H e x a n e 45 0 8 / 0 002 4 5 0 8 / 0 002 53 82/0 002 44 9 5 / 0 001 53 0 5 / 0 001 32 12/0 001 33 8 7 / 0 002 Me t h a n o 1 75 2 6 / 0 002 75 2 6 / 0 002 53 7 1/0 003 74 95 / 0 00 1 83 10/0 001 - -H. 0 99 9 8 / 0 005 99 9 9 / 0 006 100 0 1 / 0 006 100 0 1 / 0 003 100 0 1 / 0 0 0 2 - -n - H. 178 5 0 / 0 001 178 50/0 001 186 0 8 / 0 002 156 7 6 / 0 001 182 6 6 / 0 001 - 160 3 9 / 0 002 A r g o n 24 3 1/0 003 24 43 / 0 004 5 44/ 0 004 24 22/0 002 16 2 3 / 0 0 0 1 42 59 / 0 001 5 0 5 / 0 0 0 3 * The c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . oi T a o l e 7 3 : I n t e r n a l E n e r g y f o r R e g i o n I I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC* LHCI I I LHCA S72 RK BWR BACK Me t h a n e 2 14/0 006 2 14/0 007 9 01/ 0 009 2 27/0 004 18 9 8 / 0 004 32 5 0 / 0 003 0 8 2 / 0 007 P r o p a n e 15 84/0 005 15 6 6 / 0 006 143 60/0 007 15 0 1 / 0 003 7 12/0 003 17 3 6 / 0 002 9 2 9 / 0 005 n-Bu t a n e 27 5 1/0 023 27 5 2 / 0 028 220 33/0 018 12 37/0 008 1 1 8 5 / 0 008 32 5 4 / 0 006 1 1 18/0 0 1 3 Me t nano1 99 9 3 / 0 00 1 99 9 3 / 0 001 98 68/0 002 99 92/0 001 99 9 5 / 0 001 - -H i O 20 6 9 / 0 0 1 0 28 6 9 / 0 013 136 0 4 / 0 002 30 0 8 / 0 001 1 6 1/0 00 1 - -n-H. 730 76/0 009 730 7 7 / 0 01 1 100 87/0 013 803 0 5 / 0 005 1549 24/0 005 - 525 5 6 / 0 008 A r g o n 99 98/0 005 99 9 8 / 0 006 99 98/0 007 99 9 8 / 0 003 99 9 8 / 0 003 99 9 5 / 0 002 101 8 5 / 0 005 T a b l e 74: I n t e r n a l E n e r g y f o r R e g i o n IV RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC » L H C I I I LHCA S72 RK BWR BACK M e t h a n e 22 0 5 / 0 003 20 2 3 / 0 004 51 85/0 004 16 7 9 / 0 . 0 0 2 14 9 9 / 0 002 10 0 3 / 0 002 12 6 4 / 0 003 E t h a n e 19 49/0 004 19 4 9 / 0 005 14 95/0 006 20 7 7 / 0 . 0 0 3 25 3 8 / 0 003 8 53/ 0 002 17 0 6 / 0 0 0 5 P r o p a n e 28 32/0 004 2S 3 2 / 0 005 45 0 7 / 0 006 28 2 6 / 0 . 0 0 2 33 5 4 / 0 003 2 1 6 7 / 0 002 25 29/0 004 n-Bu t a n e 17 47/0 002 17 4 7 / 0 003 32 12/0 003 17 3 4 / 0 . 0 0 1 28 14/0 001 4 37/ 0 001 10 69/0 003 n - P e n t a n e 2 1 47/0 001 21 4 7 / 0 001 17 6 9 / 0 002 2 1 6 9 / 0 . 0 0 1 35 2 1/0 001 2 26/0 0 0 0 1 1 5 24/0 002 n - H e x a n e 50 36/0 006 50 3 6 / 0 008 147 32/0 008 51 6 0 / 0 . 0 0 4 58 7 7 / 0 003 5 1 6 8 / 0 002 52 0 7 / 0 0 0 6 Me t h a n o 1 150 34/0 003 150 3 4 / 0 002 153 9 7 / 0 . 0 0 3 146 3 9 / 0 . 0 0 2 1 17-28 / 0 002 - -H.O 207 53/0 016 207 5 3 / 0 0 2 0 5.15E8 / 0 . 0 2 3 177 6 2 / 0 . 0 0 8 147 . 13/0 002 - -n-H. 107 94/0 005 107 9 4 / 0 005 109 3 2 / 0 . 0 0 6 108 0 0 / 0 . 0 0 2 1 1 1 .40/0 003 - 108 7 6/0 0 0 5 A r g o n 36 76/0 0 1 0 38 6 2 / 0 01 1 33 1 1/0 014 37 18/0.006 32 . 9 9 / 0 007 37 . 4 0 / 0 004 29 9 2 / 0 01 1 t T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . to T a b l e 7 5 : I n t e r n a l E n e r g y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND L H C I L H C I I L H C I I I S72 R K BWR B A C K M e t h a n e 293 13/0 092 179 5 0 / 0 120 179 . 5 1/0 105 76 0 6 / 0 05 1 136 0 7 / 0 05 1 204 15/0 036 207 8 3 / 0 087 £ t h a n e 82 3 4 / 0 089 77 0 8 / 0 1 IB 69 . 9 2 / 0 100 87 0 1 / 0 049 82 3 4 / 0 089 83 0 2 / 0 035 85 9 9 / 0 0 8 3 P r o p a n e 240 4 5 / 0 108 3 1 1 7 5 / 0 143 3 12. 3 1/0 12 1 350 14/0 059 3 17 9 5 / 0 059 354 8 7 / 0 042 134 3 0 / 0 107 n-Bu t a n e 3 2 4 5 75/0 04 3 3239 0 9 / 0 0 4 5 3238 . 9 1/0 051 2373 3 4 / 0 025 2565 8 2 / 0 025 1 13 4 6 / 0 016 197 3 1/0 037 n - P e n t a n e 96 32/0 002 29 3 2 / 0 003 16 67/0 002 7 9 0 / 0 001 16 8 3 / 0 002 2 4 2 / 0 001 8 72/ 0 002 n - H e x a n e 97 5 5 / 0 0 1 0 1 15 56/0 0 1 3 1 16 44/0 01 1 13 1 6 7 / 0 006 107 0 8 / 0 005 128 9 1/0 004 1 19 6 4 / 0 0 1 0 Me t h a n o 1 100 0 0 / 0 004 100 0 0 / 0 004 100 0 0 / 0 006 t o o 0 0 / 0 002 100 0 0 / 0 002 - -H i O 803 4 8 / 0 035 1389 2 5 / 0 044 1389 2 5 / 0 04 1 .;1835 7 2 / 0 0 2 0 14 13 6 9 / 0 019 - -n-H, 11118 8 /0 184 1 1 135 2 /0 238 10729 2 /0 176 11733 8 /0 104 17034 .2 /O 102 - 1164 1 7 /O 175 A r g o n 1 18 90/0 1 13 80 7 2 / 0 144 80 9 1/0 134 59 14/0 062 68 . 7 0 / 0 06 1 72 17/0 04 5 38 . 9 4 / 0 107 T a b l e 7 6 : f u g a c i t y f o r R e g i o n I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC* L H C I I I LHCA S72 RK BWR BACK Me t h a n e 2 13/0 002 2 17/0 002 2 12/0.001 746 9 3 / 0 002 10 2 6 / 0 001 8.09/0 001 0 50/ 0 002 E t h a n e 2 4 8 / 0 001 2 4 8 / 0 001 2 4 8 / 0 002 852 0 4 / 0 001 8 50/ 0 002 5 .49/0 001 6 0 2 / 0 002 P r o p a n e 3 4 4 / 0 001 3 3 6 / 0 002 3 33/0 001 747 0 6 / 0 001 18 0 6 / 0 002 3.2 1/0 002 5 0 4 / 0 002 n - B u t a n e 3 5 4 / 0 001 3 5 4 / 0 001 3 55/0 001 692 36/0 001 26 7 3 / 0 001 3 3 . 0 5 / 0 0 0 0 * 3 4 8 / 0 001 n- P e n t a n e 5 8 1/0 00 0 * 5 8 1/0 001 5 76/0 001 843 0 3 / 0 001 45 13/0 001 1.60/0 0 0 0 * 1 7 9 / 0 001 n - H e x a n e 1 4 33/0 003 14 6 4 / 0 002 14 51/0 002 1 . 17E5 /0 001 139 8 4 / 0 002 1 1 .37/0 002 3 6 0 / 0 0 0 3 M e t h a n o l 100 6 7 / 0 002 100 6 7 / 0 001 100 4 4 / 0 001 1258 .00/0 001 236 7 2 / 0 001 - -H , 0 1.97E6 /0 005 1 .97E6 /0 007 1 . 98E6 /0 006 2.01E6 /O 001 1 .98E6 /O 006 - -n-H i 3 0 4 5 0 1 / 0 001 3045 0 1 / 0 002 3045 22/0 002 688 1 . 6 9 / 0 001 2655 . 3 9 / 0 .002 - 3434 .78/0 .002 A r g o n 274 18 .4/0 003 2 7 4 2 0 . 0 / 0 .003 274 18.4/0 003 4.82E5 /O .003 23088 .9/0 .004 16 182.3/0 003 2 7 5 9 1 .3 /O .004 f The c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . T a b l e 77: F u g a c i t y f o r R e g i o n II RMS-"/. E r r o r / C P U t i m e In ( % ) / s e o COMPOUND LHC* L H C I I 1 LHCA S72 RK BWR BACK Me t h a n e 0 . 9 8 / 0 . 0 0 0 * 0. 9 8 / 0 . 0 0 1 0. 9 8 / 0 . 001 4 . 5 5 / 0 . 001 0. 6 5 / 0 . 0 0 1 0. 3 1/0. 001 0. 9 3 / 0 . 001 E t h a n e 1 . 2 6 / 0 . 0 0 1 1 . 2 6 / 0 . 001 1 . 2 4 / 0 . 001 5 6 3 / 0 . 001 1 6 9 / 0 . 001 0. 17/0. 001 1 0 0 / 0 . 001 P r o p a n e 0 . •16/0 . OOI 0 . 4 9 / 0 . 001 O. 46 / 0 . 0 0 1 4 8 7 / 0 . 0 0 1 1 0 5 / 0 . 0 0 1 O 9 6 / 0 . o o o * 4 10 4 8 / 0 . 001 rt - Bu t a n e 1 1B/0. 001 1 . 18/0 001 1 18/0 0 0 0 * 5 8 7 / 0 . 001 1 9 4 / 0 001 0 3 7 / 0 0 0 0 * 1 8 1/0 001 n - P e n t a n e 1 5 6 / 0 0 0 0 * 1 . 56 / 0 001 1 5 6 / 0 001 5 89 / 0 001 2 63 / 0 001 0 25/ 0 0 0 0 * 0 9 7 / 0 001 n - H e x a n e 2 9 1/0 002 2 . 9 1/0 003 2 9 1/0 002 7 4 1/0. 002 4 10/0 002 2 40 / 0 002 1 7 2 / 0 003 Me t hano1 83 14/0 003 83 . 14/0 003 83 14/0 002 82 5 1/0 002 82 86 / 0 003 - -H,0 4 1 7 8 / 0 005 4 1 7 9 / 0 007 1 1 0 2 / 0 006 1 1 0 2 / 0 004 1 1 0 2 / 0 0 0 5 - -n i l i 307 3 22/ 0 001 307 3 2 2 / 0 00 1 3073 22/0 002 3257 5G/0 001 3 0 2 5 0 2 / 0 002 - 3 17 1 29/0 002 A r g o n 27734 2 /0 004 27735 1 / o 004 27734 2 /O 004 2 8 7 5 5 3 /O 003 2 7 6 8 7 0 /O 004 2 7 0 3 9 2 /O 003 2 7 6 0 1 8 /O 004 T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l s e c o n d . T a b l e 78: F u g a c i t y f o r R e g i o n I I I RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI L H C I I L H C I I I S72 RK BWR BACK Me t h a n e 20 . 6 8 / 0 007 20 6 8 / 0 009 29 73/0 008 4537 4 3 / 0 006 39 4 5 / 0 007 4 1 6 4 / 0 007 2 1 .95/0 0 0 9 P r o p a n e 37 .11/0 .006 37 10/0 .007 67 0 1 / 0 007 674 . 5 8 / 0 005 25 . 17/0 00 6 40 . 3 7 / 0 004 40 . 12/0 007 n - B u t a n e 5 1 .01/0 .001 5 1 0 1 / 0 .030 2 . 96E 12 /0 012 1 0 3 5 1 4 . 0 / 0 014 54 . 9 7 / 0 .015 150 .70/0 013 5 .39/0 .019 Me t hano1 93 . 9 7 / 0 .001 93 7 3 / 0 .002 94 . 20/0 .001 93 .70/0 .001 93 .72/0 .002 - -HiO 63 1 .07/0 .002 63 1 0 7 / 0 .002 34423 .2 /O .002 2856 . 9 4 / 0 .001 1263 .23/0 .002 - -n - H i 2 8 2 3 . 3 6 / 0 .011 2823 36/0 .012 3556 . 12/0 .011 8 199 . 52/0 .009 2 2 9 9 .27/0 .010 - 3 197 .60/0 .013 A r g o n 3.34E 14 /0 .005 3 . 34E 14 /0 .006 3 . 34E 14 /O .006 3 . 35E 14 /O .004 3.34E 14/0 .006 3 . 33E 14 /O .004 3 . 33E 14 /O .006 ro co T a b l e 7 9 : F u g a c i t y f o r R e g i o n IV RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC • L H C I I I LHCA S72 RK 8WR BACK M e t h a n e 2 54/0 004 2 . 54/0 0 0 5 2 5 5 / 0 005 3 38/0 0 0 3 2 5 0 / 0 004 2 4 4 / 0 003 2 5 3 / 0 0 0 5 E t h a n e 0 27/0 006 0. 2 7 / 0 0 0 6 0 25/ 0 006 1 9 5 / 0 004 0 4 2 / 0 006 0 54/ 0 004 0 3 4 / 0 006 P r o p a n e 2 96/0 004 2 . 9 6 / 0 006 3 0 0 / 0 . 0 0 5 3 8 4 / 0 004 3 0 4 / 0 004 2 8 7 / 0 004 2 9 6 / 0 0 0 6 n-Bu t a n e 0 46/0 003 0. 46/0 003 0 3 5 / 0 003 2 4 0 / 0 002 0 8 2 / 0 002 0 13/0 002 1 0 1 / 0 003 n - P e n t a n e 0 86/0 0O1 0. 8 6 / 0 002 0 6 2 / 0 001 3 0 2 / 0 001 1 4 8 / 0 002 0 14/0 002 0 5 5 / 0 002 n - H e x a n e 2 42/0 007 2 . 42/0 008 1 6 9 / 0 007 4 51/ 0 0 0 6 3 0 2 / 0 007 2 23/0 006 1 7 9 / 0 008 Me t h a n o ] 96 10/0 003 96 . 10/0 0 0 3 96 0 1 / 0 003 96 0 7 / 0 002 96 0 9 / 0 003 - -H.O 0 64/0 0 2 0 0 64/ 0 023 1 18/0 0 2 0 1 .39/0 014 0 7 4 / 0 016 - -n-H. 3472 39/0 005 3472 39/0 007 3475 2 9 / 0 006 35 15 .68/0 005 3457 9 4 / 0 005 - 3512 8 2 / 0 007 A r g o n 2 5 3 9 1 2 /0 012 25392 4 /0 014 25368 6 /0 012 2564 1 . 