THERMODYNAMIC PROPERTIES OF A 5 MOLE PERCENT PROPANE IN METHANE MIXTURE V. L. dBhirud and J. E. Powers August 1969 Tulsa, Oklahoma Department of Chemical and Metallurgical Engineering

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TABLE OF CONTENTS INTRODUCTION........................ 1 Composition...... 1 Sources of Data.................... 2 Calorimetric Data.................. 2 Volumetric Data....... 2 Calculation Procedures................. 4 Enthalpy H, Isobaric heat capacity, Cp. and Isothermal Throttling coefficient,...... 4 Entropy, S, Fugacity, f, and Specific Volume, V.. 7 Skeleton Tables of Thermodynamic Values... 10 Thermodynamic Diagrams................. 12 Acknowledgments................ 12 NOTATION..... e... 23 BIBLIOGRAPHY........................ 27 APPENDICES I. RE.VIEW OF LITERATURE................ 32 Calorimetric Data.. *...... 32 Volumetric Data................. 38 II. BASIC INTERPRETATION OF CALORIMETRIC DATA. 43. Isobaric Determinations.............. 44 Single Phase.................. 44 General Procedure........ 44 Interpretation of Data at 1000 psia..... 48 Interpretation of data at 500 psia... 52 Extrapolation with respect to temperature 58 Two-Phase (Enthalpy Traverse)......... 61 Isothermal Determinations (Single Phase)...... 64 III. THERMODYNAMIC CONSISTENCY CHECKS AND FINAL ADJUSTMENT OF VALUES OF C AND................. 67 Thermodynamic Consistency Checks.......... 67 Adjustment of Values of i AH. 70 Adjustment of Values of Cp and ~.......... 71 i1

IV. PREPARATION OF PTH TABLES AND DIAGRAMS......... 72 Selection of Arbitrary Reference States...... 7 Determination of' H. for Pure Components as Ideal Gases at +200~F.......... 74 Propane.................. 74 Impurities................ 74 Me thane............ 75 Calculation of a for the Mixture as Ideal Gas at +2000F.................... 82 Incorporation of Isobaric and Isothermal Enthalpy Differences at Elevated Pressures.82 Corrections to Obtain Complete Agreement at -280~F. 83 Interpolate to Intermediate Values of Temperature and Pressure.................... 84 V. EQUATIONS USED IN CALCULATING VALUES OF ENTROPY AND FUGACITY....................... 85 The Basic Property Relation............ 85 Isobaric Change in Entropy............. 86 Isothermal Change in Entropy.. 87 Absolute Entropy Values....... 88 Fugacity Function.. 92 VT. PRIMARY INTERPRETATION OF DATA FROM THE LITERATURE TO ESTIMATE THE VOLUMETRIC BEHAVIOR OF THE 5% MIXTURE.. 94 Isotherms at 91.6 and 200 F............ 94 Isotherms at -147.4 and -270F........ 96 VII. CALCULATIONS OF ENTROPY DIFFERENCES.101 Isobaric Differences in Entropy....... 101 Isothermal Entropy Differences......... 103 VIII. THERMODYNAMIC CONSISTENCY CHECKS FOR ENTROPY ANT) ADJUSTMENT OF VALUES OF ENTROPY, COMPRESSIBILITY FACTOR, Z, AND SPECIFIC VOLUME, V, CALCULATION OF HFJGACITY..... 109 Thermodynamic Consistency Checks... 0 109 Corrections to Obtain Thermodynamic Consistency.. 110 Calculation of Fugacity............. 111 11i

IX. PREPARATION OF MOLLIER DIAGRAM AND SKELETON TABLE OF' THIERMODYNAMIC PROPERTIES FOR THE MIXTURE. 112 Absolute Entropy of the Mixture at One Atmosphere and 91. 6~' F......... 112 Tables and Diagram of Thermodynamic Properties for the Mixture................... 113 X. SIMPSON' S RULE......:115 Area Under Given Curve............... 115 Numerical Integration to Obtain Area Under a Curve......... 0.,....117 Proofs and Comments on Simpson's Rule....... 3.18 Procedure for Curve Fitting Using Simpson's Rule... a @ m. e e e. 120 *_ * *

LIST OF TABLES 1. Composition of Nominal 5% Propane in Methane Mixture................... 2 2a. Tables of Thermodynamic Data Values at the Temperatures of Isothermal and Isenthalpic Determinations (T - 200 OF).......... 13 2b. Tables of Thermodynamic Data Values at the Temperatures of Isothermal and Isenthalpic Determinations (T = 91.60F).......... 14 2c. Tables of Thermodynamic Data Values at the Temperatures of Isothermal and Isenthalpic Determinations (T -270F).......... 15 2d. Tables of Thermodynamic Data Values at the Temperatures of Isothermal and Isenthalpic Determinations (T -147.40F).16 3a. Tables of Thermodynamic Data Values at the Pressures of Isobaric Determinations (Low Pressures)........... 17 3b. Tables of Thermodynamic Data Values at the Pressures of Isobaric Determinations (High Pressures).18 AI.1 Mixture Compositions........... 33 AI.2 Extremes in Compositions Reported by Manker (47) and Mather (50)............ 34 AI.3 Conditions of Integral Isothermal Determinations by Dillard (25)............ 35 AI.4 Conditions of Reported Joule-Thomson Values of Budenholzer et al (9)........... 36 AI.5 Conditions of Reported Joule-Thomson Measurements by Head (32'.............. 37 AI.6 Conditions of Reported Volumetric Data of Reamer, Sage and Lacey (56).......... 39 AI.7 Conditions of Reported Volumetric Data of Huang et al (37)........ 40 iv

AII.1 Information Obtained at the Maximum in Cp(T)...................48. AII.2 Constants of Equation (AIII.6) Expressing Specific Heat in the Immediate Vicinity of the Bubble Point............... 57 AII.3 Values of Cp for the 5.18% C in C1 Mixture at 500 psia as Determined by In erpretation of Calorimetric Data at Other Conditions..... 58 AII.4 Values of (PC /~T) Used to Extrapolate Cp(T) to -280~F.P.P 61 AII.5 Em.pirical Data Obtained From Interpretation of Enthalpy Traverses.......... 64 AIV.1 Thermodynamic Consistency Checks of Published Calorimetric Data for Methane at Low Temperatures.................... 76 AIV. 2 Maximum Percentage Adjustments in Published Calorimetric Data for Methane at Low Temperatures................... 80 AIV 3 Enthalpies of Pure Components as Ideas Gases at 2000F Relative to Saturated Liquid at -2800 ~F. 81 AV.1 Absolute Entropies of Pure Components as Ideal Gases at 250C........ 89 AX.1 Comparison of Coefficients of the Series Giving Area Under a Curve Using Simpson's Approximation...................119

LIST OF FIGURES 1. Temperatures and Pressures of Calorimetric Determinations on a Nominal 5% Propane in Methane Mixture (47,50).5............. e 3 2. Check of Thermodynamic Consistency of Experimental Enthalpy Determinations.... e. 6 31. Check of Thermodynamic Consistency of Entropy Differences Calculated from Calorimetric and Volumetric Data. 9 4. Pressure-Temperature-Enthalpy Diagram For Nominal 5 Percent Propane in Methane Mixture (Low Temperatures)............... 19 5. Pressure-Temperature-Enthalpy Diagram For Nominal 5 Percent Propane in Methane Mixture (High Temperatures)............... 20 6. Mollier Diagram for Nominal 5 Percent Propane in Methane Mixture (Low Temperatures)...... 21 7. Mollier Diagram for Nominal 5 Percent Propane in Methane Mixture (High Temperatures)..... 22 8. Temperatures and Pressures of Volumetric Data for Methane-Propane Mixtures as Reported in the Literature................. 41 9. Isobaric Heat Capacity, C, at 1000 psia Illustrating Method of In erpreting Calorimetric Results -.46 10. Isobaric Heat Capacity, Cp, at 1000 psia Above the Critical Point Illustrating Sharp Maximum.. 50 11. Check for Bias in Interpreting Isobaric Data Near the Maximum in C p(T) at 1000 psia.... 51 12. Check of Interpretation of Isobaric Data Near the Maximum in C (T) at 1000 psia After Correcting for Bias. 53 13. Plot of Excess Isobaric Heat Capacity, C, for the Purpose of Estimating C for the 5~ Iixture at 500 psia Where Direct Me surements Are Not Available................ 56 vi

14. Experimental Heat Capacities with Values Calculated from B-W-R Equation of State...... 59 15. Calorimetric Data Obtained Within and Through the Two-Phase Region.. 63 16. Isothermal Throttling Coefficient,, at -27 ~F Illustrating Method of Interpreting Isothermal Calorimetric Results.............. 65 17. One Complete Check of Thermodynamic Consistency of Calorimetric Data Used to Evaluate Enthalpy Differences......... 69 18. Check of Thermodynamic Consistency of Published Calorimetric Data and Calculated Enthalpy Departures at Low Temperatures. 77 19. Published Data on the Isobaric Heat Capacity of Methane as Saturated Liquid.79 20. Typical Plot of Compressibility Factor, Z, vs lnP at 91.6~F as Used to Evaluate ZdlnP.... 97 21. Typical Plot of Isobaric Density,p, for Liquid Methane-Propane Mixture for Interpolation fo Temperatures of Isothermal Calorimetric Data.. 98 22. Typical Plot of Specific Volume, V = l/p, of Liquid Methane-Propane Mixture for Interpolation to Composition of Mixture..99 23. Typical Plot Used to Evaluate Second (Residual) Term in Equation (V.9) in the Vicinity of the Peak in Cp(T) at 1000 psia........102 24. Typical Plot Used to Evaluate Second (Residual) Term in Equation (V. 9) in the Two-Phase Region. 104 25. One Complete Check of Thermodynamic Consistency of Calorimetric and Volumetric Data Used to Evaluate Entropy Differences.. 105 26. Typic-al Plot Used to Evaluate VdP in the Compressed Liquid Region....... 107 27. Area Under a Curve Using Trepezoidal Rule.. 116 28. Area Under a Curve Using Simpson's Rule.... 116 29. Simpson's Rule Used for Curve Fitting..... 121 vii

INTRODUCTI ON The continuing need for accurate thermodynamic data for natural gas mixtures and the components of natural gas has been accentuated by recent developments in the area of natural gas processing at cryogenic conditions including liquification. Accurate values of enthalpy and entropy for pure components and mixtures are required not only for direct use in design of heat exchangers, compressors and expanders but also to be used as a basis for testing and comparing various methods of prediction. Therefore the objective of this investigation was to develop accurate values of thermodynamic properties for one mixture approximating natural gas. This objective was to be met making maximum use of accurate calorimetric data and the basic definitions of classical thermodynamics so as to yield the most meaningful numbers. COMPOSITION The primary sources of calorimetric data are the theses of E. A. Manker (47 ) and A. E. Mather (50). The composition of mixtures used in these companion investigations varied somewhat during the course of the investigations and moreover did not correspond directly to any mixture for which volumetric data were available as indicated in Appendix 1. The following composition was taken as an average of those mixtures for which calorimetric data are available in relative abundance: -- -

-2TABLE 1 COMPOSITION OF NOMINAL 5% PROPANE IN METHANE MIXTURE Composition Methane, CH4 0.9464 Ethane, C2H6 0.0006 Propane, C3H8 0o.0518 Nitrogen, N2 0.0006 Oxygen, 02 0.0002 Carbon Dioxide, C02 0.0004 1.0000 SOURCES OF DATA Calorimetric Data The basic calorimetric data used in preparing the tables of thermodynamic properties are the isobaric determinations of Manker (47) and the isobaric, isothermal and isenthalpic determinations of Mather (50). These determinations extend from -240 to +2600F at pressures between 100 and 2000 psia. The temperatures and pressures at which actual measurements were made are indicated on Figure 1. Other published calorimetric data were considered but did not enter directly into the calculations (See Appendix 1). Volumetric Data No volumetric data has been reported for the composition listed above. Extensive volumetric data have been reported for methane (1,27,44,58,65,66), propane (3,15,23,26,36,55,61)

20 0 -- ~~~~~~~~~~~~~~~~~~19 1 I I I I ~~~~~~NOMINAL MOLE FRACTIONS C3 H 0.05 CH4-0.95 17U~., I o --— t —-- ---- 1600 1500 - o- 40 —---- - 1400 1300 1200 c 1100 c w:: KM 000 0 a 900 / 800 700 6OC 500:- C 400 300 200/ // 100/ 0CI I I I I IIIII -250 -200 -150 -100 -50 0 50 100 150 200 250 TEMPERATURE - OF Figure 1. Temperature and Pressures of Calorimetric Determinations on a Nominal 5% Propane in Methane Mixture. (47,50)

-4and data are also reported for seven mixtures of these two components (16,56. The range of temperatures and pressures of volumetric data for the mixtures corresponding to those for the calorimetric data are presented later (Appendix 1). CALCULATION PROCEDURES Enthalpy, H, Isobaric Heat Capacity, Cp, and Isothermal Throttling Coefficient, ~. Values of Cp and 0 for this mixture (47,50) as well as three tables of H and three PTH diagrams (48,50,5!) have been presented. There were reasons to question the interpretation of the basic data (70) even in the last of the three published tables both in the low temperature region and in the region just above the critical point for the mixture. Therefore, all data were reprocessed using improved techniques. The basic methods of determining values of C from basic p isobaric data and Q from isothermal throttling data has been covered in detail elsewhere (46,47). The improved procedures are summarized in Appendix II. Likewise the methods of selecting bases for H and calculating values of H in the two-phase as well as the single-phase region has been discussed elsewhere (46,47) and are summarized in Appendix IV. The values of Cp, Q and H reported in the skeleton table of thermodynamic data comprising the major part of this report are smoothed values which are thermodynamically self consistent. The procedure followed to insure this desirable feature is outlined below and presented in greater detail in Appendix II.

Graphical procedures are used. Smoothed curves are drawn through the basic isobaric and isothermal results to obtain a first estimate of Cp and O respectively in the single phase region (Appendix II). A modified form of Simpson's rule (Appendix II) is applied to insure the best average fit of the experimental data. Values of AH between experimental isotherms — p and AH between experimental isobars are determined by again applying Simpson's rule over small intervals. The resulting numbers are summarized on Figure 2. Enthalpy is a thermodynamic property and therefore, has the characteristics of an exact mathematical function. As a result the algebraic sum of all enthalpy differences around any closed loopi AHi should be exactly equal to zero. The actual sum divided by the sum of the absolute values of the enthalpy differences, percentage _ AHi deviation J2 xAH.i is an excellent measure of the consistency of the data. The results of such determinations are listed within each closed loop on Figure 2. Adjustments were made on individual values of AHi as required to make AHi. — 0 for all loops. These adjustments were made within the limits of precision of the basic data [Cp - +0.5%5; - +1.0%]. The amount of the adjustment and the corrected values are presented in Figure 2. As indicated the magnitude of the corrections was generally much less than

117.143 Btu/lb 115.023 80.096 2000iiiiiii14i (+0.36) Btu/lb (+0.194) (-0.384) CO ~ ~ ~ ~ J,n ~ -0~~~0. 1 21'- c ~ c',, 0 o~ o -0.05%~. i~ ~ ~~~ — ~ a C nH Oo lo~~~~~~~~~~~~~~00C 0-.031 Btu/Ib 114.027 +0.21 4 4.0_2_7 +0_._21 nm.oO 1700 -, o 70 (+0.073) +0.118% o oO O Y. A~i ~ o~ + 00 7,A H i o+I II cA 0 ~'&., i i + 000 ___ L~~~~~~~~~.i.....J, -- -0 673 u~c~ -0.0 11IO/o O or'o~ z, -0.280% u~ I 7 cn o~~~~~~~~~~~~~~C)o ~~~~~~~~~~0. I - I 280%.( 0O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~0 + 15007 - 137.144 110.842 74.702.~15oo......... (+0.379) (-0.6) (-0.274) LLI LU Lr\ %-0 ~~~~~00 0U, U 092 M +0o3 - +0.039 63 cr\ o~~~~~~~~,\C o o -O 313% 1 o % +.252% uaI +0.023% 2' 1000 -175.126 89.424 69.028 1000 1 l ) ~(-0.35) (-0.5) (-0.335) UU U) U'\ M v, u\ o~~~~~~~~~~~~~~-035 II ) C +1 06 +0.215 N U.I N — IUra_ o -7I (208.601) 69.928 63.514 O' ~~~~~~~~~~~~~I + ~~~(0 8. 0 ) _ _ _ _ _ _ _ _ _ _ _ _ _ 59. 4 528 6. 7 04 (+0.170) (-0.10) o -0.331 +0.401 -. C> 0. 1 9 4 % +0 271% o CV -o-0194%,~ o 59.44 58.7O (o.oo) (o.oo) -147.4 -27 91.6 2000 F TEMPERATURE OF ISOTHERMAL DETERMINATIONS (OF) Figure 2. Check of Thermodynamic Consistency of Experimental Enthalpy Determinations.

-7-:.ihe indicated precision. F'inally, the smoothed curves for C p and 9 were adjusted so as to be consistent with the corrected values of AH and AH Again such adjustments were made well -p -P within the limits of precision of the basic data. Values of C and 0 reported in the skeleton tables and the adjusted P smoothed. curves were used to generate the intermediate values of H listed there. As a result it is believed that the values of C are accurate to better than 1 —0.2%. those for Q are better than 4+0.5% and the H values are believed to accurate to better than 1 Btu/lb over the entire range of pressure and temperature. These statements must be tempered somewhat in various regions. Greater percentage errors are to be expected in the vicinity of, O. Higher errors for both C and 0 also occur in the p immediate vicinity of the critical point for the mixture where CP and 0 change very rapidly with both temperature and pressure (See Fig-ures A-II-1, A-II-8) and at low temperatures below pressures of 1000 psia where complete data are lacking as described in Appendix II (See also Figure 1). Entropy, S, Fugacity, f, and Specific Volume, V. Two different advanced methods of interpolation were used to estimate values of specific volume f'or a binary mixture of' methane and propane, which will be discussed in forthcoming publication. Interpolation was made primarily on Z in the gaseous region and on V in the liquid region. These were considered to be preliminary values and are not included as such in the skeleton tables.

