THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF TEE COLLEGE OF ENGINEERING THE EQUILIBRIUM OF SOLID CARBON DIOXIDE WITH ITS VAPOR IN THE PRESENCE OF NITROGEN Richard E. Sonntag A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 October, 1960 IP-469

Doctoral Committee: Professor Gordon J. Van Wylen, Chairman Professor Herbert H. Alvord Assistant Professor Bernard A. Galler Professor Joseph J. Martin Professor G. Brymer Williams

ACKNOWTEDGMENTS The author heartily expresses his gratitude to Professor Gordon J. Van Wylen, Chairman of the Doctoral Committee, for his constant encouragement and counsel during the course of the project. The valuable assistance of Professors Herbert H. Alvord, Bernard A. Galler, Joseph J. Martin, Edgar F. Westrum, and G. Brymer Williams is gratefully acknowledged. The advice of Professor John A. Clark and Mr. Herman Merte, Jr., and the assistance of Mr. Gene E. Smith are also appreciated. The financial aid and support of the National Science Foundation, the Mechanical Engineering Department, and the Horace Rackham Graduate Fellowships and Scholarships are readily acknowledged. Use of the facilities of the Heat Transfer and Thermodynamics Laboratory of the Mechanical Engineering Department and those of the Computing Center is appreciated. The author also expresses his thanks to Miss Joyann Munson for the typing and to the Industry Program of the College of Engineering for the final preparation and reproduction of the manuscript. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENT............................................. ii LIST OF TABLES.............................................. vi LIST OF FIGURES............................................. vii NOMENCLATURE....................................... ix I. INTRODUCTION................................. 1 A. Purpose..................................... 1 B. Previous Work................................... 1 II. DERIVATION OF THEORETICAL EQUATIONS.................... 7 A. General..........................................7 B. Pure Substance Solid-Vapor Equilibrium........... 7 C. Solid-Vapor Equilibrium in the Presence of an Inert Gas................................... 8 D. Pure Gas Fugacity........................... i.10 E. Fugacity of a Component in a Gas Mixture......... 11 F. Enhancement Factor............................... 12 III. EQUATIONS OF STATE.................................... 15 A. General.........................................15 B. Beattie-Bridgeman Equation of State.............. 15 C. Virial Equation of State......................... 20 1. Form of Equation and the Lennard-Jones (6-12) Potential....................... 20 2. Methods for Gas Mixtures.................. 23 IV. SOLUTION OF THEORETICAL EQUATIONS...................... 31 A. Beattie-Bridgeman Equation of State.............. 3 B. Virial Equation of State........................ 44 1. General................................... 44 2. Numerical Solutions...................... 48 V. EXPERIMENTAL APPARATUS AND INSTRUMENTATION.............. 53 A. General......................................... B. Construction................................. 55 1. Cryostat.................................. 55 2. Equilibrium Vessel........................ 59 5. Heaters.................................. 64 4. Heat Exchanger........................ 66 iii

TABLE OF CONTENTS (CONT'D) Page C. Accessory Equipment............................. 67 1. Gas Supply............................... 67 2. Pressure Regulators and Valves............ 69 D. Instrumentation.............................. 69 1. Temperature Measurement.................. 69 2. Pressure Measurement...................... 70 3. Gas Analysis.............................. 70 4. Flowmeters............................... 72 VI. EXPERIMENTAL ERRORS................................. 77 A. Temperature Measurement........................ 77 1. Potentiometer-Galvanometer............... 77 2. Thermocouples..................... 78 B. Temperature Control.............................. 79 C. Pressure......................................... 80 D. Gas Analysis....................... 81 E. Throttling Valves................................ 83. Flow Rates.......8..............8.....3.. G. Attainment of Equilibrium........................ 84 H. Solid Carbon Dioxide Filter................. 85 I. Gas Purity.................................... 85 VII. TEST PROCEDURES........................87 A. Temperature-Pressure Range..................... 87 B. Run Procedure.............................. 87 C. Criteria for an Acceptable Run................. 90 VIII. EXPERIMENTAL RESULTS.......................... 91 A. General.....................................91 B. Results and Discussion.......................... 93 IX. COMPARISON OF PREDICTED AND EXPERIMENTAL RESULTS...... 109 A. Numerical Comparisons.......................... 109 B. Concluding Remarks............................... 122 APPENDIX A - ITERATIVE METHODS............................ 123 1. Newton Method for One Equation in One Unknown.... 123 2. The Newton-Raphson Method for Two Equations in Two Unknowns.............................. 124 iv

TABLE OF CONTENTS (CONT'D) Page APPENDIX B - FLOW DIAGRAM FOR BEATTIE-BRIDGEMAN EQUATION SOLUTION S............................. 127 APPENDIX C - FLOW DIAGRAM FOR VIRIAL EQUATION SOLUTIONS..... 129 APPENDIX D - EXPERIMENTAL DATA........................ 131 BIBLIOGRAPHY................................................ 143 v

LIST OF TABLES Table Page I Values of P1, vl.................................. 43 II Beattie-Bridgeman Constants....................... 45 III Force Constants for the Lennard-Jones (6-12) Potential........................................ 49 IV Pure Gas Virial Coefficients Using the LennardJones (6-12) Potential........................... 49 V Interaction Coefficients Using the Method of Ewald............................... 50 VI Critical Constants............................... 51 VII Interaction Coefficients Using the Method of Prausnitz.................................. 51 VIII Mole Fraction of Carbon Dioxide as a Function of Temperature and Pressure..................... 100 IX Comparison of Original and Check Runs.............. 101 X Enhancement Factor e as a Function of Temperature and Pressure........................... 105 XI Comparison of Predicted and Experimental Results... 118 vi

LIST OF FIGURES Figure Page 1 Schematic Diagram of Experimental Equipment......... 54 2 Cryostat Cross-Section............................ 56 3 Outside of Cryostat................................ 58 4 Inside of Cryostat, Assembled..................... 58 5 Equilibrium Vessel Cross-Section.................... 60 6 Equilibrium Vessel Core............................. 61 7 Equilibrium Vessel Tube Coil.................... 61 8 Equilibrium Vessel, Assembled....................... 63 9 Equilibrium Vessel with Heaters..................... 63 10 Main Heater Sleeve Construction..................... 65 11 Heat Exchanger Fittings........................... 68 12 Galvanometer Julius Suspension...................... 71 13 Potentiometer-Galvanometer Installation............. 71 14 Schematic Diagram of Gas Analyzer................. 73 15 Gas Analyzer Installation..o............o.......... 74 16 Experimental Apparatus Overall Views...........o... 75 17 Mole Fraction yl vs. Pressure at 140o, 150, 160~K.. 95 18 Mole Fraction Yl vs. Pressure at 1700, 180, 190~K.. 98 19 Mole Fraction yl vs. Temperature at Constant Pressure........................................ 103 20 Enhancement Factor c vs. Pressure at Constant Temperature.o............................ 104 21 Minimal Pressure vs. Temperature................... 107 22 General Solutions at 140~K.................... 112 vii

LIST OF FIGURES (CONT'D) Figure Page 23 General Solutions at 150 K........................ 113 24 General Solutions at 1600~K........................ 114 25 General Solutions at 170 ~K........................ 115 26 General Solutions at 180 ~K....................... 116 27 General Solutions at 190~K........................ 117 viii

NOMENCLATURE A Helmholtz function A Beattie-Bridgeman constant a Beattie-Bridgeman constant B Second virial coefficient B Beattie-Bridgeman constant B* Reduced second virial coefficient b Beattie-Bridgeman constant bo Lennard-Jones force constant C Third virial coefficient C* Reduced third virial coefficient c Beattie-Bridgeman constant D Fourth virial coefficient Dl, D2,..., D22 Functions, Beattie-Bridgeman equation solution E Lennard-Jones force constant F Arbitrary function f Fugacity f Function defined by (3-22) G Gibbs function g Molar Gibbs function k Arbitrary constant k Boltzmann's constant N Avogadro's number n Number of moles P Pressure ix

NOMENCLATURE (CONT'D) R Gas constant (.08206 LIT-ATM/GM-MOLE-~K) r Molecule separation S Entropy s Molar entropy T Temperature T* Dimensionless temperature U Internal energy V Volume v Molar volume y Mole fraction B ~ Second virial in Beattie-Bridgeman equation 7 Third virial in Beattie-Bridgeman equation b Fourth virial in Beattie-Bridgeman equation e Enhancement factor QB Second virial generalized function QC Third virial generalized function cp Intermolecular force a Lennard-Jones force constant w Acentric factor Subscripts 1 Component 1 (Carbon Dioxide) 2 Component 2 (Nitrogen) 3 Combination of 1 and 2 C Critical state x

NOMENCLATURE (CONT'D) i i-th component or counter R Reduced M Mixture Superscripts g Gas phase s Solid phase Cl ~ Saturation state * PIdeal gas state (over property) - Partial molar property xi

I. INTRODUCTION A. Purpose The purpose of this investigation is to determine the effect of gaseous nitrogen on the solid-vapor equilibrium of carbon dioxide in the temperature range 140-190~K, at up to 100 atmospheres pressure. The experimentally determined gas-phase composition is to be compared with that predicted by three different methods that have been used in the past on similar systems. Temperatures as high as 1900 are of particular interest in this study in order to be able to fully evaluate the simplifying assumptions made in one of the two solutions determined for each of the three equations of state. The upper limit of pressure has been chosen so as to give a gas phase density somewhat greater than that for which the equations of state are intended, especially at the lower temperatures. B. Previous Work A great deal of information has been published in past literature regarding the increase in apparent vapor pressure of a condensed phase when exposed to an inert gas pressure. It is perhaps somewhat more realistic to refer to this effect as the solubility of the condensed phase in the inert gas, the gas being termed inert because in many systems it dissolves in the condensed phase to only a very small degree. In many of the references mentioned below, this is the case, and in any event the gas phase composition was of principal interest. -1

-2In 1880, Hannay and Hogarth(l) studied the solubility of solid potassium iodide in alcohol and other vapors. More recently, Pollitzer and Strebel(2) determined the effects of nitrogen or hydrogen on the carbon dioxide liquid-vapor equilibrium, and also that of air, nitrogen or hydrogen on water liquid-vapor at various pressures and temperatures. Barlett(3) also presented results of the solubility of water in hydrogen, and also other gases. Braune and Strassmann(4) investigated the effect of carbon dioxide on the vapor pressure of iodine at up to fifty atmospheres total pressure. In 1934, Saddington and Krase(5) studied water liquid-vapor in the presence of nitrogen at as high as three hundred atmospheres. Perkins(6) considered the influence of helium, argon or nitrogen on the decomposition of Ba C12. 8NH3(s). In 1941- Kritchevsky and Koroleva(7) determined the effects of nitrogen, hydrogen, methane or carbon dioxide on the liquid-vapor methanol equilibrium at up to seven hundred atmospheres. In the same year, Wiebe and Gaddy(8) worked with the system water-carbon dioxide at high pressures. The following year, Kritchevsky and Gamburg(9) investigated benzene in nitrogen gas. Gratch(lO) considered the effect of air on solid carbon dioxide, a system very similar to the present one, but at pressures up to fifty atmospheres. In 1948, Diepen and Scheffer(ll) published results of the remarkable influence of ethylene gas on naphthalene, in which the apparent vapor pressure was as much as 25,000 times its value for pure naphthalene. Ipatev, et. al.(12) studied the benzene equilibrium, as in Reference 9, but under the influence of hydrogen gas. In 1950, Webster(l3) found the effect of air on the water solid-vapor and liquid-vapor equilibria. He continued his work(l4) on carbon dioxide-air as in (10), but at pressures

-3as high as two hundred atmospheres. In much of this work, the air was in the liquid rather than the vapor phase. In a particularly interesting study, Dokoupil, Van Soest, and Swenker(15) determined the influence of hydrogen on the solid-vapor systems nitrogen, carbon monoxide, and nitrogen-carbon monoxide at up to fifty atmospheres, and also made some theoretical predictions of the gas compositions. Another investigation of interest is that of Ewaldl6) who determined the solubility of solid xenon in nitrogen, hydrogen, and helium, and also solid carbon dioxide in hydrogen and helium. He also made theoretical calculations of the composition, but by methods different from those of the preceeding reference. The systems discussed above were all of a similar nature to the present one, although in many of these the condensed phase was liquid rather than solid. There have also been a number of similar investigations carried out on systems comprised of more complex organic molecules. Inasmuch as the present study involves carbon dioxide, a brief mention will also be made of recent work in the area of two-component two-phase equilibria, even though that type of system is not under consideration here. The majority of this recent work involving carbon dioxide has been the phase equilibrium with one or more hydrocarbons. Poettman and Katz(17) presented the phase diagrams for carbon dioxide with propane, butane and pentane, respectively. Clark and Din(l8) were concerned with carbon dioxide-acetylene, including three-phase equilibrium. Haselden, Newitt and Shah(l9) investigated the systems carbon dioxide-ethylene and carbon dioxide-propylene, and in 1953 Bierlein and Kay(20) Studied carbon dioxide-hydrogen sulfide. The following year,