1 /O 0 1 0 2 5 3 8 0 1 / o 012 252 17 8 6 / 0 0 1 0 25368 . 1 /O 0 1 5 T a b l e 8 0 : F u g a c i t y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI LHCI I L H C I I I S72 RK BWR BACK Me t h a n e 16 47/0 105 7 36/0 134 7 . 5 1/0 1 16 5 5 6 . 5 6 / 0 . 0 8 8 7 0 0 / 0 100 6 9 5 / 0 0 8 6 6 3 2 / 0 124 E t h a n e 16 14/0 100 3 5 4 / 0 130 3 . 6 1 / 0 108 3 2 4 . 0 1 / 0 . 0 8 4 16 14/0 100 1 6 6 / 0 0 8 2 2 2 6 / 0 1 18 P r o p a n e 14 37/0 1 19 5 0 0 / 0 156 4 . 9 4 / 0 13 1 226. 16/0. 101 4 5 3 / 0 1 15 2 90 / 0 0 9 9 1 1 2 5 / 0 138 n - B u t a n e 338 43/0 045 336 98/0 047 336 . 9 6 / 0 0 5 0 2.97E7 / 0 . 0 3 7 1497 3 5 / 0 043 656 8 7 / 0 03 7 20 6 6 / 0 053 n - P e n t a n e 19 92/0 003 6 63/0 003 6 . 5 0 / 0 003 1 0 2 . 1 1 / 0 . 0 0 2 5 10/0 002 3 5 5 / 0 0 0 3 6 6 1 / 0 003 n - H e x a n e 3 87/0 01 1 5 82/0 0 15 4 . 75/0 012 20.8 1/0.009 4 5 3 / 0 009 4 33/ 0 0 0 9 2 2 5 / 0 013 Me t h a n o 1 - - - - - - -H . 0 3 2 2 0 6 2 /O 035 3 2 2 3 0 7 /O 046 3 2 2 3 0 7 /O 042 2. 1BE5 / 0 . 0 3 0 32457 .9 /O 034 - -n-H. 9 .69E5 /O 2 1 1 9 . 7 1E6 /O 267 9.70E6 /O 189 1 .01E7 /O. 179 9.74E6 /O 203 - 9 . 7 5 E 6 /O 25 0 A r g o n 1.52E5 /O 128 1.63E5 /O 160 \ .63E5 /O 147 1 . 37E6 /O. 107 1 . 6 9 E 5 /O 124 1 . 7 1E5 /O 105 1 .72E5 /O 151 T a b l e 8 1 : I s o b a r l c H e a t C a p a c i t y f o r R e g i o n I RMS-*/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * L H C I I I LHCA S72 RK BWR BACK Me t h a n e 68 56/0 0 0 1 55 6 5 / 0 001 286 8 1/0 001 59 5 9 / 0 001 40 0 9 / 0 001 6 1 88/0 0 0 0 * 75 3 5 / 0 0 0 3 P r o p a n e 93 6 5 / 0 002 92 14/0 003 1026 B3/0 002 38 14/0 001 53 4 3 / 0 001 7 17 0 2 / 0 001 6853 9 3 / 0 0 0 5 i v B u t a n u •17 53/0 003 44 9 5 / 0 003 12 16 99/0 002 29 87/0 0 0 1 42 2 2 / 0 OO 1 93 8 4 / 0 OO 1 134 12 5 /O 0 0 5 H.O 103 18/0 003 06 3 4 / 0 003 455 0 4 / 0 002 36 52/0 002 70 19/0 OO 1 - -n-H. 49 6 1/0 001 42 6 0 / 0 001 176 32/0 OOI 43 49/0 001 229 3 2 / 0 0 0 1 - 154 5 9 / 0 0 0 2 p-H, 50 58/0 001 50 5B/0 001 50 55/0 001 50 88/0 001 50 5 6 / 0 OOI - -A r g o n 49 20/0 001 47 19/0 001 45 0 6 / 0 OOI 56 6 2 / 0 o o o i 32 3 7 / 0 0 0 0 t 170 85/0 0 0 0 t 69 8 3 / 0 002 4 I n e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d . T a b l e 82: I s o b a r i c H e a t C a p a c i t y f o r R e g i o n II RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * L H C I I I LHCA S72 RK BWR BACK Me t h a n e 66 . 4 2/0 OOI 6 1 84/0 001 7 7 . 8 9 / 0 001 6 1 77/0 001 60 6 0 / 0 OOI 50 0 0 / 0 0 0 0 t 695 . 2 9 / 0 0 0 3 P r o p a n e 39 . 0 1/0 001 39 B 1/0 00 1 39 . 8 0/0 001 39 8 1/0 001 39 8 1/0 OOI 39 8 0 / 0 0 0 0 * 2 . 3 8 E 1 0 /O 002 n - B u t a n e 40. 52/0 001 40 52/0 001 40. 5 3 / 0 00 1 40 5 2 / 0 OOI 40 52 / 0 OOI 40 52 / 0 0 0 0 * 2.OOE10 10 0 0 2 H.O 65 . 6 5 / 0 003 56 24/0 003 63 . 9 6 / 0 002 50 0 2 / 0 002 50 6 1/0 OOI - -n-H. 44 43/0 001 42 5 7 / 0 001 48 . 2 9 / 0 OOI 42 28/0 001 39 0 4 / 0 001 - 2 19. 7 9 / 0 00 2 p - H i 59 70/0 001 59 8 1/0 001 63 . 4 0 / 0 001 59 56/0 001 56 7 I/O 001 - -A r g o n 59 13/0 01 1 56 0 7 / 0 001 75 3 6 / 0 001 56 6 1 / 0 001 54 6 8 / 0 001 42 43/0 0 0 0 t 8 9 7 . 7 0 / 0 0 0 2 * T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d ro oo o T a b l e 8 3 : I s o b a r i c H e a t C a p a c i t y f o r R e g i o n I I I RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHC * LHCI I I LHCA S72 RK BWR BACK Me t h a n e 42 4 1/0 012 39 8 7 / 0 012 6 0 . 3 8 / 0 009 51 76/0 007 24 5 6 / 0 007 97 8 1/0 004 58 6 9 / 0 027 P r o p a n e 95 5(5/0 027 95 30/0 027 1 . 1 IE 10 /0 02 0 34 7B/0 015 45 9 4 / 0 0 1 5 739 86/0 008 3684 5 2 / 0 059 n -Bu t a n e 45 67/0 02 4 45 0 1 / 0 025 1 . 1 1E7 /0 018 25 7 1/0 0 1 4 18 6 0 / 0 0 1 3 10 1 22/0 008 359 8 7 / 0 054 M e t h a n o l 173 6 0 / 0 0 0 1 173 6 0 / 0 001 26103.8 /0 001 125 56/0 001 87 8 7 / 0 001 - -H.O 58 0 9 / 0 032 58 0 9 / 0 032 276.2 1/0 024 16 57/0 018 32 4 1/0 018 - -n-H. 33 0 4 / 0 009 3 1 2 8 / 0 009 139.27/0 007 35 9 7 / 0 005 277 7 7 / 0 005 - 183 36/0 02 1 p-H : 43 15/0 012 39 8 5 / 0 0 13 8 8 . 4 2 / 0 . 0 0 9 33 7 7 / 0 007 2 1 2 0 4 / 0 007 - -A r g o n 45 34/0 005 42 4 5 / 0 005 3 3 . 6 2 / 0 .004 54 28/0 003 27 9 7 / 0 003 177 7 1/0 002 6 1 32/0 012 T a d e 84: I s o b a r i c H e a t C a p a c i t y f o r R e g i o n IV RMS-% E r r o r / C P U t i m e t n ( % ) / s e c COMPOUND LHC* LHCI I I LHCA S72 RK BWR BACK M e t h a n e 98 . 32/0 013 47 3 8 / 0 014 79. 6 9 / 0 0 1 0 42 2 8 / 0 . 0 0 7 46 5 6 / 0 007 44 5 7 / 0 004 1274 5 6 / 0 028 E t h a n e 29 . 7 9 / 0 004 27 6 6 / 0 005 3 1 . 70 / 0 003 27 7 7 / 0 . 0 0 2 27 3 5 / 0 002 24 9 4 / 0 001 1744 0 5 / 0 0 0 9 P r o p a n e 2 1 . 0 4 / 0 007 19 5 2 / 0 008 13. 4 2 / 0 . 0 0 5 19 5 0 / 0 . 0 0 4 19 2 9 / 0 004 17 3 8 / 0 002 1482 55/0 015 n - B u t a n e 18 . 7 5 / 0 008 15 6 4 / 0 0 0 9 25. 35/0 006 15 6 9 / 0 . 0 0 4 15 6 0 / 0 004 14 2 7 / 0 0 0 2 1289 0 4 / 0 017 H.O 33 . 5 5 / 0 013 33 5 5 / 0 0 1 3 32. 8 1 / 0 0 1 0 30 9 8 / 0 . 0 0 8 31 0 8 / 0 007 - -n-H. 32 15/0 005 26 8 0 / 0 0 0 5 37 9 5 / 0 004 26 5 9 / 0 . 0 0 3 23 6 4 / 0 003 - 476 60/0 0 1 0 p-H. 54 37/0 008 50 2 5 / 0 0 0 9 70 0 5 / 0 006 50 12/0.005 47 4 0 / 0 005 - -A r g o n 66 4 5 / 0 01 1 45 50/0 012 79 4 4 / 0 008 45 5 0 / 0 . 0 0 0 * 44 6 4 / 0 006 40 19/0 003 156 1 10/0 024 * T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d to CO T a b l e 8 5 : I s o b a r l c H e a t C a p a c i t y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI LHCI I L H C I I I S72 RK BWR BACK Me t h a n e 46 l 7 1/0 123 499 0 5 / 0 123 60. 0 3 / 0 127 60 3 5 / 0 068 28 17/0 066 66 7 0 / 0 037 6 160 0 6 / 0 265 E t h a n e A 1 4 3 / 0 032 24 7 5 / 0 032 24 . 7 5 / 0 033 28 15/0 018 27 6 2 / 0 017 28 3 6 / 0 0 1 0 27 16 9 1/0 0 7 0 P r o p a n e 868 20/0 086 855 4 9 / 0 085 68 . 2 7 / 0 087 27 6 5 / 0 048 36 5 6 / 0 045 549 2 7 / 0 025 3737 0 4 / 0 185 n - B u t a n e 47 90/0 0B3 40 5 3 / 0 082 34 . 8 0 / 0 084 22 14/0 045 19 7 5 / 0 04 4 66 3 5 / 0 025 74 1 4 8 / 0 178 M e t h a n o l 1 16 23/0 004 16 1 2 3 / 0 004 16 1. 23/0 004 1 16 5 2 / 0 003 8 0 , 7 5 / 0 002 - -H>0 1362 80/0 066 1343 8 2 / 0 065 55 7 9 / 0 067 29 6 2 / 0 036 36 2 1/0 035 - -n-H i 82 5S/0 19 1 8 1 10/0 19 1 37 76/0 196 38 2 I/O 107 80 6 2 / 0 103 - 8 . 6 6 E 5 /O 4 1 1 p-H. 24207 6 /0 152 24201 2 /0 153 1701 2 1/0 159 1701 19/0 085 1702 13/0 082 - -A r g o n 185 .75/0 1 16 180 . 2 9 / 0 1 17 46 0 7 / 0 120 46 3 2 / 0 064 42 2 9 / 0 062 56 9 7 / 0 034 10980 .7 /O 25 1 T a b l e 86: I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC' L H C I I I LHCA S72 RK BWR BACK P r o p a n e n-Bu t a ne H.O n-H. p-H. 148 . 3 9 / 0 . 0 0 3 7 7 . 9 I/O.003 1754 .66/0.002 325 . 36/0.001 64 . 4 8/0.001 140. 1 1/0.003 6 9 . 8 4 / 0 . 0 0 3 5 8 6 . 2 7 / 0 . 0 0 3 2 5 3 . 1 2 / 0 . 0 0 1 6 4 . 4 8 / 0 . 0 0 1 1 4 9 3 . 8 9 / 0 . 0 0 3 1 7 3 7 . 3 1 / 0 . 0 0 3 17 16.44/0.004 3 1 2 . 6 5 / 0 . 0 0 1 6 4 . 3 0 / 0 . 0 0 1 3 9 . 4 6 / 0 . 0 0 1 24 . 6 5 / 0 . 0 0 3 47 .89/0.002 2 9 2 . 1 1 / 0 . 0 0 1 64 . 4 7 / 0 . 0 0 0 + 3 1 6 . 8 2 / 0 . 0 0 2 156 .92/0.002 7 5 . 6 9 / 0 . 0 0 2 5 1 3 5 . 8 1 / 0 . 0 0 0 + 64 .5 1/0.000* 3 7 5 4 . 7 4 / 0 . 0 0 2 2 4 8 0 . 9 0 / 0 . 0 0 2 1 . 67E7 / 0 . 0 0 6 7.33E6 / 0 . 0 0 6 3 . 13E5 / 0 . 0 0 2 * The c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d ro CO ro l a u l e 8 7 : i s o m e t r i c M e a t C a p a c i t y f o r R e g i o n I I R M S - % E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND LHC • LHC1 I I LHCA S72 RK BWR BACK P r o p a n e n - B u t a n e M.O f 1 - M i p- H , 2 2 . -19/0 . 00 1 IG.B6/0.OOI 8 0 4 6 . 3 6 / 0 . 0 0 2 2 6 9 . 4 3/0.001 1 2 6 . 2 4 / 0 . 0 0 1 2 2 . 4 9 / 0 . 0 0 2 16.85/0.002 1 4 0 5 . 7 2 / 0 . 0 0 3 2 1 3 . 5 2 / 0 . 0 0 1 126.21/0.001 2 2 . 4 7 / 0 . 0 0 1 16.88/0.002 B 0 2 7 . 0 4 / 0 . 0 0 4 334.8 3/0.001 2 2 4 . 8 6 / 0 . 0 0 1 2 2 . 4 9 / 0 . 0 0 1 16.85/0.001 1 7 0 S . 0 9 / 0 . 0 0 2 2 0 3 . 9 7 / 0 . 0 0 0 4 1 3 5 . 8 9 / 0 . 0 0 0 * 22 . 4 9 / 0 . 0 0 0 * 16.85/0.001 47 .61/0.002 3 4 5 . 5 3 / 0 . 0 0 0 ' 2 1 9 . 0 4 / 0 . 0 0 0 * 22 . 4 8 / 0 . 0 0 1 16.84/0.001 2 . 8 2 E 1 0 /O.003 2 . 24E 10 / 0 . 0 0 3 5 5 4 0 4 . 0 /O.002 The c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d T a b l e 88; I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n I I I RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC* L H C I I I LHCA S72 RK BWR BACK P r o p a n e n - B u t a n e n-H. p-H. 1 4 9 . 0 2 / 0 . 0 2 5 75 . 16/0.024 2 0 4 . 7 6 / 0 . 0 0 9 2 4 1 . 0 2 / 0 . 0 1 2 1 4 8 . 2 5 / 0 . 0 3 0 7 3 . 4 3 / 0 . 0 2 8 1 9 6 . 5 6 / 0 . 0 1 1 2 2 8 . 6 4 / 0 . 0 1 4 1.96E10 /0.036 1.97E7 / 0 . 0 3 3 2 5 7 . 7 7 / 0 , 0 1 3 1 7 8 . 6 9 / 0 . 0 1 7 29.7 1/0.020 1 6 . 6 0 / 0 . 