Special equations were developed to make maximum use of the calorimetric data in determining both isobaric and isothermal differences in entropy as discussed in detail in Appendix VII. This is especially important because the interpolated values of the specific volume were not nearly accurate enough to yield meaningful values of either first or second derivatives such as and V. (See Appendices VI and VII). As was done with enthalpy determinations, values of ASp were calculated between each pair of experimental isotherms and AST values were evaluated between each pair of experimental isobars. The values thus calculated are summarized on Figure 3. Entropy is also an exact function and therefore, WAS. percentage 1 -1 deviation Si x 100 (2) is a good measure of the thermodynamic consistency of the data. As indicated by the values of percentage deviation listed within the loops on Figure 3 the interpolated volumetric data are in remarkable agreement with the smoothed calorimetric data. Adjustm.e nts tnhe volumetric values were made so as to make ASi O0 for each individual loop. All such adjustments were made within the anticipated l.imits of acn-curacy of tbhe interpolated volumetric values (+-0*.5%). The c:orrections made to ASp and AST are presented on Figure 3. Note that the corrections are more or less random with respect to sign and are usually considerably less than +0.5o indicating excellent agreement between the calorimetric and interpolated vourn. etri.c values.

0.314854 0.237174 0.132225 2000 ao,%r.- -0o000145 r M,i~~~0.'0 0 1 -4'l 0;%000 +oo -0.027% o oo + ~~~~~.. + XAS'' + o+ -o0 - i'+ I+ mm +.000047.00023 co'', u -- -o +. 0 0 024 CD -oINO I.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -." _ Btu/IbOF o~~~ -0-045 C)~ - + 0.235606 +. 0 68% 1 o+ 00.v 0.366139 \~ 0 0280163 0.12340 150000 -o + I T.IAS N -o,04N +/ 0.10 o 006A \i....o o.; 0.00011970 002+30C0147 C)~ ~ ~ ~ ~ ~~~~~~~Oc.... 0 C') LC' I~~~~~~~~~~~~0- CD + 366 3 +.00 0.92%6 00 0 23 0.1 1 L/)o0.0 0 (1)~~~~~~~~~~~~~ 004 ZD ~,LJJ..+ ~~~~~~00 00'-~ tn 0~l 0'-I I + 0 + I + ~~~0.3643 028844 0.1137667 1 0\ 0000 "' ~, ~.o\O~ o +0. 0 001 197 -.v - r._ +.000147 co o0 a: r- - V\ CO 00 M O M in 0 oMO~ c" on ~ L~Oq v, I oa +0.009% o oo +0.115% "v, I olo ~- ~ + 0 + oC- ~.+ o> CD a0 + 0 O 0-e~~~~~~ + +I + O000.648844 0.113766 1000 " N \DN 0 0c'oCO) +0.000125 13 " r-I. r-,~~L" oo, —- 0 Ir r ~ 0 ~) +0.031% 0 - O O, O + I + 0I 00 M 1 -ncV\~ O. 14.782 -. 104954 +o o'~+ 001 5 13 u~lr ~o -1~47.4 -2 ~7 9~ ~6 + 0.147% TEMPERATURE (OF) ~+ I+ I + 14.7.... -147.4 -27 91.6 200. TEMPERATURE (~F) Figure 3. Check of Thermodynamic Consistency of Entropy Difference, Calculated from Calorimetric and Volumetric Data. -9

-10The values of Z and V were modified so as to be consistent with the corrected values of ASp and AST. The corrected values of Z and V were used to calculate values of entropy at equal intervals of pressure along each experimental isotherm. This.procedure yields values of Q, V and S that are thermodynamically consistent with values of H determined from calorimetric data only. The consistency of the values should be better than 1.part in 1000. The bases selected for the entropy determination are _ -= 0 for each pure component as a perfectly oriented crystal at absolute zero on the temperature scale as described in detail in Appendix IX. As a result the values presented are for absolute entropy according to the third law of thermodynamics. The smoothed and corrected values of V (and Z) were used to calculate values of fugacity, f, of the mixture at temperatures corresponding to each experimental isotherm according to fundamental' definitions of classical thermodynamics as described in detail in Appendix V. It seems reasonable that these values are also thermodynamically- consistent with other reported values to 1 part in 1000. Skeleton Tables of Thermodynamic Values The follow ing tables contain the values developed from published calorimetric and volumetric data by the procedures described in preceeding sections. The first four tables are for the temperatures of the isothermal and isenthalpic determinations as indicated on Figure 1. For each temperature, values of Q, H, V, f and S are listed at at'nospheric pressure and at intervals of 50O psia from zero to

-112000 psia. The bases for enthalpy are H - 0 for all compounds (including C02) as saturated liquids at -2800F. These choices are consistent with previously published values for methane (40,41), propane (70) and their mixtures (47,48,50,51,70,73). The bases for entropy are S = 0 for all pure compounds as perfectly oriented crystals at absolute zero. Therefore, the values of S are absolute values in accordance with the third law of thermodynamics. The pair of fold-out tables are principally for the pressures of the experimental isobars as indicated on Figure 1. Values at zero and atmospheric pressure are also included. These values are based primarily on tabular values listed by API Project 44 (1,58). At each pressure above atmospheric, values C H and S are listed at intervals of 10~F between -280 and is00F and at each of the temperatures of the isothermal and isenthalpic determinations. Values of H and S are listed within the two-phase region but values of C are not reported in this p region. Values are reported only where experimental determinations were made or, in some cases, such as the low temperature region, where considerable care was taken in estimating these data by crossploting. Therefore, blanks in the table indicate lack of data at those particular conditions of pressure and temperature. As indicated previously the values listed in the skeleton tables correspond in general to conditions of experimental determinations as represented on Figure 1. These values therefore

-12represent smoothed experimental values and are expected to be accurate on the order of 1 part per 1000. Therefore, these values can be used to advantage to test methods of' prediction for all the properties. Thermodynamic Diagrams The values from the skeleton tables were plotted on large graph paper and smoothed so as to yield both PTH and Mollier (S, H) diagrams (See Figures 4 and 5). Both are presented as pairs of 11 x 17 diagrams so that values of H can be read to better than 1 Btu/lb and S can be read to be tter than 0.05 Btu/lb~F over the entire range of temperature and pressure. A CKNOTWLEDGMENTS Richard J. Giszczak made significant contributions to this report by carrying out the interpolations of the volumetric data to yield values at the desired temperatures and compositions. This work was supported by a special grant from the Natural Gas Processors Association. The manuscript was typed by Diana Frinkel.

TABLES OF THERMODYNAMIC DATA Table 2a Mixture of Methane and Propane Containing Approximately 5.2 Percent Propane Values at the Temperatures of Isothermal and Isenthalpic Determinations T = 200~F P 0H V f S psia Btu/lb psi Btu/lb ft /lb psi Btu/lb~R 0 -0,02146 465.670. 0 14.7 -0.02144 465.356 27.46 14.868 2.697828 50 -0.02140 464.603 8.0377 49.59 2.558466 100 -0.02134 463.537 4.009 98.61 2.478786 150 -0.02129 462.474 2.6671 147.27 2.431625 200 -0.02123 461.412 1.9963 195.64 2.397760 250 -0.02117 460.353 1.5934 243.73 2.371185 300 -0.02112 459.296 1.3244 291.56 2.349234 350 -0.02105 458.241 1.1315 339.11 2.330484 4o0 -0.02100 457.188 o0.9858 386.32 2.314078 450 -0.02096 456.138 0.8730 433.18 2.2994 1 500 -0.02090 455.090 0.7831 479.73 2.286308 550 -0.02086 454.043 0.7100 526.0 2.274246 600 -0.02081 452.999 0.6489 571.98 2.263147 650 -0.02076 451.958 0.5972 617.68 2.252844 700 -0.02071 450.920 0.5528 663.10 2.243215 750 -0.02067 449.885 0.5144 708.23 2.234168 800 -0.02061 448.854 0.4810 753.10 2.225634 850 -0.02058 447.827 0.4516 797.70 2.217544 900 -0.02050 446.804 0.4254 842.06 2.209843 950 -0.02043 445.784 0.4021 886.18 2.20249-5 1000 -0.02036 444.770 0.3811 930.07 2.195467 1050 -0.02028 443.75Q 0.30Q 97jy 2.188744 1100 -0.02020 442.734 0.42 1b.3 2.182270 1150 -0.02012 441.723 0.3273 1059.57 2.176046 1200 -0.02002 440.718 0.3131 1102.29 2.170028 1250 -0.01991 439.718 0.3001 1144.82 2.164216 1300 -0.01980 438.724 0.2880 1187.15 2.158589 1350 -0.01967 437.737 0.2768 1229.27 2.153126 1400 -0.01953 436.757 0.2664 1271.21 2.147838 1450 -0.01937 435.784 0.2567 1312.93 2.142696 1500 -0.01920 434.820 0.2477 1354.46 2.137698 1550 -0.01900 433.869 0.2403 1395.98 2.132894 1600 -0.01879 432.928 0.2325 1437.33 2.128158 1650 -0.01855 431.997 0.2251 1478.51 2.123536 1700 -0.01829 431.077 0.2183 1519.55 2.119028 1750 -0.01802 430.170 0.2118 1560.44 2.114635 1800 -0.01770 429.276 0.2058 1601.22 2.110359 1850 -0.01739 428.396 0.2000 1641.85 2.106180 1900 -0.01705 427.533 0.1946 1682.34 2.102102 1950 -0.01672 426.687 0.1895 1722.71 2.098124 2000 -0.01638 425.859 0.1846 1762.97 2.094248 13

TABLES OF THERMODYNAMIC DATA Table 2b Mixture of Methane and Propane Containing Approximately 5.2 Percent Propane Values at the Temperatures of Isothermal and Isenthalpic Determinations T = 91.6~F P ~ H V f S psia Btu/lb psi Btu/lb ft /lb psi Btu/lb~R 0 -0.03040 406.970 14.7 -0.03042 406.523 22.94 14.67 2.60088 50 -0.03044 405.452 6.6985 49.50 2.46082 100 -0.03050 403.932 3.,3291 98.11 2.380356 150 -0.03054 402.410 2.2081 146.07 2.332404 200 -0.03059 400.886 1.6476 193.44 2.297738 250 -0. 03064 399.359 1.3106 240.25 2.270356 300 -0.03069 397.83 1.0851 286.66 2.247525 350 -0.03074 396.297 0.9233 332.30 2.227974 400 -0. 03080 394.76 0.8016 377.22 2.210779 450 -0. 03085 393.220 0.7073 421.60 2.195355 500 -0.03091 391.677 0.6324 465.34 2.181354 550 -0.03097 390.131 0.5703 508.45 2.168477 60oo -0.03104 388.583 0.5197 551.03 2.156536 650 -0.03110 387.033 0.4768 593.11 2.145372 700 -0.03116 385.480 0.4394 634.61 2.134875 750 -0.03123 383.922 0.4070 675.52 2.124952 800 -0.03129 382.360 0.13789 715.87 2.115532 850 -0.03135 380.793 0.3542 755.69 2.106545 900 -0.03140 379.223 0.3326 795.00 2.097933 950 -0.03145 377.651 0.3133 833.83 2.089661 1000 -0.03148 376.077 0. 2955 872.17 2.081701 1050 -0.03151 374.499 0.2186 910.00 2.074029 1100 -0.03152 372.922 0.2634 947.13 2.066613 1150 -0.03151 371.346 0.2504 983.78 2.059449 1200 -0.o314 369.771 0.2387 1020.0 2.052485 1250 -0.03144 368.197 0.2280 1055.74 2.045718 1300 -0.03137 366.626 0.2182 1091.11 2.039127 1350 -0.03127 365.059 0.209Q 1126.08 2.032695 1400 -0.03115 363 5.497 0.2005 1160.68 2.026433 1450 -0.03101 361.441 0.1927 1195.31 2.020270 1500 -0.013084 360.392 o0.1850 1229.13 2.014291 1550 -0.013062 358.852 0.1787 1262.67 2.008440 1600 -0.03036 357.327 0. 1724 1295.86 2.002735 1650 -055.815 0.1666 1328.72 1.997158 150 -0.03005 1554.15319 0.1613 1361.29 1.991692 ~1750 0.02915 352. 847 0.1564 1393.62 1.986357 1800 -0.02844 351.418 0.1516 1425.70 1.981199 1850 -0.02752 350.029 0.1469 1457.49 1.976124 1900 -0.02658 348.685 0.1427 1489.01 1.971264 1950 -0.02559 347.391 0.1387 1520.28 1.966563 2000 -0.02459 346.147 0.1350 1551.35 1.962023 14

TABLES OF THERMODYNAMIC DATA Table 2c Mixture of Methane and Propane Containing Approximately 5.2 Percent Propane Values at the Temperatures of Isothermal and Isenthalpic Determinations T = -27~F P 0 H V S psia Btu/lb psi Btu/lb ft3/lb Btu/lboR 0 -o. o4559 347.530 14.7 -. o4589 346.858 50 -0.04670 345.223 100 -0.04791 342.858 150 -0.04907 340.434 200 -0.05033 337.948 250 -0.05164 335.399 300 -0.05305 332.781 350 -0.05447 330.093 400 -0.05598 327.332 450 -0.05749 324.495 500 -0.05921 321.578 550 -. o6096 318.575 600 -0.06288 315.479 650 -0.06479 312.287 700 -0.06681 308.997 750 -0.06883 305.606 800 -0.07084 302.115 850 -0.07296 298.520 900 -0.07487 294.824 950 -0.07679 291.033 1000 -0.07850 287.153 0.1645 1.898613 1050 -0.07970 283.193 0.1565 1.886046 1100 -o.08010 279.160 0.1474 1.873576 1150 -0.07990 275.196 0.1395 1.861264 1200 -0.07899 271.221 0.1312 1.849175 1250 -0.07707 267.31 0.1238 1.837426 1300 -0.07434 263.534 0.1165 1.836123 1350 -0.07111 259.898 0.1102 1.815302 14oo -o.06727 256.438 o.1046 1.805018 1450 -0.06293 253.183. 09867 1.795323 1500 -o.o5838 250.15 0.09419 1.786275 1550 -0.05351 247.356. 08986 1.777851 1600 -. 04908 244.791 o.o8586 1.770063 1650 -0.04474 242.446 0.08297 1.762844 1700 -0.04081 240.307 0.08041 1.756161 1750 -0.03729 238.354 0.07833 1.749955 1800 -o. 03406 236.571 0.07625 1.744186 1850 -o0.03104 234.943 0.07432 1.738821 1900 -0.02822 233.462 0.07256 1.733832 1950 -0.02529 232.124 0.07096 1.729209 2000 -0.02247 230.930 0. 06936 1.724952 15

TABLES OF THERMODYNAMIC DATA Table 2d Mixture of Methane and Propane Containing Approximately 5.2 Percent Propane Values at the Temperatures of Isothermal and Isenthalpic Determinations T = -147.4~F P 0 H V S psia Btu/lb psi Btu/lb ft3/lb Btu/lb~R 500 -0.0023 112.977 0.04805 1.448738 550 -0.0020 112.868 0.04783 1.446971 600 -0.0018 112.771 0.04765 1.445248 650 -0.0016 112.684 0.04751 1.443561 700 -0.0014 112.607 0.04733 1.441912 750 -0.0011 112.540 0.04716 1.440299 800 -0.0009 112.488 0.04700 1.438740 85C -0.0007 112.446 O. 4685 1.437218 90C -0.0005 112.414 0.04671 1.435732 950 -0.00035 112.390 O. 04656 1.434274 1000 -0.0002 112.377. 04642 1.432856 1050 -0.0000 112.372. 04626 1.431469 1100 0.0002 112.377 0.04610 1.430119 1150 O.0003 112.39 o.o4596 1.428798 120)0 o. 0004 112.409 0. 04581 1.42750 1250 0.0005 112.433 0.04568 1.426222 1300 0.0006 112.461 0.04552 1.424963 1350 0.0007 112.494 0.04539 1.423724 1400. ooo8 112.532 0.04527 1.422504 1450 0.00095 112.576 0.04514 1.421307 1500 0.0011 112.627. 04498 1.420136 1550 0.0012 112.683 0.04471 1.418987 1600 0.0013 112.745 0.04437 1.417867 1650 0.0014 112.812. 04369 1.416782 1700 0.0015 112.885 0.04301 1.415736 1750 0.0016 112.962 0.04248 1.414717 1800 0.0017 113.045 0.04197 1.413733 1850 0.0018 113.132 0.04146 1.412776 1900 0.0019 113.224 0.04100 1.411851 1950 0.00205 113.322. o4065 1.410959 2000 O. 0022 113.427 o.o4o045 1.410098 16