-4Donnelly and Katz(21) considered the liquid-vapor and three-phase loci of carbon dioxide-methane, with very interesting results. In 1959 Sobocinski and Kurata(22) worked with the same system as (20) but also included a three-phase study. Regarding the mutual solubility of liquid carbon dioxide with other liquids, Thiel and Schulte(23) published some results in 1920. Quinn and Jones(24) later presented a more complete discussion on the subject, and Francis(25) published mutual solubilities for 261 different liquids with liquid carbon dioxide in 1954. None of these included nitrogen, however, due to the low critical temperature of nitrogen. Only a fairly small number of investigations have been concerned with the properties of the carbon dioxide-nitrogen system. Of these, two have been found which deal with the solubility of solid carbon dioxide in liquid nitrogen, those of Ishkin and Burbo(26) and also Fedorovao(27) Abdulaev(28) and also Mills and Miller(29) considered the liquid-vapor phase diagram of this system, but the results were quite incomplete. From the theoretical viewpoint, that of predicting gas phase composition in the condensed phase-inert gas type of system, Dokoupil(15) used a simplified solution involving the Beattie-Bridgeman equation of state with very good results, although pressures were limited to fifty atmospheres. In 1955 Robin and Vodar(30) performed calculations for the data of References 3, 4, 5, 7, 8, 9, 12, and others, using a Statistical Mechanics virial-form of equation, and assuming the Lennard-Jones potential to be valid. The equation was limited to only the second virial coefficient, but results were generally quite good, except for systems

-5involving either water as the condensed phase or hydrogen as the inert gas. The same year, Webster(S1) discussed his work(131,4) from a theoretical standpoint. Still in 1953, Ewald, Jepson and Rowlinson(52) made calculations for the data of References 11, 14, and others, also using a virial-form equation of state with the Lennard-Jones potential and empirical combination of the critical constants, resulting from the theorem of corresponding states. Their equation included terms through the third virial coefficient, and the results were reasonably good, at least for systems of fairly simple molecules, Ewald(16) later improved upon this method by using intermolecular force constants determined from experimental virial coefficients, instead of estimating them by empirical relations with the critical constants. He then made semi-empirical combinations of these pure substance force constants to enable calculation of the interaction coefficients for gas mixtures. Prausnitz(33) presented a new method for determining interaction coefficients for the second and third virial terms, his method based on the principle of corresponding states and tables compiled from generalization of mixture behavior. Prausnitz considered an isotherm of (11) as one application of his method, with good correlation. Ziegler(34) has made very extensive calculations for the solidgas and liquid-gas equilibria of methane-hydrogen, using the methods of Dokoupil,(15) Ewald,(16) and others.

-6The methods and procedures discussed briefly here will be presented in detail in Section III, in order to obtain solutions of the theoretical equations of the following section.

II. DERIVATION OF THEORETICAL EQUATIONS A. General For a pure substance, the thermodynamic property relation dU = TdS - PdV + GldnI + G2dn2 +... (2-1) becomes dU = TdS - PdV + gdn (2-2) Consider the Helmholtz and Gibbs functions: A = U - TS (2-3) G = U + PV - TS (2-4) Differentiating (2-3) and (2-4), -dA dU - TdS - SdT (2-5) dG = dU + PdV + VdP - TdS - SdT (2-6) Substituting these relations into (2-2) yields dA = - SdT - PdV + gdn (2-7) dG = - SdT + VdP + gdn (2-8) B. Pure Substance Solid-Vapor Equilibrium Expanding Equation (2-8), d(ng) = ndg + gdn = -nsdT + nvdP + gdn or dg - -sdT + vdP — (2-9) It has been shown, according to the methods of Gibbs, that for a solidvapor equilibrium at a given temperature and pressure (the sublimation pressure P) gs = gg (2-10) -7

-8If the vapor behaves ideally, = RT (2-11) P Then, from (2-9) and (2-11) at constant temperature, dg = RT dP = RTd (in P) (2-12) P In order to preserve this form for a real gas, the quantity fugacity is introduced by the definition: dg = RTd (in f) (2-13) with the realization that a real gas approaches ideal behavior as P ->0, so that Lim (f/P) 1 (2-14) Therefore, the criterion for equilibrium can be written fs = fg = f (2-15) where the superscript 1 refers to the value at the vapor (sublimation) pressure of the pure substance. C. Solid-Vapor Equilibrium in the Presence of an Inert Gas Denoting the substance appearing in both phases by the subscript 1, the requirement for equilibrium in this case is S = ^ (2-16) where denotes the partial free energy of component in the gaseous phase. For component 1 of the mixture, Equation (2-13) is written dGl = RT d(ln fl) (2-17)

-9where the fugacity of 1 approaches the partial pressure as the total pressure tends to zero, or Lim (fl) = i (2-18) P- O P Therefore, the criterion for this equilibrium is that fl = f (2-19) Equating Equation (2-9) with (2-13) written for solid 1 at constant temperature, dgl = RTd (in fl) vldP (2-20) Equation (2-20) is now integrated from the vapor pressure P1 to the pressure of the system, giving IS P RT n = v dP (2-21) If P1 But, substitution of Equation (2-19) results in rS P RT in 1 = v dP (2-22) fi P which is the equilibrium equation for the system under consideration. In order to calculate the percent composition of the gas-phase as fixed by (2-22), it will be necessary to determine expressions for 1 and fl in terms of the behavior of the gas mixture and pure substance 1, respectively. In both cases, T and V will be selected as independent variables, the reasons for which will become obvious later.

-10D. Pure Gas Fugacity (component 1) Taking a cross partial differential in Equation (2-7), avTn an TV (2-23) Or, at constant T,n: dg ~ - () dV (2-24) an T,V Substituting Equation (2-13) into (2-24), RTd (in f) ~ - (-P) dV (2-25) an T,V which may also be written as RTU (ln fV) = - [(n) - dV (2-26) on T,V V Equation (2-26) is to be integrated from an arbitrarily low pressure P * to the vapor pressure P. From this point on, the subscripts 1 will be included to denote that pure substance. As P* is made to approach zero, note that the lower limit on the left-hand side of (2-26) tends to (flV) - (PV1)* = nlRT (2-27) and the lower limit on the right-hand side of (2-26) goes to V -, o as P* e-O (2-28) so that RT ln fV P(. RTV ]v (2-29) nlRT T V ni.L L 0 T jV

-11But, since V,/nl = vI at P1, (2-30) 0 P00 a a RT in( f) f 1 [() - T]dV - RT In Plv (2-31) [(-) (2a5) i av n n T,V V RT giving an expression for f? in terms of the behavior of pure substance 1. E. Fugacity of a Component in a Gas Mixture Substituting (2-5) into (2-1), dA = - SdT - PdV + Gldnl + G2dn2 (2-32) Taking a cross partial differential,'(|1) m = E ( ) (2-3355) T)n T nl T,V,n2 Or, at constant T,n: dG1 = - (a) dV (2-34) =nl T,V,n2 Substituting Equation (2-17) into (2-34), RTd (ln fn) - ( )TVn dV (2-35) As before, this may be rewritten as RTU. (in fV) - n [( T n) T]dV (2-56) Rd!nVl T,Vn2 Equation (2-36) is to be integrated from an arbitrary P* to the pressure of the mixture P. As P* - 0, the lower left-hand limit of (2-36) is (flV) - (yPV)* = ylnRT (2-37) Integrating, Tf~v V ap RT RT ln - J [V(-) - dV (2-58) YlnRT anl1 T,Vn2 v

-12But, V/n = v of the mixture (2-39) Therefore, oo RT ln( = [( ) - - ]dV - RT in - (2-40) yiP V anl T,V,n2 V RT relating fl to the behavior of the gas mixture. The reason for selecting volume (rather than pressure) as an independent variable in the derivations of the two preceding sections: is now quite apparent. In order to evaluate Equation (2-31) it will be necessary to employ an equation of state for pure 1, and for Equation (2-40), an equation of state for the gas mixture will be required. In both cases, one of the accurate pressure-explicit equations of state can be used. Had pressure been used as an independent variable, the resulting equations analogous to (2-31) and (2-40) would have necessitated the use of volume-explicit equations of state, which are inherently less accurate than the pressure-explicit forms. F. Enhancement Factor Before substituting (2-31) and (2-40) into the equilibrium Equation (2-22), it will be convenient to introduce a dimensionless parameter called the enhancement factor e, which is defined by the following conditions: E = Yl/Yl(ideal) (2-41) where, for Yl(ideal): a) the gas mixture behaves ideally, so that = = ylP

-13b) increasing pressure from P1 to P has no effect on the fugacity of the solid, or f = fl 1 1 c) Pure 1 at P1 behaves ideally or P 0a fl P 1 1 Therefore, Yl(ideal) p (242) and e = ylP/Pl (2-43) Now, substituting Equation (2-31), (2-40), and (2-43) into (2-22) gives the equilibrium equation, which upon rearrangement may be written O0 Pvz 8P) RT RT in E = RT In — P + J [( - ]dV'pVL l V 6n T,V V 0o P - F *-P R ]dV+ vdP (2-44) - CnTiI T,V,n2 v P The enhancement factor is a convenient parameter for determining deviation from the ideal behavior given by the assumptions a) - c) above. It should be recalled that in Equation (2-44), the first integral refers to pure substance 1, the second to the gas mixture, and the third to solid 1. For given values of temperature and pressure, Equation (2-44) specifies one equation in two unknowns, yl and v. Note that yl is included in e and will also appear upon integration of the integral for the gas mixture. The second relation between y1 and v comes from the gas mixture equation of state itself, thereby permitting solution of the set for the given T and P.

III. EQUATIONS OF STATE A. General In order to be able to obtain solutions of the theoretical Equation (2-44) and thereby predict gas-phase composition for a given temperature and pressure, it will be necessary to use an equation of state to evaluate the integrals. Consequently, an equation will be required for pure carbon dioxide and also one for the gas mixture. The former is not at all a difficult task, as its use lies in an integration from zero to the vapor pressure of carbon dioxide, which is a small number at the temperature of interest. Accurate equations of state for gas mixtures are a different matter, however, and particularly at high densities. Beattie and Stockmayer(35) have given an excellent review on equation of state development up to 1940, including that from Statistical Mechanics. A great deal of progress has been made since that time, especially in the area of high density equations, mixtures, and theoretical correlations from Statistical Mechanics. B. Beattie-Bridgeman Equation of State The Beattie-Bridgeman equation of state (36, 37, 38) is most commonly presented in the form p RT (l-c)(v+B') - (3-1) v2 v2 -1~5

-16where: A' = A(l-a/v) (3-2) B' - B(l-b/v) (3-3)' = c/vT3 (3-4) in which A, B, a, b, c are the five constants evaluated empirically from P-V-T data for a given substance. While the above form of Equation (3-1) gives some insight into the nature of the Beattie-Bridgeman equation, showing the constants as corrections to the ideal-gas equation, it is perhaps more useful, especially for this study, to present the equation in a virial form, RT P 7 6 = - + + +- + (3-5) where P, 7, 8 can be called the second, third, and fourth virial coefficients, and are given in terms of the Beattie-Bridgeman constants by P - BRT - A - cR/T2 (3-6) 7; Aa - BbRT - BcR/T2 (3-7) 56 BbcR/T2 (3-8) Care must be taken not to confuse these virial coefficients B, 7, 8, with those for an equation of state written in terms of the compressibility factor Pv/RT. The Beattie-Bridgeman equation of state has been applied to many gases, including all the common ones, with excellent correlation up to approximately 0.8 times the critical density. At densities greater than the critical, however, the equation begins to fail very

-17badly, since it has been empirically fitted in the lower density region only. This equation has also been successfully applied to mixtures of gases, through empirical combination of each of the five constants for the pure gases. Although mixture representation has not been found to be as accurate as for pure gases, it is, nevertheless, very good, and comparable to other methods of algebraic representation of similar complexity. Since the mixture of the present investigation consists of only two components, further discussion will be limited to binary mixtures. Of course, extension of the methods to more complex mixtures follows. Consider a constant k as part of an equation of state used to represent a given pure gas. If it is assumed that this constant has numerical values kl and k2 for the two gases comprising the mixture, then k for such a mixture can be given by 2 2 kM = lk + 2yl2k3 + y2k2 (3-9) Where k3 is termed the interaction coefficient, the value of which remains to be determined as some function of kl and k2 only. Three logical combinations of kl and k2 to try in attempting to find a good value for k3 are the following: Linear combination (arithmetic mean) k = -(kl+ k2) (-10) 3 2~~~~~~~~~~~~~(-0

-18Square-root combination (geometric mean) 1/2 k3 = (klk2) (3-11) Cube-root or Lorentz combination k (kl/3 I kl3)3 (3-12) 3 ~8 1 2 For the linear combination, substitution of (3-10) into (3-9) yields kM = ylkl + y2k2 (3-13) or, generalizing, kM = Yiki (3-14) i For the square-root combination, substituting (3-11) into (3-9) gives: kM y kl + 2Yl2(klk2 + y2k2 (3-15) or, generalizing, kM= iy Yi 2)2 (3-16) Finally, for the Lorentz combination, substituting (3-12) into (3-9) results in: k =y2k + 1 Y (kl/ + kl/3) 3 + y2k (3-17) or, generalizing Equation (3-17), M 8 Zz Yijk1/3 ^ +k/)(3-18) Mk = Z Z y yj(k/ i + k/3) (3-) M 8ij'