0 1 9 2 4 2 . 6 9 / 0 . 0 0 8 2 6 3 . 8 7 / 0 . 0 1 0 77 . 0 3 / 0 . 0 1 5 3 1 . 5 1 / 0 . 0 1 5 3 9 0 4 1 . 9 / 0 . 0 0 6 5 0 5 8 . 9 2 / 0 . 0 0 7 3357 .9 1/0.022 2 6 9 7 7 . 9 / 0 . 0 1 9 2.00E8 / 0 . 0 6 1 1 .60E9 / 0 . 0 5 7 1.05E5 /0.021 T a b l e 8 9 : I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n IV RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC" L H C I I 1 LHCA S72 RK BWR BACK P r o p a n e n - B u t a n e n-H. p-H. 2 8 . 0 2 / 0 . 0 0 7 2 8 . 3 9 / 0 . 0 0 8 8 1 . 3 8 / 0 . 0 0 4 1 2 0 . 6 9 / 0 . 0 0 8 2 2 . 6 7 / 0 . 0 0 8 1 7 . 8 7 / 0 . 0 0 9 6 0 . 9 7 / 0 . 0 0 6 1 0 4 . 0 3 / 0 . 0 1 0 14 . 6 9 / 0 . 0 1 0 3 9 . 4 5 / 0 . 0 1 1 1 0 7 . 9 1 / 0 . 0 0 6 1 9 0 . 7 1 / 0 . 0 1 2 2 4 . 8 8 / 0 . 0 0 6 2 1 .36/0.007 6 3 . 1 3 / 0 . 0 0 4 1 0 6 . 4 3 / 0 . 0 0 6 2 2 . 6 9 / 0 . 0 0 4 1 8 . 3 1 / 0 . 0 0 5 8 3 . 2 4 / 0 . 0 0 3 1 2 4 . 9 0 / 0 . 0 0 5 1 5 . 5 2 / 0 . 0 0 6 10.90/0.007 7 . 1 8 E 5 / 0 . 0 1 7 1.05E6 / 0 . 0 1 8 10524.2 / 0 . 0 1 2 co co T a b l e 90; I s o m e t r i c H e a t C a p a c i t y f o r R e g i o n V RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI LHCI I L H C I I I S72 RK BWR BACK P r o p a n e n - B u t a n e n-H i p-H, 1 2 7 0 . 4 6 / 0 . 0 8 4 6 5 . 9 1/0.081 1 3 7 . 5 7 / 0 . 1 8 9 3 6 0 9 9 . 6 /0.152 1 2 5 9 . 8 5 / 0 . 0 9 9 62 . 3 4 / 0 . 0 9 4 1 2 2 . 5 6 / 0 . 2 4 3 3 6 0 1 1 . 8 /O.194 1 6 9 . 7 0 / 0 . 0 9 9 55 . 12/0.095 6 5 . 2 3 / 0 . 2 2 1 2 1 5 5 . 3 4 / 0 . 175 174 . 10/0.062 2 6 . 5 1 / 0 . 0 6 1 68 . 8 0 / 0 . 140 2156. 12/0. 1 12 1 5 5 . 6 5 / 0 . 0 4 9 29 .93/0.047 6 5 4 . 3 9 / 0 . 1 1 0 2 1 6 9 . 9 3 / 0 . 0 8 9 2 5 7 5 . 2 7 / 0 . 0 6 7 3 6 8 6 . 2 2 / 0 . 0 6 4 1.55E8 /O.194 1 .04E6 /O. 186 1.04E6 / 0 . 4 3 0 T a b l e 9 1 : P r o p a n e - T h e r m o d y n a m i c P r o p e r t i e s a l o n g t h e C r i t i c a l I s o t h e r m RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC S72 RK BWR BACK Hr 2873 6 6 / 0 0 0 1 3387 27/0 0 0 0 + 3 105 9 9 / 0 001 3630 34/0 001 3373 0 9 / 0 0 0 1 ST- 42 0 9 / 0 0 0 1 42 7 6 / 0 002 42 97/0 001 42 77/0 001 101 5 5 / 0 003 A T 8 5 0 4 8 / 0 001 836 8 8 / 0 001 836 88/0 001 834 85/0 001 109 9 9 / 0 001 Gr 34 1 27/0 001 355 75/0 002 355 75/0 001 355 73/0 001 353 8 4 / 0 001 U r 2 1 1 0 9 / 0 ooo * 261 75/0 001 234 30/0 OOI 283 90/0 0 0 0 * 262 2 7 / 0 002 t / P 2556 13/0 002 2556 15/0 001 2556 13/0 001 2539 13/0 002 2542 4 0 / 0 002 + The c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d T a b l e 9 2 : J o u 1 e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I RMS-% E r r o r / C P U t i m e t n ( % ) / s e c COMPOUND LHC' L H C I I I LHCA S72 RK BWR BACK M e t h a n e n-H, p-H, 2 9 . 9 9 / 0 . 0 0 1 5 8 2 . 4 0 / 0 . 0 0 1 4 7 9 . 4 4 / 0 . 0 0 1 2 9 . 9 9 / 0 . 0 0 1 49 1 . 54/0.001 4 0 6 . 6 7 / 0 . 0 0 1 12 1. 2 6 / 0 . 0 0 2 26 . 59/0.001 23 . 7 5 / 0 . 0 0 1 31 .78/0.001 6 4 0 . 2 6 / 0 . 0 0 1 5 4 0 . 3 0 / 0 . 0 0 1 4 3 . 8 0 / 0 . 0 0 1 2 0 9 3 . 9 8 / 0 . 0 0 1 2 1 1 3 . 8 6 / 0 . 0 0 1 4 0 . 3 9 / 0 . 0 0 1 4 1862.8 / 0 . 0 0 3 3 17 19.0 / 0 . 0 0 2 CO T a b l e 9 3 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n II RMS-% E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHC * LHCI I I LHCA S72 RK BWR BACK Me t h a n e P r o p a n e n-Bu t a n e n - H i p-H. 8 1 .83/0.001 3 5 . 0 5 / 0 . 0 0 0 4 •43.3 1/0. OOO 4 22 .94/0.001 2 1 . 5 7 / 0 . 0 0 1 24 . 12/0.001 3 5 . 0 5 / 0 . 0 0 0 4 4 3.3 1/0.000 4 2 2 . 9 4 / 0 . 0 0 1 22.2 1/0.001 79.4 1/0.002 4 6 . 2 1 / 0 . 0 0 0 4 9 . 5 8 / 0 . 0 0 0 i 12.84/0.001 5.58/0.001 24 . 2 8 / 0 . 0 0 1 34 .47/0.001 4 3.4 1/0.000 4 20. 15/0.001 18 .67/0.001 18 .52/0.001 4 0 . 5 6 / 0 . 0 0 0 4 5 0 . 0 8 / 0 . 0 0 0 4 2 5 . 5 2 / 0 . 0 0 1 2 6 . 7 6 / 0 . 0 0 1 16 . 3 8 / 0 . 0 0 1 2 3 . 3 9 / 0 . 0 0 0 4 2 7 . 5 2 / 0 . 0 0 0 4 89 1 9 . 7 8 / 0 . 0 0 3 19378.7 /0.0O1 17 3 2 1 . 0 / 0 . 0 0 1 1 2 5 0 . 7 3 / 0 . 0 0 2 i 4 T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d T a b l e 9 4: J o u 1 e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n I I I RMS-% E r r o r / C P U t i m e I n ( % ) / s e c COMPOUND LHC* L H C I I I LHCA S72 RK BWR BACK Me t h a n e H.O n-H. p-H. 22 .84/0.012 23 .62/0.001 195 1 .68/0.009 1 4 6 2 . 2 0 / 0 . 0 1 2 22 .84/0.014 2 3 . 6 2 / 0 . 0 0 1 1601 .7 1/0.008 1 2 0 0 . 9 3 / 0 . 0 1 1 2 0 . 7 6 / 0 . 0 1 6 7 3 . 7 0 / 0 . 0 0 2 5 3 3 . 5 9 / 0 . 0 1 3 3 9 9 . 9 5 / 0 . 0 1 7 1 4 . 3 4 / 0 . 0 0 9 24 .08/0.001 21 1 5 . 3 1 / 0 . 0 0 7 1 6 3 0 . 8 6 / 0 . 0 0 9 5 1 . 2 9 / 0 . 0 0 7 2 7 . 9 1 / 0 . 0 0 1 9 9 3 5 . 8 7 / 0 . 0 0 6 4 2 9 4 . 6 5 / 0 . 0 0 7 1 0 7 . 7 5 / 0 . 0 0 6 1 . 14E5 / 0 . 0 2 7 1 . 2 3 E 5 /0.021 to CO T a b l e 9 5 : J o u 1 e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n IV RMS-'/. E r r o r / C P U t i m e In ('/.)/sec COMPOUND LHC • LHCII I LHCA S72 RK BWR BACK Me thane 1 5 . 4 6 / 0 012 14 15/0 014 42 . 9 6 / 0 017 12 . 53/0 0 1 0 8 7 2 / 0 008 9 0 4 / 0 006 1064 1 0 /O 028 E thane 9 . 0 5 / 0 0 0 0 * 9 0 5 / 0 0 0 0 * 25 . 0 5 / 0 0 0 0 * 8 75/0 0 0 0 * 9 8 8 / 0 0 0 0 t 14 15/0 0 0 0 * 2 0 7 4 6 4 /0 001 Propane 1 3 7 3 / 0 0 0 1 13 7 3 / 0 002 3 1 . 8 5 / 0 0 0 2 13 8 7 / 0 001 2 1 0 3 / 0 001 8 3 1/0 001 22 16 7 0 / 0 0 0 3 n-Bu t ane 16 10/0 0 0 1 16 10/0 001 40. 56/0 002 15 9 4 / 0 001 23 1 1/0 001 17 7 8 / 0 001 1 .00E6 /0 00 3 H J O 28 5 5 / 0 003 28 5 5 / 0 004 32. 45/0 004 27 5 1 / 0 002 40 0 3 / 0 0 0 2 - -n - H i 12 8 1/0 004 12 8 1/0 0 0 5 9 . 2 7/0 006 10.76/0.004 22 2 9 / 0 003 - 1557 5 0 / 0 01 1 p - H i 12 4 4 / 0 008 13 0 8 / 0 009 20 4 1/0 01 1 10 3 0 / 0 006 21 6 4 / 0 005 - -Neon 23 2 6 / 0 00 1 20 6 2 / 0 001 45 7 6 / 0 0 0 2 2 1 9 9 / 0 001 23 . 3 0 / 0 001 - -Argon 1 0 6 3 / 0 .000 * 15 . 8 8 / 0 .000 * 12 0 7 / 0 0 0 0 * 16 5 1/0 .000 * 14 . 16/0 .000 * 34 .67/0 .000 * 9308 .45/0 .001 i T h e c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n e a r e s t m i l l i s e c o n d T a b l e 9 6 : J o u l e - T h o m s o n C o e f f i c i e n t s f o r R e g i o n V RMS-'/. E r r o r / C P U t i m e In ( % ) / s e c COMPOUND LHCI LHCI I LHCII I S72 RK BWR 8ACK Me thane 184 . 4 5 / 0 13 1 5 1 .72/0 156 5 1 .03/0 150 13.03/0 095 6 1 . 6 7 / 0 074 254 3 5 / 0 064 2 .81E5 /O 27 1 E t hane 119. 18/0 002 1 4 6 . 6 0 / 0 002 146.60/0 002 150.38/0 001 1 4 6 . 8 9 / 0 001 140 OB/0 001 2 . 85E7 /O 004 P ropane 59 . 24/0 0 0 1 4.48/0 00 1 4 .48/0 001 4 . 9 9 / 0 0 0 0 1 1 1 .75/0 0 0 0 * 5 64 / 0 0 0 0 r 2 6 2 4 7 . 9 /O 001 H i 0 6 0 . 0 8 / 0 o o a 4 7 . 8 7 / 0 0 10 4 7 . 8 7 / 0 009 4 9 . 7 7 / 0 006 4 0 . 3 4 / 0 004 - -n •• H i 3 6 2 E 1 5 /0 195 2 4 8 E 1 5 /O 248 2 48E15 /O 231 1 .08E15 /O 142 2 . 25E 15 /O 1 1 1 - 9 . 39E 17 /O 422 p-H, 3 8 8 E 1 5 /O 155 2 6 8 E 1 5 /O 197 2 68E15 /O 183 1 .05E15 /O 1 14 2 . 6 IE 15 /O 089 - -Neon 5 8 . 9 0 / 0 005 16.80/0 007 16.5 1/0 006 17 .00/0 004 1 7 . 2 8 / 0 003 -A r g o n 137.09/0 006 5 6 . 9 9 / 0 007 3 9 . 4 6 / 0 007 4 2 . 0 6 / 0 004 4 6 . 9 6 / 0 003 43 4 8 / 0 003 67 143.6 /o 012 t I ne computer rounds the CPU time to the n e a r e s t m i l l i s e c o n d . CO Ch T a b l e 97: I n v e r s i o n C u r v e - RMS-% E r r o r / C P U t i m e i n ( % ) / s e c COMPOUND LHCI LHCI I L H C I I 1 S72 RK BWR M e t h a n e P r o p a n e n - B u t a n e p-H, A r g o n 3 1 . 2 3 / 0 . 0 0 5 58 . 5 3 / 0 . 0 0 9 53 .69/0.009 3 9 . 4 9 / 0 . 0 0 4 3 7 . 0 2 / 0 . 0 0 3 12.94/0.005 4 1 . 3 5 / 0 . 0 1 0 25 . 9 3 / 0 . 0 1 0 3 9 . 2 8 / 0 . 0 0 3 2 1 . 3 8 / 0 . 0 0 3 12.94/0.005 4 0 . 3 9 / 0 . 0 0 9 2 5 . 8 6 / 0 . 0 0 9 3 8 . 5 9 / 0 . 0 0 3 1 7 . 8 7 / 0 . 0 0 3 1 2 . 6 1 / 0 . 0 0 5 1 0 0 . 4 4 / 0 . 0 0 9 58 .27/0.009 6 1 .70/0.003 29. 12/0.003 5 . 6 2 / 0 . 0 0 4 44 .6 1/0.008 2 5 . 5 2 / 0 . 0 0 9 8 5 . 8 8 / 0 . 0 0 3 22 . 2 1/0.003 37 . 7 5 / 0 . 0 0 8 54 . 5 3 / 0 . 0 2 0 8 0 . 13/0.020 4 5 . 9 5 / 0 . 0 0 5 T*L.I» u t; LHC • q ( l . ] ) / C r * u o i k i £ q u * t l o n Conp4r i *on lo r n -h , B H S - X ' E r r o r / C P U t i n . in ( X ) / i « c LHC • q O . l ) Gr*DO»k1-0»uD«r1 C qua t1 on CALC . a t c i O N | OECION 1 1 «E cion 1 1 1 SCGIOH I V REGION V REGION 1 BI010: I I SEC 1 ON I 1 1 DEC ION I V KEGION V V 10 . it/0 114 3 36/0 1 It t l t/0 31S 0 76/0 1 It 11,01/1.111 16.67/0.136 36 . 61/0.131 13 . 43/0 279 5. 3 1/0.Ill 15.05/5. 463 V 22Bb3 • 7/0 001 1 1 I/O 00 1 1764 97/0 00 1 0 41/0 OOOl 14 .16/0.00? 24105.22/0.OOOl 1 I 93/0.001 1117. 14/0 001 3 . 19/0.0001 54.26/0 006 I It 96/0 003 0 7t/0 003 31 01/0 0 0 6 0 36/0 003 IO.47/0.091 IC.14/0.OOJ 3 53/0.001 11. 66/0 0O5 1 . 93/0.003 9.32/0 09 4 nr 2 21 65/0 001 174 32/0 OOI 3 1 1 • 0/0 0 0 6 1 I t 13/0 OOI 1.39E6 /0.097 341.33/0.001 170 46/0.001 311 77/0 005 1 17 40/0.001 3.36E6 /O 1 13 i r &l«4 16/0 00 1 1 1& 16/0 OOI 6 9 6 91/0 o o t 117 44/0 004 13640.1 IO.167 763.12/0.OOI 166 37/0.OOI 646 51/0 009 1 16 33/0.0O5 I4S52.4 / o i a 3 *r 42/0 00 1 195 34/0 00 1 l i t t I/O 00! 