TABLES OF THERMODYNAMI( Table 3a Mixture of Methane and Propan( ApproximatelY 5.2 Percent Values at the Pressu, of Isobaric DeterminaP = 0 psia P = 14.7 psia P 250 Psia P = 400 psia C P H H S CP H S C P H S C P Btu/lb'F Btu/lb Btu/lb Btu/lboR Btu/lb-F Btu/lb Btu/lboR Btu/lboF Btu/lb Btu/lboR Btu/lb' Temperature Temperature OF OF -28o u.46o 228.97 0.7765 1.759 0.995950 0.7749 2.46 o.994819 -28o 0 77O -270 o.460 233-' 57 0.7779 9.459 1.037546 0.7759 10.26 1.036952 -270 0:77 2 -26o o.461 238' 17 236.25 2.115641 0.7795 17.259 1.07752 0.7771 18.o6 1 07693 -26o 0. 7754 -250 o.462 242-77 24o.75 2.1-3759 0.7826 25-159 1.116039 1:11547 0 0.78oo 25.96 -250.7770 -24o o.462 247-37 245.25 2.158550 0.7887 33-159 1.153237 0.7857 33.86 1.152226 -24o 0.7834 -230 o.463 252.07 24 85 2.179201 0.796o 41-359 1 lN684 0.7922 41.86 1.187803 -230 0-789C -220 o.463 256.67 2N-45 198784 Mo8o 49.659 1:22 993 0.8030 4g.96 1.222286 -220 0.7991 — -210 o.464 261.27 259.05 2.217563 o.8242 58-059. 1.259310 o. 8169 5 o6 1.255359 -210 0.8124 -200 o.464 265.87 263-75 2.236o o.8435 66.659 1.29303 0.8332 66.26 1.287524 -200 o.826P -190 o.465 270.47 268.45 2.25374o 0 862 75.459 1.32623 0.8485 74.66 i-319236 -190 o.8419 -18o o.466 275-07 273-15 2.27o837 o:88 7 84.459 1.35877 o. 8724 83.26 1.350 2 -180 o.864-, -170 o.466 27-77 277-95 2.287'94 92.o6 1-38N15 -170 0.892 5 0.9150 93.659 1.391 87 0.9028 -16o o.467 28.47 282-75 2.303871 196.059 1.73836 0.9478 101-56, 1. 413622 -16o 0.931 —, -150 o.468 289.17 287-55 2.319605 221.059 1.82o46 1.035 111.46 1.446072 -150 0.983-: -14,(.4 o.468 290-37 288.85 2.323781 233.459 1.83428 115-16 1.457954 -147.4 l.oooh -i4o o.469 293.87 292-35 2.334845 237-759 1 87351 154.48 1.58283- -14o l.o62LI -130 o.470 298-57 297-15 2.349619 251-159 1:91478 222-34 1.79076 -130 1.254c -120 o.471 303.27 301-95 2.363948 262.259 1.9'4793 243-03 1.85205 -120 -110 2.37 97605 o. 4 72 307-97- 3o6.75 7862 271-959 1 256.46 1.89o62 -110.-loo o.473 312.67 311-55 2.39138 28o.959 2:ool4l 267.26 1.92076 -100 - 90 0.474 317-37 316-35 2.4o4530 289.859 2.0258o 276.84. 1.94677 - 90 - 8o o.475 322-17 321.15 2,417329 298.959 2.05oo6 286.11 1.97103 - 80 - 70 0.477 326.87 325-95 2.42979b 308-559 2.075011 295.49 1.99519 - 70 - 6o o.478 331.67 330-75 2.441951 0.566 318-759 2.loo86o 305-49 2.02025 60 - 50 o.48o 336.47 335-55 2. 453806 0.557 323-559 2.112698 315-78 2.o4544 50 - 40 o.482 341.2-7' 34o.45 2.465616 0.550 328.459 2.1-24512 o. 4go 322.62 M61902 40 o.645-- 30 0.484 346.07 34R-R5 2.477147 0-541 33.459 2.136276 0.510 327.49 2.07334 30 o.632r 27 o.485 347-57 34 - 5 2.480623 0.54,, 335-359 2.139752 0.527 329.0 2.076946 27 o.630C - 20 0.487 350-97 350-95 2.488413 0.53 338.959 2.148ooo 0.564 332-51 2.084965 - 20 o.621'. - 10 0.489 355-77 355.85 2.499424 o 9.61o,, 0 o.491 2 53 344.159 2.159684 0.564 337:g7 2.097199 - 10 0 4 36o.77 36o.75.510192 0:534 49-359 2.171112 0.563 343 3 2.110019 0 o.6ol,., 10 0 494 365-67 365.6-5 2.520732 O'.534 354-5~ 2.182509 0.563 349-59 2.122355 10 0.594: 20 0:497 370-57 370-55 2.531050 0.535 39,659 2.19366T 0.562 355-35 2.134432 20 0.58( 0.500 2-54115 0.536.059 2.2o4802 375-57 375.45 3 5 0.562 361.01 2.146059 0.58 o 0.503 38o. 7 38o.45 2.551257 0.537 37o.459.2.2157o8 0.562 366.67 2.157451 o 0.582'.50 0.506 385-27 385.45 2.561169 0.537 375.859.2.226599 0.561 372.23 2.168425 50 0.58ol' -6o 0.509 390-77 390.45 2.570879 0.538 381-359 2.237 -8' 0.561 377.'79 2.179186 6o 0.579'1 70 0.512 395.87 395.45 2.58o4o4 0-54o 386-959 2.247945 0.561 383-25 2.189558 70 0.5781, 80 0.516 4oo.97 40-0.489 2.58975 0.543 392-559 2.228412 0-56o 388.61 2 199554 80 0-578 90. 0.51-9 4o6.17 405-721 2.99421 0 546 398.159 2.2 8687 0-56o 393.87 2:209189 go 0-577 91.6 0-52o 4o6.97 4o6.523 2. oo88,, 0:546 399-359 2.27o866 0-56o 394-76 2.210779 91.6 0-57T 100 0.523 411-37 410-.917 2.608778 100 0.577. ilo 0.527 416-51 416.113 2.617974 110 0.578( 120 0.531 421.87 421.4o8 2.627183 120 0.5791 N 0 - 535 427-17 426.7o4 2.636235 Ho 0.58o' 0.539 432-57 582( 150 0.543 437-97 437.'495 2.654210 1 0:584' - 50 16o 0.547 443-37 442.87'6 2.662960 16o 0.587 170 0.552 448.87 448.383 2.671773 170 0.589. 18o 0.556 454.47 453-983 2.680592 18o 0.592' 190 0-56o 46o.07 459.680 2.689268 190 0.5941 465.67 465-356 2.697828 200200 0.565 0.598. 210 0.570 471-37 470-956.2.7o6249 210 o.602( 220 0.574 477-07 476.656 2.714694 220 o.605 I.n- n- I.n- — /- - 1. nnn n <nn,

TABLES OF THERMODYNAMIC DA' Table 3b Mixture of Methane and Propane C ApproximatelY 5.2 Percent Pri Values at the Pressures of Isobaric Determinatic P = 1000 psia P 120Q Psia P P = 1100 psia CP H S CP H' S CP H S Cp Btu/1b0F Btu/lb Btu/lboR Btu/lb"F B`tu/1b Btu/lboR Btu/lboF Btu/lb Btu/lboR Btu/1b0F Bti Temperature Temperature O.F OF -28o 0.7694 5.442 0.990754 -280 0-760 7 -270 O'.7704 13.1 6 1.032348 -270 0.76 o 15, —:26o 883 -26o 0.7714 20.8 0 1.071 0.7650 23, -250 0.7745 28-564 1.109558 -250 0.7660 30, -24o 0.7765 36.318 1.145629 -240 0.7680 8. -23b 0.7805 44.113 1.180278 -230 0.7720 Z -220 0.7885 51-958 1.21366o. -220 0.7765 53. -210 0.7925 59-863 1.245907 -210 0.7825 61. -200 o.8o15 67-848 1.277217 -200 0.7875 69. -190 o.8145 75-933 1.307711 -190 0.7955 77. -18o o.8316 84.148 1.337566 -180 0.8070 85, -170 o.8516 9.2.564 1.367094 -170 o.8215 93. -16o o.8726 101-135 1.96150 -16o o.8390 -101. -150 0.9030 109-993 1. 25195 -150 0.858o lio. -147.4 0.9062 112-377 1.432856 -147.4 o.8630 112. -14o 0.9231 119.145 1.454261 -14o 0.8789 119, -130 0.9780 128.676 1.483573 -130 0.9055 127. 120 1.0494 138.8o6 1.513815 -120 0.9376 137. -110 1.1412 149-714 1.545463 -110 0.9787 146. -100 1.2974 161.82Q 1.579576 -100 1.0288 156, - go 1.6277 176.26o 1.619o8o i.2939 170.978 1.598847 - go 1.0930 167. - 8o 2.4152 196.028 1.6716 1 1.6o48 188-525 1.648935 1.4920 184.858 1.635853 - 8o 1.1697 178. - 70 2.4631 220-723 1.735M 2.1077 207-585 1.698431 1.7332 201. o69 1.677944 - 70 1.2605 190. - 6o 1.9321 242.64 1.791286 1.9966 228-361 1.751010 1.8523 219.261 1.723975 - 6o 1.4557 203. -'50 1.4700 259.4N 1.832686 1.7554 247-196 1.797520 1.7402 237.444 1.768go4 - 50 1. 299 217. - 40 1.22006 272.746 1.864843 1.3725 262.488 1.834382 1.4710 253-505 1.807620 - 4o 1.4319 232. - 30 1.0609- 284.1ol 1.891536 275-.498 1.864997 1.2288 266.924 1' 839202 - 30 1.3607 246. - 27 1.022.0 287-15 1.898613 279.201 1.873576 1.1768 271.227 1.849175 27 1.3292 250. - PO 0.9487 294.2N 1.9149og 20 1.2621 259. - 10.0.8731 303-353 1.935299 10 1.1761 271. 0 o.8169 311-785 1.953827 0 1.0970 282. 10.0-7756 319-729. 1.970904, 10 1.0205 293. 20 0.7438 327-315 1.986864 20 0.9439 303. o.7180 334.623 2.001923 0 o.8752 312. 4 0.6986 341.697 2.o16207 o o.8285 320. 50 o.6852 348.607 2.02989o 50 0.7977 328. 6o o.6702. 35'5-361 2.o43023 60 0-774 3 6 70 o.6603 362.003 2.055678 70 0.7554 34.4 8o o.6513 368-566 2.o67943 80 0.7385 351. 90 o.6444 375-050 -2-079837 go 0.7256 359. 91.6 o.6426 376-077 2.081701 91 6 0.7242 360. 100 o.6389 381.467 2.ogi4o8 100 0.7159 366. 110 0.6359 387.836 2.102685 110 0.7074 373. 120 0.6339 594.-175 2.11 715 120 0.7oo4 380. N9 o-6339 4oo-514 2.1N559 N90 o.6930 387. 1 0.6339 406.842 2.135201 1 o.6855 94. 150 o-6329 413-170 2.14 665 150 o.6795 01. 16o o.6319 419.490 2.155943 160 o.675o 4o8. 170 o.6319 425.81o 2.166o6o 170 o.6715 414. 18o o.6326 432-130 2.176o16 180 0.669o 421. 190 0.634 438.450 2.185815 190 o.666o 428. POO n - 6-4P UL4-770 9 1 QP)467 POO o.6696 4,4 -

PRESSURE (PSIA) 0 0 0 0 0 00 0 0 0 0P U 0 0 0 0~C 0 0 00 0 0 0 0 00 0 00 0 0 J_ ooooo ooooo I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t-~~~~~~~~~~~~~~~~~~~~~~~~~~~~ili: I I.i;; i 1-ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J -4 - 2 8 4: i j -4 4 t-1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i- -- t~~~~~~~ C~~~~~~~~~~~~~~~~~~~~~J ~ ~ ~~~~~~1 2 5 f T-H -r7~~~~~~~~~~~~~~~~~~~~~ —1LT'- ~I1. i.ri iiIIi Ip~~~~~~~~~~~-1 2 501f 0 ~ ~ ~ ~ ~ ~ ~ ~~0i Oq 1.;.1. i.;:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ iI -I.~ i:! a,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 ~~~~~~-220~ 0~~~~~~~~~~~~~~-9 03 R~~~~~~~~~~~~~~ v, 0 ULI/CIII _176~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L I 4 -1.. (D Xil~ F= IV~~~~~~~~~~~~~~~~~~~~1 -170 0 ~~~~~~~~~~~~~-160a (D....... I fill I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: _ ~.,:i. i.I.1 -i; F-I -lo~~, ~~;ct —-t-~L I 0 -l" 77771 ~ ~ ~ t~C-t- IF~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~it -rttii -: —— ~!- -- -— ~ -f~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (D CO8 i-i — i O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~T C-f ~~~1 ~~~~~~~3 Ft~~~~~~~~~~~-i-~~~~~~~~~a~~~~~; L ~~~~~~~~~ii~~~~~~~f~~~~~-:~~~~~~~t110 0V~-t (D _TTi —' F —J~~~~~~~~~~~~~~~~~~~~~~~~~~~ij~i II- i 0 ~ ~ ~ ~ ~ ~ i ITP~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (D A~~~~~~ OD~~~~~~~~~~~~~~~ 0> co 0 4~~~~~~~~~~~~~ SF ii bbbbbis I-~~~~~~~~~~~~~r ~~~O C:It —-1i'-i0 I (O ( 44I i ~ao,I 0 A rl+-L_+L_ VI~~~~~~~~~ci

PRESSURE (PSIA) 0 0 0 0 0 00 0 0 0 0P U 0 0 0 0~C 0 0 00 0 0 0 0 00 0 00 0 o o o o o oo o o o o oo o o o _ 0 ~ t ~ l:II.;;i1i - -- -4 - 280 4~_ i j -4 4 t-1 t~~~~~~~~~~~~~~~~~~~~~~~~~~~ f T-H -r7~~~~~~~~~~~~~~~~~~~~~ —1LT'- ~I1. i.ri iiIIi I ~~~~~~~T~ 0 T~~~~~~~~0i 24Uiti —i il 0~20 4-H.- L iiif- ii- - O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -_4-fI 1A~~~~~1 Z: CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_'=JUICII _7 0z ~i iP 0~~~~~~~~~~~~~~-5 CD. il~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J IV -140~~~~~~~~~~~~~~Tl -0 0~~~~~~~~~~~~~~10 L' (3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~EJ i#~~~~~~~~~~~~~~~~~~~~~4-4 (D co -171 iCJ ~~t; —~ —I~~~;::-:::-: I:::::-. -_.-:r-~:cl:::-.: —-f -~i4i (D 4+44~~~~~~~~~~~~~~~~~~~~~~4 15 0 4L- I - - - - - -~~~~~~~~~~~~~~~~~~~~~~~~~~~-t -rtt — tj-:-l —~-:i -- t: -~ _U1,, ~~~~ ~ ~ ~ 0 (D( - O ili c 140ii~0on — i O H: iittii-i i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4C\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~III CD'AH1 C ff- 0 T-i (D T-t- 77 F f I 17 - - - ---- 4710 i I —-i 0 Temperature#I;n, i-t;~~~~~~~~~~~~~~O II I T~ ~ -I ~l:i A~~~~~~~~~~~~~9. OD 7~~~~~~~~~~~~~ 0 oil) I I 71I~ ~ ~~~ ~~~~5 r > o >ru~Jr 0 > r T coP r U~~~~~~~~~ SF ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ co L -Aif

1.9 + i -4T i I I -- - --- - ------ --- ---- _Ft_1 1 1 4 IL +H Methane Propane COMPONENT MOLE FRACTION 444CH4 0.9464 44_Tlw H in C2Hr,, 0.0006 + ifl C31-18 0.0518 I I L I - I 1.7 it -H4-Hl+ N2 O.U-UU6 + 02 0.0002 T C02 0.0004 1.6 BASES: H 0 For Pure Components I Os Saturated Liquids at -280'F S=O For Pure Components 1.5 as Perfectly Oriented Crystals at OOR 4 +4 I T — The, University of Michigan 0 1969 1 T 7 T i.4 ------------ 4- I 1 1 1 1 M + i.3 tt yo +4 i.2 t -t-;a iling, I 11 IL I I I i L A_ -T I II.0 44 1 -4-1 Q 4_4 i~ - 0.9 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 21

2.8 I IM Methane'Propane IIAIII II L COMPONENT MOLE FRACTION 2.7 ++H 0 CH4 0.9464 IlF H6, 0.0006 C2 0.0518 C3H8 4E ch 2.6 N2 0.0006 I I I It'll 1111111 -44-14 1 tHill 02 0.0002 I I till I I i l l I i t A i I l l' I I C02 0.0004 H+44+ -BAS 11111.1 2.5 S H =0 For Pure Components U as Saturated Liquids at -280OF cj S 0 For Pure Components t as Perfectly -Oriented Crystals at OOR 24 1 H The University of Michigian I t i l l i t I 1969 LL Or M 4d#, _j 2.3 4 I II I I I T M I I LE.! W =-L -ILL fH 2.2 7r F I I I F I -4+4 it 2.1 I it I 101T - - - ------- -.......... iH 0 2.0 LA11 I I HillI I 1.9.............................. - - ----- Ill I; I is 4_4 di i.8 210 220 230 240 250 260. 270 280 29 300 310 320 330' 340 350 360 370 380 390 400 410 420 430 44

NOTATION a,b constants used in Equation (II.6) a. coefficient in a polynomial A area under a c(.urve B second virial coefficient in volume polynomial virial equation (Units of V) Bm sercond virial coefficient of a mixture B' second virial equation in pressure polynomial virial equation (Units of P) B second virial coefficient; of component 1 B second Virial coefficient of inter'acf tion between a molecul of component i and a molecule of component 2 B2 second virial coefficient of component 2 C third virial (coeff'icient of volume polynomial virial equation (Uni ts of V2) C' third virial ctoefficient in pressure polynomial virial equation (Units of p2) isobaric heat capacity - (HcH/T) p, (Btu/lb) pm CpIX isobaric heat capacity of a mixture (Btu/lb) C psl isobaric specific heat of' saturated liquid (Btu/lb' Cp excess isobaric heat capacity (Btu./lb) Jfugacity (psia) (> specf.'i tic tree energy (Btu/lb) 1-] speci:tfi.c enthalpy (Btu/lb) _iR speci.-fic e'nthalpy at the beginning (Btu/lb).T.i s pecific enthalpy of ith component (Btu/lb) Tm specific enthalpy of the mixture (Btu/lb)

H~ speci.tfic enthalpy at zero pressure (Btu/lb) HT specific enthalpy at zero pressure and temperature T (Btu/lb) IGM ideal gas mixture n1 integer P pressure (psia) pressure at the beginning (psia) R gas constant RGM real gas mixture S specific entropy (Btu/lb~:F) s_ specific entropy of ideal gas (cal.,/gm mole C) Si specific entropy in. the i th state (Btu/lb(~F) T temperature (~F or ~R) TB temperature at the beginning ( ~F or ~R) Tj temperature in ith state (~F or ~R) TS bubble point ( )F) U spec ific internal energy (Btu/lb) V specific volume (fQt3/lb or cc,/g) x mole fract.ion or mass fraction or an independent variable dependent'~ variable PV (:: ompressibility factor RT compressibil]ity fac-tor of a mixtu-re Greek Notation t!. change in the specific enthalpy between two states (Btu/lb) AHR change in specific enthalpy between two temperatures at E constant pressure (Btu/lb) AHT change in the specific erntlhalpy between two pressures at constant temperature (Btu/lb)

-25N..vap specitic enthalpy of vaporization (Btu/lb) AS. change in the specific entropy between two states (Btu/lb~F) AS change in specific entropy at constant pressure between two temperatures (Btu/lb~F) ASST change in specific entropy at constant temperature between two pressures (Btu/lb~F),p Joule-Thomson coefficient fugacity coefficient ( -f/p) {P density (g/cc or lb cu ft) isothermal throttling coefficient -( Sub s c r i p ts B conditions at start i ~ component in a mixture or a running variable 5j a running variable m mixt ure 0> outlet conditions.p pressure s.]J. saturated. liquid Tp tempe r a ture vap vaporization econdition 1. inlet condition or component. ].] component I 12 interaction condition between a molecule of component 1 and a molecu]le of component 22 component 2 or outlet condition 22 c(omponent 2 35 component three or a state 3