-19A great deal of work has been done in the past in order to determine which combination of the three methods given here should be used for the five constants in the equation of state (39,..,} 42), and others. In particular, those of (15, 41, 42) suggest a square-root combination for the constants A and c, a Lorentz combination for B, and linear combination for both a and b, especially for gaseous equilibrium systems. In order to test the various methods used previously for mixtures, a computer program was written and executed using the IBM 704 computer facilities of The University of Michigan.(43) The ten most commonly suggested methods of combination were tried, and compared with experimental data for various mixtures of nitrogen and carbon dioxide (44,..., 47), and also for mixtures of other gases. The method suggested above gave good correlation, and while it did not show the lowest standard deviation (in pressure) at low densities, it was among the best, and held reasonably well to higher densities than any of the other methods tried, a factor which is of prime importance in this investigation. There are many references in the literature to the use of the Beattie-Bridgeman equation as applied to gas mixtures. Of these, (15, 41, 48, 49, 50) are of interest in that they dealt with application to high-pressure equilibrium calculations, including mixture fugacity determination. In particular, Reference 49 referred to the prediction of gas-phase composition for the iodine solid-vapor or liquid-vapor equilibrium in the presence of air, carbon dioxide, or

-20hydrogen. Also, as mentioned in Section I, Reference 15 made use of the Beattie-Bridgeman equation for similar calculations in the systems nitrogen-hydrogen, carbon monoxide-hydrogen, and nitrogen-carbon monoxidehydrogen. C. Virial Equation of State 1. Form of Equation and the Lennard-Jones (6-12) Potential The virial equation of state was used by Kammerlingh Onnes,(5) although simply as a curve-fitting technique. It was not until some time later that much success was achieved in relating the virial coefficients to intermolecular forces and distances using the methods of Statistical Mechanics. The virial equation is written as a power series in pressure or reciprocal volume, although the latter is more common and generally gives a better correlation with experimental data. The equation is normally written in terms of the compressibility factor, as: Pv + B + C + D +(-19) E l 1_l+B^.. (3-19) RT v v2 v3 where B, C, D, the second, third and fourth virial coefficients, respectively, are temperature functions but are not the same as those of the Beattie-Bridgeman Equation (3-5), differing in each case by the factor RT. Considerable effort has been put forth in the area of determining virial coefficients during the past 25 years, from both the theoretical and empirical standpoints. Almost no success has been realized beyond the third virial coefficient C, and so higher order terms will not be considered further here.

-21From the methods of Statistical Mechanics, the second and third virial coefficients can be given(52)53) in terms of the intermolecular potential energy cpij(r) of a pair of molecules i and j, of separation rij by the relations B -2 tN f [EI ( _ l]r2dr (3-20) 0 C = -8 J fl2fl3f2 rl2rl3r23 drl2drl3dr23 (3-21) 3 (Integral over all r12> r153 r23, which form a triangle) where fj(rij) = [ Eij/kT - 1] (3-22) and N is Avogadro's number. The assumption of additivity of forces has been made here, and also that the molecules have spherically symmetric force fields. Solution of (3-20) and (5-21) for the virial coefficients will, of course, require an expression for the intermolecular potential function cp(r). The Lennard-Jones (6-12) potential, References 53, 54, 55, is such an expression for non-polar molecules, relating the potential energy to the constants E and a in the form p(r) = 4E[(o/r)12 - (a/r)6] (3-2) For a given substance, a is the separation for which the potential energy is zero, and E is the maximum attractive energy between two molecules, occurring at r = a(2)1/6. In (3-23), for large separation of the molecules (r > a), the inverse sixth-power attractive component dominates. On the other hand, for r ~ ca, the inverse twelfth-power repulsive term is dominant.

-22It should be pointed out that the exponents 12, 6 of (3-23) are empirical, and by no means universal. Some calculations have been made using other exponents, particularly for a (6-9) potential, but results seem to correlate better for the (6-12) form, at least for the second virial coefficient. Integration of (3-20) for the second virial can be performed by substitution of (3-23). In this operation, it will be convenient to incorporate a set of dimensionless parameters, defined as: T* = kT/E (3-24) B*(T*) = 3B B (3-25) -2 Na3 o The result of the integration is then (2j+1) B*(T*) = Z bj (T*) (3-26) j0O Where the coefficients bj are given by (2)j +2 ~(2j-1) (3-27) bj=I... r (3-27) j 4j'l 4 The values for B* (T*) have been evaluated and compiled.(56,53) A similar procedure is followed for the third virial coefficient, by substituting (3-23) into (3-21) and integrating, also making use of (3-24) and the function c* (T*) = (3-28) bo

-23This integration results in _(j+l *(T*) = c cj (T*) (5-29) j=0 The coefficients cj of (3-29) are not simple functions, but instead are complex integrals. The values of C* have been evaluated by numerical procedures,(57' 58, 59) and are also compiled in Reference 55. Some experimental determinations of the virial coefficients and equations for nitrogen and carbon dioxide from P-V-T data are given in References 60,..., 65. Virial coefficients are determined from the low density compressibility data according to the following procedure: The equation of state (3-19) is rearranged into the form (P - 1) v B + c(v) (3-30) For a given temperature, a plot of (Pv 1) v vs, 1/v RT will yield B as the intercept at (l/v) = 0 and C as the slope at (l/v) = 0, both values being for that temperature only. The difficulty with this method of determining virial coefficients is that extremely accurate compressibility data at low densities is required. Force constants for the Lennard-Jones (6-12) potential may be computed from the second virial coefficient as found from compressibility data, or by a similar procedure using viscosity data. 2. Methods for Gas Mixtures In the case of gas mixtures, the equation of state has the same form as (3-19), but the virial coefficients must now represent the mixture,

-24and are therefore functions of mole fractions as well as the temperature. It has been shown(66) by methods similar to those for pure substances that the n-th virial coefficient is a polynomial of n-th degree in the mole fraction. That is, BMX = yiYjBi (3-31) im CMIX = YiYjYkCijk (3-32) ij k Where the Bij, Cijk in which all subscripts are not the same are termed interaction coefficients. The remaining ones, those for which all subscripts are identical, are simply the coefficients for the pure substances comprising the mixture. The interaction coefficients can be represented by equations analogous to (3-20), (3-21), in which the intermolecular potential function is that between molecules of different species. In the case of a binary mixture, which is of principal interest here, for components 1 and 2 the Equations (3-31) and (3-32) may be expanded into 2 2 BMIX = ylBll 2yly2B12 + y2B22 (3 2 2 3 cMIX = YlClll + 3YlY2C112 + 3YlY2C122 + Y2C222 (3-34) The problem in representing mixtures now reduces to that of determining the various interaction coefficients appearing in (3-33), (3-34). The B11, Cll1 are those for pure component 1, etc. The method following (3-30) for experimentally determining virial coefficients cannot readily be followed for mixtures, as sufficiently accurate low-density P-V-T data is lacking. Even in cases for

-25which this can be done, it must be remembered that virial coefficients so determined are valid for that exact composition only, and are therefore used in turn to calculate the interaction coefficients. Until such a time when extensive mixture data are available, it seems reasonable to combine the pure substance intermolecular force constants semi-empirically, as suggested in References 16, 53, 62, 67,..., 70, and the resulting combined constants used with the LennardJones potential to calculate the interaction coefficient B12 of Equation (3-33). The most generally accepted method of combination is 1/2 El2 = (EE2) (3-35) 1 (3-36) a12 =. (al + 2) ( 6) From the definition of the parameter bo in (53-25) Equation (3-36) may be given as b = 1/5 1 1/5 bo = [!(bo~) + (bo) ] (3-37) 012 2 [ b) 2 The interaction coefficient can now be computed by using E12 and bol2 as for a pure substance in (3-24), (3-25), and tables of the function B* (T*). A similar method is needed to be able to evaluate the two interaction coefficients of (3-34) for the third virial coefficient. Rowlinson, et al.(70) and Ewald(l1) have used the following relation for the two third-order interactions, n n 1 - (3-38) Eln2 = E1 + ) boln2 = [(1 - 3)(boQ)1/3 + (3)(bo) ] (-9)

-26in which n is either 1 or 2. Specification of these parameters now enables calculation of both C112 and C122 by using the Lennard-Jones potential tabulations for C* (T*) and Equations (3-24), (3-28) as for a pure substance. It is obvious that the above procedures for computing the interaction coefficients requires a knowledge of the force constants for the pure substances comprising the mixture. A method similar in nature to that above has been suggested in References 32, 53, 71, and 72, this being particularly useful in cases for which the pure substance force constants are unknown. In this case, empirical relations between the force constants and critical constants are used, assuming the theorem of corresponding states. The relation used by Ewald(32) is kTc/E = 1.28 (3-40) vc/bo = 1.46 (3-41) The corresponding numbers given by Hirschfelder(53) for spherical nonpolar substances are 1.30 and 1.33, respectively. Examination of force constants determined from virial coefficient data shows considerable deviation of the values from either of these sets, even for simple molecules. The rule is still useful as an approximation, however, where other data are lacking. It is apparent that relations of the form (3-40), (3-41) permit calculation of the pure substance virial coefficients, but in order to use these relations to estimate interaction coefficients, it will be necessary to combine the pure gas critical constants in some manner to

-27give the pseudo-criticals Tc12- vc12, etc., which can in turn be used with (3-40), (3-41) to specify (E/k)12, bo12, etc. Guggenheim(71) has shown that, assuming the principle of corresponding states, the relations TC12 (TC1 TC2) (3-42) =1 1/5 1 1/5 3 V12 = [ e) + (c) ] (3-43) are analogous to (3-35), (3-37). Similarly, for the interaction coefficients in the third virial coefficient (3-34), Ewald(32) suggests that, an an approximation 1 _ n n 3 3 (3-44) Tln2 = T1 C2 (-44) n 1/ n 1/3 3 VCln2 [(1 - )(vcl) + ()(V c2)1 (-45) where, again, n is either 1 or 2. The relations (3-44), (3-45) are seen to be identical in form to (3-38), (3-39) of the preceding method. Equations (3-40),..., (3-45) now permit estimation of all the coefficients appearing in (3-33), (3-34), by using the tabulations for B* (T*) and C* (T*), as was done previously. As mentioned before, this method will certainly be inferior to the preceding one, but has proved quite useful in calculations involving substances for which the LennardJones intermolecular force constants were not known. The final method of estimating interaction coefficients to be discussed here is a somewhat different approach to the problem, and is due to Prausnitz.(33) His method has been based on extension of the work of Pitzer, et.al.(73'74) to mixtures of gases.

-28Pitzer has correlated the properties of non-polar pure gases through a revision of the theory of corresponding states. In doing so, he found it necessary to introduce the acentric factor W, a third correlating parameter (in addition to reduced temperature and pressure), this being defined by = -loglO PR(sat at TR = 0.7) - 1.00 (3-46) Since this work dealt with pure gases only, the three interaction coefficients of (3-33), (3-34) cannot be predicted by Pitzer's correlation alone, Guggenheim and McGlasham,(71) among others, had previously suggested means for predicting the coefficient B12, using only the twoparameter theory of corresponding states and the definitions of (3-42), (3-43). Prausnitz and Gunn(75) later used Pitzer's principles, along with an extension of the acentric factor correlation of Pitzer to determine the critical pressure and temperature for mixtures. Prausnitz developed these principles to enable prediction of the virial coefficients for mixtures. His procedure can be summarized as follows: For a pure substance i, the second virial coefficient is correlated in reduced form as Bi (= T )i (3-47) vci ci where QB, a function of the reduced temperature and acentric factor, is tabulated from generalized behavior. For a mixture, the reduced

-29interaction coefficient of the second virial is given in a similar form, B12 - (T'12) (3-48) vc12 c12 Calculation of the interaction coefficient is now seen to depend on knowledge of the parameters Tc12, o12, and vcl2. From Reference 75, it was decided that the latter two should be combined linearly, so that 2 s= 1^( + a2) (3-49) c12 = (Vc + v) (3-50) It was also found that Tc12 could not be expressed in the above manner, but could instead be correlated well by a square-root combination with a correction factor, or T12 = kl2(Tcl Tc2) (5-51) where the factor k12 < 1, and is a function of the critical volume ratio. For (Vcl/Vc2) = 1, k12 = 1.0, and it decreases with an increase in the ratio to approximately 0.44 at (vcl/vc2) = 7. A similar method is used to estimate the two interaction coefficients appearing in (3-26) for the third virial, the correlation being given as Cln2 c T =c(' Wln2) (5-52) vcln2 TCln2 where n is either 1 or 2. Approximate values for 0c are given in the reference as a function of the two parameters, these values being based

-30on Pitzer's generalized tables and additional volumetric data. Again, it is necessary to empirically combine the parameters for the pure gases to obtain Tcln2, l.n2, and. also Vcln20 It is suggested that these be given by Tc = (Tc Tn T2) (3-53) Cln2 C1 cn C2 n12 - (CO1 + Wn + -2) (3-54) v 2 (v + v + ) (5-55) Vln2 3cl n vc2 While the values so obtained for Cln2 will certainly be less accurate than those for B12, it should be remembered that in the equation of state the third virial coefficient is of much lesser importance than is the second. Prausnitz2 method for determination of the virial coefficients has been tested in numerous casest and found to give fairly good correlation for mixtures at moderate densities.