139 ( I/O 004 4696.66/0.161 273.74/0.OOI 116 16/0.001 163 62/0.ooe 110.14/0.004 4725.4 1/0 154 lir 11)4 01/0 OOI 15 J 23/0 OOI 100 OS/0 o o t 1 IS 10/0 006 1741.11/0.It! 167.40/0.OOI 160 41/0.001 103 14/0 001 1 19 71/0.005 4093.57/0 157 ur is] 5 9 / 0 001 163 15/0 00 1 742 61/0 006 101 27/0 002 10721.2 /0.092 (63.10/0.001 166 73/0.OOI 1 19 4 I/O 006 IOS 23/0.003 12052.1 /O 106 f/P 32115 00/0 o o i 3 16a 04/0 00 1 3033 e i / o 008 3414 13/0 004 9.7IEI 10.Ill 3656.64/0.OOI 1346 34/O.OOI 3596 11/0 009 3514 25/0.004 S.16EC 10 164 Cp 54 33/0 O O O l 43 24/0 00 I 62 00/0 0 0 6 2i 69/0 003 36.44/0.091 40.76/0.OOOl 42 31/0.0001 30 59/0 004 36 59/0.002 37.49/0 076 Cv 1 9 7 14/0 001 176 65/0 OOOl 136 32/0 0 0 6 67 6 6/0 003 63.96/0. 100 3 6 3.61/0.OO1 111 IS/0.001 363 9 1/0 0O6 57 46/0.003 66.04/0 1 12 » 4 7 1 as/o OOOl 27 41/0 OOOl 1546 60/0 004 17 03/0 002 3.38!16 la.103 390.67/0.OOI 36 11/0.OOOl 1641 55/0 .006 36 31/0.OOOl 100.60/0 1 15" l Tti* conpu i t r round* tn« CPU t i n * to tut n « * r * i t a l l ) . i»cond * Iii* Gr * lie t K t - Cttutj* r » «qu« i ion u o n not product, m y c r o n o v t r p o l n i i . A l l jou I • - Thomson cot f f I c.l nn t » h*v» nagat lv * i i g n a . »r>d » r * t m n t o r t c o n t U t r i d to D» nor* I n . c c u r i t i than than* of the LHC • qw»l I on. T a b l e 9 9 : LHC e q ( 3 . 3 ) f o r p-Hi RMS-0/. E r r o r / C P U t i m e i n ('/,)/sac LHC e q ( 3 . 3 ) CALC . REGION I REGION I I REGION I I I REGION IV REGION V V 10. 2 3 / 0 106 3 . 6 7 / 0 107 8.97/0 290 1 . 54/0 1 19 18.90/5 774 P 2 2 4 4 3 . 7 0 / 0 00 1 1 . 19/0 00 1 1 7 6 8 . 87/0 OOI 1 . 3 1/0 0 0 0 * 5 1 .08/0 009 T 17 . 16/0 003 0 . 8 1/0 003 2 1.71/0 006 1 . 1 1/0 002 9.46/0 092 Cp 50. 5 8 / 0 000* 59 . 2 9 / 0 0 0 0 * 3 5 . 7 5 / 0 007 50. 13/0 004 1 7 0 1 . 2 0 / 0 079 C v 64 . 4 4 / 0 001 130. 3 7 / 0 00 1 18 1.60/0 006 103 . 3 3/0 005 2 155 . 2 8/0 0 8 0 390 . 8 7 / 0 001 26 . 3 1/0 0 0 0 * 1 160.02/0 006 16 . 6 3/0 004 3.64E 16 /O 082 I Trie c o m p u t e r r o u n d s t h e CPU t i m e t o t h e n o a r e s t m i l l i s e c o n d CO co Tanlo lOO: I. i e I mezs-Howe 1 I-Campbo I I E q u a t i o n C o e f f i c i e n t s COMPOUND P q d m ( r e g i on i l l ) ii ( reg ion I I I ) in ( r e g i o n I V ) n r e g i o n 1V) Me t Hano 0 . 2 'J a 3 0 7 3 0 9 0 . 1273275 1 -0 52600106 -0 0036562 1 17 -5 5592962 0 0 6 1 -1 4 1 6 1 1 E t bane 0 3036379 0 8 1 872B9 0. 16450488 - o 54012257 - 1 0794276 P r o p a n t 0 . 3 2 G 2 0 8370 0. 17577069 -7 4220744 -0 063262845 5 1483243 1 5480747 n-Bu tane 0 3 - 1 1 9 - 1 8 0 85446 1 0 184 12330 - 15 078340 -0 25653637 -3 8904542 -2 4223220 n-Pentane 0 3 5 5 5 3 0 84872 0 18797425 -8 0690685 -0 033330732 - 1 9923638 - 1 6449802 n - Ho x a nt." 0 3 6 1 3 3 1 0 8O0908 0 18349053 - 13 711188 -0 10360140 - 10 507508 -0 0976299 Me thano 1 0 3 6 1 3 3 1 0 735 136 0 222 13833 - 103 649 13 -0 14 146328 - 10 845764 -0 . 1 1507029 t-butano 1 0 5 0 9 9 8 0 89228 0 243927 17 137 89800 2 1 .30099G H< 0 0 5096 7 7 0 720508 0 2 1220687 3 2736112 -0 00014152201 1 1282078 -0 .8948037G H.S 0 3 3 19 1 0 773706 0 16797844 - 12 126348 -0 43 123385 -2 48 12437 - 1 .45G8014 n-H, 0 0-4 i a o 0 26397 0 0178 1888 15 830657 0 018854944 -7 4422 126 - 1.9496076 p-H, 0 0 3 9 1864 0 27676 0 01694486 15 368758 0 0036014809 - 17 189830 -7 .0703367 Neon 0 1 8 7 19 3 0 7404 0 10675371 3 0076908 0 .0075366123 -5 . 2834 198 -3 .21 1000 1 Argon 0 2 4 1 1 0 7668 0 13286672 0 .86287268 -0 .0076551974 -3 . 3463545 - 1 .4 001787 to CO T a b l e l O I : T e m p e r a t u r e R a n g e s ( i n d e g r e e s K e l v i n ) f o r t h e A p p l i c a t i o n o f E q u a t1 o n ( 3 . 3 ) o n Hr, S r . A r . G r . U r a n d f / p COMPOUND r e g i o n I r e g 1 o n 11 r e g 1 o n I I I r e g i o n IV r e g i o n V M e t h a n e 5.0 5.0 10.0 1 .0 L=17.84.U=18.52 E t h a n e 1 .0 1 .0 - 1 .0 L=26.70.U=27.37 P r o p a n e 5 .O 5.0 lO.O 1 .0 L=29.83.U=30.12 n - B u t a n e 1 .O 1 .0 10.o 1 .0 L = 32. 13.U = 3 2 . 0 9 n - P e n t a n e 1 .0 1 .0 - 1 .O L=33.10.U=21.93 n - H e x a n e 10.0 1 .0 - 1 .0 L = 22. 18.U = 22. 10 Me t h a n o 1 1 .0 1 .0 5.0 1 .0 L=5.O.U=5.0 t - b u t a n o I - - - - L= 19.47,U=19. 14 H,0 2 . 6 2.6 2 . 6 2 . 6 L=2.6.U=3.4 H,S - - - - L = 24.80.U=25.88 n-H; O.O 0.0 1 .0 O.O L=25.62.U=3500.0 p - H . - - - - L=25.62.U=3500.0 N e o n - - - - L=6.36.U=6.52 A r g o n 1 .0 1.0 14 . 49 1 .0 L= 14.49.U=15.15 290 N o t e 1: F o r c o m p o u n d s t h a t h a v e n o t b e e n t e s t e d , s e t t h e t e m p e r a t u r e r a n g e b e t w e e n 1.0-5.0 t o p r o v i d e t h e b e s t a c c u r a c y . N o t e 2: B o t h u p p e r a n d l o w e r r a n g e l i m i t s h a v e b e e n p r o v i d e d f o r r e g i o n V. T a b l e 102: T e m p e r a t u r e R a n g e s ( 1 n d e g r e e s K e l v i n ) f o r t h e A p p l i c a t i o n o f E q u a t i o n (3.3) o n Cp a n d C v COMPOUNO r e g i o n I r e g i o n I I r e g i o n I I I r e g i o n IV r e g i o n V M e t h a n e 17 . 84 17.84 17.84 17.84 L= 17.84,U=18.52 E t h a n e - - - 2 6 . 7 0 L » 1 . O . U = 1 . 0 P r o p a n e 29 .83 2 9 . 8 3 2 9 . 8 3 2 9 . 8 3 L=10.0.U=10.0 n - B u t a n e 3 2 . 1 3 32. 13 32. 13 32 . 13 L=10.0,U=10.0 n - P e n t a n e - - - - -n - H e x a n e - - - - -M e t h a n o 1 - - 5.0 - L=S.O.U=S.O t - b u t a n o I - - - - L » 2 . 6 . U = 3 . 4 H.O 2 . 6 2 . 6 2 . 6 2.6 -H.S - - - - -n-H, 5.0 5.0 5.0 S.O L=5.0.U=5.0 p-H, 5.0 5.0 5.0 5.0 L=5.O.U=5.0 N e o n - - - - -A r g o n 1 4 . 49 14.49 14.49 - -N o t e 1: F o r c o m p o u n d s t h a t h a v e n o t b e e n t e s t e d , s e t t h e t e m p e r a t u r e r a n g e b e t w e e n 1.0-5.0 t o p r o v i d e t h e b e s t a c c u r a c y . N o t e 2: B o t h u p p e r a n d l o w e r r a n g e 1 1ml t s h a v e b e e n p r o v i d e d f o r r e g i o n V. T a b l e 1 0 3 : T e m p e r a t u r e R a n g e s ( I n d e g r e e s K e l v i n ) f o r t h e A p p l i c a t i o n o f E q u a t i o n ( 3 . 3 ) on J o u 1 e - T h o m s o n C o e f f i c i e n t s a n d t h e I n v e r s i o n C u r v e COMPOUND ^ - r e g i o n I ^ - r e g i o n 11 ^ - r e g i o n I I I ^ - r e g i o n IV p - r e g i o n V I n v e r s l o n C u r v e Me t n a ne 1 .0 1 . 0 5.0 5 . 0 L - 1 . 0 , U - l . 0 L= 1 7 . 84 , U= 18 . 52 E t n a n e - - - 1 . 0 L=1.0,U= 1 .0 -P r o p a n e - 5.0 - 5 .0 L = 5.0,U=5 .0 L=I0.0,U= 10.0 n - B u t a n e - 5 .0 - 5 .0 - L=10.0.U= 10.0 n - P e n t a n e - - - - - -n-Mexane - - - - - -Me t h a n o 1 - - - - - -t - b u t a n o 1 - - - - - -H i O - - 2 . 6 2.6 L = 2.6 ,U = 3 . 4 -H . S - - - - - -n - H . 20.0 1 .0 20.0 1 .0 L=5,0,U=5 .0 -p - H . 20.0 1 .0 20.0 1 . 0 L = 5.0,11 = 5.0 L=1.0,U =1.0 N e o n - - - 1 .0 L = 6 . 3 6 , U « 6 . 5 2 -A r g o n - - - 5.0 L * 5 . 0 , U « 5 . 0 L = 5.0,U=5.0 N o t e 1: F o r c o m p o u n d s t h a t h a v e n o t b e e n t e s t e d , s e t t h e t e m p e r a t u r e r a n g e b e t w e e n 1.0-5.0 t o p r o v i d e t h e b e s t a c c u r a c y . N o t e 2: B o t h u p p e r a n d l o w e r r a n g e l i m i t s h a v e b e e n p r o v i d e d f o r r e g i o n V a n d t h e I n v e r s i o n c u r v e . rO U3 T a b l e 104: P-V-T I n p u t O a t a f o r R e g i o n I 2 9 2 COMPOUNO NPTS T n b ( K ) T c ( K ) P c ( a t m ) u [ 6 1 ] A T r DATA REF . M e t h a n e 30 1 1 1 .42 190 .55 45 .803 0 .008 0 . 478 - 1 . OOO [ 7 0 ] P r o p a n e 19 231 . 1 1 3 7 0 .O 42 .0823 0 . 152 0 .512 - 1 . o o o [ 7 0 ] n - B u t a n e 18 272 .65 425 . 16 37 . 4 6 3 5 0 . 193 0 642 - 1 . o o o [ 7 0 . 7 1 •1 n - P e n t a n e 17 309 22 469 . 77 33 . 2 987 0 . 25 1 0 667- - 1 . o o o [ 7 0 . 72 '] n - H e x a n e 26 34 1 87 507 . 87 29 . 9 135 O . 2 9 6 O 538- -1 . o o o [ 7 0 . 73 '] Me t h a n o 1 15 337 86 513 15 78 . 4601 0 559 0 542- 1 . o o o [ 7 0 . 74 *] H,0 1 16 373 15 647 3 0 2 18 3 0 7 0 344 0 422- 1 . o o o [ 7 5 ] H,S 32 2 12 875 373 539 88 8 7 3 3 0 100 0 517- 1 . o o o [ 7 6 . 7 7 ] n-H, 24 2 0 38 33 23 12 9 8 7 9 -0 22 0. 42 1 -1 o o o [ 7 0 ] p - H , 23 20 . 28 32 98 12 7 6 3 8 -0 22 0. 424- 1 o o o [ 7 0 ] N e o n 2 1 2 7 . 0 9 44 4 26 1929 0 OOO 0. 5 6 3 - 1 o o o [ 7 0 ] A r g o n 70 8 7 . 29 150. 86 49 3 4 6 - 0 . 0 0 4 0. 5 5 5 - 1 o o o [ 7 0 ] T O T A L 411 * V o l u m e t r i c d a t a b y I n t e r p o l a t i o n u s i n g A i t k e n ' s m e t h o d N o t e : c o e f f i c i e n t s f o r t h e L e e - K e s l e r e q u a t i o n a r e t a k e n f r o m r e f [ 4 0 ] c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 6 1 ] , t a b l e 3-6. T a b l e 105: P-V-T I n p u t D a t a f o r R e g i o n I I COMPOUNO NPTS T n b ( K ) T c ( K ) PC ( a t m ) U [ 6 1 ] A T r DATA R E F . M e t h a n e 30 1 1 1 .42 190 .55 45 .803 0 .008 0 . 