-26Superscripts E excess value 0 zero pressure value

-27BT BLIOGC-RAP HY ]. API Research Project No. 44., "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds," A and M College of Texas, College Station, Texas (1953) 2. Barnard, A.J., S.P. Luthra, D.M. Newitt and M.U. Pai, Proc. Roy Soc. (London) A209, 143 (1951) 3. Beattie, J.A., W.C. Kay and J. Kaminsky, J. Am. Chem. Soc. 59, 1589 (1937) 4 Beenakker, J.J.M., B. Van Eijnbergen, M. Knoester, K.W. Taconis and P. Zandbergen, Symp. Thermophys. Prop., Papers, 3rd, Lafayett, Ind., p. 114, 1965 5. Benedict, M.. G.B. Webb and L.C. Rubin, J. Chem. Phys., 8, 334 (1940) 6. Benedict M., G.B. Webb, and L.C. Rubin, J. Chem. Phys., 10, 7747 (1942) 7. Bi(her, L.B., and D.L. Katz, Ind. Eng. Chem. 35 754 (1943) Brewer, J., Project No. 975-01, Air Force Office of Scientific Research, Arlington, Virginia, Dec. 1967 9. Budenholzer, R.A., D.F. Botkin, B.H. Sage and W.N. Lacey, Ind. Eng. Chem., 534, 878 (1942) 10. Budenhol-zer, R.A., B.H. Sage and W.N. Lacey, Ind. Eng. Chem., 35, 12124 (1943) 11. Canjar, L.N. and F.S. Manning, Hydrocarbon Process. Petrol. Refiner, 41 No. 8 121-3 (1962) 12. Canjar, L.N. and F.S. Manning, "Thermodynamic Properties and Reduc.ed Correlations for Gases", Gulf Publishing Co. Houston, Texas 1967 13. Canjar, L.N., N.R. Patel and F.S. Manning, Hydrocarbon Process. Petrol. Refiner, 2_1, No. 11, 203-24 (1962) 1.24 Canjar, L., V.M. L-'ejada and F.S. Manning, Hydrocarbon Process. Petrol. Refiner, -11, No. 9, 25- 4- (1962) ibid No. 10 1249-50 (1962) 15. Chu, J.C., N.F'. Muxeller., R.M. Busche, and A.S. Jennings, Pet. Processing, 1203 (1950)

-2816. Collins, S.C. and F.G. Keyes, Proc. Am. Acad. Arts Sci., 72, 283 (1938) 17. Collins, S.C. and F.G. Keyes, J. Phys. Chem. 43 5 (1939) 18. Colwell, J.H., E.K. Gill and J.A. Morrison, J. Chem.Phys. 39, 635 (1963) 19. Curl, R.E., Personal Communications 20. Cutler, A.J.B. and J.A. Morrison, Trans. Farad. Soc., 61, 429 (1965) 21. Dana, L.I., A.C. Jenkins, J.N. Burdick and R.C. Timm, Refrig. Engr. 12, 387 (1926) 22. Dantzler, E.M., C.M. Knobler and M.L. Windsor, J. Phys. Chem. 72, No. 2 676 (1968) 23. Deschner, W.W., and G.G. Brown, Ind. Eng. Chem. 32 836 (1940) 24. Dillard, D.D., M.S. Thesis, Oklahoma State Univ. (1966) 25. Dillard, D.D., W.C. Edmister, J.H. Erbar and R.L. Robinson, A.I.Ch.E. J., 14 923 (1968) 26. Dittmar, P., F. Schultz and G. Strese, Chemie-Ing. Techn. 34, 437 (1962) 27. Douslin, D.R., R.H. Harrison, R.T. Moore and J.P. McCullough, J. Chem. Eng. Data 9, 358 (1964) 28. Ernst, G., Dr. Ing. Dissertation, Universitat Karlsruhe, Germany 1967. 29. Eucken, A. and W. Berger, Z. ges Kalte-Ind., 41 145 (1934) 30. Finn, D., Ph.D. Thesis, Univeristy of Oklahoma (1965) 31. Frank. A. and K. Clusius, Z. Physik. Chem., B36, 291 (1937) 32. Head, J. F., Ph.D. Thesis. University of London. 1960 355. Helgeson, N.L. and B.H. Sage, J. Chem. Eng. Data 12, 47 (1947) 34. Hestermans, P., and D. White, J. Phys. Chem. 65 362 (1961) 55. Hoover, A.E., I. Nagata, T.W. Leland, Jr. and Riki Kobayashi, J. Chem. Phys. 48 No. 6, 2633 (1968)

-2936. Huang, E.T.S., Ph.D. Thesis, University of Kansas, (1966) 37. Huang, E.T.S., G.W. Swift and F. Kurata, AIChE J., 13 846 (1967) 38. Huff, J.A., and T.M. Reed, III, J. Chem. Eng. Data, 8, 306 (1963) 39. Hujsak, K.L., H.R. Froning, and C.S. Goddin, Chem. Eng. Prog. Symp. Ser., 59, No. 44, 88 (1963) 40. Jones, M.L., Jr., Ph.D. Thesis, University of Michigan (1961) 41. Jones, M.L., D.T. Mage, R.C. Faulkner and D.L. Katz, Chem. Eng. Prog. Symp. Ser. 59 (44), 52 (1963) 42. Kemp, J.D. and C.J. Egan, J. Am. Chem. Soc. 60 1521 (1938) 4>. Klaus, R.L. and H.C. Van Ness, Personal Communications 44. Kobayashi, Riki, Personal Communications 45 Lane, Ralph E.,"The Extension of Jenkins' Fifth-Difference Modified Osculatory Interpolation Formula for Unequal Intervals," Unpublished Report, Military Physics Laboratory, University of Texas. 46. Mage, D.T., Ph.D. Thesis, University of Michigan, (1964) 47. Manker, E.A., Ph.D. Thesis, University of Michigan (1964) 48. Manker, E.A., D.T. Mage, A.E. Mather, J.E. Powers and D.L. Katz, Proc. Ann. Conv., Natl. Gas Process. Assoc., Tech. Papers 43, 3 (1964) 49. Manning, F.S. and L.N. Canjar, J. Chem. Eng. Data 6 364-5 (1961) 50. Mather, A.E., Ph.D. Thesis, University of Michigan (1967) 51. Mather, A.E., V.F. Yesavage, J.E. Powers and D.L. Katz, Proc. Ann. Conv., Natl. Gas Process. Assoc., Tech. Papers, 52. Mather, A.E., V.F. Yesavage, J.E. Powers and D.L. Katz, Proc. Ann. Conv., Natl. Gas Process. Assoc., Tech. Papers, 46, 13 (1967)

-3053. Mickley, H.S., T.K. Sherwood, and G.E. Reed, "Applied Mathematics in Chemical Engineering," McGraw-Hill Book Company, Inc., New York, 1957 54. Pena, M.D., and E.Cerverar, Thermodynamics Symposium, Hydelberg, W. Germany, Sept. 1967, No. 3, 10 55.. Reamer, H.H., B.H. Sage and W.N. Lacey, Ind. Eng. Chem. 41, 482 (1949) 56 Reamer, H.H., B.H. Sage and W.N. Lacey, Ind. Eng. Chem. 42, 534 (1950). See corrections, Ind. Eng. Chem. 42, 1258 (1950) 57. Roebuck, J.R., Phys. Rev. 2, 299 (1913) 58. Rossini, F.D., et al., "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,"t Carnegie Press, Pittsburgh, Pas., 1953 59. Sage, B.H., H.D. Evans and W.N. Lacey, Ind. Eng. Chem. 31 763 (1939) 60. Sage, B.H., E.R. Kennedy and W.N. Lacey, Ind. Eng. Chem. 28, 601 (1]936) 61. Sage, B.H., J.G. Schaafsma and W.N. Lacey, Ind. Eng. Chem. 26, 1218 (1934) 62. Sahgal, P.N., J.M. Geist, A. Jambhekar, and G.M. Wilson, Intern. Advan. Cryogen. Eng. 10, 224 (1965) 63. Sehgal, I.J.S., V.F. Yesavage, D.L. Katz and J.E. Powers, Hydrocarbon Process. 8, (8) 137 (1968) 64. Stull, D.R., E.F. Westrum, Jr., and G.C. Sinke, "The Chemica.l Thermodynamics of Organic Compounds," John Wiley and Sons, Inc., New York, 1969 65. Vennix, A.J., Ph.D. Thesis, Rice University (1966) 66. Vennix, A.J., T.W. Leland and Riki Kobayashi, Submitted for Publication in J. Chem. Phys. 67. Wiebe, R. and N.J. Brevoort, J. Am. Chem. Soc. 52 622,(193(

-3168. Wiener, L.D., Paper presented at -;8th National AIChE Meeting, Dallas, 1966 69. Yarborough, L. and W.C. Edmister, AIChE J. 11, 492 (1965) 70. Yesavage, V.F., Ph.D. Thesis, University of Michigan (1968) 71. Yesavage, V.F., A.W. Furtado, D.L. Katz and J.E. Powers, "Enthalpy Data for a Mixture Contcaining 77 Mole Percent Propane in Methane and Comparison with Prediction," To be published in AIChE J. 72. Yesavage, V.F., A.W. Furtado, and J.E. Powers, Proc. Ann. Conv., Natl. Gas Process Assoc., Tech. Papers, 471,3(1968) 73. Yesavage, V.F., D.L. Katz and J.E. Powers, Fourth Symposium on Thermophysical Properties, A.S.M.E., New York (1965) 74. Yesavage, V.F., D.L. Katz and J.E. Powers, J. Chem. Eng. Data 14, No. 2 (1969) 137 75. Yesavage, V.F., D.L. Katz and J.E. Powers, "Enthalpy Data for a Mixture Containing 51 Mole Percent Propane in Methane" To be published in AIChE J. 76. Yesavage, V.F., A.E. Mather, D.L. Katz and J.E. Powers, Ind. Eng. Chem. 56 35 (1967)

-32Appendix I Review of Literature A number of investigators have reported both calorimetric and PVT data for methane, propane and their mixtures. Brief resumes of the most pertinent references are given in this Ap~pendix. Calorimetric Data Although a number of different investigators have reported calorimetric data for methane (9,18,29,31,134,39,40,41,46,62,68), propane (21,28,30,33,59,60,69,70,73,74), and their mixtures (9,24,25,32,47,48,50,51,52,70,71,54) the results of E.A. Manker (47) and A. E. Mather (50) were used almost exclusively for the following reasons: 1) The data of these two investigators were obtained with. mixtures of almost identical composition. 2) These experimental investigations cover an extensive range of both temperatures and pressures - certainly those of primary interest to the natural gas processing industry (-240~F to +250~F at pressures up to 2000 psia). 3) These data are thermodynamically consistent to +0.2%. 4) No other data reported in the literature were obtained at exactly the same composition.

-33Manker (47) and Mather (50).- The calorimetric data used in making all calculations were obtained directly from the theses of Manker (47) and Mather (50). Manker reported isobaric data over a range of temperature from -2240 to +90OF with some relatively crude Joule-Thomson determinations at 60 ~F up to 2000 psia. Mather obtained supplementary isobaric data between 90 and 2500F at pressures between 500 and 2000 psia, isothermal results between 100 and 2000 psia at -27, 91.6 and 2000F and isothermal and isenthalpic data at -147~F between 600 and 1800 psia. The temperatures and pressures of experiments made by these investigators are illustrated in Figure 1. The average compositions of the mixtures as reported by these two investigators are summarized in Table AI.1. TABLE AI.1 Mixture Compositions Reported by Manker (47) and Mather (50) Component Manker (47) Mather (50) Methane, CH4 0.9464 0.9463 Ethane, C2H 0.0006 0.0006 Propane, C 3H8 0.0518 0.0510 Nitrogen, N2 0.0006 0.0007 Oxygen, 02 0.0002 0.0001 Carbon Dioxide, CO2 0.0004 0.0003 1.0000 1.0000

-34There was some variation in composition during the course of the investigatiors as determined by periodic chromatographic determinations. The extremes of such variations are indicated in terms of the propane mole fractions in Table AI.2. TABLE AI.2 Extremes in Compositions of Manker (47)and Mather (50) Mole Fraction Propane Investigator Lowest Value Average Value Highest Value Manker (47) 0.0491 0.0518 0.0521 Mather (50) 0.051 0510 0.052 The greatest variation in composition occurred during runs made through the two-phase region. No attempt was made to correct for variation in composition for individual runs. The average composition reported by Manker (47) as given in Table AI.1 was used as the base composition because the majority of Mather's runs were made at temperatures relatively far removed from the critical region and therefore, small variation in composition would be expected to have a negligible effect. Dillard, et al (25).- Results from a series of isothermal determinations with methane and methane-propane mixtures have been reported by Dillard (24) and Dillard, et al (25). The equipment was designed to yield integral measurements of

isothermal changes in enthalpy between elevated pressures and atmospheric. Results were reported for methane and two methanepropane mixtures as indicated in Table AI.3.'iABLE AI. 3 Conditions of Integral Isothermal Determinations of Dillard (25) Mole Fraction Inlet Temperature Propane in Methane Pressures Range (psia) (OF) 0.0000 500 to 2000 150 0.051 500 to 2000 90 150 200 0.126 500 to 2000 90 150 200 Replicates were reported for most runs. The precision oiDf the experiments as indicated by analysis of the replicates is -+3)0%. The mean of all the determinations made at one set of c:eonditions is consistent with data obtained by Mather (50) w Jt}iin + 5%. Mather (50) made use of these results to justify use of calculated values based on. the BWR equation of state (5,6) in publishing PTH diagrams for binary-1 mixtures containing 11.7 and 28.0 mole percent propane in methane. The results

reported by Dillard et al (25) are considered to be at least one order of magnitude less accurate than those of Manker and Mather and therefore, were not given further consideration in preparing this report. Budenholzer et al (9).- Joule-Thomson data for mixtures of methane and propane have been reported. The range of composition, temperatures and pressures of these reported values is summarized in Table AI.4. TABLE AI.4 Conditions of Reported Joule-Thomson Values of Budenholzer et al (9) Mass Fraction Temperature (~F) Pressure (psia) Methane high 1 ow high 1ow 0.2458 310 70 1500 0 0.4934 310 70 1500 0 0.7552 310 70 1500 o As indicated in Table AI.4, Budenholzer et al have reported values of the Joule-Thomson coefficient for mixtures of methane and propane. In addition, Sage, Kennedy and Lacey (60) have published values for propane and Budenholzer et al (10) report values for methane so that these data could be used to interpolate with respect to composition.

-37Joule-Thomson data, i, can be used with values of i.sobaric heat capacity, Cpto calculate the isothermal throttling coefficient,,. p 1 - kC (AI.1) Values of the isothermal enthalpy departures are obtained by integration. P (H-H~)T dPT (AI.2) P=O Comparisons made by several investigators (49,50) have illustrated that the Joule-Thomson data covered in Table AI.4 are thermodynamically inconsistent with values calculated from PVT data and published values of Cp (11), as well as the results of Manker (47) Mather (50),Dillard et al (25) and values calculated from the BWR equation of state (5,6). Therefore, no attempt was made to incorporate these experimental data on the Joule-Thomson coefficient in the results of this report. Joule-Thomson data for mixtures of methane and propane hlave also been reported by Head (32). The range of composition temperatures and pressures of these reported values is summarized in Table AI.5. TABLE AI. 5 Colnditions of Reported Jou]le-Thlomson Measurements of Head (32), Fraction Methane Temperature (K) K Pressure (atm) o.489, high low |-..... hi*h low 360 258 40v0o6 1.71

i'or a mixture containing approximately 51 mole percent p:ropanre in methane at 152..2"F the values of lJ reported by t -iad (32) are consistent to within 3% with values interpolated to that composition from data published by Budenholzer et al (9) as reported by Yesavage, Katz and Powers (75). However, values of Q calculated using Equation (AI.1) disagree with the values of Yesavage et al by -8% at zero pressure and +9% at 600 psia (71). These calculated values based on the data of Head (32) also disagree with values of O calculated from PVT data for the mixture (69) and with data calculated by the BWR equation of state (5,6). Therefore, no attempt was made to incorporate the Joule-Thomson values of Head into the enthalpy and entropy values of this report. Volumetric Data A great deal of volumetric data has been reported for pure methane and pure propane. Only two investigators have reported PVT data for mixtures. The published results for the mixtures will be reviewed first and then the data for pure methane and propane used for purposes of interpolation with respect to composition will be mentioned. Reamer, Sage and Lacey (55,56). These investigators reported smoothed values of volumetric data on several binary mixtures of methane and propane. The compositions and range of temperatures covered by this report are listed in Table AI.6.

-59TABLE AT. 6 Conditions of Reported Volumetric Data of Reamer, Sage and Lacey (56) Temperature Pressure (OF) (psia) Low 40 200 High 466 10,000 Composition Mixture Mole Fraction Mole Fraction Me thane Propane 1 0.10 0.90 2 0.20 0.80 3 0.30 0.70 o. 40o o. 60 5 0.50 0.50 6 0.60 0.40 7 0.70 0. 30 8 o 0.80 0.20 9 0.90 0.10 Huang et al (317). The conditions corresponding to values reported for methane-propane mixture are reported in'ABLIEL AI.7

-40TABLE AI.7 Conditions of Reported Volumetric Data of Huang et al (37) Temperature Pressure (OF) (psia) Low -238 50 High 100 5,000 Composition Mixture Mole Fraction Mole Fraction Methane Propane 1 0.221 0.779 2 0.500 0.500 3 0.753 0.247 The ranges covered by the volumetric data of Reamer, Sage and Lacey (56) and Huang et al (37) are indicated in Figure AI.1. Volumetric Data for Methane. None of the volumetric data listed above are for the composition of the mixture investigated by Manker (47) and Mather (50) In fact, all mixtures reported contain substantially more propane than 5 mole ~. Therefore, it was necessary to utilize volumetric data for methane to interpolate with respect to composition to the desired value. The published data for the volumetric behavior of methane is much too extensive to be reviewed in its entirety. Instead, use was made of the most recent compilations of volumetric data.