IV. SOLUTION OF THEORETICAL EQUATIONS A. Beattie-Bridgeman Equation of State For convenience, the Beattie-Bridgeman Equation (3-5) is repeated here as p RT + + + (4-1) v v2 v3 v where = BRT -A - cR/T2 (4-2) 7 = Aa - BbRT - BcR/T2 (4-3) = Bbc R/T2 (4-4) Using the definition of enhancement factor, (2-43), Equation (2-44) may be written in a convenient form for solution, R iP RT s 00 RT ln ylVl - [ (n)T_ R ]dV - | vsdP aP RT + f [(T-V) - ]dv = o (4-5) V l T,V,n2 Now, writing (4-1) for pure I as nRT n2 1 n3y1 n4i p=.... + + v1 + (4-6) Then, 6p RT 2npl 3n2 l 4n351 = — + 2 + - + (4-7) (n T,V V V2 V V4 so that the first integral of (4-5) becomes 70 P RT 21i 37i 4 1 [(-)V ] dV = - + + (4-8) -31

-52The second, integral of (4,-5) involves solid, 1, and can be evaluated by use of the relation vl(PT) = vl(0,T) [ - k1P (4-9) Where k1 is the isothermal compressibility of solid lo Therefore, P p 7 s 2 )] J VldP vl[(- (P -P1 ) (4p-10) 1 If it can be assumed that kl is negligible, P f vldP = v Pp (-4-11) 1 In order to evaluate the third integral, of (4-5), Equation (4-1) must be written for the gas mixture, nRT n2.M n3YM. n48M P n: M+ + (4-12) Then, =-+ (n2 M)] +, (nM)] IRT 1 _ 1 l T,V,n2 + n (n )1 + [ n ) 1 6 (n4 + 1 [ (n4& (4-13) and the integral is 00 a P RT 1 r- (n2\ 1 [ _ (n3) f [( ) -]dV VI- 2V [ (n 2M)] + (n7M)] V Tn+ T,V,n2 1 - (nlM ) (4-14) +;3[a nl h)

-33The first term of (4-14) contains a (n2M) = 2npM + n2 aM (4-15) 6n1 6n, But, 2npM = 2n[BMRT - AM - MR/T2] (4-16) and n2 M = n[RT (n aM) (n aM) R (n ) ]( 6nl Tn n (n 72- nl (4-17) Substituting (4-16) and (4-17) into (4-15), 6BM, 6AM) (n2M) = n[RT(2BM + n -) - (2AM + n -) - (2cM + n ] (4-18) Tn, Similarly, the second term of (4-1.4) contains i-(n3YM) = 3n2yM + n3 YM (4-19) 6nl 6n, But, 3n2rM = 3n2[AMaM - BMbM RT - BMCM R/T2] (4-20) and n3 YM = n2[AM(n aM) + a(n M) RT B M )( aSn^I d dni 6ni nl ( nl - nRT B(n - RThM(n B BMn - M(n M(n ) (4-21) Substituting (4-20) and (4-21) into (4-19), 2 / 6 \ PaM 6AM\ a (n3yM) = n2[(3AMaM + AMn + aMn A) 6dn-i, bM jBM - RT(3BMbM + BMn a + bMn an) - R(5BMcM + BMn + C+ c BM) ] (4-22) T2 dn^ 1n,

-34Finally, the third term of (4-14) contains a (n46M) = 4n53M + n4 |M (4-23) anl an But 4n3 M = 4n3 [BMbMM R/T2] (4-24) and n4 M = n3 R [BMbMn M + BMcMn + bMcMn (4-25) 6n, l (4 -25 ) Substituting (4-24) and. (4-25) into (4-23), i.(n4) = n3 R 4BMbMcM + BMbMn CM + BMcMn abM + bMcMn aBM] (4-26) Now, substituting (4-18), (4-22), and. (4-26) into (4-14), 00 "', RrI: 1 6BM, 6AM~ I[ (dR) - I iv = [RT(2BM +n (2AM + n V 6n_ T,V,n2 V n R dCM\- 1 -a, AM\ - (2cM + n CM) + AM M + AM) T- 6n! 2v-2[3A~a M + A~n n + aMn n - RT(53BMbM + BMn 1bM + bMn BM) R /~, - ScM aBM - (3BMCM + BM cMn a) + (4-27) 1 R aCM bM BM] + 3[T ] [4BMbMcM + BMbn n + BMcMn + bMcMn1 n In order to solve (4-27), it will be necessary to decide on the methods of pure gas constant combination to be used for the five constants, in order to reduce the terms AM, BM, etc.,, and their derivatives to expressions involving the pure gas constantso

-35The method of combination to be used here is that discussed in Section III, that being square-root combinations for the constants A and c, cube-root for B, and linear combinations for a and b. Thus, AM = (Z yiA/)2 = A2 + 2y1(A5-A2) + yl(A1+ A2-2A) (4-28) where, by definition, A3 = (A1A2) (4-29) Also, BM: d YiYj (B/ i B 3) BM = y(B. i 0 = B2 + 2yl(B3-B2) + y2(B1+B2-2B3) (4-30) Where B3 = (B/3 + B/) (4-31) As for AM, CM = (Z yci/ )2 = c2+ 2y (c-c2) + y2y(c.+c2-2c3) (4-32) where c3 = (clc2) /2 (4-33) Finally, aM =Z yiai = a2 + 2y1(a3-a2) (4-34) where a = -(a +a2) (4-5)

-56and bM = yib - b2 + 2y (b3-b2) (4-36) where b3 =(bl+b2) (4-37) Next, taking the partial derivatives required in Equation (4-27), n M = 2[(A3-A2) + yY(A1+2A2-3A3) - l(A1+A2-2A3)] (4-38) an1 n 4M = 2[(B -B2) + Yl(B1+2B2-3B3) - yl(B1+B-2B) ] (4-39) Onl ~n cM 2[(c3-C2) + yl(c1+2c2-3c3) - y((cl+c2-2c3)] (4-4 onl n M = 2[(a3-a2) - Yl(a3-a2)] (4-41) n b = 2[(b5-b2) - yl(b3-b2)] (4-42) Substituting Equations (4-28), (4-30), (4-32), (4-34), (4-36), (4-38),..., (4-42) into (4-27), Equation (4-27) can be reduced, after considerable algebra, to the form 6P RT 2 [(nl T,V,n2 V - = [3 + Yl(l - 3)] 3vv 2+ R yl[ (l-yl ) +(l-y1) ](Bl+B2-2B3) (C1+c2-2c 3)} + 2 {8223-62+yl(4(233-75223+642+4122) + y3 (4133 -12 v27+15 23 - 613+3122+ 612-1 2) (4-4() + y1(-8sl33+1.2233-13 223+5ll3+36l22-25112+1052-l )

-37+ Y (46133-462335+46223 4 113-122+ 112-352+351) ) (1 —y1)(bl-b) (Bl+B2-2B3) (cl+c2-2c)} (4-4 Cont'd) Where, for convenience, 1 = B1 RT - A1 - cl R/T2 (4-44) P3 = B3 RT - A3 - c3 R/T2 (4-45) = A1 a1 - RT (Blbl) - R (B1 cl) (4-46) T2 2 = A2 a2 - RT (B2b2) - R (B2c2) (4-47) Y13 = (A3al + Ala3) - RT(B3bl + Blb3) - R-(B3Cl + Blc3) (4-48) Y5 (A3a2 + A2a3) - RT(B3b2 + B2b3) - R-(B3c2 + B2c3) (4-49) 6 = Blblcl R/T2 (4-50) 62 = B2b2c2 R/T2 (4-51) 112 = (Blblc2 + Blb2cl + B2blcl) R/T2 (4-52) 6122 = (Bb2c2 + B2blc2 + B2b2cl) R/T2 (4-5) S11, = (Blblc3 + Blb3cl + B3blcl) R/T2 (4-54) 6133 = (Blb3c3 + B3blc3 + B3b3c1) R/T2 (4-55) 6223 = (B2b2c3 + B2b3c2 + B3b2c2) R/T2 (4-56) 233 (B2b3c3 + B3b2c3 + B3b3c2) R/T2 (4-57) The quantities defined above in (4-44) through (4-57) are functions of the gas constants and temperature onlyo It should be mentioned at this point that it is possible that Equation (4-43) could be slightly reduced in length, but it was left in this form partly for convenience in attempting certain other combinations

-38of coefficients. Also, any additional work of reduction did not seem Justified, inasmuch as the solution was programmed for the IBM 704, and presented no problem. The reduction of Equation (4-5) is now completed by the substitution of Equations (4-8), (4-11), and (4-43), resulting in RT In Yl1v -(Dl) - vl (P-P) + y 3 + Y1(D2)} V V + 2-{(D3) + yl(D4) + y2(D5) - (D21) Y2 [(1-y1)2+(1-y1)3]} 2v1 T2 2 4 + {(D(6) + yl(D7) + yY(D8) + y3(D9) + yl(D10) + (D22) Y (1-y)} = 0 (4-58) where for further convenience, = 21 3 rl 4 5(4 ) a1 2(v~)2 VI) 5 (D2) = 1 - 3 (4-60) (D3) = 2 723 - 72 (4-61) (D4) = 2 (2 713 - 2 725 + 72 - l7) (4-62) (D5) = -4 713 + 2 723 - y2 + 5 71 (4-63) (D6) = 8223 - 62 (4-64) (D7) = 4 8233 - 7 5223 + 6 52 + 6122 (4-65) (D8) = 4 133 - 12 233 + 15 223 - 6113 + 3 6122 + 6112 - 12 62 (4-66) (D9) = -8 6133 + 12 6233 - 13 6223 + 5 5113 + 3 6122 - 2 6112 + 10 62 - 61 (4-67) (D10) = 4 613 - 4 2 + 4 223 - 4 113 - 6122 + 6112 - 3 62 + 3 561 (4-68)

-39all of which are functions of the various P, 7, 8 parameters defined in (4-44),..., (4-57), that is, functions of the gas constants and temperature only. Also, (D21) = R (B1 + B2 - 2B3) (cl + c2 - 2c3) (4-69) (D22) = R (bl - b5) (B1 + B2 - 2B3) (cl + c2 - 2c3) (4-70) which are functions of only the gas constants, and will be identically zero if either the B's or c's are combined linearly. Considering T and P as the independent variables, Equation (4-58) gives one equation in two unknowns, y1 and v. It still remains to reduce Equation (4-1) written for the mixture, so as to obtain a second equation relating v and y1 for specified values of P and T. To achieve this reduction, Equations (4-28), (4-30), (4-32), (4-34), and (4-36) are substituted into (4-2), (4-3), and (4-4) to find the virial coefficients for the mixture. After considerable reduction, the resulting expressions are M= 2 + 2l 3 - 2) + 2 (1 + 2 - 2) (4-71) where = B B2RT - A2 - c2 R/T2 (4-72) Similarly, M = 72 + 2y1 (725 - 2 72) + y(2 71. - 4 523 + 5 72 + 2y (- 713 + 23 - 2 + 71) 2 (yj) (l-y)2 (B+B2B3) (cl+c2-2c5) (4-73)

-40Finally, 8M= 82 + 2y(86223 - 2 2) + y2(-10 8223 + 4 8233 + 8122 + 15 62 - 2 63) + 2y3(7 6223 - 4 6233 4 - 8122 + 8113- 6 82 - 81) + y (- 6 8223 + 4 8233 + o122-28113+ 6 22 + 3 81) + (Y (l-y) (bl-b3) (B1+B2-2B3) (cl+c2-2c3) (4-74) Where 83 = B3b3c3 R/T2 (4-75) Substituting (4-71), (4-73), and (4-74) into (4-1) and rearranging, RT + -P2 + yl(D11) + y2(D12)} v v2. 2 +L 2 y3~ (D15) - (D21)23(1-y1)2} + 1 {72 + yl(D13) + yl(Dl4) + y(D15) - (D21) T2 +- {f2 + yl(D16) + y1(D7) + y(D1l8) + yl(Dl9) here, again 2(D22) for convenience2 = 0 (4-76) Where, again, for convenience, (Dll) = 2 (P3-2) (4-77) (D12) = p + P2 - 23 (4-78) (D13) = 2(723 - 2 72) (4-79) (D14) = 2 713 - 4 723 + 5 72 - (4-80) (D15) = 2(- 713 + 23 - Y2 + 71) (4-81) (D16) = 2(6223 - 2 62) (4-82) (D17) = - 10 6223 + 4 8233 + 122 + 15 2 -2 23 (4-83) (D18) =2(7 6225 - 4 6233 - 6122 + 113 -6 62 681) (4-84) (D19) = - 6 6223 + 4 625533 + 8122 - 2 6113 + 6 62 + 3 81 (4-85)

-41Thus, the set of equations to be solved, i.e., (4-5) and (4-1), is now written as Equations (4-58) and (4-76). For a given P and T, P1, vl, and v1 are known. All the D terms are known in terms of the pure gas coefficients and T, leaving the two equations in the two unknowns yl and v. A simultaneous solution of the two will yield the desired values for yl, the mole fraction of carbon dioxide in the gas phase. For low temperatures, Yl and Pi will be very small and v1 very large. Therefore, terms involving the factors yl, Pl, or l/vl can be assumed negligible (with the exception of the first term of Equation (4-58) containing the product Y1 vl), so that (4-58) reduces to 2iTI V'2'23 - 1/2'2 2(F.223 -82) RT in ^yl - vs P + 0+P 72 - 12 + 2 - ) 0 (4-86) v v v2 5 v3 and (4-76) reduces to RT +2 + Y2 + 82 - p = 0 (4-87) V V It is apparent that this set is a good deal simpler to solve than the general case, since (4-87) is not a function of yl, so a simultaneous solution is unnecessary. The simplified solution as given by the set of Equations (4-86), (4-87) is that used by Dokoupil,(15) and others, where the assumption that yl is small was made early in the derivations, and no general solution was considered. Comparison of this solution with the general one (4-58), (4-76) will give an indication of the error associated with this assumption.