4 7 8 - 1 .000 [ 7 0 ] E t h a n e 18 184 .52 305 .5 48 . 4 8 7 4 O .098 0 . 49 1 -1 .OOO [ 7 0 ] P r o p a n e 19 23 1 . 1 1 3 7 0 .0 42 .0823 0 . 152 0 5 1 2 - 1 . OOO [ 7 0 ] n - B u t a n e 33 272 .67 425 . 16 37 . 47 O . 193 0 64 1- 1 .000 [ 7 1 ] n - P e n t a n e 29 309 19 469 .65 33 . 25 O 25 I 0 6 5 8 - 1 .000 [ 7 2 ] n - H e x a n e 44 34 1 89 507 87222 29 92 16 0 296 0 53 8 - 1 .OOO [ 7 3 ] M e t h a n o 1 45 338 0 5 13 15 78 46 O 559 O 5 3 2 - . OOO [ 7 8 ] t - b u t a n o 1 13 356 483 508 872 4 1 769 1 0 618 0 70 1 - .000 [ 7 9 ] H,0 1 16 373 15 647 30 2 18 3 0 7 O 344 0 42 2 - . o o o [ 7 5 ] H, S 32 212 875 373 539 88 8 7 3 3 O 100 0. 5 17- . o o o [ 7 6 . 7 7 ] n-H, 24 20. 38 33 23 12 9 8 7 9 -O 22 0. 42 1 - . o o o [ 7 0 ] p-H, 23 20. 28 32 98 12 7 6 3 8 - o . 22 0. 42 4 - . o o o [ 7 0 ] N e o n 2 1 27 . 0 9 44 . 4 26. 1929 0. OOO 0. 4 7 3 - .000 [ 7 0 ] A r g o n 70 87 . 29 150. 86 49 . 346 - o . 004 o . 555-1 . o o o ( 7 0 ) T OTAL 517 N o t e : c o e f f i c i e n t s f o r t h e L e e - K e s l e r e q u a t i o n a r e t a k e n f r o m r e f [ 4 0 J c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 6 1 ] . t a D l o 3 - 6 T a b l e 106 : P - v - T I n p u t D a t a f o r R e g i o n I I I COMPOUND NPTS Tnb ( K ) Tc K ) P c a t m ) u [61] A T r A P r DATA R E F . S A T ' N R E F . Me t h a n e 156 1 1 1 42 190 55 45 803 0 ooa 0 . 499 - 0 . 945 0 . 0 0 4 3 - 0 862 [70] [70] P r o p a n e 1 19 23 1 1 1 370 0 42 0823 0 152 0 622 -0 9 19 0 0 2 3 5 - 0 938 [70] [70] n - B u t a n e 16 272 65 425 16 37 4635 0 193 0 690 -0 925 0 527 - 0 7 90 [70] [70] n - P e n t a n e 24 309 22 469 .77 33 . 2987 0 251 0 624 -0 837 0 296 - 0 889 [70] [70] n - H e x a n e 23 34 1 87 507 .87 29 . 9 135 0 296 0 676 - 0 95 1 0 330 -O 990 [70) [70] Me t hano1 46 337 86 5 1 3 . 15 78 . 4601 0 559 0 58 1 - 0 953 0 0 6 4 7 - 0 836 [74] [78] H i 0 256 373 15 647 .30 2 18 . 307 0 344 0 422 -0 978 0 00004 5 -0 .904 [75] (75] H i S 17 2 12 875 373 .53889 aa .8733 0 100 0 743 - 0 922 0 153 -0 957 [76] [ 7 6 , 7 7 ] n - H i 45 20 38 33 . 23 12 . 9879 - o 22 0 434 - o 993 0 0 7 6 2 - 0 9 12 [70] [70] p - H i 46 20 28 32 .98 12 .7638 - o 22 0 424 - 0 970 0 0 7 7 3 - 0 928 [70] [70] N e o n 26 27 09 44 . 4 26 . 1929 0 000 0 563 -0 901 0 0 3 7 7 - 0 942 [70] [70] A r g o n 103 87 29 150 .86 49 .346 - 0 004 0 563 -0 994 0 0 2 0 0 - 1 000 [70] [70] TOTAL 877 N o t e : c o e f f i c i e n t s f o r t h e L e e - K e s l e r e q u a t i o n a r e t a k e n f r o m r e f [40] c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 6 1 ] , t a b l e 3-6 VJO CO T a b l e 107: P-V-T I n p u t D a t a f o r R e g i o n IV COMPOUND NPTS Tnb ( K ) T c ( K ) P c ( a t m ) o [61 ] A T r A P r DATA REF. SAT'N R E F . M e t h a n e 130 111. 42 190. 55 45 803 0. 008 0 . 5 2 5 - 0 . 997 0. 0 0 4 3 - 0 . 8 6 2 [ 7 0 ] [ 7 0 ] E t h a n e 100 184 . 52 305 5 48 4874 0. 098 0 . 6 5 5 - 0 . 982 0. 0 2 0 3 - 0 . 8 14 [ 7 0 ] [ 7 0 ] P r o p a n e 13 1 23 1 1 1 370 0 42 082 3 0. 152 0 . 6 2 2 - 0 . 973 0. 0 0 2 3 - 0 . 8 2 1 [ 7 0 ] [ 7 0 ] n - B u t a n e 53 272 67 425 16 37 4 7 0 0 0. 193 0 . 6 5 9 - 0 . 997 0 0 2 6 7 - 0 . 9 6 1 [ 7 1 ] [71 ] n - P e n t a n e 32 309 19 469 65 33 2500 0. 251 0 . 6 6 0 - 0 979 0 0 3 0 1 - 0 . 8 4 2 [ 7 2 ] [ 7 2 ] n - H e x a n e 148 34 1 89 507 87222 29 9216 0 296 0 . 6 7 8 - 0 995 0 . 0 0 2 3 - 0 . 9 1 0 [ 7 3 ] [ 7 3 ] Me t hano1 56 338 0 5 13 15 78 4 6 0 0 0 559 0 . 6 6 0 - 0 985 0 0 0 0 8 6 - 0 . 8 6 7 [ 7 8 ] [ 7 8 ] t - b u t a n o l 76 356 4 8 3 3 3 508 87222 4 1 7691 0.6 18 6.720-0 993 0 0 2 3 9 - 0 . 5 7 0 [ 7 9 ] [ 7 9 ] H i O 409 373 15 647 30 2 18 307 0 344 0 . 4 5 3 - 0 978 0 0 0 0 0 4 5 - 0 . 6 7 8 [ 7 5 ] [ 7 5 ] H i S 37 2 12 875 373 5 3 8 8 9 88 .8733 0 100 0 . 7 4 3 - 0 922 0 01 1 2 - 0 . 4 5 9 [ 7 6 ] [ 7 6 . 7 7 ] n-H: 19 20 38 33 23 12 .9879 -0 22 0 . 6 6 2 - 0 963 0 0 7 6 0 - 0 . 7 6 0 [ 7 0 ] [ 7 0 ] p - H : 20 20 . 28 32 . 98 12 .7638 -0 22 0 . 6 6 7 - 0 9 7 0 0 . 0 7 7 3 - 0 . 7 7 3 [ 7 0 ] [ 7 0 ] N e o n 18 27 . 09 44 . 4 26 . 1929 0 0 0 0 0 . 5 6 3 - 0 .901 0 .00 3 8 - 0 . 3 7 7 [ 7 0 ] [ 7 0 ] A r g o n 101 87 . 29 150 . 86 49 .346 -0 004 0 . 0 0 2 - 0 .990 0 .597-0.994 [ 7 0 ] [ 7 0 ] TOTAL 1330 N o t e : c o e f f i c i e n t s f o r t h e L e e - K e s l e r e q u a t i o n a r e t a k e n f r o m r e f [ 4 0 ] c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 6 1 ] , t a b l e 3-6 T a b l e 108: P-V-T I n p u t D a t a f o r R e g i o n V COMPOUND NPTS Tnb ( K ) T c ( K ) P c ( a t m ) u [61 ] A T r A P r DATA REF. Me t h a n e 9 19 111.42 190.55 45.81 0.008 0. 4 9 9 - 5 .248 0 . 0 0 4 3 1 - 2 1 . 5 4 7 [ 7 0 ] E t h a n e 7 6 0 184.52 305 . 5 48 .49 0.09B 1.015-1 .637 0 . 0 2 0 4 - 1 0 . 1 7 7 [ 7 0 ] P r o p a n e 408 231.11 369 . 99 42 .08 0. 152 1 .000-1 .622 0 . 0 0 2 3 5 - 1 4 . 0 7 1 [ 7 0 ] n-Bu t a n o 274 2 7 2 . 6 5 425 . 16 37 .47 0. 193 0 . 7 3 1 - 1 . 2 0 2 1 .054- 17.914 [ 7 0 ] n - P e n t a n e * 136 5 1 3 0 9.22 309 . 19 469 .77 469 . 65 33 . 30 33 . 25 0. 25 1 0 . 6 2 4 - 0 . 8 3 7 1 .001 - 1 .2 7 8 1 . 186-5 . 928 0 . 0 3 0 1 - 2 1 . 0 5 3 [ 7 0 , 7 2 ] n - H e x a n e * 153 23 1 34 I ,89 3 4 1 . 8 9 507.87 5 0 7 . 8 7 2 2 2 29 . 92 29 . 92 0. 296 0 . 6 7 6 - 0 . 95 1 1 . 0 1 2 - 1 . 3 2 0 1 .320-9 . 898 0 . 0 0 2 2 7 - 1 .365 [ 7 0 , 7 3 ] Me t hano1 1 18 3 3 7 . 8 6 5 13. 15 78 .46 0.559 0 . 5 8 1 - 0 . 9 5 3 1 .271 - 13.062 [74 ] t - b u t a n o l 26 356 .48 5 0 8 . 8 7 2 2 2 4 1 . 769 1 0.6 18 1 .04 1- 1 .096 0 . 0 2 3 9 - 0 . 6 5 2 [ 7 9 ] H i O 8 10 373 . 15 647 . 30 2 18.31 0. 344 0 . 4 2 2 - 1 . 6 5 8 0.00004 52-4 .52 1 [ 7 5 ] H i S 150 212.88 3 7 3 . 5 3 8 8 9 8 8 . 8 7 3 3 0 . 100 0. 188- 1 . 598 0.01 13-7.657 [ 7 6 ] n-H. 7 19 20 . 38 33.23 12 . 99 -0.22 0.48 1- 15.047 1 .064-75 . 988 [ 7 0 ] p-H. 7 17 20. 28 32 .98 12 . 76 -0. 22 0 . 4 8 5 - 15 . 16 1 0 . 0 7 7 3 - 7 7 . 3 2 2 [ 7 0 ] N e o n 5 13 27 .09 44 . 4 26 . 19 0.000 0 . 6 7 6 - 6 . 7 5 7 0 . 0 0 3 7 7 - 7 . 5 3 6 [ 7 0 ] A r g o n 1320 87 . 29 150.86 49 . 35 -0.004 0. 597-8 . 6 17 0 . 0 2 0 0 - 2 0 . 0 0 0 [ 7 0 ] TOTAL 7305 * Two d i f f e r e n t d a t a s o u r c e s w e r e t o c o v e r r e g i o n V more e x t e n s i v e l y N o t e : c o e f f i c i e n t s f o r t h e L e e - K e s l e r e q u a t i o n a r e t a k e n f r o m r e f [ 4 0 ] c o e f f i c i e n t s f o r t h e BWR e q u a t o n a r e t a k e n f r o m r e f [ 6 1 ] , t a b l e 3-6 cn T a b l e 109: Thermodynamic P r o p e r t y Input Data f o r Region I COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) u 6 1 ] mo 1ec. wt . [61] ATr DATA REF . Methane 30 1 1 1 42 190 65 4640845 0 0 00009916644 [70] 0 008 16 043 0 586- 1 000 [73-*] E thane 28 184 52 305 38 4893881 7 0 00015825860 0 098 30 070 0 604- 1 000 [73**] P ropane 32 23 1 1 1 369 .96 4257290 9 0 00020043223 0 152 44 097 0 S25- 1 000 [73**] n-Butane 10 272 67 425 . 16 3796647 750 0 0002 54 90000 0 193 58 124 0 659-0 997 (71 ] n-Pentane 9 309 19 469 .65 3369056 25 0 00030400000 0 . 25 1 72 15 1 0 660-0 979 (72] n-Hexane 44 34 1 89 507 .87222 30316 1 4 2 0 00036856562 0 296 86 178 0 538- 1 000 [73] Me thano1 45 337 86 5 13 . 15 79634 1 5 9 - 0 . 559 32 04 2 0 532- 1 000 [78*] H;0 1 16 373 15 647 . 30 22120000 0 - 0 . 344 18 015 0 422-1 000 [73**,75] n - H i 28 20 268 33 . 180 13 13000 0 0 00006414912 -o .22 2 016 0 420- 1 000 [82* ] Argon 69 87 30 150 .86 4898000 0 0 000074570932 -0 .004 39 .948 0 555- 1 000 [83*] TOTAL 411 f u g a c i t i e s c a l c u l a t e d u s i n g thermodynamic i d e n t i t y f u g a c i t i e s i n t e r p o l a t e d Note: c o e f f i c i e n t s , D1j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 rO T a b l e 110: Thermodynamic P r o p e r t y Input Data f o r Region II COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e ) u [61 ] mo 1ec. wt . [6 1] ATr DATA REF. Me thane 13 1 1 1 42 190 . 55 4641000 0 0 00009884092 [70] 0 008 16 04 3 0 577-0 997 [70,73**,84] E thane 19 184 52 305 . 5 49 13000 0 0 00014171990 0 098 30 070 0 589-0 995 [70.73**,84 ] Propane 18 23 1 1 1 370 .0 4264000 0 0 0019598670 0 152 44 097 0 622-0 994 [70.73** , 84 ] n-Butane 10 272 67 425 . 16 3796647 750 0 00025490000 0 193 58 1 24 0 659-0 997 (71 ] n-Pentane 9 309 19 469 . 65 3369056 25 0 00030400000 0 25 1 72 151 0 660-0 979 [72] n-Hexane 44 34 1 89 507 . 87222 30316 14 2 0 00036856562 0 296 87 178 0 582- 1 000 [73] Methano1 45 337 86 513 . 15 79634 15 9 - 0 559 32 042 0 532-1 000 [78*] HrO 1 16 373 15 647 . 30 22120000 0 - 0 344 18 015 0 422- 1 000 [73**,75] n-H. 28 20 268 33 . 180 1313000 0 0 000064 14912 -0 . 22 2 016 0 420- 1 000 [82*] Argon 69 87 30 150 . 