3000 D D' C C METHA NE - PRO PA NE M I XT URE S ABCD HUANG (37 ) 2500 j A'BC'D'REAMER, SAGE AND LACEY ( 56 ) AEFGH SATURATION CURVE FOR I5% PROPANE IN METHANE MIXTUR, 2000 V) lol C- I I 4Z- 1500 / F / 500 E A' B' A 0 - 300 -200 -100 0 100 200 300 400 500 T-TEMPERATURE (F) Figure AI.l Temperatures and Pressures of Volumetric Data for MethanePropane Mixtures as Reported in the Literature

-42API Project 44. A recent supplement of API Project 44 (1,58) lists values of the volumetric data for gaseous methane from -115 to 2140~F at pressures up to 15,000 psia. These values were used for the purposes of interpolation primarily as a matter of convenience. Vennix et al. The API supplement does not include the published data of Vennix (65,66) and of Douslin et al (27). The data of these two recent investigations are remarkably consistent (+0.02f) at the temperature of overlap (0~C) and have been combined to yield a single equation of state (44). Values calculated using this equation of state were found to be in very good agreement with values interpolated from the API values. Volumetric Data for Propane. The interpolation procedure used to obtain volumetric data for a mixture containing 5 mole X propane is relatively insensative to the values selected for pure propane. Whereas extensive consideration of all published volumetric data is planned subject to availability of funds, values for propane published by Reamer, Sage and Lacey (5')were used in connection with this report because they appear to be reasonably consistent with other published data and were in a form that is convenient for interpolation.

Appendix II Basic Interpretation of Calorimetric Data The calorimetric results of Manker (47,48) have been interpreted twice in the past to yield tables and graphs of enthalpy values (48.50,51) and the combined results of Manker (47) and Mather (50) have been so interpreted once (50,51). Unfortunately eventhe latest interpretation (50,51) appears to be in question for two principle reasons: 1) The isotherms at low temperatures as drawn by Mather (50) exhibit a sharp curvature at low pressures that is not in evidence in the PTH diagrams for methane (40,41), nitrogen (46), propane (70) or any of their mixtures (47,50,70,71,72,73D,74,'75) which have been produced from results obtained in the Thermal Properties of Fluids Laboratory. 2) The correlational efforts of Yesavage (70) were very successful in fitting the enthalpy data for all five binary mixtures of methane and propane except the 5% propane in methane mixture at low temperatures where deviations of more than 4 Btu/lb were noted. In an attempt to resolve these apparent discrepancies a complete reinterpretation of the original data was undertaken to insure that the resulting enthalpy values (and the entropy data obtained therefrom) would be as meaningful as possible. The purpose of this Appendix is to provide detailed descriptions

-44of the procedures used to reinterpret the basic data to provide the bases for an improved estimate of the enthalpy behavior of this mixture with particular emphasis on changes which were incorporated in the analysis. Isobaric Determinations Manker (47) reported the results of 525 isobaric determinations both in the single-phase region and within the twophase region. Mather (50) presented 28 isobaric data points in the single-phase region at elevated temperatures. Single-Phase General procedure. In making measurements in the singlephase region, determinations are usually made with a common inlet temperature and temperature increases varying from 10 to 1000F. In processing the data to yield smoothed values of Cp(T), mean values of C are calculated from the measured isobaric increases in enthalpy and temperature. (HT2- HT1) P p T= T (II.1) CTp T-T 2 1 Values thus calculated are plotted over the temperature interval T1 - T2. Typical values appear on Figure AII.1 as solid horizontal lines extending over the experimentally measured temperature intervals.

-45For isobaric enthalpy differences measured with a common inlet temperature, T1, a difference in enthalpy between two outlet temperatures, T2 and T3, can be calculated: (HT3 HT2)p (T- HT p - HT P )P (II.2) Such differences can be used to calculate additional values of C. Typical values are plotted as dashed lines on p Figure AII.1. Point values of C can be determined from a plot of C vs T by applying the identities H H CP (T- Ti) (II.3) J i p j Cp dT (II.4) Ti where Ti and Tj are used to indicate any two experimental values of temperature. Thus the integral of the function C (T) between T. and T. must equal the experimentally measured values of HT.- HT A Tj i and the product Cp(Tj - Ti). Graphical procedures were applied to determine Cp(T) by an iterative procedure. The interative procedure employed was based entirely on graphical procedures. A draftsman's spline was used to plot a first approximation of Cp(T). Integrations were made between each experimentally determined temperature interval as indicated by Equation (II.4). Simpson's rule (See Appendix X) was used to obtain integral values both accurately and rapidly. Adjustments

2.6 2.4 2.2 2.0 1 DATA POINTS m% -— ~.6 L- DIFFERENCED DATA. 1.6 m 1.4 1.2,o 1.0 0.8 Q6 MOLE FRACTION PROPANE.05 0.4 - MOLE FRACTION METHANE.95 0.2 -280 -240 -200 -160 -120 -80 -40 0 40 80 TEMPERATURE OF Figure AII.1 Isobaric Heat Capacity, C, at 1000 psia Illustrating Method of Interpreting Ca orimetric Results.

-47were made in the curve representing Cp(T) until the deviations between the experimental values of enthalpy difference and the values determined by graphical integration showed random variation. In general. the agreement between experimental and calculated values was on the order of +0.2% or:better. After the iterative graphical procedure described above yielded the desired preliminary relation for Cp(T), Simpson's rule was applied to evaluate isobaric enthalpy differences over temperatures intervals other than those noted experimentally, In particular, enthalpy differences were evaluated over uniform intervals of 100F to aid in preparation of the tables and graphs and between temperatures corresponding to isothermal determinations made with this mixture (See Figure 1). The latter calculations are used to check the thermodynamic consistency of the two types of data (isobaric and isothermal) obtained for this mixture as will be described in detail in Appendix III. The general procedure described above was applied to determine approximations of Cp(T) and isobaric enthalpy differences at all pressures from 1100 psia to 2000 psia over the temperature interval studied at each pressure (See Figure 1). Special care was taken in interpreting the data over the temperature intervals in which C exhibits a maximum. Table AII.l summarizes information on the peak values (including that at 1000 psia).

-48TABLE AII. 1 Information Obtained at the Maximum in Cp(T) Pressure Temperature C at Peak oF Btu/lb~F 2000 -25.5+1.0 1.139 1700 -35.5+11.0 1.2572 1500 -44. 5+1. O 1. 434 1200 -60.5+0~.6 1.8502 1100 -67.5+0.5 2.12 1000 -72. 5+0.2 2.564 Additional efforts were expended at pressures of 1000 psia and below and to extrapolate the reported values upward in temperature to +300 ~F and downward to -2800F. Interpretation of the data at 1000 psia. Over much of the temperature range of experiments at 1000 psia (-240 to -2050F) it was possible to obtain good agreement between calculated and experimental values using the general graphical procedure described in the preceeding section. Manker (47,481) reported an unusually heavy concentration of data points in the immediate vicinity of the maximum of Cp(T) at 1000 psia. This peak is quite sharp (Cp 2.5 Btu/lb~F at the peak) and large values of Cp are in evidence between -120 and -30~F (See TFi'igure AII.1). Thus the enthalpy difference between these two temperatures is exceptionally large and interpretation of the data over this range of temperature exerts a disprooortionate

-49influence on the enthalpy values calculated for low temperatures and low pressures where rather large discrepancies were noted in earlier interpretations. Therefore, special efforts were taken to interpret the data in the vicinity of the peak. A plot with expanded scales for both temperature and C was made. The working drawing measured 70 in. by 30 in. A reproduction in reduced form is presented as Figure AII.2. As in the general case, a curve representing Cp(T) was drawn using a spline and the results of integrations by Simpson's rule were used to make adjustments and obtain a better estimate of the function Cp(T). A more sensitive procedure was used in making the next adjustment: 1) The adjusted curve for Cp(T) was integrated over small temperature intervals and enthalpy differences were calculated corresponding to each experimental temperature interval (HTo- HT ) 0 Pi calc 2) Difference between the calculated and experimental enthalpy differences were evaluated A(AH) E(HT -HT) -(HT -T ) o 1 cal0 o i expt and plotted against (HT -HT ). The first plot thus preO i calc pared is presented as Figure AII.3. Note that all differences are positive indicating a bias. 3) A dotted line was drawn through the differences and through the origin exhibit random scatter instead of a bias. The curve representing Cp(T) was then adjusted to satisfy this condition.

2.5 5.18 Mole% C3 IN C1 I000 psia 2.0 LL 0 a _ m I - i 1% T 0.n 1.0 -130 - 110 -90 -70 -50 -30 T- TEMPERATURE (~F) Figure AII.2 Isobaric Heat Capacity, C, at 1000 psia the Critical Point Illustrating Sharp Max -50o

0.40 1 1 1 1 1 I 5.18 MOL. %C3 in CI 1000 psia TI = -99. 8 F FIRST ESTIMATE AND CORRECTION 0.30 a13. 7 -51

4) Simpson's rule integrations were again performed using the adjusted curve of Cp(T) and new values of (HT -HTI) o i calc determined for direct comparison with the experimental values. The resulting plot is presented as Figure AII.4. By using this special procedure it was possible to represent the experimental data by means of a smoothed curve of Cp(T) well within the limits of precision of the experimental data (+0.3%). Interpretation of the data at 500 psia. Although both Manker (47) and Mather (500) give indication in their theses that enthalpy traverses were made at 500 and 600 psia and Mather (50) presents the results of a check of thermodynamic consistency that incorporates the enthalpy travers at 500 psia, no data for these runs are included in Manker's thesis. The data for these runs were included in the Proceedings of the NGPA (600 psia) (48). The results were not included in Manker's thesis because of unresolved problems with respect to the fl-owmeter calibrations associated with these runs. Another problem associated with attempting to establish the enthalpy behavior of the mixture at low temperatures and pressures below 1000 psia can be understood by considering Figure 1; the isobaric run made between -250 and -1800F at 500 psia does not overlap the enthalpy traverse through the two phase region nor does it intersect the isothermal run made at -147.4~F. Estimation of C is difficult near the twophase region because Cp changes in value relatively rapidly

0.40 I I l I 5.18 MOL % C3 0.30 in CI 1000 psia T= -99. 8F SECOND ESTIMATE 0.20 * 0. 10.) cu 0.00 Iri -0.P0 ~II ~ ~ ~ T Figure AII.4 Check of Interpretation of Isobaric Data Near the Maximum in Cp(T) at 1000 psia after Correcting for Bias. -553

-54in this region. Considerable effort was expended in an attempt to establish the enthalpy behavior at 500 psia and low temperatures because the results of previous interpretations may have been misleading. Resolution of the problems associated with estimation of the enthalpy change across the two-phase region at 500 psia will be described later in this Appendix because it involves use of the isothermal data. The method of establishing values of C up to the bubble point at 500 psia is presented in the p paragraphs that follow. As indicated in Figure 1, isobaric data were obtained in the temperature range from -245 to -175~0F. The bubble point is at -128~F at this pressure. As indicated in the previous section the results obtained from -1400F across the two-phase region at 500 psia are suspect. Therefore, a procedure was developed to estimate Cp(T) at 500 psia at low temperatures which does not rely on the values reported by Manker (47). Reliable values of Cp(T) are available at 500 psia over the temperature of interest for methane (40,41), propane (70), and four other mixtures of methane and propane [11.7% C3(50), 28.0 C3(50), 50.6% C3(70) and 76.6% C3(70)]. Therefore, it is possible to plot C vs composition at various values of T at 500 psia and interpolate to obtain a good estimate of C for the 5% mixture. Interpolations with respect to composition are aided by working with excess quantities; in this case the excess heat capacity, CE; P

C C - 5 p Cpm i xiCp (II.5) where Cpm refers to the mixture and Cpi refers to the pure components, methane and propane in this case. A typical plot is presented as Figure AIII.5. Note that the values are plotted vs mass fraction C in C1 because it was found that the plot 3 1 was much less skewed when thus plotted than when plotted against mole fraction. Smoothed values of CE were read from the plot at 5.18 mol % C3(0.1306mass fraction C3) and values of Cpm were calculated using Equation AII-5 making use of the published C p values for methane (40,41) and propane (70) Experimental data are reported by Manker (47) in the single-phase region near the bubble points at 250,400,650 and 800 psia. Two experiments were reported in the liquid phase at 250 psia and four values are given at each of the other pressures. It was assumed that C is a linear function of p temperature near the bubble point. The data were individually fitted to the equation Cp = a + b(TS - T) (II.6) where TS refers to the bubble point temperature at the given pressure. The results are summarized in Table AII.2.

0.08 I I I I I I I I I METHANEPROPANE 0La. 0L O\SYSTEM 0 -o'"~~~~~~~~~~~~~ ~500 psia 0.06 0 170~F m I \ A 180~F ~~~~, ~ ~~~~~~~~~~~~0 190 OF >0 T ~a. 0.1% I - 0 () 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 MASS FRACTION C3 IN C: Figure AII.5 Plot of Excess Isobaric Heat Capacity, CE for the Purpose of Estimating Cp for the 5% Mixture at 500 psia Where Direct Measurements are not Available.

-57TABLE AII.2 Constants of Equation (AII-6), Expressing Specific Heat in the Immediate Vicinity of Bubble Point Pressure T a b S (psia) (Btu/lb~F) (Btu/lb F2) 250 -167 1.06 0.015 400 -141.65 1.23 0.0216 500 -128.1 1.36* 0.025* 650 -111.87 1.60 0.033 800 - 96.0 2.22 0.0758 * interpolated value Plots were made of a and b versus pressure and the values estimated at 500 psia are reported in Table AII.2. The straight line relation was employed to estimate Cp(T) at 500 psia between -150 and -1280F. Comparison with the values reported in the NGPA Proceedings (48) at 500 psia between -140 and -128OF indicates that the experimental values for this enthalpy traverse as reported were too low by about 5% probably as a result of an error in the calibration of the flowmeter. The values of C in the range -240 to -128~F p determined as described above are summarized in Table AII.5.

-58TABLE AII.3 Values of Cp for the 5.18% C3 in C1 Mixture at 500 psia as Determined by Interpretation of Calorimetric Data at Other Conditions. Temp. (~F) -240 -230 -220 -210 -200 -190 Cp Btu 0.772 0.782 0.795 0.809 0.826 o.844 Temp. (~F) -180 -170 -16 0 -10 -150 - 1 30 -128.1 C p Btu 0.866 0.894 0.928 0.972 1.037 1.182 1.36 Extrapolation with respect to temperature. It was decided to report values between -2800F and 4-3000F although experimental determinations were limited to the interval from -245~F to +2500F. It was felt that such extrapolations could be made accurately. Extrapolation to elevated temperatures was based on blending experimental values with ones calculated using the BWR equation of state. The nature of the agreement is illustrated on Figure AII.6. The C values at elevated temperatures p reported in the thesis of Mather (50) were obtained by blending the results of BWR calculations with experimental values and therefore these values were used directly in this region.

1.00 Methane - Propane Mole Fraction C3H8 0.052 Mole Fraction CH2 0.948 Pressure 2000 PSIA Data Points Difference Points Mather (967).90 Experimental Data of o Tabulated Values of Manker End Manker (1964) m I 0\ * Benedict, Webb, Rubin(1940,1942) D 0 r.80 7060 -I 20 60 100 140 180 220 260 300 TEMPERATURE (OF) Figure AII.6 Experimental Heat Capacities with Values Calculated from B-W-R Equation of State.

-60The BWR equation - as well as most other published equations of state - generally yields erroneous results at high densities especially at low temperatures. On the other hand Cp(T) tends to approach a constant value at low temperatures as indicated in Figure AII.1. Therefore, an empirical approach was used to extrapolate Cp(T) to lower temperatures. By definition T Cp(T2) - Cs(T1) f( p dTp (II.7) Values of (aCp//T)p were approximated by taking differences in tabulated values / C p C (T)- Cp(T1) (II.8) P 2 1 T Values of differences thus calculated were plotted at each pressure and smoothed. The resulting values are listed in Table AII. 4. In estimating the values of (bCp/6T)p at low temperatures p. P as indicated by underlined values, note was made of the trends -in (b2Cp /T2)p with temperature and of the observation that Cp is essentially independent of' pressure at low' temperatures (4o,70).

-61TABLE AII.4 Values of (aCp/CT)p Used to Extrapolate Cp(T) to -280~F ( Cp/6T)p in Btu/lb(OF)2 x 10 tressure (psia) Temperature 500 1000 1500 2000 (OF) -160 0.635 -170 0.44 o.186 0.175 0.13 -180 0.335 0.170 0.155 0.11 -190 0.25 0.15 0.12 o0.o8 -200 0.20 0.13 0.09 0.o06 -210 0.16 0.10 0.07 0.05 -220 0.12 0.08 0.01 0.04 -230 0.10 0.o6 0o.o05 0.03 -240 0o.o8 0.05 o.o4 0.02 -250 o.o6 0.04 0.03 0.01 -260 0.04 0.03 0.02 0.00 -270 0.02 0.02 0.01 0.00 -280 0.01 0.01 0,00 0.00 Two-Phase (Enthalpy traverses) Manker (41) reported isobaric determinations made near the boundaries as well as within the limits of the two-phase region. Experimental data have been reported at 250, 400, 650 and 800 psia (47) as well as at 500 and 6oo psia (48) (See Figure 1). As indicated in the previous section of this Appendix, the values reported at 500 and 600 psia are suspect probably because of an error in the flowmeter calibration.

-62The data at 250, 400, 650 and 800 psia were interpreted to yield enthalpy values both within the two-phase region and the single-phase region. near the bubble- and dew-points. The procedure used also yielded values of the bubble- and dew-points. Experimental enthalpy traverses at each pressure were made by feeding liquid at constant temperature to the calorimeter and adding varying amounts of electrical energy until the outlet material was determined to be completely vaporized. By making minor corrections for the fact that the inlet temperature does vary slightly it is possible to calculate enthalpy differences from a common inlet temperature. Plots of such enthalpy differences are illustrated on Figure AII.7. Note the breaks in the curves indicating both bubble- and dew-points for this mixture at the various pressures. Values of bubbleand dew-points determined in this manner are listed in Table AII.5. Isobaric enthalpy differences between the bubble- and dew-points can be read directly from these plots and values thus determined are also listed in Table AII.5. Enthalpy differences within the two-phase region were also noted at intervals of 10~F at pressures of 250, l400, and 800 psia for use in preparing the final PTH diagram and tables. The data at 650 psia was not suitable for this purpose because few points were reported within the two-phase region except very near the bubble- and dew-points. The method used to incorporate these results in preparation of the PTH diagram will be discussed in detail in Appendix III.