-42The procedure for determining the two different solutions will be to first solve the simplified set for a given pressure and temperature by using the Newton iterative method to find v in Equation (4-87). This value is then substituted into (4-86) to find the mole fraction of carbon dioxide. The Newton method is outlined in Appendix Al. The initial guess for v to begin the iteration is generated by converting the Beattie-Bridgeman Equation (4-1) to the approximate volume explicit form, and using the value determined from that. Equation (4-1) is converted by multiplying by the quantity (v/P), giving RT' 7 8 v = + -- + + (4-88) P Pv Pv2 P;v3 In order to make (4-88) explicit in volume, the v's appearing on the right-hand side of the equation are replaced by the ideal-gas law (RT/P), resulting in RT B 7p _P2 v. + + 4-7p- + -p- (4-89) P RT (RT)2 (RT)3 While (4-89) is not as accurate as the Scatchard volumeexplicit form, in which the v's on the right-hand side of (4-88) are replaced by similar equations, it is considerably better than the ideal gas law, and more than sufficient as an initial trial to ensure convergence in the iteration. 0 S 1 For both solutions, values of Pi, vl, and v1 are required for D S each temperature. P1 and vl were read into the program, having been taken from Reference 24. The values of P1 had been calculated from the Plank-Kuprianoff Equation and checked against the data of Reference 76.

-43At each temperature, v1 was calculated from (4-89) using P1. This was felt to be sufficiently accurate, inasmuch as the vapor pressure was D S always very small. The numerical values for P1, vl, are listed in Table I. TABLE I 0 s VALUES OF P1, vl s { Lit T (~K) P1 (ATM) v ( ) 140 0.00184 0.0270 150 0.00841 0.0272 160 0.0514 0.0274 170 0.0993 0.0276 180 0.274 0.0278 190 0 679 0.0280 The pure-gas Beattie-Bridgeman constants used in the calculations are those of Reference 37, and are listed in Table II. TABLE II BEATTIE -BRIDGEMAN CONSTANTS Units: Atm., Lit/Mole, ~K Constant Comp. 1 (C02) Comp. 2 (N2) A 5.0065 1o3445 B o010476 0,05046 a 0.07132 0,02617 b 0.07235 -oo00691 c 660 x 104 4.2 x 104 _................,

-44Upon completion of the simplified solution, the Newton-Raphson iterative method is used to solve the set of Equations (4-58), (4-76) for the general solution, the principle being outlined in Appendix A2. Inasmuch as this method requires reasonably good initial guesses in order to ensure convergence, the values for y1 and v determined by the simplified solution are used for the initial trial. In the program written for the IBM 704 computer to calculate both sets of results, the general solution was not executed in cases for which the simplified one gave a value of y1 less than 0.001, inasmuch as it was felt that the two solutions would prove to be identical. Examination of the results confirms this opinion. The flow diagram for the computer program is given in Appendix B, and results are compared with the experimental in Section IX. B. Virial Equation of State 1. General For the solution of the theoretical equilibrium Equation (2-44) using the virial equation of state (3-19), the virial equation is rewritten in a pressure-explicit form p RT RT + RT C (4-90) v v2 v3 As before, Equation (2-44) is put into the form of (4-5). Then to evaluate the first integral of this equation, (4-90) is written for pure substance 1 as nRT n2RT B1 n3RT C1 P - + V + V3- (4-91)

-45Therefore, P) RT 2nRTB1 3nRTCi (4-92) n T,V + + The first integral of (4-5) is thus found to be 00 J [( P RT 2RTBi 3RTC (4-93) a 3 T V - ]Dv = 9 + 3)2 As for the Beattie-Bridgeman solution, the isothermal compressibility of solid 1 is neglected, so that the second integral of (4-5) results in Equation (4-11). For the third integral of (4-5), Equation (4-90) is written for the gas mixture as n RT n2RTBM n3RTCM ( 4) p =~ + V2 + (4-94) V V2 V3 where BMI CM are given by Equation (3-33), (3-34), respectively. Differentiating (4-94) with respect to nl, - V d n RT RT RT an,) V --- + an (n2BM) + R a (n3CM) (4-95) 6nl T V n2 so that the third integral is (a RT RT (n BM)] (n CM)1 (4-96) V nl T,,Vn2 V [5n4j From Equation (3-33), 2 2 2 (4-97) n2BM = nB1 + 2nn2B + nB2 (4-97) Therefore a (n2BM) = 2nlB1 + 2n2B12 = 2n [YlB + Y2B12] (4-98) anl

-46Similarly, from (3-34) nCM = nCl + + 3+ nn2C11 2 + nnnC2 (4-99) and.. (n3CM) = 3nlC + 6nln2C112 + 3n2C122 an+ = n2[ylC + 2Y2C + Y2C122] (4-100) Now,substituting (4-98) and (4-100) into (4-96)?cVn2 - ]dV vP R 2RT (y1B1 + Y2B12) IV T 12 V nl T,V,n2 v + RT (Y2C + Y 112 + 122 (4-101) 2# 1 + 2yly2C112 + y2c122) At this point, Equation (4-5) can be solved by substitution of (4-93), (4-11), and (4-101), giving for the equilibrium equation, y1v3~~ ~ 2v Yn — 2B1 3C1_ 1 (P-P1) + - (yB1 + Y2B12) ir Vy 2(v1) RT v +,L (ylCl + 2yly2C112 + 2 C122) = (4-102) 2v2 or, eliminating Y2, 0 S in lV 2B1 -C1 V- (p-p1) + 2 [B12 + Yl(B1-B12)] v v1 2(v)2 RT v + 2v- [C122 + 2y1(C112-C122) + y1(Cl+C122-2C112)] = 0 (4-103) 2v^ Equation (4-103) gives one equation in the two unknowns yl, v. The second relation between the two comes from the equation of state for the mixture (4-94) and the coefficient Equations (3-33) and (3-34). The

-47equation of state becomes; RT RT 22 + (1L 2YlY2B12 + B2) V v2 R T (Y1 + 3y + 3 2112 + 3YY2CL22 + C2g)- P o (4-io4) v3 But, y2 can be eliminated from (4-104), yielding RT + [B2 + 2y(B2-B2) + y +B) ] RT 2 + v [C2+3Y (C122-C2) + 3Y1 (C12+C2-2C122) + y(C1-C2+3C22-3C112)] P = (4-1.05) For a given pressure and temperature, solution of the set (4-103), (4-105) for v and the desired y1 will first necessitate specification of the various pure gas virial coefficients, interaction coefficients and also v1 at the temperature under consideration. As was done with the Beattie-Bridgeman equation of state, a simplification of this general solution can be made at low temperatures, in which case y1 and P1 become very small and v1 very large, so that (4-103) reduces to y1v1 Pv+ 2B 12 5 (4in Y ll +.. + 3C122 = 0 (4-106) v RT v 2v2 and, similarly, (4-105) becomes RT RTB2 RTC2p --- + + - P = 0 (4-107) v v2 v3 Once the four coefficients have been specified, the simplified set (4-106), (4-107) can be solved quite easily, since y1 does not appear in the latter equation.

-482. Numerical Solutions Two methods of predicting virial coefficients will be considered in the solution of both the simplified set of Equations (4-106), (4-107), and also the general set (4-103), (4-105), these methods having been discussed in some detail in Section III C. In these calculations, an equation of state is used as the second relation between the dependent variables [in addition to the integrated form of the equilibrium Equation (2-44)]. In the majority of the references cited in Sections I and III, the mole fraction of the condensed component in the gas phase was always very small, and so only the simplified solution was determined. Inspection of the equation of state for this solution (4-107) shows that, as would be expected, the equation reduces to that for pure component 2. Therefore, in many cases those authors simply eliminated the equation of state and substituted experimental specific volumes of component 2 directly into the equilibrium Equation (4-106), to find the gas phase composition. These solutions then reduce to a test only of the assumptions made in integrating the theoretical equations and the methods for predicting interaction coefficients. For each of the methods to be used here, those of Ewald.(6) and Prausnitz,(3J) the same pure gas virial coefficients will be used. These were calculated using the Lennard-Jones (6-12) potential with force constants taken from experimenntally determined coefficients,(5) these constants being given in Table III.

-49TABLE III FORCE CONSTANTS FOR THE LENNARD-JONES (6-12) POTENTIAL (E/k) ~K 0 lit mole CO2 (1) 189.0.1139 N2 (2) 95.05.06378 These constants were used with the equations of Section III C1 and the tables of Hirschfelder.(53) Values of the virial coefficients so determined are listed in Table IV. TABLE IV PURE GAS VIRIAL COEFFICIENTS USING THE LENNARD-JONES (6-12) POTENTIAL Units: atm, ~K, Lit/mole T B1 C1 B2 C2 140 -0.48605 -0,02631 -0.07944 +0.00224 150 -0.43118 -0.01224 -0.06905 +0.00213 160 -0.58571 -0.00399 -o0o6019 +0.00203 170 -0.34740 +0.000o96 -005256 +0,00193 180 -0.31469 +0.00393 -0~04592 +0.00185 190 -0.28667 +0.00578 -0,04009 +0.00178 The first method of determining the three interaction coefficients required in the general. solution is that of Ewald, this method making use of the semi-empirical combinations (3-35), (3-37), (3-38), (3-39), and also the Lennard-Jones potential tabulations of B*(T*) and C*(T*) from Reference 535 The resulting values are given for the six temperatures in Table V.

-50TABLE V INTERACTION COEFFICIENTS USING THE METHOD OF EWALD Units: atm, ~K, lit/mole T B12 C112 C122 140 -0.20327 +0.00204 +0.00361 150 -0.17989 +0,00384 +0.00364 160 -0.15960 +0,00476 +0.00357 170 -0.14369 +0,00519 +0o00346 180 -0.12877 +0,00535 +0o00333 190 -0.116o6 +0.00535 +0.00320 The earlier method of Ewald(32) is not considered in the solutions, inasmuch as it is known to be less accurate than the later method of Reference 16 given above. The second method of estimating the three interaction coefficients is that of Prausnitz. This procedure requires the acentric factors of Pitzer,(73) and also critical temperatures and volumes of the two pure substances, tabulated in Table VI, according to Reference 77. TABLE VI CRITICAL CONSTANTS Pc (atm) Tc(~K) Vc (lit/mole) CO2 (1) 72.9 304.2 0.0942 N2 (2) 33.5 126.1 0.0901 2 _-

-51The coefficients are then computed using Prausnitz' generalized tables.(3) As mentioned in Section III, the values of C112, C122, were difficult to determine accurately, so the values listed in Table VII have been smoothed from the original calculations. TABLE VII INTERACTION COEFFICIENTS USING THE METHOD OF PRAUSNITZ Units: atm., 0K, lit/mole T B12 C1l2 C122 140 -0.2315 -0o00172 +0o00376 150 -0.2000 -0o00017 +0o00402 160 -0.1749 +0 00120 +0.00410 170 -0.1540 +0.00240 +0.00400 180 -0 1360 +0o 00335 +0.00370 190 -0.1216 +0o00378 +0o00350 For either of the two methods under consideration, the seven coefficients required are now known as functions of temperature, as are 0 s a P1 v'. The only variable remaining to be specified as vl, and this may be easily calculated by using an approximation to the equation of state, since P1 is very small. Both sides of Equation (4-90) are multiplied by (v/P) and the v's remaining on the right-hand. side are approximated by the ideal gas law, leaving RT PC v + B + - (4-108) P RT or, for component 1 at pressure Pi, ~ RT PlC1 vi L+ B1 + (l (4-109) P1?PRT

-52As was done with the Beattie-Bridgeman equation solutions, Equation (4-107) is solved for v by trial and error using the method of Appendix Al, the initial trial for v being computed from (4-108). The value of v satisfying (4-107) is then substituted into (4-106) to find y1. The value of vs in this equation is that listed previously in Table I, and v1 is determined from (4-109). As before, in cases for which the simplified solution gives a value of yl > 0.001, the general set of Equations (4-103), (4-105) is then solved simultaneously for v and yl, using the method of Appendix A2. The solutions v, yl of the simplified equations are used as the initial trials in the iterative procedure for the general solution. The flow diagram for the program written to determine these solutions for either of the mixture methods discussed above is listed in Appendix C. Numerical results for both methods are compared with those of the Beattie-Bridgeman equation and the experimental results in Section IX.