86 4898000 0 0 000074570932 -0 004 39 948 0 555- 1 000 [83*] TOTAL 371 * f u g a c i t i e s c a l c u l a t e d u s i n g thermodynamic i d e n t i t y ** f u g a c i t i e s i n t e r p o l a t e d Note: c o e f f i c i e n t s f o r the BACK e q u a t i o n , D i j and C, are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [ 4 0 ] , t a b l e 3-6 ro T a b l e 111: Thermodynamic P r o p e r t y Input Data f o r Region III COMPOUND NPTS Tnb' (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e ) u [61 ] mo 1ec. Wt. [6 1 ] ATr APr DATA REF. Methane 15G 1 1 1 42 190 . 55 4641000 0 0. 00009884092 0 .008 16 043 0 499 -0 945 0 00431-0.862 [70] Propane 12 1 23 1 1 1 370 .0 4264000 0 0. 0019598670 0 . 152 44 097 0 622 -0 973 0 0235-0.938 [70] n-Butane 599 272 638 425 . 16 3796000 0 0. 00025510000 0 . 193 58 124 0 317 -0 988 0 00263-1.000 [80] Methanol 32 337 86 513 . 15 7963415 9 - 0 . 559 32 042 0 58 1 -o 922 0 0126-0.628 [74] H.O 256 373 15 647 . 30 22120000 0 - 0 . 344 18 015 0 422 -0 978 4 52E-5-0.904 [75] n-H. 217 20 390 33 . 180 1313000 0 0. 00006414560 -0 . 22 2 016 0 422 -0 964 0 00774-1.000 [82] Argon 124 87 30 150 . 86 4898000 0 0. 000074570932 -0 .004 39 948 0 563 -o 961 0 0163-0.918 [83] TOTAL 1505 Note: c o e f f i c i e n t s f o r the BACK e q u a t i o n , D i j and C, are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [40], t a b l e 3-6 CO T a b l e 112: Thermodynamic P r o p e r t y Input Data f o r Region IV COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) ( J [61 ] mo 1ec. Wt . [6 1 ] ATr APr DATA R E F . Methane 83 1 1 1 42 190 55 4641000 0 0 00009884092 0 008 16 043 0 630 -0 997 0 0218-0.961 [70. 73* *.84 ] E thane 107 184 52 305 5 4913000 0 0 00014171990 0 098 30 070 0 622 -0 995 0 0206-0.949 [70. 73* *,84] Propane 96 231 1 1 370 .0 4264000 0 0 0019598670 0 152 44 097 0 649 -0 995 0 0238-0.951 [70. 73** ,84 ] n-Butane 53 272 67 425 . 16 3796647 750 0 00025490000 0 193 58 1 24 0 659 -0 997 0 0267-0.961 [71] n-Pentane 32 309 19 469 .65 3369056 25 0 00030400000 0 251 72 151 0 660 -0 979 0 0301-0.842 [72] n-Hexane 147 34 1 89 507 . 87222 3031614 2 0 00036856562 0 . 296 87 178 0 678 -0 995 0 00227-0.910 [73] Methanol 56 337 86 513 . 15 79634 15 9 - 0 . 559 32 042 0 660 -0 985 5 89E-5-0.0589 [78* ] H*0 409 373 15 647 . 30 22120000 0 - 0 . 344 18 015 0 453 -0 978 4 52E-5-0.678 [73* *,75] n-H; 1 10 20 390 33 . 180 1313000 0 0 000064 14912 -0 . 22 2 016 0 452 -0 964 0 00774-0.774 [82* ] Argon 253 87 30 150 . 86 4898000 0 0 000074570932 -0 .004 39 948 0 563 -0 994 0 00204-0.919 [83* ] TOTAL 1346 * f u g a c i t i e s c a l c u l a t e d u s i n g thermodynamic i d e n t i t y ** f u g a c i t i e s i n t e r p o l a t e d Note: c o e f f i c i e n t s f o r the BACK e q u a t i o n , D i j and C, are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [40], t a b l e 3-6 T a b l e 113: T h e r m o d y n a m i c P r o p e r t y I n p u t D a t a f o r R e g i o n V COMPOUND NPTS T n b (K) T c ( K ) Pc ( P a ) Vc (cu.m/mo1e) u [ S I ] m o l e c . W t . [ 6 1 ] A T r A P r DATA REF. Me t h a n e 2 169 111.42 190.55 464 1000.0 0 . 0 0 0 0 9 8 8 4 0 9 2 0.008 16.04 3 0 . 4 9 9 - 7 . 8 7 2 0 . 0 2 1 8 - 2 1 . 8 3 3 [ 7 0 , 7 3 * *,84 I E t h a n e 2 0 8 5 184 . 52 305 . 5 4 9 1 3 0 0 0 . 0 0 . 0 0 0 1 4 1 7 1 9 9 0 0.098 3 0 . 0 7 0 1 .002-4.9 10 0 . 0 2 0 6 - 1 0 . 3 1 2 [ 7 0 , 7 3 * * , 8 4 ] P r o p a n e 2 5 1 3 231 . 1 1 3 7 0 . 0 4 2 6 4 0 0 0 . 0 0 . 0 0 1 9 5 9 8 6 7 0 0. 152 44. 0 9 7 1.005-4.054 0 . 0 0 2 3 5 - 6 . 6 5 4 [ 7 0 , 7 3 * * , 8 4 ] n - B u t a n e 1025 272 .638 272 . 6 7 425 . 16 3796647 . 75 3 7 9 6 0 0 0 . 0 0 . 0 0 2 5 5 1 0 0 0 0 0 0 . 0 0 0 2 5 4 9 0 0 0 0 0. 193 58.124 0 . 3 1 9 - 0 . 9 8 8 1.002-1.646 1 . 0 5 4 - 1 8 . 4 4 0 0 . 0 0 2 6 3 - 1 8 . 6 8 9 [ 7 1 , 8 0 ] n - P e n t a n e 51 3 0 9 . 1 9 4 6 9 . 6 5 3369056 . 25 0.0OO3O4OOO00 0. 251 72.151 1.001-1.278 0 . 0 3 0 1 - 2 1 . 0 5 3 [ 7 2 ] n - H e x a n e 231 341 .89 5 0 7 . 8 7 2 2 2 3031614 . 2 0 . 0 0 0 3 6 8 5 6 5 6 2 0. 296 87 . 178 1.006-1.159 0 . 0 0 2 2 7 - 1 . 3 6 5 [ 7 3 ] M e t h a n o l 100 3 3 7 . 8 6 5 1 3 . 1 5 7 9 6 3 4 1 5 .9 - 0. 559 32.042 0 . 5 8 1 - 0 . 9 2 2 1 . 2 5 6 - 1 2 . 5 5 7 [ 7 8 * ] H.O 8 10 3 7 3 . 1 5 6 4 7 . 3 0 22 120000.0 - 0. 344 18.015 0.422-1 .658 4 . 5 2 E - 5 - 4 . 5 2 1 [ 7 5 * ] n-H; 4401 2 0 . 3 9 0 3 3 . 1 8 0 1 3 1 3 0 0 0 . 0 0.000064 149 12 -0. 22 2.016 0 . 4 5 2 - 9 0 . 4 1 6 7 . 7 4 E - 4 - 7 7 . 3 5 1 [ 8 2 * ] A r g o n 2668 87 . 30 150.86 4 8 9 8 0 0 0 . 0 0 . 0 0 0 0 7 4 5 7 0 9 3 2 -0.O04 39.948 0 . 5 9 7 - 7 . 2 9 2 0 . 0 0 2 0 4 - 2 0 . 4 1 6 [ 8 3 * ] TOTAL 16053 * f u g a c i t i e s c a l c u l a t e d u s i n g t h e r m o d y n a m i c i d e n t i t y ** f u g a c i t i e s i n t e r p o l a t e d N o t e : c o e f f i c i e n t s f o r t h e BACK e q u a t i o n , D i j a n d C, a r e t a k e n f r o m r e f e r e n c e s [ 4 5 , 4 6 ] c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 4 0 ] , t a b l e 3-6 00 o o 301 T a b l e 114: E n t h a l p y a n d E n t r o p y D e p a r t u r e F u n c t i o n D a t a - r e g i o n I COMPOUNO H b a s e ( J / m o 1 e ) S b a s e ( J / m o l e - K ) R e f . C o n d i t i o n H". S 3 r e f . Me t h a n e - - - [ 7 3 ] E t h a n e - - - [ 7 3 ] P r o p a n e - - - [ 7 3 ] n - B u t a n e - - - -n - P e n t a n e - - - -n - H e x a n e - - - -M e t h a n o 1 - 2 2 0 5 8 5 . 1 6 3 1 4 0 3 3 1 0 3 . 0 4 0 0 0 0 9 5 s a t . V a t 298 K [ 7 8 . 8 2 " ] H.O - 2 7 4 9 3 7 . 7 9 2 3 2 4 4 3 6 3 . 6 6 2 4 7 3 4 9 s a t . L a t 2 7 3 . 5 K n-H, - 5 9 . 1 8 4 4 2 8 6 1 1 2 . 0 6 2 7 3 5 1 5 f l u i d a t 133.15K.1 a t m . [ 7 3 ] A r g o n - 7 6 9 0 . 7 7 6 9 7 6 4 7 . 5 0 4 9 8 1 2 0 b a c k c a l c . U s i n g U r a n d A r r e s u 1 t s o f r e f . [ 4 5 ] [ 7 3 ] N o t e : H r = ( H - H r e f ) - ( H b a s e - H r e f ) + H b a s e - H ° = H - H ° S r = ( S - S r e f ) - ( S b a s e - S r e f ) + S b a s e - S 0 = S - S ° I n many I n s t a n c e s t h e d e p a r t u r e f u n c t i o n s H r a n d S r w e r e unknown, w h i l e t h e l i t e r a t u r e p r e s e n t s e n t h a l p i e s a n d e n t r o p i e s a s ( H - H r e f ) a n d ( S - S r e f ) . U n f o r t u n a t e l y t h e r e s e a r c h e r s w o u l d s o m e t i m e s n e g l e c t t o s t a t e t h e v a l u e s o f H r e f a n d S r e f . I t was t h e r e f o r e n e c e s s a r y t o r e d e f i n e b o t h t h e e n t h a l p i e s a n d e n t r o p i e s . T h i s was d o n e b y a r b i t r a r i l y s e l e c t i n g a v a l u e o f ( H b a s e - H r e f ) a n d ( S b a s e - S r e f ) f r o m t h e same r e f e r e n c e s u c h t h a t H b a s e a n d S b a s e w e r e known. F i n a l l y t h e I d e a l g a s e n t h a l p y a n d e n t r o p y a t t h e c o r r e s p o n d i n g t e m p e r a t u r e s w e r e d e t e r m i n e d . B y s o l v i n g t h e two e q u a t i o n s a b o v e , t h e r e s i d u a l f u n c t i o n s w e r e s e t u p f o r u s e i n t h i s w o r k . * T h e i d e a l g a s e n t h a l p y a t 0 K 13 a l s o r e q u i r e d f r o m r e f [ 8 2 ] , t a b l e 3. 302 T a b l e 115: E n t h a l p y a n d E n t r o p y D e p a r t u r e F u n c t i o n D a t a - r e g i o n I I COMPOUNO H b a s e ( J / m o 1 e ) S b a s e ( J / m o l e - K ) R e f . C o n d i t i o n H ° . S" r e f . Me t h a n e - - - -E t h a n e - - - -P r o p a n e - - - -n - B u t a n e - - - -n - P e n t a n e - - - -n - H e x a n e - - - -M e t h a n o 1 - 2 2 0 5 3 5 . 1 6 3 1 4 0 3 3 103.04OO009S s a t . V a t 298 K [ 7 8 . 8 2 * ] H,0 - 2 7 4 9 3 7 . 7 9 2 3 2 4 4 3 6 3 . 6 6 2 4 7 3 4 9 s a t . L a t 2 7 3 . 5 K n-H, -59. 1 8 4 4 2 3 6 1 1 2 . 0 6 2 7 3 5 1 5 f l u i d a t 133.15K.1 a t m . [ 7 3 ] A r g o n - 7 6 9 0 . 7 7 6 9 7 6 4 7 . 5 0 4 9 8 1 2 0 b a c k c a l c . U s i n g U r a n d A r r e s u 1 t s o f r e f . [ 4 5 ] [ 7 3 ] N o t e : H r = ( H - H r e f ) - ( H b a s e - H r e f ) + H b a s e - H ' = H - H ° S r = ( S - S r e f ) - ( S b a s e - S r e f ) + S b a s e - S ° = S - S ° I n many i n s t a n c e s t h e d e p a r t u r e f u n c t i o n s H r a n d S r w e r e u n k n o w n , w h i l e t h e l i t e r a t u r e p r e s e n t s e n t h a l p i e s a n d e n t r o p i e s a s ( H - H r e f ) a n d ( S - S r e f ) . U n f o r t u n a t e l y t h e r e s e a r c h e r s w o u l d s o m e t i m e s n e g l e c t t o s t a t e t h e v a l u e s o f H r e f a n d S r e f . I t was t h e r e f o r e n e c e s s a r y t o r e d e f i n e b o t h t h e e n t h a l p i e s a n d e n t r o p i e s . Th<3 was d o n e b y a r b i t r a r i l y s e l e c t i n g a v a l u e o f ( H b a s e - H r e f ) a n d ( S b a s e - S r e f ) f r o m t h e same r e f e r e n c e s u c h t h a t H b a s e a n d S b a s e w e r e known. F i n a l l y t h e i d e a l g a s e n t h a l p y a n d e n t r o p y a t t h e c o r r e s p o n d i n g t e m p e r a t u r e s w e r e d e t e r m i n e d . By s o l v i n g t h e two e q u a t i o n s a b o v e , t h e r e s i d u a l f u n c t i o n s w e r e s e t up f o r u s e i n t h i s w o r k . * T h e i d e a l g a s e n t h a l p y a t 0 K i s a l s o r e q u i r e d f r o m r e f [ 8 2 ] , t a b 1e 3. T a b l e 116: E n t h a l p y a n d E n t r o p y D e p a r t u r e F u n c t i o n D a t a - r e g i o n I I I COMPOUND H b a s e ( j / m o 1 e ) S b a s e ( J / m o l e - K ) R e f . C o n d i t1 o n H ° . S° r e f . M e t h a n e -83068 7 8 8 5 8 166 - 1 . 