O- l I I' I I' I I I ENTHALPY TRAVERSES -20- 5.18 MOL. % C3 inC' o 250 psia 40 400 psia -~40 ~v 650 psia A 800 psia -60 -80 0 -0 I-200 0 20 40 60 80 100 120 140 160 180 200 220 240 260 ENTHALPY ABOVE INLET TEMPERATURE- (Btu/lb.) Figure AII.7 Calorimetric Data Obtained Within and Through the Two-Phase Region.

-64TABLE AII.5 Empirical Data Obtained From Interpretation of Enthalpy Traverses Pressure Bubble Point Dew Point vap psia.F.F Btu/lb 250 -166.5 -60.5 221.0 400 -141. 7 -48. 0 194.3 500 -128.1 -41.4 178.5* 650 -111.8 -40.3 147.4 800 - 95.2 -37.5 116.1 * Calculated (See Thermodynamic Consistency Checks) Isothermal Determinations (Single-Phase) Mather (50 ) reported the results of isothermal calorimetric determinations in the single-phase region for this mixture at -147.4, -27.0, 91.6 and 2000F. (See Figure 1). The basic procedures used to interpret the data to yield values of - (lH// P)T (II.9) and isothermal enthalpy differences are similar to those used to interpret the isobaric data as described in the previous section. A typical plot of (Hp2- HP1)T - E P2 1 P (II.10) is illustrated in Figure AII.8. An iterative graphical procedure involving use of Simpson's rule for integration was used to yield ~ = f(P). The values of Q0 at low pressure listed by Mather (50) were used to define the curve between 0 and 100 psia.

0.09 0a 5.18 Mole %C3 in Ci a3 I 0.03 - L_ 0.07 0 0.01 I I I I I I I I I I -i 0.06 -rJ 0

-66A curve for 0 = f(P) was generated over the entire range of experimental pressures so as to yield random variation in differences between experimental and calculated values of isothermal enthalpy differences. The resulting curves were then integrated to yield values of enthalpy difference for every 50 psi interval to aid in making the final tabulation and between experimental isobars for use in making the thermodynamic consistency checks to be described in Appendix III. At -147.4O~F the data extend only from 509 to 1791 psia. In interpreting these data extrapolations were made to 500 and 2000 psia but not beyond these pressures. The values of Cp, 0 and isobaric and isothermal enthalpy differences determined by the iterative graphical procedures described in this Appendix were considered to be good estimates subject to small corrections as required to obtain thermodynamic consistency of all experimental results. The tests for thermodynamic consistency and adjustment of values are described in Appendix III.

-67Appendix III Thermodynamic Consistency Checks and Final Adjustment of Values of C and / P Thermodynamic Consistency Checks. Enthalpy is a state thermodynamic property and is therefore represented by an exact mathematical function. This fact provides a severe test of the thermodynamic consistency of the isobaric and isothermal data obtained for this mixture. The exactness of the enthalpy function is readily expressed in the form of a mathematical restriction. If one begins at a thermodynamic state designed by TB and PB such that HB = f(TB,PB) (III.1) and considers a number of changes in enthalpy, AH. corresponding to sequential changes in state such that the final state is identical to the initial state, then the algebraic sum of such changes must be identically equal to zero. ZAH. 0 (III.2) If independent experimental data are used to evaluate the sequential. enthalpy differences in such a closed loop experimental errors will almost always yield a non-zero value for their algebraic sum. A measure of the experimental error on a percentage basis is provided by the equation percentage _ -Ix 100 deviation - 11x 100 (lIa.j)

-68The isobaric and isothermal determinations carried out with this mixture were made so as to provide a number of checks of the thermodynamic consistency of the data. With reference to Figure 1, note that isobaric data are reported at 1500 psia from -240 to +150~F and at 1000 psia from -240 to +40~F. In addition, isothermal determinations were made at -147.4~F between 500 and 1000 psia and at -27~F between 100 and 1950 psia. A "loop" of four independent experimental determinations is thus formed by the isobaric determinations at 1000 and 1500 psia between -147.4 and -27~F and the two isothermal determinations at -147.4 and -27~F between 1000 psia. This particular loop is illustrated as Figure AIII.1 In carrying out the interative graphical procedures to yield Cp(T) and Z(P) as described in Appendix II, values of AH. corresponding to the four individual sides of the loop of Figure AIII.1 were determined. zaHi = [+(137.144)+(36.64)-(175.126)+(-0.25)] Btu/lb -1.092 Btu/lb Therefore, the net result of experimental error for this particular closed loop is -1.092 Btu,/lb as expressed within the box enclosed within the loop. In expressing this error on a percentage basis the sum of the absolute values of all differences is taken as denomenator as expressed by Equation (III.3) z IHii -= [(137.144)+(36.64)+(175.126)+(0.25)] Btu/lb 349.16 Btu/lb

(Hp)e p = 137.144 Btu/lb -P expt 1500 (correction = +0.379 Btu/lb) ~ ]~~~~~AHi 4-J 4 =3 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ [.n 0 Ib. o~~~~~~~~~~~~~~~ -1.092 lb. ~ ~ %cv\ l I ~~ O + 0 w~~~~~~ I, II IAHw~~~ 1-i C X 100= - o w X. ~ -X -,, x. - i.4AJ ~~~~~~~II CC: Q 4-I _ U U- %) L-l~ -0.313O/o 1- <- 0 0 U (AH) = 175.126 Btu/lb p expt 1000 (correction =-0.35 Btu/lb) -147.4 -27.0 TEMPERATURE (OF) Figure AIII.1 One Complete Check of Thermodynamic Consistency of Calorimetric Data Used to Evaluate Enthalpy Differences.

7 0Therefore, the average percentage error of this set of four experimental determinations is given by percentage _ -1.092 Btu/lb deviation -349.16 Btu/lb 100 = This result is also included within the box enclosed by the loop in Figure AIII.l. The result of all checks of thermodynamic consistency of the enthalpy determinations are summarized in Figure 2. Associated with each leg is the value of AH. determined from the raw data -1 as described in Appendix II. Each completed loop encloses a box containing the value of experimental deviation expressed both as actual error (Btu/lb) and percentage error. Adjustment of Value of AH. The enthalpy differences of each leg were adjusted that that AH. 0= for each closed loop. Note the adjustment of legs that are common to two loops effects the error in both loops. In all cases an attempt was made to limit such corrections to the anticipated experimental errors associated with the precision of the data. Experienlce has shown this to be about +0.3% for the isobaric data. A value of +0.8% was used as the expected precision of the isothermal data because of the limitations of the differential pressure balance of Roebuck (57) which was used in making the isothermal measurements. The values of the corrections are listed for the one loop on Figure AIII.1, and for all loops in Figure 2.

-71Tn some cases it was necessary to apply corrections in excess of the anticipated limits on precision as indicated above. Corrections of about 0.5% were made to the isobaric data between -27.0 and 91.6~F at pressures of 1000 and 1500 psia. As noted with reference to Figure 1 there are regions over which no data were obtained over these ranges of conditions. Thus some of this larger correction was probably required to adjust for incorrect estimates where data were lacking. A correction of 1% was made on the isothermal leg at -27.00F between 1000 and 1500 psia. This correction is made in the region of the maximum in 0 (See Figure AII. ). Adjustment of Values of Cp and Q As indicated earlier, the experimental values of AH. li.sted on Figures AII.1 and Figure 2 were determined by graphical integration of the functions C (T) and 0(P) as estimated by iterative graphical procedures described in Appendix II. The adjusted values of AHi as developed to satisfy the condition AHi 0 for each loop are therefore inconsistent with the functions C (T) and V(P) as originally P developed from the basic experimental data, i.e. these original estimates are thermodynamically inconsistent by approximately - 0.2% on the average. Therefore, the functions Cp(T) and 0(P) were adjusted so as to be consistent with the adjusted values of AH.. The corrected values of C and Q0 are shown — i.p on all figures and are listed at temperatures and pressures corresponding to experimental conditions on Tables IIa,b,c,d and IIIa,b. Extrapolated values are included with corrections.

-72App endix IV Preparation of PTH Tables and Diagrams In preparing the final table and diagram of enthalpy values for the mixture every attempt was made to incorporate accurate data reported in the literature for the pure components as a final check ofh the accuracy of the results reported by Manker (47) and Mather (50). The following steps are followed in order: 1) Arbitrary reference states are selected for each pure component in the mixture. 2) Data for the pure components at relatively low pressures are used to determine the enthalpy of each component as an ideal gas at zero pressure and at an elevated temperature corresponding to one of the sets of isothermal_ calorimetric determinations reported for this mixture. 3) The enthalpy of the mixture in the ideal gas state at zero pressure and temperature of the isothermal determination is calculated assuming no heat of mixing under these conditions. 4) The isothermal and isobaric enthalpy differences determined by interpretation of the calorimetric data at elevated pressures as described in Appendices II and III are used to calculate values of enthalpy at pressures and temperatures of experimental determinations.

5) The thermodynamic consistency of the data at low pressures from the literature and at elevated pressures from the Thermal. Properties of'Fluids Laboratory is checked by determining the enthalpy of the liquid mixture at low temperature and.pressure. Adjustments are then made to satisfy the condition that AHi = 0 around the closed loop of the enthalpy diagram. In preparing the last published PTH diagram for this mixture, Mather (50) chose to "close the loop" by drawing isotherms at low temperatures and pressures which exhibited unusual (and unacceptable) curvature. Jones (40) had a similar problem in attempting to obtain such closure when working with the enthalpy data for pure methane. He chose to resolve this discrepancy by applying a correction of 4 Btu/lb to the values at low pressures as calculated using data from the literature. It was decided to take exceptional care with respect to every step of the reprocessing of the data in order to obtain values of the highest accuracy obtainable from the data. This Appendix is devoted to detailed descriptions of the procedures that were followed. Selection of Arbitrary Reference States The specification of arbitrary reference states for each pure component is a matter of convenience. It was therefore, specified that H 0 for pure liquid methane and propane at their saturation pressures at -280~F. This choice was made for two reasons:

-74. 1) The choice is consistent with reference states selected for methane (40,4,1), propane (70) and several mixtures (47,50,51,70) by previous investigators. 2) All values at -280~F and above will be positive, i.e.> 0. It was necessary to consider the effects of minor impurities in preparing the final PTH diagram. For consistency, it was specified that H = at -280~F and the corresponding vapor pressures for the impurities N2, 02, C2H6 and CO2 in the liquid state. Therefore, all pure components were specified to have the same reference condition; H - 0 for saturated liquid at -2800F. Determinption of H. for Pure Components as Ideal Gases at +200~F Relatively little effort was spent to determine H for the minor components in the mixture: Propane. Yesavage (70) gave detailed consideration to published data from the literature and calculated a value of H 298.7 Btu/lb for pure propane at 2000F and zero pressure. He found this value to be consistent with interpretation of his calorimetric data at elevated pressures and therefore, this value was accepted as published. Im.purities. Values of H at 200~F and zero pressure relative to the pure saturated liquids at -2800F for nitrogen and oxygen were calculated from values listed by Canjar and Manning (12). Carbon dioxide does not exist as a pure liquid at -280~F and therefore, a value was estimated from the values of N2 and

02 by assuming that H at 200~F and zero pressure is a linear function of molecular weight. A value for ethane was obtained as the arithmetic average of the ones for propane and that for methane (the determination of which is described in detail in the pargraphs to follow). The estimation of Hi for CO2 and C2H6 as ideal gases at 2000F is admittedly relatively crude but the total mole fractions of these two components is.001 so that extended efforts to estimate these values was not considered to be justified. Methane. As noted previously, both Jones (40) and Mather (50) had difficulty in their attempts to establish the thermodynamic consistency of their data with published data for the pure components at low pressures. The fact that both investigators had similar difficulties led' to the conclusion that the published data for methane might be somewhat in error. Therefore. the published datawere themselves checked for thermodynamic consistency. Values of enthalipy differences for methane at low pressures are available from a variety of sources. These include values of Cp of the saturated liquid (20, 31,34), calorimetric determinations of the heat of vaporization (18,31), values of enthalpy departures for the vapor H~- H)(1) and the enthalpy behavior of methane at zero pressure, H~. (58). These data can be used to calculate enthalpy differences between two fixed states. Such differences should be independent of the data used to evaluate the difference thereby providing a test of thermodynamic Consistency.

The various paths selected to evaluate the enthalpy difference between saturated liquid at 1000K and the ideal gas state at 150~K are illustrated on Figure AIV.1. Each such difference is evaluated as the sum of the enthalpy difference required to increase the temperature of the saturated liquid from 100~K to the temperature at which vaporization occurs. ( Cps ldT), the heat of vaporization at that temperature (4Hvap), the enthalpy change associated with the isothermal change from a saturated vapor to the ideal gas state at zero pressure (-(H~- H)) and finally the change in enthalpy of the ideal gas methane from the temperature of vaporization to 150~K(H0 150K - H T) TABLE AIV. 1 THERMODYNAMIC CONSISTENCY CHECKS OF PUBLISHED CALORIMETRIC DATA FOR METHANE AT LOW TEMPERATURES T T (t Cpsidt aHvap H~-H'H0 -H H 0 -H lO-150 -T -150 -100 ~K Ca/1 gmol Cal/9mol Cal/gmol Cal/gmol Cal/gmol 100 0 2030 13.8 397.1 2440.9 (-1.3) (-4.8) (-0.8) 110 129.3 1963 26.0 317.7 2436.0 (+1.1) (-2.5) (-0.6) 1-20 261.7 1888 45.7 238.3 2433.7 (+1.8) (-1.0) (-0.5) 130 399.0 1800 74.3 158.9 2432.2 (0) (+2.1) (-0.3) 14o 542.8 1698 113.9 79.5 2434.2 (-1.0) (+l.o) (-0.2) 150o 693.7 1577 167.1 0 2437.8 (-2.0) (-1.8)

U) U) w G // 150 ~K 140 ~K.0 OK' 120 OK H-Enthalpy of Methane above Saturated Liquid at -280 OF = 100 OK Figure AIV.1 Check of Thermodynamic Consistency of Published Calorimetric Data and Calculated Enthalpy Departures at Low Temperatures.

-781. ('Cps.l.dT). Values of the heat capacity of saturated liquid methane, Cps.1., as reported in the literature (20,31,34) were plotted vs T as illustrated in Figure AIV.2. A smooth curve was drawn through the p-oints giving primary emphasis to the results of Frank and Clusius (31), Hestermans and White (34) and Cutler and Morrison (20). The integral was determined by Simpson's rule for every interval of 10~K. The results are summarized in the second column of Table AIV.1. 2. (Hap)H Values of AH listed as the third column -yap -v a p of Table AIV.1 are based primarily on the calorimetric data reported by Hestermans and White (34). These data have been shown to be in good agreement with values calculated from data on the vapor pressure and volumetric behavior of methane (34). The values listed at 100~K is the lowest possible value within the combined limits of accuracy of the independent results of Hestermans and White (14), Frank and Clusius (1) and Colwell, Gill and Morrison (18). 3. (H~ - H). Values of (H_~ - H) obtained from the latest API tabulation for methane (1,58) are included as column four in Table AIV.1. No value was listed for 100~K and therefore, a value was estimated by differencing the tabular values. 4. (H 50- HE). A value of' 397.1 cal/gmol was determined directly from values listed in the API 44 tables published in 1952 (1,58). This value was found to be inconsistent with the values of C0 listed in the same publication but, nonetheless, P

16.0 16.0 I I I I I I I I l l I l/ CUTLER AND MORRISON ( 20 ) METHANE 15.5 HESTERMANS AND WHITE (34 ) WIEBE AND BREVOORT( 67) 15.0 0 14.5 - o o\ E'I135|A,,| 1 o 140 -- 13.5 i. 12.5 l.I 12.0 95 100 105 110 115 120 125 130 135 140 145 150 155. T-TEMPERATURE (K) Figure AIV.2 Published Data on the Isobaric Heat Capacity of Methane as Saturated Liquid. Figure AIV. 2 Published Data on the Isobaric Heat Capacity of Methane as Saturated Liquid.

-80 - C~ values were used for purposes of interpolation. The values P thus determined are listed in the next-to-last column in Table AIV. 1. The sum of all differences is listed in the last column of Table AIV.l. The values differ by as much as 8.7 cal/gmol or about 0.4%. All values were made to equal 2434 cal/gmol by applying the corrections included in the parentheses in Table AIV. 1. Maximum percentage adjustments are indicated in Table AIV.2. TABLE AIV.2 MAXIMUM PERCENTAGE ADJUSTMENTS IN PUBLISHED CALORIMETRIC DATA FOR METHANE AT LOW TEMPERATURE Type of Data Adjustment 1. dT 0.9% ~ ps.l. 09% 2. ap 0.1% -yap 3. H~ - E 35. % 4. H0 - H 0.2% 150 -T The large percentage correction of H~-H is justified on the basis that this value was calculated using the BWR equation of state. at low temperatures where large errors of extrapolation have heen noted in other systems (62,72,76). Note also that little or no corrections are required for (H~-H) at temperatures of 1300~K and above.