V. EXPERIMENTAL APPARATUS AND INSTRUMENTATION A. General In experimental. work of this nature, three types of systems have been used successfully in the past, namely the static, flow, and circulation methods. These types and the relative advantages and disadvantages of each are discussed in Reference 15 and 78. The flow method of achieving equilibrium was selected for this investigation, inasmuch as it is this type of system in which the data should find application, and further to avoid errors introduced in the collecting of samples, as is encountered in a closed system. In addition, the flow-type system allows for continuous gas-phase analysis, a great factor in assuring the achievement of equilibrium, constancy of conditions, and true compositiono In the experimental system (Figure 1), a high-pressure gas mixture consisting of a small percentage of carbon dioxide in nitrogen is cooled down to the desired temperature, during which process most of the carbon dioxide is frozen out of the mixture. Some of the solid will no doubt deposit on the walls of the tubes, and the remainder is trapped in a filter and retained at the equilibrium point. It is necessary that absolutely no solid particles pass through the trap, inasmuch as the solid would sublime on leaving the cryostat and give an erroneously high carbon dioxide concentration in the gas phase, which is to be analyzed after being warmed to ambient temperature. The heat exchanger serves the dual purpose of cooling the supply gas, so as to avoid large temperature gradients in the cryostat, -535

-54NITROGEN OR HIGH- PRESSURE HIGH - PERCENT GAS MIXTURE CO2 SUPPLY SUPPLY EXHAUST REULATORS^) FLOWMETER FLOW CONTROL THROTTLE BACK PRESSURE HEAT CONTROL \ Y EXCHANGER ANALYZER EXHAUST CONSTANT ANALYZERTEMPERATURE ANALYZER -- ^ / CRYOSTAT FLOWMETER ANALYZER PICK -UP UNIT AMPLIFIER a OUTPUT SOLID CO2 TRAP -- TEMPERATURE - CONTROL Figure 1. Schematic Diagram of Experimental Equipment

-55and bringing the cryostat outlet gas up to approximately room temperature. The gas is then throttled to low pressure, analyzed, and exhausted. Although the flow rate through the system is small, only a small percentage of the total is to be passed through the analyzer, to avoid compression of the gas in the sample cell, which might affect the reading. The needle valve permits accurate regulation of the analyzer flow, to be held constant at the same value at which the analyzer has been calibrated. The analyzer back pressure valve performs a similar function in controlling analyzer pressure. The back pressure valve on the main exhaust line is necessary to enable achievement of the proper analyzer flow at low total flow rates, and also to prevent surging at low flow rates. The second supply manifold is not used during the data runs, but is instead an auxilliary. The high-concentration carbon dioxide supply may be used as desired prior to the actual run, in order to ensure a good deposit of solid in the cryostat. Pure nitrogen is used to purge the system both during the initial cooling and also after runs in which the system becomes plugged with solid. B. Construction 1. Cryostat The cryostat (Figures 2, 3) consists of a solid cylindrical piece of styrofoam 22, 21 inches in diameter and 17 inches high. The inside is hollowed out to a depth of 11 inches and diameter of 9 inches to contain the equilibrium vessel, leaving insulating walls 6.inches thick. The cryostat is also fitted with a styrofoam lid 6 inches thick.

-56INSTRUMENTATION WI RES TUBES TUBE GUARD \-,u.. [ - -FI BERGLASS / HEATEGUARD MAIN. GUARD — BLANKET/ / HEATER SLEEVE VENT LID " ^A/I'ON I LIQUID V^?a^_ o: ^/// NITROGEN ~o // //, —-S.S.LINER STYROFOAM Figure 2. Cryostat Cross-Section

-57It was originally planned to use the styrofoam cryostat directly as the liquid nitrogen container, as was done in References 79, 80, 81, but when this was attempted the interior walls cracked badly, at some points being as much as 2 inches deep. Therefore, in the final design, a 9-liter stainless steel beaker is used as a liner to hold the liquid nitrogen, which is transferred from a 50-liter storage container at such a rate as required to hold the liquid level constant. The styrofoam now serves only as support and insulation, as discussed in Reference 82. The liquid transfer system is seen in Figure 5. Liquid level is indicated by a styrofoam float fitted inside the beaker, with the float pointer protruding through the top of the cryostat lid. It should be mentioned that little further cracking of the walls was observed after insertion of the liner. It is not known whether the cracks would have become more severe had the liner not been used. The equilibrium vessel is suspended in the liquid nitrogen inside a brass can supported from the top by a stainless steel ring (Figures 2, 4), and is insulated from the liquid in order to reduce heat transfer rates. This cooling tendency is counter-balanced by three variable-output heaters. Of these, two surround the sides and bottom as an integral part of the insulation between the vessel and liquid nitrogen, their output being individually regulated so as to hold the block temperature uniform and constant. The third heater is a guard to protect the inlet and outlet tubes of the vessel. The cryostat is located at the end of the long horizontal heat exchanger, the inlet and outlet tubes passing through the side wall of

,..- -L >.) I -'jf"O.a )joop 3 - 1. on'3, p 01 W U 0 —,',.'V I JI:~~~:'~::~:~'-:: ~~: -:::~~~~~~~~~~~i:'.i~~~~~~~~~~~~~~~~~:.........::::::::'ii: ijililii~iiiiiiiiii...:::::::::;.....~~~~~~~~~~~~~~~~::::'':::'~::::::::::::::..I...... ~::: i:::-~: -::-la i:j!:i~i_::::::ll-liil iii....ii:::::-~-lill:'::::: I...:::::::''':::'''':' ili:l i i::i iill......::::.....iil:::::::::::::::::::::::::: i-_:i:-l~i l::::i:......::: i~ili~j~llillj'jlilill-:l:i:::::i~iili::...... ii: iliij iliiiij liiiii~iiiii l:::::::::::::::...::::::::::::: I~iiiiiiiiiiiiiiiiil....i: i -.....:::::::::::::::::::::::::::::::::::: 11::: i1:::: "::~.......:l::: l~::::::i ~s~~~3~sol~~t~~x 3,fc~~ apysrt~~~n g............ ~~'~ a~~ii~~i~~4 ~ 8~~:~~i ~ ~ ~ I-:-:: ~~~i~~'~~:~ ~~~:~~:~:.... % 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~............... -iiiiiiiiiiiiii I~~~~~~~~~......:n- F~~ii:~ii':iiiiiiiX::iiii.............. ~~~~~~::::::::::::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~-.............. ~i~iiiiiiiiii~ii~iiisiiiiiiiiii~;~:~~i:~~..... ii:i......... i~i...........:.:..:~::;:-'-:::..................... r~~~~~~~~~~~~~~~~~~~~~~~~~....... i':~~~~~;~~~~~:::::li::ll::ii............... ~~~~~; ~ ~ ~...........:........................-..........~~~~~~~~~~~: Ijii a~~~~~~~~~~~~~~....~... i:: ~ ~ ~ ~ ~........... -i ~ ~........................~... I.... ~~::::1:p-ns*~s ~ ~ ~ ~ ~ ~..........:-............. r.: ~ ~ ~ ~ ~ ~ ~ ~............ L........ 8:::I~~li~~~i~ L~:~~;..g-.......-................... iii 1:Il~rl:: ~l~i... -......................................~: ~~i~~:: i: 5 j~~~A uj~~t'~: F..............:...j....................................................... --.................................................... F"~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~........................ ~~~~~~~~~~~~~~~.................... LI B ~~~~~~~~~~~~~~~~~~~..........................

-59the cryostat. During operation of the equipment, the cryostat is enclosed in a heavy sheet steel box, shown in Figure 3, as a safety precaution. 2. Equilibrium Vessel The equilibrium vessel, shown in cross-section in Figure 5, is the heart of the experimental apparatus, and must perform two vital functions. The flowing gas must be brought to the equilibrium temperature, at which point the solid carbon dioxide present in the strear must be trapped and retained, while the gas passes through and leaves the cryostat. It is strictly necessary that no point in the system be allowed to cool below the equilibrium point temperature, and also that no solid particles be permitted to pass through the filter (one of the major difficulties encountered in using the flow system). The vessel should consist of a long flow path to allow sufficient time for equilibrium to be attained, yet in as compact a volume as possible. It should also have a large mass and high thermal conductivity, so as to minimize temperature gradients, and still have a rapid thermal response to permit accurate temperature control. The flow passages must be of sufficient strength to withstand the high gas pressure. With these views in mind, the core of the equilibrium vessel is constructed of a cylindrical brass block fitted at the top with a thin stainless steel sleeve, as seen in Figure 6. This central brass block does not extend to the top of the vessel, so as to avoi3 having the relatively warm inlet gas enter the vessel at a point very close to the equilibrium point. The gas flows through a 5/8" O.D. x.049" wall

-6oTC - 10 GUARD HEATER OUTLET TUBE INLET TUBE -~ <'-w- i-'"^ * MAIN HEATER TC -9 9A-'l * /la/- - I t m. SS SLEEVE TC - 3 -- _ // y/ i ----,,,,, —.Al^ /-COPPER SLEEVE BRASS CAN 0 —-- / _! ^ X-^~ 0 - RING SEAL.0/// * ^ltfc I I y ^^^y^ A / BRASS CORE TC -I (GAS)PLU TC- 5 TCT-6 BOTTOM HEATER, *C * X - - ~ ~ - - - ~ -~ ~- ~, INSULATION Figure 5. Equilibrium Vessel Cross-Section

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-62SS tube tightly coiled around, the block and sleeve giving a flow path of approximately twelve feet (see Figure 7)~ Thermal conductivity has been sacrificed at this point in favor of high strengtho The tube is silver soldered to a shoulder at the bottom part of the block, the flow path leading to the center of the block and, up vertically through a tightly packed, filter of spun glass. The gas leaves the equilibrium vessel through a 3/8" SS tube sealed, by a silicone rubber O-Ring seal in combination with a Teflon back-up ringe A heavy copper sleeve is fitted, over the coils and around the bottom of the vessel, so as to minimize any temperature gradientso The completed, unit (Figure 8) is 4-1/2 inches in diameter, 6-3/4 inches high, and weighs approximately 25 poundso Special fittings are silver-soldered to the inlet and outlet tubes to allow for packing gland connections to the heat exchanger tubes, and also the gas temperature thermocouples and the equilibrium pressure tap. All of these glands are Conax high-pressure packing glands having Lava sealants with the gland pipe threads sealed to the fittings by Permacel tapeo Provision is made for measuring the temperature at twelve points in and around the equilibrium vessel, to assure that no point becomes too cold and to permit minimization of temperature gradients through individual heater control. The equilibrium gas temperature is measured by thermocouple Noo 1, which is sealed at the top fitting and hangs down through the vertical outlet tube It is held in place at the equilibrium point by a special plexiglass ring, to avoid the possibility of having the wires touch the tube, block, or even each other

Figure~~~~~~~~~~~~~~~~~.,.......... Figu: l:reu Vessel.; As-semb.led, Figure 9. EcluJ..]..J. brium Vessel ith...eat.ers.............................. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~-::_i:::::::::: i::-...........::::~~............. ~ ~ ~ ~ ~ ~ ~ ~ ~:~::j:::~i...............................~~~~~~~~~~:........... F'igure 9, li~~~~~~~~caui~~~li~~bri.~~~~u ll Vcssc~~~~~ w~~~.th I~~~ca~.......