19201271 s a t . L a t 133.15K [ 7 3 ] P r o p a n e - 107630 8 5 8 6 4 9 4 6 0 9 7 5 0 2 7 8 8 s a t . L a t 5 0 F [ 7 3 ] n - B u t a n e - 121699 7 5 0 1 3 9 6 0 0 08 167925 s a t . L a t 5 0 F [ 7 3 ] M e t h a n o l - 192094 5 7 0 8 0 0 0 0 0 0 p e r f e c t c r y s t a l a t OK [ 7 8 ] H,0 -274937 7 9 2 3 2 4 4 3 63 6 6 2 4 7 3 4 9 s a t . L a t 2 7 3 . 5 K n-H, -59 1844286 1 12 0 6 2 7 3 5 15 f l u i d a t 133.15K.1 a t m . [ 7 3 ] A r g o n - 7690 7 7 6 9 7 6 47 5 0 4 9 8 1 2 0 b a c k c a l c . U s i n g U r a n d A r r e s u 1 t s o f r e f . [ 4 5 ] [ 7 3 ] N o t e : H r = ( H - H r e f ) - ( H b a s e - H r e f ) + H b a s e - H ° = H - H " S r = - ( S - S r e f ) - ( S b a s e - S r e f ) * S b a s e - S 0 =S-S 0 I n many I n s t a n c e s t h e d e p a r t u r e f u n c t i o n s H r a n d S r w e r e u n k n o w n , w h i l e t h e l i t e r a t u r e p r e s e n t s e n t h a l p i e s a n d e n t r o p i e s a s ( H - H r e f ) a n d ( S - S r e f ) . U n f o r t u n a t e l y t h e r e s e a r c h e r s w o u l d s o m e t i m e s n e g l e c t t o s t a t e t h e v a l u e s o f H r e f a n d S r e f . I t was t h e r e f o r e n e c e s s a r y t o r e d e f i n e b o t h t h e e n t h a l p i e s a n d e n t r o p i e s . T h i s was d o n e b y a r b i t r a r i l y s e l e c t i n g a v a l u e o f ( H b a s e - H r e f ) a n d ( S b a s e - S r e f ) f r o m t h e same r e f e r e n c e s u c h t h a t H b a s e a n d S b a s e w e r e known. F i n a l l y t h e I d e a l g a s e n t h a l p y a n d e n t r o p y a t t h e c o r r e s p o n d i n g t e m p e r a t u r e s w e r e d e t e r m i n e d . B y s o l v i n g t h e two e q u a t i o n s a b o v e , t h e r e s i d u a l f u n c t i o n s w e r e s e t u p f o r u s e i n t h i s w o r k . 304 T a b l e 117: E n t h a l p y a n d E n t r o p y D e p a r t u r e F u n c t i o n O a t a - r e g i o n IV COMPOUNO H b a s e ( J / m o 1 e ) S b a s e ( j / m o l e - K ) R e f . C o n d i t i o n H ° . S° r e f . Me t h a n e - - - -E t h a n e - - - -P r o p a n e - - - -n - B u t a n e - - - -n - P e n t a n e - - - -n - H e x a n e - - - -Me t h a n o 1 - 2 2 0 5 8 5 . 1 6 3 1 4 0 3 3 1 0 3 . 0 4 0 0 0 0 9 5 s a t . V a t 298 K ( 7 8 . 8 2 * ] H.O - 2 7 4 9 3 7 . 7 9 2 3 2 4 4 3 6 3 . 6 6 2 4 7 3 4 9 s a t . L a t 2 7 3 . 5 K n-H< - 5 9 . 1 8 4 4 2 8 6 1 12.062735 15 f l u i d a t 133.15K.1 a t m . ( 7 3 ] A r g o n - 7 6 9 0 . 7 7 6 9 7 S 4 7 . 5 0 4 9 8 1 2 0 b a c k c a l c . U s i n g U r a n d A r r e s u l t s o f r e f . ( 4 5 ] ( 7 3 ] N o t e : H r = ( H - H r e f ) - ( H b a s e - H r e f ) + H b a s e - H ° = H - H ° S r = ( S - S r e f ) - ( S b a s e - S r e f ) + S b a s e - S 0 = S - S ° I n many i n s t a n c e s t h e d e p a r t u r e f u n c t i o n s H r a n d S r w e r e u n k n o w n , w h i l e t h e l i t e r a t u r e p r e s e n t s e n t h a l p i e s a n d e n t r o p i e s a s ( H - H r e f ) a n d ( S - S r e f ) . U n f o r t u n a t e l y t h e r e s e a r c h e r s w o u l d s o m e t i m e s n e g l e c t t o s t a t e t h e v a l u e s o f H r e f a n d S r e f . I t was t h e r e f o r e n e c e s s a r y t o r e d e f i n e b o t h t h e e n t h a l p i e s a n d e n t r o p i e s . T h i s was d o n e b y a r b i t r a r i l y s e l e c t i n g a v a l u e o f ( H b a s e - H r e f ) a n d ( S b a s e - S r e f ) f r o m t h e same r e f e r e n c e s u c h t h a t H b a s e a n d S b a s e w e r e k n o w n . F i n a l l y t h e I d e a l g a s e n t h a l p y a n d e n t r o p y a t t h e c o r r e s p o n d i n g t e m p e r a t u r e s w e r e d e t e r m i n e d . By s o l v i n g t h e two e q u a t i o n s a b o v e , t h e r e s i d u a l f u n c t i o n s w e r e s e t u p f o r u s e In t h i s w o r k . * T h e i d e a l g a s e n t h a l p y a t O K i s a l s o r e q u i r e d f r o m r e f [ 8 2 ] , t a b l e 3. 305 T a b l e 118 E n t h a l p y a n d E n t r o p y D e p a r t u r e F u n c t i o n O a t a - r e g i o n V COMPOUNO H b a s e ( J / m o 1 e ) S b a s e ( J / m o l e - K ) R e f . C o n d i t i o n H". S ' r e f . Me t h a n e - - - -E t h a n e - - - -P r o p a n e - - - -n - B u t a n e - - - -n - P e n t a n e - - - -n - H e x a n e - - - -M e t h a n o I - 1 9 2 0 9 4 . 5 7 0 8 0 0 O 0 0.0 p e r f e c t c r y s t a l a t OK [ 7 8 ] H,0 - 2 7 4 9 3 7 . 7 9 2 3 2 4 4 3 6 3 . 6 6 2 4 7 3 4 9 s a t . L a t 273.5 K. [ 7 3 ] n-H, - 5 9 . 1 8 4 4 2 8 6 1 1 2 . 0 6 2 7 3 5 1 5 f l u i d a t 133.15K.1 a t m . [ 7 3 ] A r g o n - 7 6 9 0 . 7 7 6 9 7 6 4 7 . 5 0 4 9 3 120 b a c k c a 1 c . U s i n g U r a n d A r r e s u 1 t s o f r e f . [ 4 5 ] [ 7 3 ] N o t e : H r = ( H - H r e f ) - ( H b a s e - H r e f ) * H b a s e - H " = H - H ° S r = ( S - S r e f ) - ( S b a s e - S r e f ) + S b a s e - S 0 = S - S 0 I n many i n s t a n c e s t h e d e p a r t u r e f u n c t i o n s H r a n d S r w e r e u n k n o w n , w h i l e t h e l i t e r a t u r e p r e s e n t s e n t h a l p i e s a n d e n t r o p i e s a s ( H - H r e f ) a n d ( S - S r e f ) . U n f o r t u n a t e l y t h e r e s e a r c h e r s w o u l d s o m e t i m e s n e g l e c t t o s t a t e t h e v a l u e s o f H r e f a n d S r e f . I t was t h e r e f o r e n e c e s s a r y t o r e d e f i n e b o t h t h e e n t h a l p i e s a n d e n t r o p i e s . T h i s was d o n e b y a r b i t r a r i l y s e l e c t i n g a v a l u e o f ( H b a s e - H r e f ) a n d ( S b a s e - S r e f ) f r o m t h e same r e f e r e n c e s u c h t h a t H b a s e a n d S b a s e w e r e known. F i n a l l y t h e i d e a l g a s e n t h a l p y a n d e n t r o p y a t t h e c o r r e s p o n d i n g t e m p e r a t u r e s w e r e d e t e r m i n e d . By s o l v i n g t h e two e q u a t i o n s a b o v e , t h e r e s i d u a l f u n c t i o n s w e r e s e t up f o r u s e i n t h i s work. T a b l e 119: Cp and Cv P r o p e r t y Input Data f o r Region I COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) 0 [61 ] mo 1ec. wt . [61 ] ATr DATA REF. Methane 29 1 1 1 S3 1 190 . 555 4595000 0 0.000098917 0 008 16 043 0 509 -0 996 [85] P ropane 57 23 1 068 369 . 85 4247500 0 0.0001999967 0 152 44 097 0 23 1 -0 987 [86] n-Butane 58 272 638 425 . 16 3796000 0 0.0002550977 0 . 193 58 124 0 3 1 7 -0 988 [80] H , 0 6 1 373 1 5 647 . 15 22055000 0 - 0 . 344 18 0 1 5 0 422 -0 998 [87] n-H i 2 1 20 390 33 . 180 1313000 0 0.00006414912 -o . 22 2 0 1 6 0 420 -0 964 (82] p-H, 22 20 268 32 .976 1292800 0 - -0 .22 2 0 1 6 0 4 18 -1 000 [82] Argon 18 87 30 150 . 86 4897900 0 0.000074570932 -0 .004 39 948 0 556 -0 920 [83] T o t a l 266 Note: c o e f f i c i e n t s , D1j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 Tab l e 120: Cp and Cv P r o p e r t y Input Data f o r Region II COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) o [61 ] mo 1ec. wt . [61] ATr DATA REF, Methane 29 1 1 1 63 1 190 . 555 4595000 0 0 0000989 17 0 .008 16 043 0 509 -0 996 [85] Propane 24 231 068 369 . 85 4247500 0 0 0001999967 0 . 152 44 09 7 0 512 -0 972 [86] n-Butane 25 272 638 425 . 16 3796000 0 0. 0002550977 0 . 193 58 1 24 0 527 -0 993 [80] H i O 6 1 373 15 " 647 . 15 22055000 0 - 0 . 344 18 015 0 422 -0 998 [87] n - H , 2 1 20 390 33 . 180 13 13000 0 0 00006414912 -0 . 22 2 016 0 420 -0 964 [82] p - H i 22 20 268 32 .976 1292800 0 - -0 . 22 2 016 0 4 18 -1 000 [82] Argon 20 87 30 150 . 86 4897900 0 0 000074570932 -o .004 39 948 0 556 -o 966 [83] T o t a l 202 Note: c o e f f i c i e n t s , D1j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 UJ O C h Table 121: Cp and Cv P r o p e r t y Input Data f o r Region III COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) VC (cu.m/mo1e) u 61 ] mo 1ec. wt . (61 ] ATr APr DATA REF . Methane 284 1 1 1 63 1 190 . 555 4595000 0 0 . 000098917 0 008 16 043 0 499 -0 97 1 0 00544-0 . 979 (61 , 85 ] Propane 620 23 1 068 369 . 85 4247500 0 0 0001999967 0 152 44 097 0 243 -0 973 0 00235- 1 .000 [61 . 86 ] n-Butane 57 1 272 638 425 . 16 3796000 0 0 0002550977 0 193 58 124 0 329 -0 988 0 00263-1 .000 (61 , 80] Methano1 32 337 86 5 1 3 . 15 79634 15 9 - 0 559 32 042 0 58 1 -0 922 0 0126-0. 628 (61 . 