Tabular values of H~ from API 44 were used together with the result of the above interpretation to determine the enthalpy of methane as an ideal gas at 2000F. The results are summarized as follows: Btu/lb H as saturated liquid at -280~F 0 Latent heat of vaporization at -2800F 227 72 Effect of Pressure on Enthalpy (5 -* 0 psia at -280~F) 1.01 Increase in temperature as an ideal gas -280 ~F to -1900 ~F 44.47 -190 ~F to +2000F 203.33 476.53 In contrast Mather used a value of 477.79 Btu/lb. The values of H. resulting from these considerations are summarized in Table AIV. D. TABLE AIV. 3 Enthalpies of Pure Components as Ideal Gases at 200OF Relative to Saturated Liquid at -280~F Mole Molecular Fraction BtHlb Weight Methane 0.9464 476.53 16.042 Ethane 0.0006 (437.5) 30.068 Propane 0.0518 398.68 44.094 Nitrogen 0.0006 195.6 28.016 Oxygen 0.0002 192.6 32.016 Carbon Dioxide 0.0004 (184) 44.011

-82Calculation of H for the Mixture as lIeal Gas at +200~F At zero pressure gases behave ideally in that there is little or no heat of mixing. Therefore, the enthalpy of the mixture in the ideal. gas state at 2000F can be calculated by applying the equation H x.. (AIV.1) -m i (AIV. Note that xi is listed as mole fraction whereas H. is 1 given on a mass basis in Table AIV.2. The value calculated by applying Equation (AIV.1) together with the values listed in Table AIV.2 is 465.67. Incorporation of Isobaric and Isothermal Enthalpy Differences at Elevated Pressures. The base temperature of 2000F was selected because isothermal data were obtained at this temperature. Making use of isothermal enthalpy differences evaluated as descrihed in Appendixes II and III, values of enthalpy of the mixture as an actual gas were determined at intervals of 50 psia which includes every pressure at which isobaric determinations had been made. At these pressures the isobaric enthalpy differences were utilized to establish values of enthalpy over the entire temperature range at elevated pressures. Because of the adjustments which had been made'to the isothermal and isobaric enthalpy differences as described in Appendix III, all enthalpy values were consistent at the intersection of isothermal and isenthalpic determinations. However, in spite

-83of' the fact that considerable care had been used in interpreting the data at elevated pressures, slight additional corrections were required to obtain complete agreement with published data at -2800F. Corrections to Obtain Complete Agreement at -280~F The procedure described above yielded values of enthalpy for the mixture at -280~F at pressures of 2000, 1500, 1000 and 500 psia. Cutler and Morrison (20) report data on the heat of mixing of methane and propane at 100~K (-280~F)and these data indicate that the liquid mixture at low pressures has an enthalpy of 0.268 Btu/lb. It is possible to draw a curve connecting the individual values at 200, 1500, 500 and about zero psia, but such a curve does not appear to have the correct curvature at low pressures. The points at 2000, 1500, and 1000 are fit very well with a straight line but the extension of this line misses the point at 500 psia by about 1 Btu/lb and the point at saturation pressure by 2.5 Btu/lb. An arbitrary decision was made to consider the isotherm at -280~F to be a straight line. A final adjustment of 0.54 Btu/lb was made on the values at 2000 psia arnd 1000 psia with somewhat smaller corrections at 1500 psi..a. These corrections applied over the entire temperature ir-terval for which the enthalpy changed by about 430 Btu/lb so that the final correction was 0.13%. Uniform adjustments of thiS amount was applied to values of C so that the final tabulated values of C and H are consistent. p --

i.-nterpolation to Intermedi.ate Values of Temperature and Pressure.........e..,z.._.,. i..em-pera~ure an..._... - A large scale (20" x 8'?) graph was prepared and all adjusted values of er.thalpy plotted. Isotherms were then drawn through points of common temperature. The data at 1700 psia was readily inc(orporated because it joi.ned with isothermal data at -270'F. S iglht extrapol.ation of' the data at 11.00 and 1200 psia was requA.:ired in order to incorporate these data. Experimental. d-l;at obtained with partial vaporization of thle mixture was use.::d to determine the location of the iso-therms within the two-phkase region. The dat'a obtained at 800 psia was readily irlcorporated because it intersected the isotherm at -27iF. The isobar at 650 psia ].ikewise intersected this isottlerm.'but very few poin. ts were obtained within the two-phase region at th-is pressure, The data at 800 and 650 psitl w-ere used to gOuide th.e extrapolati.on of the curves at l-.0)0 and. 250 psia to -27'~I-E. The extrapolations are illustrated on. Figure AII.7. Tabulated values of enthal py are presented on Tables IIa and lId and IIIa, TIb.'I'.he PTH diagrams were photo reduced and are presented, as Figures 4a Cand 4bb.

Append ix V'iiquations Used in. Calc.ul ating Values of Entropy and Fugacity. Th.e availability of an extensive table of adjusted values of enthalpy made it possible to c(alculate isobaric entropy differences quitle precisely. These values also permitted utilization of' published volumetric data to maximum advantage in calculating isothermal. entropy differences. As a matter of convenience, tihe basis for al.l entropy values was taken to be a pure, perfectly oriented crystals at absolute zero. The equations used in makirg the calcu-laations are summarized in this Ape nd. ix. The:- Basiic Proper-ty PRelation Entropy is rela- ated to other thermodynamic properties by th-ie fundamental Gi. b'bs rel ation. In the absence of significant ch.',anges in. surface area, and othier miscel.l.aneous effects, this relation is written. as dU - TdS - Pd.V (V.1) Inor rpor at pa ing t.he definition of enthalpy yiel ds an equava Ient expressiornt d(li - ldS -- VdP (V.1) Similarly incorporatiorn of the definition of Gibb's free energy, G., G H i- TS (v.4)

y ields dG -SdT + VdP (V.5) These relations serve as the basis for al]l calcu ations of entropy differences made for this report using calorimetric and volumetric data for the mixture. Isobaric Changes in Entrop If pressure is held constant Equation (V.3) reduces to dH - TdS (v.6) Thus isobaric entropy differences are evaluated as -T SrL )p= fdTf P dS(V.7)'ST O~t)P TT'[' T dHI C -p -. Cd) _T T c dT (V.8) T T p T. In addition to data on Cp values of (H H have been PP _H )p have been determined as described in Appendix II. As noted there, considerable care was taken in the region of the peaks. Rather tih:an repeat, such irntegrations and run the risk of introducing computational erro:rs, Equation (V.7) was modified so as to permit maxi.mum utilization of the adjusted enthalpy val-ues. Integrating th-e last terms of Equation (V.7) by parts yields HfT, 1HT {2 EITT (S1, -OT)F'P T 1 2 1 dTr (V.9) -2- 1 2 T 1T

This is the relation that was used to calculate isobaric entropy difference so as to obtain values that are as consistent as possibl e with the tabulated enthalpy values. Isothermal Changes in Entropy:n evaluating isothermal entropy differences it is common practice to utilize Equation (V.5) to obtain the Maxwell's identity (apa )T (I )P (V.10) and then make use of' published volumetric data to evaluate isothermal differences in entropy P2 -AS f () dP (V.11) This procedure requires differentiation of volumetric data and a resulting loss of' accuracy. An alternative procedure was developed to permit maximum utilization of the calorirnetric data which were obtained with this mixture. Under isothermal conditions Equation (V.D) can be rewritten as d E Vd P ~~~d S I- - ~ ~(V.12) -T T T -= Inltegration serves to yield values of isothermal entropy differences in terms of isothermal. enthalpy differences and integrals of the volumetric function.

(.2 -9T T T- - dPT (V1) P Values of the enthralpy differences were obtained from the tables of adjusted values. Volumetric data from the literature were used to evaluate the integral term. For liquids, V is relatively independent of pressure and therefore, the integration was carried oult graphically using values of V directly. In t;he gaseous region V changes greatly with P and graphical integration is awkward. Therefore, a simple transformation was made. The compressibility factor Z is defined by PV z R- (v..14) Therefore ) (Hp -Hpf ) (Sp2- Sp IT T T 1 (V.15) Thus isothermal entropy differences were evaluated by application of Equation (V.12) with integration of the volumetric contribution being determined by either Equation (V.13) or (V.]5) as appropriate. Absolute Entropy Values In order to tabulate values of entropy it is necessary to -e-stab:l ish a value at some condition of temperature and pressures. It is especially conveinent in working with mixtures to have all mixtures referred to common bases for the pure components. Further convenience is provided if the bases are selected to

-89yield values of absolute entropy in accordance with the third law of thermodynamics. This procedure was followed in calcula ting absolute values of entropy for the mixture. Val.ues o:f absolulte entropy for pure compoonelnts as ideal gas at atmosph eric pressure and 25~"C have been tabulated recently by Stull, Westrum arid Sinke (624). Values taken from these tables are listed in Table AV. 1. TABLE AV. 1 Absolute Entropies of Pure Components as Ideal Gases at 25~C Component x. mol S~ (cal/gmolOK) frac. ti on Me th-iane O.92464 4 P4 5l 2 11 -th L ne 0.0 06 54 0- () 5 Prop ane ) 0.051]8 64) 5I. Wfi;rogen OO. 0006 2)-5. (67 Oxyg en 0.0C)0 249.003 CO o.ooo04 51.07..00()0.Th'he' ab solu-el -e ernt.ropy f' ot, t;.he mixture under study as a mixt.ure oQt' ideal. gasets at -lt, )''C is e. aleulated by- applying the equa-' Ex. lnx. (v. 1_6) (:M) l ( x }- t- XV wh:-.r'ein tlhe second ternm C.oikes:in.Crlo accoultr tihe entropy of ltiixi.ng of ideal gases.

Applying Equation (V.16) together with the values iis ted in Table AV.1 yields _S -298 = 45.3547 cal/gmol-e~K. Using the molecular weight of the mixture, 17.525, yields SM-298 = 2.5876 Btu/lb 0F. Next corrections must be applied to account for the fact that the mixture is not composed of ideal gases even at a pressure as low as one atmosphere and a temperature as high as 25~C. Equation (V.12) applied for a real gas mixture. An equivalent expression can be written for an ideal gas mixture. P ~(S2-~~ S i _VdP (v.17) (S2-.)IGM T-'I For a mixture of ideal gases, enthal py is independent of pressure at constant temperature and therefore, Equation (V.17) reduces to Pe (S - Sl)IGM R dlnP (V.18) Written in the same form by combining Equations (V.12) and (V.1)4) the expression for the actual gas mixture is given by P0 (S2 -1 _S1 i)RGM G- ZmdlnP (V. 19) Pt Subtracting the two equations yields

-91 - (H - H (S2RGM -2IGM) (SiRGM - IGM) T (Z - 1) - R ( Zl)P dP (V. 20) P1.At zero pressure the entropy of the actual gas mixture is equal to that of an ideal gas mixture. Therefore, setting P1 - O eliminates the second term on the left of Equation (V.20) and yields a value for the enthalpy departure, (H~- H2) on the right hand side. Further P2 is set equal to 1 atmosphere 1 (H - H1 atm )RGM (Zm-l) S:T -_~~~~~~~~~~~ so R.RdP SRGM,1 atm IGM T - (V.21) In evaluating the difference the enthalpy departure -(H at. )9 GMis determined from the detailed analysis of ti- enthalpy behavior. The volumetric contribution at this ].ow pressure is -readily evaluated by application of the virial equation of state PV B C. R- = 7 + - ---- (V.22) Expressed in a different form Z. 1 B'P + C'P6- + (V.23) The coeffic-ients of these two series are related iri t., "tema t i c1 ally 1 B' - B/RT C' - (C-B2)/(RT)2 (V.24) At low pressure Equation (V.23) reduces to Z - 1 + B'P (V.25)

This rel ation can be utilized to evaluate volumetric term iri Equation (V.21) at low pressures -(_H-H )H - So -s (Ho-H1 atm)EGM m -RGM, 1 atm S~IGM -= RI BdP 0 (V.26) In estimating the second virial coefficient for the mixture it is considered that it is a binary mixture of methane and propane. 2 2 B - x B + 2x x B +2 2 (V. 27) mix 1 1 x2 ~ x2 2 V.27) Equations (V.16) and (V.26) are used in combination with the value of S0 to determine the absolute entropy of the mixture at f25~C and 1 atm. Values of absolute entropy at other temperatures and pressures are then determined by applying Equation (V.9) to calculate isobaric entropy differences and Equations (V.13) to evaluate isothermal entropy differences The Fugacity Function Fugacity is defined by the equations dGT. RTdlnf (V.28) f -e-.1 as P -O 0 At coristant temperature Equation V. 5 reduces to dGT - VdPT (V.29) TLte re fore RTdl nf --- VdP (V.30) T1 -' T"3

-93Incorporating the definition of Z [Equation (V.14)] and subtracting dlnPT from both sides of Equation (V.30) yields T din ( ) (Z-l)dlnPT (V.31) Incorporating the limit expressed by Equation (V.30) permits one to evaluate values of fugacity coefficient: = (f/p) at any pressure and thereby determine f. The integral term is difficult to evaluate graphically at low values of pressure. The substitution of Equation (V.25) is appropriate in such cases. Therefore, the integral expression is evaluated in two parts, one fromP - 0 to 1 atm and the second applies at higher pressures.

-94Appendix VI Primary Interpretation of Data from the Literature to Estimate the Volumetric Behavior of the 5% Mixture. As indicated in Appendix V, accurate volumetric data are required in order to determine the isothermal effect of pressure on enthalpy. The procedures used to estimate the volumetric behavior of the mixture will be described in this Appendix. The sources of volumetric data for methane, propane and thleir mixtures are reviewed in Appendix I. The four isothermal runes at -147.4, -27.0, 91.6 and 200~F were divided into two sets of two each in keeping with the availability of data at high temperatures from several sources (27,-55,56,65,66) and at low temperatures from only one source (36,37). Isotherms at 91.6 and 200~F Reamer, Sage and Lacey (55,56) report values of Z for nine methanre-propane mixtures over a fairl y wide range of temperatures and pressures, none of which coincides with the compositions of the. binary under study. The methane data tabu-.lated by API Project )L44 (1,58) and the propane data of Reamer, Sage and Lacey (55) were used to permit interpolation to the c.mIposit ion of the mixture. The choice of the propane values is not critical in attempting to establish the volumetric belhavior of a mixture.ont-ainring 95 mole percent methane. Conversely, the values used for methane will exert a significant influence. Therefore,

the values obtained from the API tables (1,58) were checked against values calculated using the Vennix-Kobayashi equation of state for methane (44). The results of these computations were furnished by Kobayashi (44) and were found to be in excellent agreement with the values used in the calculations (+0.1%). Tabulated values of Z from the three sources mentioned above were interpolated using a large digital computer. First interpolations were made of methane, propane and the nine mixtures to yield values at the two desired temperatures 91.6 and 200~F. Next irlterpolations were made with respect to composition to 5.18 mole percent propane in methane. The values thus obtained were considered to be preliminary values suitable for calculation of a first approximation of entropy differences as wil b.:)e discussed in Appendix VII. Use of digital computers to interpolate tabular values is sub ject to considerable error. Therefore, two independent methods were used; the method of Lane (45) and that of Klaus and Van Ness (43). Both do not necessarily go through the data points but do insure continuity of both first and second derivatives over thie entire range of the table. Cons-iderable modification of the L,anr merthod was mTequiqred. as'w.1i] be desc. riibed ien. a subsequent public- a tLionr-.'Th'e me thod of TI'/ d-aus and Va.n eiess was'used to check the val.idity o:f tite val]ues c al. cut lated b:y thie Lane method. Good as;rzeerrtmen.t was obtairled whlichl was taken to mean that both interpolatioun mnethods probab]. y yie] d re:Liab. e values

-96The interpolated values of Z were plotted vs lnP to facilitate evaluation of the integral expressions in Equations (V.14) and (V.28). A typical curve is shown on Figure AVI. 1. Isotherms at -147.4 and -27.0~F The only data for mixtures at these low temperatures are the relatively crude values (+1%) reported by Huang (36) and Huang and Kurata (37). These contributors also report values for propane and methane. These data for the pure components was used for purposes of interpolation with respect to composition because it was felt that use of more accurate values for methane (27) was not justified. The data are reported at temperature intervals of 20~C from -150~C to 0~C and at intervals of 1000 psia up to 5000 psia with some intermediate values at 500 psia and at the bubble point of some mixtures. Data are tabulated for four mixtures. The comparitive paucity of tabulated values in the range of primary interest (-147-.4 and -27.0~F up to 2000 psia) required that graphical procedures be used for interpolation. Therefore, density was plotted vs temperature at constant pressure and composition as illustrated in Figure AVI.2. Values read from such plots at -147.4 and -27.0~F were plotted as isobars with composition as ordinate. A typical plot is presented as Figure AVI.5.

1.00 0.98 0.96 0.94, 0.92 U<I ~ 1% 0.90(n 0.88 cn 0r a. o 0.86 0.84 0.82 5.18 MOL % 03 IN CI 0.80 078 0 1.0 2.0 3.0 4.0 5.0 6.0 70 8.0 log P (PSIA) Figure AVI.1 Typical Plot of Compressibility Factor, Z, vs lnP at 91.60F as Used to Evaluate fZdlnP. -97

.48 24.7mol.%.46 PROPANE IN METHANE.44.42 ao.40L 500 psia.40. I1 3000 psia m b~z.38 a, 1000 psia 32000 psia.34 -.32 -160 -140 -120 -100 -80 -60 -40 -20 0 20 T- TEMPERATURE (OF) Figure AVI.2 Typical Plot of Isobaric Density, P, for Liquid MethanePropane Mixture for Interpolation to Temperatures of Isothermal Calorimetric Data.

* @ancxml jo uoTqTsoduIoD oq uoeliod.axuI gjo a@nc.xI. aued'dodc -aueqq aJ pTnbrj j$o d/T = A'oTxjoadSd Jo oaTd Iezod~RI Qm IAV GInBx, 3NVdObld NQOI1OV8J 310W-N 0'1 90 90 D 3Z' A!sd 0002 ~I I 0 i.....C I< - NAZI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I rl) _o 0 c I I

-100As illustrated on this plot, density is a strong function of composition at high concentrations of methane. Thus the interpolation to 5.18% C is subject to considerable variation 3 depending on the method of placing the draftman's spline. It was virtually impossible to estimate the volumetric behavior of the mixture at -27~F between 1000 and 1500 psia. Whereas the 5% mixture is in the single phase over the entire pressure range at this temperature, the mixtures containing 25 percent propane exists as two phases at 1000 psia. The preliminary estimates of Z and V obtained by the interpolation procedures described in this Appendix are used to obtain first estimates of the isothermal effect of pressure on entropy as described in detail in Appendix VII. These values are checked for thermodynamic consistency as described in Appendix VIII. Adjustments are made on the estimates of Z and V in Appendix VIII to obtain values which are thermodynamically consistent with tabular values of H and S. These corrected values are then used to calculate the fugacity of the mixtures at the two higher temperatures.