-64somewhere inside the tube Thus, this thermocouple is directly exposed to approximately the equilibrium temperature for a distance of seven inches, which should assist in giving an accurate readingo 3. Heaters The main heater used in the regulation of temperature is wound inside a cylindrical. sleeve which slides over the equilibrium vessel (Figure 9). The entire unit is 12 inches high and. actually contains two separate heater windingso The lower one covers the bottom 7 inches immediately around the vessel, and the upper 5 inches is merely a guard. to prevent the inlet and outlet tubes and. fittings from possibly being cooled below the equilibrium temperature. The space around the fittings and inside the heater sleeve is packed with fiberglass. This upper assembly is visible in Figure 4. The construction of this main heater sleeve is as shown in Figure 10. Two layers of asbestos paper are cemented together to form the cylindrical base, with an inside diameter of 4-1/2 inches. Then 42 feet of Chromel A Gage 28 resistance wire are coiled around the lower 7 inches of the sleeve, and cemented in place with Sauerisen porcelain cement. Another layer of asbestos paper is cemented around the assembly, on top of which is coiled a second 42 foot length of wire, to be connected in parallel with the first, Several more layers of asbestos are cemented around the outsie to ompete the iinsulation and bring the outside diameter to 5 inches, in order to fit snugly inside the brass can supporting the equilibrium vesselo

-65EQUILIBRIUM VESSEL OUTER WALL ASBESTOS PAPER LAYERS /WIRE OF OUTER COIL WIRE OF INNER COIL |' c 1 \ ^ X.-SUPPORTING CAN WALL X \ ^ <) * 3 ^PORCELAIN CEMENT,.^ 4 I 4 Figure 10. Main Heater Sleeve Construction

-66The guard heater is wrapped on the upper 5 inches of the sleeve in the same manner, except that only a single layer of resistance wire, 12 feet in length, is used on this portion. There is a second portion of this guard not contained in the sleeve, this part being connected in parallel with the first. It is U-shaped and lies directly beneath the long horizontal tubes connecting the equilibrium vessel to the heat exchanger outside the cryostat, This part is comprised of the same sandwich construction as the other, and also contains 12 feet of heater wire. In order to prevent a cold spot from developing on the bottom of the vessel, it was found necessary to cover the bottom with a small heater in the shape of a 4-1/2 inch diameter flat disc. This is of a similar construction, consisting of 12 feet of wire coiled back and forth between layers of asbestos. There is also a styrofoam disc, 1-1/4 inches thick, to insulate the vessel bottom from the liquid. The three heaters are controlled individually by 110-volt variacs, rated at 2 amperes maximum current. 4. Heat Exchanger The heat exchanger is of the concentric-tube counterflow type construction, and is comprised of 12 sets of tubes 68 inches long, giving a total length of 68 feet. All material in the exchanger is brass, and walls are extra-heavy to withstand the necessary pressures. All connections are silver-soldered. The principal factor kept in mind in the design of the heat exchanger was that there were to be no soldered joints between the two

-67passages. All joints are instead between a passage and the atmosphere as shown in Figure 11. This design eliminates the possibility of the equilibrium vessel outlet gas becoming contaminated by the inlet gas because of a tiny internal leak between the two passages. Since pressures are always at least five atmospheres, any pinhole developing would then merely cause a small leak of the gas outward to the atmosphere. At one end, connections between sets of tubes are relatively simple, as the double-length inside tube is U-shaped and fits into two of the outer tubes, as shown in Figure lla, A straight tube serves to connect the annular outer-tube passages at this end. At the opposite end, connections are considerably more difficult, as can be seen in Figure llb, Since the incoming gas passes through the inside tube, those fittings are rather elongated at one end, in an attempt to induce some of the solid being formed to deposit there instead of blocking the tube passages. The heat exchanger is heavily insulated with fiberglass blanket and enclosed inside a 10 inch steel pipe closed at both ends, for protection in case of a failure. C. Accessory Equipment 1. Gas Supply Gas mixture cylinders at 1600 psig were purchased from the Matheson Company, the concentration of carbon dioxide depending on the range of temperature and pressure to be covered. For most of the temperature range 140-160~K, the supply mixture contains 0.5% carbon dioxide, whereas for most of the range 170-190~K, 2.0% or 4.0% cylinders are used

-68Figure. Heat Exchanger Fittings! \ \ \

-69as the running supply. Normally ten such cylinders are manifolded together to prevent the cylinder pressure from dropping excessively during a highpressure run. A single cylinder containing from 6-10% carbon dioxide is sometimes used prior to the start of a data run to ensure a good deposit of solid. Caution must be exercised in this matter, however, to prevent the system from becoming plugged with solid. Nitrogen for purging the system was purchased from the General Stores of The University of Michigan. 2. Pressure Regulators and Valves A Matheson No. 3-580 high-pressure regulator is installed on the supply manifold to control the equilibrium gas pressure. A No. 1-320 low-pressure regulator is used on the nitrogen manifold for system purging, and also in those cases where the initial high-concentration supply is to be used. Two Hoke needle valves in series are used in the exhaust line to control the flow rate and throttle the gas to just above atmospheric pressure. A Matheson needle valve controls the analyzer flow rate, and two Fisher type 289U relief valves, 1/2 to 1-1/2 psig range, are used to regulate the back pressure on the two branches of the exhaust line. D. Instrumentation 1. Temperature Measurement All temperatures are measured relative to the ice point using 30-gage copper-constantan thermocouples. References (82, 83, 84) and others were consulted in the design and installation. EMF's from the

-70ten important equilibrium vessel thermocouples designated in Figure 5 are measured by a Leeds and Northrup type K53 potentiometer, in conjunction with a type 2285 reflecting galvanometer, which is hung from the ceiling by a Julius suspension as shown in Figures 12 and 13. EMF's from the remaining six thermocouples are read on a Leeds and Northrup type 8662 potentiometer. These six temperatures are as follows: equilibrium vessel outlet fitting; cryostat gas outlet tube; equilibrium pressure gage (gas); exhaust line downstream of flow control valves; main stream flowmeter; and gas analyzer exit. The thermocouples were calibrated with a specially built vaporpressure thermometer patterned after that of Scott. (8285) The pure vapor used was nitrogen or carbon dioxide, depending upon the temperature range. In addition, the thermocouples were also calibrated at the freezing point of mercury. Deviations from standard EMF were then plotted against EMF, and tables of observed EMF versus temperature prepared. 2. Pressure Measurement.Equilibrium pressures greater than 50 atmospheres are read on a calibrated 6 inch Marsh bourdon gage reading to 3000 psig. Those of 50 atmospheres and less are measured on a calibrated 6 inch Ashcroft gage reading to 800 psig. Other pressures measured are read on stock bourdon gages of various ranges. The other pressures read are downstream side of flow control throttle, main stream flowmeter, gas analyzer exhaust, liquid nitrogen storage container, and of course, upstream and downstream sides of the manifold regulator.

................................................................................................................................................ Mi.........................................................................................................-..............................................................................................................................................................................................................................................................................................................-.......................................... Figure.12. G,,-ilvanorfie-ter fulius Skis pe- its J. on............................................................................................................................................................................................................................................................................... --........................ - -..................................................................................................................................................................................................................................................................................................................................................... Compatibility:a via t ion............................................................................................................................................................................................................................................................................................................................................................

-723. Gas Analysis The equilibrium gas mixture is analyzed in a continuous-flow Beckman L/B Model 15A non-dispersive infrared analyzer, sensitized to detect for carbon dioxide. A schematic diagram of the pick-up section is given in Figure 14. Three sets of cells, of 3/16", 3/4", and 5-1/4" length are used, depending on the percentage of carbon dioxide expected. Cell windows are sapphire, and the detector is charged with carbon dioxide to 50 mm. pressure. The output signal from the detector is fed to the amplifier unit, and eventually read on a calibrated meter. The gas analyzer is connected in the exhaust line with a needle valve in order to be able to control the flow rate through the analyzer pick-up section. The exhaust line from the analyzer contains a flowmeter and a back pressure regulator to control the analyzer gas pressure. The unit as installed is shown in Figure 15. 4. Flowmeters A Matheson No. 206 tapered-tube flowmeter with a glass float, 0 - 1.2 Standard CFM, is used to measure the flow in the main exhaust line. Either a No. 201A, 0.1 - 2.0 Standard CFH, or a No. 201, 0.01 - 0.08 SCFH, tapered-tube flowmeter measures the analyzer flow rates. The overall set-up of the experimental apparatus and instrumentation is shown in Figure 16.

-73TWIN INFRARED CHOPPER MOTOR SOURCES CHOPPER SAPPHIRE WINDOWS ZERO ADJUST O O O -0 GAS SAMPLE INLET 0 DESICCANT — / | SAMPLE CELL I~g0 = REFERENCE O CELL - —' 0 0 0. E l0 O 0 00- GAS SAMPLE II q - I ~e-3 o I OUTLET DETECTOR 00 0 o 0 1 (CO2 AT 50mm) O0 0 0 IP gO0 0 00 0 O0 AC SIGNAL 2 | \ OUTPUT TO AMPLIFIER Figure 14. Schematic Diagram of Gas Analyzer

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................................................... XX:Israeli-..:...............................................................:if -Fijian;.................................................................................................................................................................................................................................................................... ---............................................................................................................................................................................................... M-0;......................... no................................................................................................ Ki Applicability.;0........... A..................... X.................... characteristic..........

VI. EXPERIMENTAL ERRORS A. Temperature Measurement 1. Potenti ometer -Galvanometer Leeds and Northrup Company specify a maximum error for the K3 potentiometer of + (0.015% + 0.5kv) in the range used. In tests run at the liquid nitrogen point over a period of several weeks, the maximum deviation from the mean in ten separate readings was found to be 0.5J.v, well within the limits given above. The potentiometer case is grounded through a 0.01|f capacitor, in order to guard against electrostatic disturbances, The potentiometer is balanced against an Eppley unsaturated type cadmium standard cell, certified to be accurate to 0.01%. Regular checks showed the standardization of the potentiometer to remain quite constant over a period of time. The sensitivity of a change in standardization on an EMF measurement was also made, a galvanometer scale deflection of + 8 mm on the standard cell circuit being required to give an error of l4v in the unknown EMF measurement, for the temperature range of interest, The standard cell balance was always held within + 2mm deflection during operation of the equipment. The reverse key on the potentiometer was also used periodically, to assure that no errors were introduced in the thermocouple-potentiometer circuitry. The Leeds and Northrup No. 2285 reflecting-type galvanometer used in conjunction with the potentiometer has a sensitivity of 0.0005.t amp/mm at a distance of 1 meter. During the operation the galvanometer -77

-78was checked regularly for zero-drift. The galvanometer hangs from the ceiling by three cables, each connected to the Julius suspension by a short spring to damp out any vibrations in the vertical direction. Rotation about the vertical axis is completely eliminated by the three dashpots filled with a heavy mineral oil, and any tendency for the light spot to move is further reduced by the design of the suspension system, which is counterbalanced to put the center of gravity of the suspended mass at the galvanometer mirror. 2. Thermocouples All thermocouples were constructed of 30 gage copper-constantan wire, the ten used to indicate the block and tube temperatures being made from the same spools. The two thermocouples used in the Conax glands to indicate gas temperature were from different spools. The thermocouples were calibrated using the vapor-pressure thermometer discussed in V Dl, along with carbon dioxide or nitrogen. The manometer used to read vapor pressure was considered accurate to 0.1 inch of water, and vapor pressures were taken from References 24, 76, 82, 86. Calibrations were also made at the freezing point of mercury. Curves of deviation versus EMF were slightly parabolic in shape and very smooth, indicating that the thermocouples are reliable to better than 0.1~C in the range covered. According to Wiebe,(87) they should be accurate to approximately + 0.05~C. Scott(85) was somewhat more conservative, but as he pointed out, his experience was based on thermocouples of widely differing origins. The only differences of

-79more than lP.v (~.04~) observed in calibration were between the two sets of thermocouples, where deviations were quite different. The curve for the two gas thermocouples, (Nos. 1,2) had smaller deviations from the standard than did the block and tube temperature thermocouples (Nos. 3,..., 10) at any given temperature, this difference reaching 14(iv at the boiling point of nitrogen. B. Temperature Control Manual control of the equilibrium vessel temperature proved to be very precise, well within the + 0,1~C limits desired. The block temperature uniformity was always more than adequate, the side temperatures being held either at the equilibrium value or 0.1~C high to preclude any possibility of a cold point. The center of the block bottom was held at the desired value by manipulation of the small heater covering the bottom. This often resulted in a temperature 0.1 to 0.2~ high at the bottom corner, due to the junction of the side and bottom heaters there. This could probably have been partially eliminated by respacing the heater coils in the bottom heater, but this was not deemed necessary. The top part of the block normally ran 0.1 to 0.20 high also, due to the warm gas entering there, but this was expected. This temperature also proved to be a function of the liquid nitrogen level in the cryostat, the top temperature increasing slightly if the level were permitted to fall. It was found that the block temperature distribution is a function of both the gas velocity and the supply gas carbon dioxide concentration. The reason for the first is obvious; the other is due to

-80the fact that freezing more carbon dioxide out of the mixture tends to warm the central block slightly. With respect to this latter point, perhaps the more conventional equilibrium vessel comprised of a tube coil in a stirred liquid would be superior to the type used in this investigation. In other respects, the type used here seems to be better. The top heater was installed to assure that the inlet and outlet gas temperature was always greater than that at the equilibrium point. The amount of difference is unimportant, except that if it becomes too large, the temperature distribution in the block will be affected-. C. Pressure For equilibrium pressures of 50 atmospheres and less, the accuracy of the pressure reading and control is of the order of + 0.1 atmosphere, as read on the calibrated Ashcroft gage. The manifold pressure regulator was certainly not this steady, but any differences were apparently damped out in the 55 feet of flow path between the regulator and equilibrium vessel. In a few runs at low pressures the regulator apparently hunted to the extent that equilibrium pressure varied as much as + 5 psig. In these cases, the run was not started until the pressure settled down to a stable value. For pressures greater than fifty atmospheres, the accuracy of the Marsh gage is considered to be approximately + 0.5 atmospheres. At five of the six isotherms studied, the fifty atmosphere point was run twice, once with each gage. In every case, the resulting equilibrium compositions were identical, or at least within the limit of experimental accuracy.