74] H,0 749 373 15 647 . 15 2205500 0 - 0 344 1 8 0 1 5 0 422 -0 9G3 2 77E-5-0 . 998 (61 . 87] n-H; 2 18 20 390 33 . 180 1313000 0 0 00006414912 -0 .22 2 016 0 422 -0 964 0 00762-0 . 985 (61 . 82 ] p-H; 29 1 20 268 32 .976 1292800 0 - -0 . 22 2 016 0 425 -0 970 0 00774- 1 .000 [82,88] Argon 124 87 30 150 . 86 4897900 0 0 000074570932 -0 .004 39 948 0 563 -0 96 1 0 0163-0. 9 19 [61 83 ] T o t a l 2889 Note: c o e f f i c i e n t s , D i j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45.46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 Tab l e 122: Cp and Cv P r o p e r t y Input Data f o r Region IV COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) u [61 ] mo 1ec. wt . [61] ATr APr DATA REF . Methane 294 1 1 1 63 1 190 . 555 4 59 5000 0 0 0000989 17 0 008 1 6 043 0 525 -0 997 0 00544-0 979 [61 . 85 ] E t hane 95 184 52 305 . 5 4913000 0 0 0001417199 0 .098 30 070 0 655 -0 982 0 0204-0.6 1 1 [61 . 70] P ropane 164 23 1 068 369 . 85 4247500 0 0 0001999967 0 . 152 44 097 0 5 14 -0 973 0 00235-0 800 (61 . 86 ] n-Butane 183 272 638 425 . 16 3796000 0 0 0002550977 0 . 193 58 1 24 0 54 1 -0 988 0 00263-0 948 (61 . 80] H:0 3 15 373 15 647 . 15 2205500 0 - 0 . 344 10 015 0 430 -0 963 2 77E-5-0 680 (6.1 . 87 ] n-H, 1 10 20 390 33 . 180 1313000 0 0 00006414912 -0 . 22 2 016 0 452 -0 964 0 00762-0 762 (61 . 82 ] p-H, 193 20 268 32 .976 1292800 0 - -0 . 22 2 016 0 455 -0 970 0 00774-0 774 [82 , 88 ] Argon 253 87 30 150 . 86 4897900 0 0 000074570932 -0 .004 39 948 0 563 -0 994 0 00204-0 .919 [61 , 83 ] T o t a l 1607 Note: c o e f f i c i e n t s . D i j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45.46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 W O -J T a b l e 123: Cp a n d Cv P r o p e r t y I n p u t D a t a f o r R e g i o n V COMPOUND NPTS Tnb ( K ) T c (K) Pc ( P a ) Vc (cu.m/mo1e) o [61 ] mo 1ec. wt. [ 6 1 ] A T r A P r DATA REF. M e t h a n e 28 1 1 1 1 1 63 1 190 . 555 4 5 9 5 0 0 0 0 0. 0 0 0 0 9 8 9 17 0 008 16 043 0 499 -3 . 254 0 0 0 5 4 4 - 2 1 7 . 6 2 8 [61 ,85] E t h a n e 734 184 52 305 . 5 4 9 1 3 0 0 0 0 0. 0 0 0 1 4 1 7 1 9 9 0 098 30 0 7 0 1 015 - 1 . 637 0 0 2 0 4 - 1 0 . 1 7 7 [ 6 1 , 7 0 ] P r o p a n e 1954 231 068 369 . 85 4 2 4 7 5 0 0 0 0. 0 0 0 1 9 9 9 9 6 7 0 152 44 097 0 232 - 1 . 893 0 0 0 2 3 5 - 1 . 6 4 8 [61 ,86 ] n - B u t a n e 187G 272 638 425 . 16 3 7 9 6 0 0 0 0 0 0 0 0 2 5 5 0 9 7 7 0 193 58 124 0 319 - 1 . 646 0 0 0 2 6 3 - 1 . 844 [ 6 1 , 8 0 ] M e t h a n o l 100 337 86 513 . 15 79634 15 9 - 0 . 559 32 04 2 0 581 -0. 922 1 2 5 6 - 1 2 . 5 5 7 [ 6 1 , 7 4 ] H.O 1494 373 15 647 . 15 2 2 0 5 5 0 0 0 - 0 . 344 18 015 0 422 - 1 . 967 2 7 7 E - 5 - 0 . 4 5 3 [61 ,87 ] n-H. 4401 20 390 33 . 180 1313000 0 0 0 0 0 0 6 4 149 12 -0 . 22 2 016 0 452 -90.4 16 0 0 0 7 7 4 - 7 7 . 3 5 1 [61 ,82] p-H. 3528 20 268 32 .976 1292800 0 - -0 . 22 2 016 0 434 -90.975 0 0 0 7 7 4 - 7 7 . 3 5 V [82 , 8 8 * ] A r g o n 2668 87 30 150 . 86 4 8 9 7 9 0 0 0 0 0 0 0 0 7 4 5 7 0 9 3 2 -0 .004 39 948 0 597 -7 . 292 0 0 0 2 0 4 - 2 0 . 4 1 6 [61 ,83] T o t a l 19566 N o t e : c o e f f i c i e n t s , D i j a n d C, f o r t h e BACK e q u a t i o n a r e t a k e n f r o m r e f e r e n c e s [ 4 5 , 4 6 ] c o e f f i c i e n t s f o r t h e BWR e q u a t i o n a r e t a k e n f r o m r e f [ 6 1 ] , t a b l e 3-6 * t h e C p ' d a t a was f i t t o a c u b i c p o l y n o m i a l s o a s t o be a b l e t o c a l c u l a t e v a l u e s a t t e m p e r a t u r e s g r e a t e r t h a n 700K. The p o l y n o m i a l e q u a t i o n i s p r e s e n t e d b e l o w , b u t s i n c e i t p e r f o r m e d p o o r l y i t i s n o t r e c o m m e n d e d f o r u s e . Cp'=4.21 1 1 8 7 3 9 2 5 6 1 5 1 + 0 . 0 2 8 0 3 5 5 2 2 2 1 4 5 4 0 * T - 7 . 5 2 8 3 9 2 0 3 E - 5 * T ' + 5 . 9 0 9 9 4 6 0 7 E - 8 * T J (ca1/mo 1 e - K ) , w h e r e T i s i n K e l v i n s . CO o co T a b l e 124: Joule-Thomson P r o p e r t y Input Data f o r Regions I and II COMPOUND NPTS Tnb (K) Tc (K) PC (Pa) Vc (cu,m/mo1e) o (6 1] mo 1ec. wt . [61 ) ATr DATA REF. Methane 29 1 1 1 63 1 190 . 555 4595000 0 0.0000989 17 0.008 16 043 0.509 -o 997 [85] P ropane 5 231 068 369 . 85 4247500 0 0.0001999967 0.152 44 097 0.625 - i 000 (86,89] n-Butane 6 272 638 425 . 16 3796000 0 0.0002550977 0. 193 58 124 0 . 692 -o 888 [80.91] n-H, 2 1 20 390 33 . 180 1313000 0 0.00006414912 -0.22 2 0 1 6 0.420 -o 964 [82- ] p-H, 22 20 268 32 .976 1292800 0 - -0 . 22 2 016 0.4 18 -1 000 (82*] T o t a l 83 Note: c o e f f i c i e n t s , D i j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45.46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 * c a l c u l a t e d from (dV/dT)/V data T a b l e 125: Joule-Thomson P r o p e r t y Input Data f o r Region III COMPOUND 'NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e ) o (61 ] mo 1ec. wt. [61 ] ATr APr DATA REF. Methane 284 1 1 1 631 190.555 4595000 0 0.0000989 17 0.008 16.043 0. 499 -o 97 1 0.00544 -0 979 [85] H, 0 26 373 15 647 . 15 2205500 0 - 0. 344 18.015 0.457 -o 535 0.00748 -0 593 [87,92] n-H, 2 18 20 390 33.180 1313000 0 0.000064149 1 2 -0.22 2.016 0.422 -0 964 0.00762 -0 985 (82* ] p-H, 29 1 20 268 32.976 1292800 0 - -0. 22 2.016 0.425 -o 970 0.00774 - 1 000 (82* ] T o t a l 819 Note: c o e f f i c i e n t s , D i j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-5 * c a l c u l a t e d from (dV/dT)/V dat a co o T a b l e 126: Jou1e-Thomson p r o p e r t y Input Data f o r Region IV COMPOUND NPTS Tnb (K) Tc K) Pc (Pa) Vc (cu.m/mo1e ) u 61 ] mo 1ec. wt . [61] ATr APr DATA REF . Methane 294 1 1 1 63 1 190 555 4595000 0 0 . 000098917 0 008 16 043 0 525 -0 997 0 00544-0.979 [85] E thane 6 184 52 305 . 5 4 913000 0 0 . 000 1417199 0 098 30 070 0 963 0 020S-0.702 [70, 90 ] P ropane 32 23 1 068 369 . 85 4247500 0 0 0001999967 0 152 44 097 0 796 -0 976 0 0406-0.812 [ 86 , 89 ] n-Butane 27 272 638 425 . 16 3796000 0 0 0002550977 0 193 58 124 0 692 -0 888 0 0363-0.363 [80, 91 ] H i O 72 373 15 647 . 15 2205500 0 - 0 344 18 015 0 667 -0 965 0 00448-0.224 [87. 92 ] n-H, 1 10 20 390 33 . 180 13 13000 0 0 00006414912 -0 22 2 016 0 452 -0 964 0 00762-0.762 [82* ] p-H* 193 20 268 32 .976 1292800 0 - -0 22 2 016 0 455 -0 970 0 00774-0.774 [82* ] Neon 26 27 09 44 . 40 2653701 75 - 0.000 20 183 0 699 -0 989 0 0268-0.772 [93 , 94** ] j Argon 7 87 30 150 . 86 4897900 0 0 000074570932 -0 . 004 39 948 0 684 -0 982 0 00204-0.408 [83 , 95 ] T o t a l 767 Note: c o e f f i c i e n t s , D 1 J and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [ 4 5 , 4 6 ] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [ 6 1 ] , t a b l e 3-6 c a l c u l a t e d from (dV/dT)/V data ** e s t a b l i s h e d by f i n i t e d i f f e r e n c e d i f f e r e n t i a t i o n UJ T a b l e 127: Jou1e-Thomson p r o p e r t y Input Oata f o r Region V COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) Vc (cu.m/mo1e) u 61 ) molec. wt . [61] ATr APr DATA REF. Methane 28 1 1 1 1 1 63 1 190 . 555 4595000 0 0. 000098917 0 008 16 043 0 499- 3 . 254 0 00544-217.628 [85] E thane 35 184 52 305 . 5 4913000 0 0. 0001417199 0 098 30 070 1 016- 1 . 236 0 0206-0.842 [70,90] Propane 12 23 1 068 369 .85 4247500 0 0 0001999967 0 152 44 097 1 02 1 0 0406-0.893 (86,89] H,0 172 373 15 647 . 15 2205500 0 - 0 344 18 015 0 457- 1 . 659 0 000538-0.162 [87,92] n-H, 4401 20 390 33 . 180 1313000 0 0 000064 149 12 -0 22 2 016 0 452- 90.416 0 00774-77.351 [82*] p-H, 3528 20 268 32 . 976 1292800 0 - -0 22 2 016 0 434- 90.975 0 00774-77.35 1 (82* ] Neon 1 15 27 09 44 . 40 2653701 75 - 0.000 20 183 1 010- 2 .045 0 0436-7.323 [93,94**] Argon 127 87 30 150 . 86 4897900 0 0 000074570932 -0 004 39 948 0 8 16-3 . 799 0 0207-4 . 137 [83,95] T o t a l 11201 Note: c o e f f i c i e n t s , D1j and C, f o r the BACK e q u a t i o n are taken from r e f e r e n c e s [45,46] c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 c a l c u l a t e d from (dV/dT)/V data e s t a b l i s h e d by f i n i t e d i f f e r e n c e d i f f e r e n t i a t i o n T a b l e 128: I n v e r s i o n Curve Input Data COMPOUND NPTS Tnb (K) Tc (K) Pc (Pa) o [61 ] mo 1ec. wt . [61] ATr APr DATA REF. Methane 27 1 1 1 67 191.06 4640685 0 0.008 16 043 1 .099 -2 460 5 786- 1 1 507 [96*] Propane 52 23 1 068 369.85 4247500 0 0. 152 44 097 0.811 -2 190 0 378- 1 1 762 [86] n-Butane 52 272 638 425 . 16 3796000 0 0. 193 58 1 24 0.823 -2 023 0 713- 1 1 837 [80] p - H i 17 20 268 32.976 1292800 0 -0.22 2 016 0.849 -6 065 0 774- 12 701 [82] Argon 18 87 284 150.86 4898050 5 -0.004 39 948 0 . 862 - 1 989 1 433- 1 1 300 [97*] T o t a l 166 c o e f f i c i e n t s f o r the BWR e q u a t i o n are taken from r e f [61], t a b l e 3-6 * v o l u m e t r i c r e s u l t s were I n t e r p o l a t e d 

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