-101Appendix VII: Calculation of Entropy Differences After the adjusted enthalpy values were obtained as described in Appendix III and preliminary interpretation of the published volumetric data was completed as described in Appendix VI, it was only necessary to utilize these results in connection with the equationr of Appendix V to evaluate isobaric and isothermal differences in entropy. Isobaric Differences in Entropy As developed in Appendix V (H - HT- H(NT ) -T HT HT1)P (ST 1 ST )p+ - dT (V.9) )2 T2 T The first term on the right side of the equation is readily evaluated at intervals of 100F and at temperatures of isothermal determinations by direct substitution of correct values of H from tables. In addition to this matter of convenience, it was felt that this relation would yield values of entropy that are very consistent with the adjusted enthalpy values because normally the integral term does not contribute more than 10 percent to the value of the isobaric entropy difference. The integral term was evaluated graphically by plotting [(_HT-HT )/T 1 vs T as i)llustrated by Figure AVII.1. This particular curve is for the region of the sharp peak illustrated in

100.. 90 80 1000 pisa 70 1500 pisa o ~60 T 0 L2000 psia 50 40 x c\j30 20 5.18 MOL% C03 IN CI 10 0 -170 -150 -130 -1 10 -90 -70 -50 -30 -10 T-TEMPERATURE (~F) Figure AVII.1 Typical Plot Used to Evaluate Second (Residual) Term in Equation (V.9) in the Vicinity of the Peak in Cp(T) at 1000 psia. -102

-103Figures AII-1 and AII-2. Note that the residual represented by the integral is a smooth function even when C exhibits a sharp maximum. Note also that Equation (V.9) applies within the twophase region of a mixture as well as in the single-phase. Therefore, the relation was applied to obtain values of S within the two-phase region. Figure AVII.2 is a plot of the residual integral across the two-phase region. Again a smooth relation is obtained. The values thus determined were tabulated at even intervals of 10~F. In addition, differences between experimental isotherms were calculated. Two typical values are included in Figure AVII.3. They are the values presented horizontally above the isobars between the two experimental isotherms. Both values are given algebraic sign s consistent with an increase in temperature. Other values are summarized on Figure 3. Isothermal Entropy Differences Equation (V.13) was applied directly to evaluate isothermal entropy differences at -147.4~F and at -27.0~F _~ 1 _~ P2 P P1 Values of the isothermal enthalpy difference were determined directly from values listed in the table of adjusted enthalpies (Appendix III).

uoyGa{ aseyq-oMtJ ayq uF (6 A) uoFqenba UF uaij (lenpsaGH) puoc3s ag4enleAI o0 psnf 401d TeoTdASL enIIAV GaUT (Jd 3fnliV3dd31J 0O- 09- OL- 06- 01i- 0~ 1- I - 00 NI0%10 1vi 8 1A9 D!sd o08 o!sd ot xv 0 puo 4uiod elqqnB * 01 o!sd d 0 8i Llil -1191~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C4

AS = 0.235606 Btu/lb~ F I I L IL.L IL 0 0 o 0 C CD\- I 00233 Btu 1bF 0 I I'-J'-' +o Io I _ | | 0_ I a. 1cd~~~~~~~ 00. o 0 o' + )*l I'l II ] 1S II II H HI - 2 7 9 1 o Figure A 3 One Check of Thermodynamic Constency of Calorimetric and Volumetric Data Used to Evaluate AS = 0.228016 Btu/lb~F 1000 -27 91.6 TEMPERATURE (OF) Figure AVII.3 One Complete Check of Thermodynamic Consistency of Calorimetric and Volumetric Data Used to Evaluate Entropy Differences.

Typical values of (Hp - Hp )/T are listed in Figure AVI.3. 2 1 They are presented vertically to the left of the experimental isotherm. Both values are given an algebraic sign consistent with an increase in pressures. Additional values are summarized in like manner on Figure 3. The integral term in Equation (V.12) was evaluated graphicallv. Plots of V vs P were made using values of V interpolated as described in Appendix VI. A typical plot is illustrated by Figure AVII.4. One value of (1/T)fVdPT is presented in Figure AVII.3. It is presented vertically to the right of the experimental isotherms at -270C. The value is given an algebraic sign consistent with an increase in pressure. However, note that a minus sign appears before the integral in Equation (V.13). Additional values are presented vertically to the right of the isotherms at -1I'7..4 and -27.0~F,in Figure 3. A slightly modified form of Equation (V.13) resulting from substitution of Equation (V.14) was used to determine isothermal entropy differences at 91.6 and 200F P2 (S, - Sp zT (Hp - Hp )/T - P ZdlnPT (vII.1) (S~H P Hp )T - 2 R T 2 1 As illustrated previously', the first term on the right hand side of the equation is evaluated using adjusted values of en thalpy. The integral term is evaluated by plotting the interpolated values of Z (Appendix VI) vs lnP. A typical

3.0 5.18 Mole % Propane in Methane 2.9 I1% T=-1474 ~F o0 2.8 1 C 271..J a2.6 2.5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 P- PRESSURE (PSIA) Figure AVII.4 Typical Plot Used to Evaluate fVdP in the Compressed Liquid Region.

plot Si; preser nted. as i r-'igure AVI.. A val.ue is presented on Figure AVI.3 vertically to the right of the 91.60F isotherm. The sign is associated with an increase in pressure. Note that the integral term is preceeded by a negative sign in Equation (VII.1). Additional values are summarized in like manner on Figure 3.

Appendix VIII'h-e rmodynami.c Cons.is tency Chec k-s for Enrtropy and Adjus tment o.[' Val]es of' EntropyE), Comrnpress.ibi li ty Fac tor Z%, and Specific Vol.ume, V, Cal.cul at ion o:f. -Fuac ty. I.nt:ropy is a -t,.bermody-nami (-. propertty as is erithalpy. Therefore. checki.s of Lit-e t,:ermodynamic c(onsi.stency of the calculated values can be made as was done withii ent-haalpy in Appendix III. The errors in the errors e tency checks canu hop)e P'ulll.y'be eliminated by cad.justing the volumetrirc values wi.t"'i r- t':he ].imits of the errors introduced experimenrltaly and resul tJ:g fr'rom the use of i n terpol] a t; on procedure s T.h e rmodynam:i. c Cons istenc Cy (he c-ks'I'he.princ:i_pl e of' the thermodynarni. c consis ten c c hec-k has'been:il l.ust:ra ted irn Appendix III. The app iL a ti. on t o e!ntropy i.s shlown wi. tb re:ference to F;'ig re AVII.. As indic.:ated'by the.first, equat.i o.o n i ncl1uded:i n th.e, boI-)x wi. t-hin the c osed 1 oop, lthe sum of:' the indi -vidual cortri.buti..ons to isobaric and iso t.hler'mal entropy di fffe rences taken.in a c-lockwi se manner with pairt icul. a'r atterti _on to s'i..gn is not zero but -0.'0(0233 Blu/lb"'t or equivalently, -0.()oL'%. Similar summaries are included in the boxes wit;h each c].osed loop of Figure 35. Nlote that the maxismunm _nconsis tency introduced by integration of the resildual isobar:ic. term and the volum.etric fulnction in tlhe isothl:ermal. legs i.s 0._l1%.70. The average deviation for t:he'? loops tested is 0. (.3%.

-110l Corrections to Obtain Thermodynamic Consistency It seemed highly desirable to make all corrections required to obtain thermodynamic consistency by adjusting the integral volumetric terms as long as this objective could be accomplished within the anticipated limits of accuracy of the interpolated values for Z and V. The corrections were made by a trial and error procedure. Typical_ values of corrections are listed vertically within parentheses under the appropriate value in Figure AVII.3. Note that a relatively large correction (-2.45%) is required at -27 ~F compared with a correction of 0.42% at 91.60F. It seemed that such corrections were probably within the limits of accuracy of the interpolated values: 1) Huang 36,37 ) indicated a probable accuracy of +1%. Interpolation could easily increase this error by a f'actor of two. Therefore, the correction of -2.46% seems reasonable. 2) It is difficult to assign a value for accuracy of the v-olumetric data at +91].6 and 200~F because of' errors introduced tDy interpolation procedures. Corrections on the order of 0.4% appear to be justi.f'i.fa-)ble. All percentage corrections are summarized on Figure 3. The other corrections applied to Huang's data are all less than the valu(e of -2.46 %presented in Figure AVII.3. The largest corr: - tioY.n at; the elevated temperatures occ-urs at 200F~ between 1000

-111and 15000~F and amounts to -0.65%. This mayy ex eed the limits of ac-curacy of' the interpolated vol.umetric data. PNote that of a change of o.oo8% in one of the isobaric enthalpy differences included in this loop would reduce the required correction on the volumetric term to 0.4%. No adjustment of the enthalpy- values were made. Instead the corrections listed on Figure 3 were accepted as proper and proportional adjustments in V and f were made. These values are listed for each isotherm on Tables IIa, b, c and d. Calculationr of YFPugacity As indicated in. Appendix V th.e:fugai.t;y coefficient, f/P, is convenient ly evaluat;ed by application of the equation I a tt P lrn(f/P, ) f B dP - + (Zm-l)dlnP (V. 30) 1 atm The second i:ntegral is readily evaluated from the plots o[' Z vs lnrlP. Equatiorn (V.27) is applied together with values orL' lth!e second vi ria.l r oe:fi:tienrft for me thane ( 35), propane ( 8, 3.58 ) and t-,.e interac tio n cooeffif'i.ierlt (8,22,38 ) to.evala -te m3. Val..ue.s of -50.29 and 2,-' m3 /gmole are obtained a-._-. & and 200~F r es pec: ti:.vely. The calcul ated values of lfauiBa ~lK't,/ y' f, are I:is.ed a1t 911.( and 200F"r in T'able IIb and IIa......i lue.. are ].:is t.ed at -D)L1l.l and -2 7~(F because of the lack of values reported by TIuang (36,37 ) at low pressures.

-112Appendix IX Preparation of Mollier Diagram and Skeleton Table of Thermodynamic Properties for the Mixture. The adjustment of volumetric data for the mixture as described in Appendix VIII yielded values of specific volume, V, fugacity, f, and isobarice and isothermal entropy differences that are thermodynamically consistent with the adjusted enthalpy values described in Appendix IV. These results were used to calculate values of absolute entropies for the mixture corresponding to experimental isobars and isotherms. A Mollier diagram was then prepared to permit interpolation of values to intermediate pressures and temperatures. The purpose of this Appendix is to describe the procedures employed in making these calculations. Absolute Entropy of the Mixture at One Atmosphere and 91.6~F The determination of the absolute entropy of the mixture at 25~C as a mixture of ideal gases was described in detail in Appendix V. The value obtained is 2.587-(6 Btu/lb~F. The correction from an ideal gas mixture to an actual one is best made at a temperature of one of the isotherms where detajiled attention is given to the estimate of enthalpy departures at low pressures. Therefore entropy of the real gas mixture was obtained at 91.6''Fi (closer to 25~C) and a pressure of one atmosphere. This involves two sLteps namely isobaric and isothermal correction to the entropy of ideal gas mixture at 25~C. Isothlermal correction is made by using Elquationl V.26.

-RGM,1 atm- IGM IGM - 0 (V.26) Values of enthalpy departure (_H- H atm) fr t eal gas mixture was read off from the larger P-T-H plot, (Figure L4a and 5b) and found to be 0.5 Btu/lb. This amounts to a contribution of -0.000926 Btu/lb~F to the entropy difference. Second virial coefficient for the mixture was calculated using Equation V.27 and found to be 62.63 cm3/gmole. Volumetric contribution in the Equation V.26 amounted to -0.0002901 Btu/lb~V. These two corrections to the ideal mixture entropy, 2.5876 Btu/lb~F yield a value of. 2,586964 Btu/lb~F for the entropy of real gas mixture at 25~C and one atmosphere. Isobaric correction was made using Equation V.9'between temperatures of 77~F and 91.60F' H H H -T 2 -T. T -i ~9 (Srp - ST )P T J]2 dT (V.9) -,, g' 2 T1Isobaric ent.halypy d i:'.f-'erenee between temperatures of 77~F and 91.6~01? was read o:l'f from the P-T —H plot and found to be 7.57 eBtu/tlb. Isobarie entropy differences according to Equation V. 9 waa found to b'te 0.0 139913 Btu/l. b "F. After making both isothermal a nd isobar i:c correct:i on val]ue for real] gas mixture erltropy at 91.60~F' ard one atmosphere was obtained as 2.60688 B l tu/] b ~F. Tables and Diagram of Thermodynamic Properties for the Mixture Adjusted values of isothermal entropy differences at intervals 50 psia at 91,.6~% as determined from ealorimetric and

-"14and volumetric data as described in Appendix VIII were used to calculate entropy values at similar pressure intervals between ]-4.7 and 2000 psia. These values are listed in Table lib. Adjusted values of isobaric entropy differences at 250, 500, 1000, 1500 and 2000 psia were utilized to determine entropy values at these pressures between -2(80 and +3000F. These values are listed in Table TIIa and IIIb. The values of absolute entropy a, the intersection of each experimental isobar and isotherm were thus determined and were thermodynamically consistent as guaranteed by the precautions taken in Appendix VIII. The values along each isotherm are listed in Tables IIa, IIb, IIc and IId. The adjusted isobaric enthalpy values at 1700, 1200 and 1100 psia in the singlephase region and at 800, 650 and 400 psia through the twophase region were tied into the isothermal values at -27~F to provide intermediate values. These values are likewise listed in Tables lila and IIIb. It can be considered that the values 1isted in Tables at conditions of experimental measurements are smoothed experime- ntal v a lues. As such it is recommended that these values be utilized primarily in testing methods of prediction enrtropies of mixtures. Two large scale (20 in. by- 35 in.) plots of the skeleton entropy-enthalpy values were made. Curves were drawn through the points on the experimental isobars and isotherms in what appeared to be a self-consistent manner. Curves were drawn both in the sirng].e- and two-phlase regions. Reduced copies of these plots are presented as Figures 5a and 5b.

-115Appendix X,qimpson' Rule Many problems in engineering i. nvol yve graphical integration. There are many procedures which are employed to facilitate such computations. Normally the interval of integration is divided into many parts and areas for each interval under the eurve is found. Le't us visualize one such a curve given in Figure AX.1. Area Under Given Curve One can divide the area to be found out into many trapezoids. If the interval of integration x- x1 is divided into n equal increments and the (n+l) corresponding ordinates are y0, YI, --- Yn then the area is given by x xl1(1) This method is known. as the'trapezoidal rule". A better method for engineers is Simpson's rule. According to this rule, assumption is made that the curve can be represented by a cubic. Sometimes it is said that Simpson's rule amounts to replacing the actual curve by a second degree parabola over the interval under consideration. Even though it is true, it can be shown as below that the integral is precisely accurate if the curve can be represented by a cubic over the interval of integration. The proof that follows is taken directly from class notes distributed by Professor R.L. Curl (19.

(x1xy1) \iit) X1 X X3h X -- Figure AX.1 Area Under a Curve Using Trepezoidal Rule B(xay2 ) y A(xt,yi xt X - (X1+ x2) x2 x --- Figure AX.2 Area Under a Curve Using Simpson's Rule -116

-117lluncl:erical Integrat-ion Tlo t ObtaIn Area Under A Curve Consider a I'urlctioon y(x) which is represented by a series of value pairs (yi,xi). We wish to find the area under this function between xl and x2. If the values (yixi) are plotted, we may implement the numerical integration technique known as Simpson's Rule by a simple graphical method. Figure AX.2 illustrates the following steps. Plot yi vs xi. and draw a smooth curve through the 1 points. 2. Locate (x1,yl) and (x2,y2) on the curve and draw a straight line AB between these points. 3. (x1 + x2)-the midpoint of the intervalsketch a vertical line from AB to the curve. 4. Locate a point at x -= - (x1 - x2) which lies 2/3 of the way from line AB to the curve. This (y) is the mean value in the interval (x1,x2). Then: 2 ydx (x2 - x1) y xl X1 Note 1. For accuracy, the distance from AB to the curve, on the graph paper, should usually be less than 10% of the distance between x1 and x2 on the graph paper. Use several intervals to accomplish this. Note 2. Method is the same for curves concave upward - rule is still 2/3 of the way from the straight line.

Proof and Comments on Simpson's -Rule Represent y(n) by a polynomial expansion about xl. y(x) _ an (x-xl) (X.1) Then x 2 n n+l 1 ydX n=o n+l (X2 xl) (X.2) 1 For our approximation 2~x [Y(X]) + Y(x2) 2 (xl+x2 y(X)+ Y(X2)A ~ (x2 - xl) 22.. 2 (X.15) 1X + x X~ 6 (x2 xl) [(xl) + y(x2) + Y 2 X4) (NOTE: This is Simpson's rule) But from (X.1) y(xl) - Y(x2) an (x2 - x n=Ox Y ( - an (_2 x n 2 n nO A |~ o(X2-l 1) -+ ~ ~X~ln+l 4 an (-x )n+l I~ ~a-(2-X) l a )Ix+ -e (x2I1)

-119+ +a + ) an(X2 xl) n — -a] + a ( -Xxl)n+ n=l Now compare coefficients TABLE AX.1 COMPARISON OF COEFFICIENTS OF THE SERIES GIVING AREA UNDER A CURVE USING SIMPSON'S APPROXIMATION Exact Approximation n 1 General Term n (n + a n n 0 a a 0 0 1 1 2 1 2 al 1 1 2 a3 2 3 a2 3 a3 a3a. 1 1 4 a 5 a4 5 4 5 E a 5 1 1 6'7 a6 - a6 etc.,

-1120This approximation is therefore, exact for a third-order polynomial function. For a pure fourth-order curve (x4), the 5 1 1 error is ) -', or about 4%, or about 4% of the fourthorder term for an arbitrary fourth-order polynomial. Procedure for Curve Fitting Using Simpson's Rule This rule of graphical integration can be used in reverse for curve-fitting. Suppose in Figure A-VII-3, we are given a data point as we encounter in the calculations and desire to find a curve the area under which in the interval x1-x2 is same as that of ABCD. Following procedure may be adopted. Generally, the nature of the curve is known in terms of whether it is concave upward or convex upward. We also assume that it can be represented by a cubic. Let EFG represent the first guess for the curve. Then one can draw a straight line EIG. At X1+ X2 x - 2 we must then have, according to Simpson's rule HI - 2 HF. This criterion can be satisfied by adjusting the position of E, F and/or G permitting one to satisfy a number of such requirements simultaneously. By a few trials, considerable experience and patience, a curve can be drawn that yields excellent agreement with all precise experimental values. Thus the eriterian of equal area can be satisfied accurately with compari - tive ease. In the curve fitting work involved in this study, this procedure of curve fitting was followed wherever applicable. Otherwise areas were made equal by counting. Accurate graphical integration is facilitated greatly by application of these procedures.

HI=2HF t\I y A Y A B C D x1 ~ (x1+x2) X2 X AIC Figure AX.3 Simpson's Rule Used for Curve Fitting. -121

UNIVERSITY OF MICHIGAN IIII 1-11:U~1:!11 II~ lI 3 9015 02229 1358 THE UNIVERSITY OF MICHIGAN DATE DUE. _, 74~q