-81The remaining pressures measured were read on bourdon gages accurate to approximately + 1% full scale, which is satisfactory. D. Gas Analysis The Beckman L/B Model 15A infrared analyzer is specified to have a maximum error of + 1% full scale, with a sensitivity of one-half that value. The particular instrument used in this study was tested, with results as follows: Zero drift in 8 hrs. + 0.1% full scale Span drift in 24 hrs. + 0.3% full scale Max. noise level + 0.05% full scale Calibrating cylinders were purchased from the Matheson Company. The zero-point was set with pre-purified nitrogen, and tests showed that small differences in the zero setting had no effect on the upscale readings. The calibrations were made at the temperature, pressure, and flow rate to be encountered in the experimental runs. On several occasions, extensive tests and recalibrations of the unit were made continuously for as long as ten hours, with the resulting output meter readings being in every case constant within + 0.5% of full scale. It was found early in the testing that the analyzer could not be run with the cover plates removed from the pick-up section (as has been done in some installations), inasmuch as the resulting air movement between the sources and cells tended to cause a drift of as muW as 1% in some instances. The readings were also found to be extremely sensitive

-82to the gain control setting, so the dial was locked prior to calibration and subsequent operation of the unito Tests were also performed to determine the effects of varying analyzer pressure and flow rate. During actual data runs, the pressure was always held at 0o6 psig, but no difference in analyzer reading could be detected by varying the pressure + 0.05 psig from this valueo Similarly, the analyzer flow rate ranged from 0.5 to 10oO SCFH, depending on cell length used, but no difference in reading was observed due to varying the flow from 0.2 to 1,5 SCFH. These tests were carried out with all three lengths of cells. Some tests were also run at much lower flow rates, down to 0.02 SCFH, but the readings were found to be more unstable and sensitive to changes in this regiono Prior to starting the equipment, the analyzer was calibrated using four or more pointso As mentioned before, the gain control dial was locked and never touched once the calibration had been startedo Recalibrations were made regularly at the conclusion of a set of runs, and also periodically during a lengthy seto No recalibration ever showed a change of as much as + 0.5% full scale. The consistency of these recalibrations also shows that the tendency for the analyzer exit temperature to increase by approximately a degree during a series of runs has no effect on the calibration curve or accuracy of the readings. Twenty-five percent of the points were checked by rerunning at different flow rates. In every case, the analysis was either identical or at least within the + 0.5% full scale considered to be the accuracy

-83of this instrument. Among these tests were check runs made at certain points using different cell lengths, in order to assure that two different cells would give the same analysis of the gas phase. These checks were also run at different flow rates in order to test the attainment of equilibrium and effectiveness of the filter. It was at one of these different-cell checks that the largest deviation of any of the checks was found, that of 160~K, 70 atm. pressure. The 5-1/4 inch cells showed a carbon dioxide mole fraction of 0.00334 + 0.00002, while 3/4 inch cells gave a reading of 0.00330 + 0.00003. It is noted that the limits of these readings do overlap, and the test was therefore considered successful. E. Throttling Valves There was some question as to whether severe throttling of the gas mixture across the flow control valves could possibly cause a change in composition across those valves. Therefore, using a mixture of known composition, a. thoroughtest was run, completely covering the entire range of flow rates and equilibrium pressures. In no case could a difference in analyzer reading of even + 0.5% be detected. F. Flow Rates The tests were normally run at such a flow rate as to give a gas velocity in the equilibrium vessel of approximately 0.1 ft/sec (3 cm sec As long as the velocity was of this order of magnitude, the results were not affected by changes in flow, so a precise measurement of flow was not required. The tapered tube flowmeter used can certainly be relied on to + 10% of the reading, which is sufficient.

-84The analyzer flow, as discussed in VI-D, is not critical in the range used, so the same accuracy as with the large flowmeter is again acceptable. G. Attainment of Equilibrium In work of this type, the important question of whether or not a true equilibrium is achieved in a run can only be answered by inspection of the behavior of the system during the run, and by resorting to indirect tests. The most important factor in determining the attainment of equilibrium was having the continuous reading gas analyzer directly in the system. At the beginning of operation, the analyzer would read some value greater than that to be expected in the experimental point, because the initial high-concentration mixture was always run at a very low pressure. When the running mixture was turned on (at the higher operating pressure), the analyzer output reading began to drop within a very few minutes and would steadily fall to the equilibrium reading, at which point it would remain for however long the test was run. The same points were run without the initial mixture, in which case the analyzer reading steadily rose from zero to above the equilibrium reading, and then fell slowly to the same valueo It is felt that the ability to continually watch the analysis results during the test run was of immeasurable benefit in obtaining good experimental data. The flow rates used were such as to give a particle an average time of over two minutes spent in the equilibrium vessel, which should be much more than a minimum time to come to equilibrium, especially in view of the relatively small change of temperature in the equilibrium vessel.

-85The rerunning of 25% of the experimental points at different flow rates, ranging in equilibrium vessel gas velocities of from 0.04 to 0.2 ft/sec, is again a strong argument for the attainment of equilibrium in the vessel, in view of the consistency of the results. H. Solid Carbon Dioxide Filter The tests run at different flow rates, discussed in the preceding sections, give a strong indication that little or no solid passed the filter. If this were not the case, it would be certain that higher flow rates would result in greater percentages of carbon dioxide in the exhaust gas. Of the 16 points checked at different flows, the lower velocity gave identically the same reading 6 times, a lower reading 6 times, and a higher reading 4 times. Of the 10 that were not identical, the difference was always within the limits considered as the gas analyzer accuracy. It should also be mentioned that for many of these, the check point was. run at a later time on a different calibration curve, so that a reading identical to the original was often impossible due to limitations in the preciseness to which the analyzer scale could be read. I. Gas Purity The gas mixtures were made up of 99.7% minimum purity extradry nitrogen (impurities principally argon) and 99.8% minimum carbon dioxide (impurities principally nitrogen and carbon monoxide). Before running, the system was purged for at least 30 minutes with nitrogen.

-86Calibrating gas mixtures for the gas analyzer were mixed of gases of the same purity as given above. The analyzer zero setting was always made with nitrogen of 99.996% minimum purity, with most of the remaining impurities substances that would not affect this reading. Moreover, as mentioned in VI-D, a small error in the zero setting will only affect the lower portion of the calibration curve, which was not encountered in the experimental work.

VII. TEST PROCEDURES A. Temperature-Pressure Range Six isotherms were studied, ranging from 140 to 190~K in ten degree intervals. The minimum pressure was five atmospheres absolute, and was incremented from ten to 100 atmospheres in ten atmosphere intervals. The five atmosphere point at 190~K was not taken because the gas-phase carbon dioxide concentration exceeded the range of the analyzer. This was not considered of prime importance, as the experimental and predicted results should conform closely at this low pressure. At 140~K, the 100 atmosphere point was not studied, inasmuch as densities greater than 1.5 times the critical were not of interest in this investigation. B. Run Procedure The cryostat required a cooling-down time of from three to four hours, due to the fact that the equilibrium vessel is insulated from the liquid nitrogen to reduce heat transfer rates. While the cryostat was being cooled, the gas analyzer was carefully calibrated at four or more points and rechecked. During the last half-hour of cooling, nitrogen was passed through the system in order to purge the lines and also cool down the heat exchanger. Once the desired block temperature was reached, the main heater was turned on and set to hold that value. This setting varied from approximately 10 watts at 140~ to 120 watts at 190~K. The small bottom -87

-88heater was then adjusted to prevent a cold spot on the bottom of the block, the power drawn by this heater varying up to two watts. When the temperatures were stable and quite uniform, the gas mixture was turned on. In some runs, the initial high-carbon dioxide concentration mixture was used first to give a good deposit of solid, but at the higher pressure points, where the mass flow rates are larger, this initial mixture was not used, in order that the system could be run for longer periods of time without plugging. The test results were found to be independent of whether or not the initial mixture was used. The running mixture was turned on, and the desired pressure and flow rates set. The heater settings often had to be readjusted at this time, because of the carbon dioxide freezing out of the mixture. The top guard heater was adjusted periodically to ensure that the inlet and outlet tubes and fittings did not approach the equilibrium temperature, the maximum output of this heater running approximately ten watts. After the temperature, pressure and flow had been stabilized, the analyzer valve was opened and the analyzer flow and pressure were set to the values at which the unit had been calibrated. Depending on the flow rate through the system, the analyzer lagged from four to ten minutes behind the equilibrium point. The output reading was observed over a period of time, while the temperature, pressure, and flow were held constant. When the equilibrium condition had been reached, the system was run for at least thirty minutes to ensure that no further change would take place, after which the readings were taken. Inasmuch as the gas phase was not analyzed by collecting a sample over an interval of time as was done by Webster,(l3'14) Dokoupil;15)

-89etc., or by periodic sampling, as by Ewald(16) and others, it was not deemed necessary to run each point a number of times and take the average of the readings. Since the gas phase was analyzed continuously during a run, it was possible to observe the manner in which the carbon dioxide percentage changed as the system approached equilibrium, at which point it remained constant for as long as desired. This was felt to be a very reliable test of the concentration, as well as of the equilibrium itself. In order to check this equilibrium, mid particularly the solid carbon dioxide filter, 25% of the data points were repeated at different flow rates. Three different sets of sample and reference cells were used in the analyzer, depending upon the concentration expected. The 5-1/4" cells were used for carbon dioxide mole fractions less than 0.004, the 3/4" cells from 0.004 to 0.020, and the 3/16" cells from 0.02 to 0.10. inasmuch as the gas analyzer accuracy is considered to be a fixed percentage of full-scale deflection, it would be desirable to have the experimental readings fall on the upper portion of the calibration curve, so as to minimize the percent error in the reading itself. Consequently, the runs were grouped according to the carbon dioxide mole fraction expected, and the analyzer calibrated with the maximum in carbon dioxide percentage as small as possible. The portion over 90% fullscale deflection was not used, however, because the output becomes quite non-linear in this region. This procedure worked quite well, especially since the observed concentrations were at least fairly close to those expected, and were generally smaller. Some points that fell particularly low on the calibration curve were rerun later using different analyzer calibration.

-90C. Criteria for an Acceptable Run The following conditions must all be met, in order for a run to be considered successful: 1) The temperature, pressure, flow rate, and analyzer readings must all be constant for thirty minutes. 2) There should be no temperature in the block more than 0.5~C above the equilibrium temperature TC-1, and none less than TC-1. 3) Inlet and outlet tube temperatures TC-9 and TC-10 must be at least 0.5~ greater than temperature TC-1. 4) The outlet fitting and upper tube temperatures TC-11 and TC-12 must be at least 5~ greater than TC-1. 5) The analyzer output should read at least 0.001 less carbon dioxide than the supply mixture. Once the system was checked out and the difficulties eliminated. the only runs discarded were those in which the system plugged with solid before condition 1 above could be fulfilled. This occurred possibly ten or twelve times during the course of the experimental work.

VIII. EXPERIMENTAL RESULTS A. General Considerable difficulty was experienced during the initial running of the equipment. The gas-phase carbon dioxide mole fractions were found to be even lower than that predicted for ideal behavior, which seemed very unreasonable, and were strongly dependent upon flow rate through the system. At this time, the apparatus was being run with only the side and bottom heaters, and it was discovered that the equilibrium vessel upper tubes and fittings were being cooled below the equilibrium temperature by the cold cryostat vent gas. This resulted, in effect, in an equilibrium temperature lower than what was being held and measured in the vessel itself, with a correspondingly smaller percentage of carbon dioxide remaining in the gas phase for a given pressure. These troubles necessitated the addition of sealed pipes on the cryostat vent, filler, and float pointer, in order to carry the cold nitrogen vapor completely out of the cryostat, these pipes being visible in Figure 4. In addition, the fittings were heavily packed with fiberglas, and a new heater sleeve designed to completely surround the fittings and tubes, as shown in Figures 4 and 9. A second portion of the heater, part of which can also be seen in Figure 4, was placed horizontally beneath the tubes leading out of the cryostat, to guard those from possibly being overcooled. In order to regulate this new guard heater properly, the four thermocouples designated TC-9 to TC-12 were -91

-92added, and read regularly. These precautions completely eliminated the cooling difficulties, and no further troubles were experienced. The ten equilibrium vessel temperatures designated TC-1 to TC-10 in the data (locations shown on Figure 5) are considered reliable to better than 0.10C, and the upper cryostat temperatures TC-11 and TC-12 to 0.25~. Temperatures TC-13 to TC-16, at the equilibrium pressure gage (gas), line downstream of the throttle, and the two flowmeters, were checks only, and are listed to 0.50. Equilibrium pressures of 60 atmospheres and above are accurate to roughly 0.5 atmospheres, and those less than 60 atmospheres to approximately 0.1 atmosphere. Accuracy of the gas-phase analysis is based upon the extensive testing of the analyzer carried on through the entire course of the experimental work, and is considered to be + 0.5% of full-scale deflection. The operating procedure of attempting to keep the experimental results reasonably high on the calibration curve results in a variation in accuracy from point to point depending on the calibration curve used in the run. These error limits are listed with the experimental data in Appendix D, and are seen to be roughly + 1% of the reading in most cases although for some of the extremely small concentrations at 1400K, they reach + 5%. In the data of Appendix D, the run designations are as follows: the letters A, B, C refer to the three sets of analyzer cells used, and the following numbers denote the calibration curve under which the point was run. For example, all points listed under run Al were made with the 5-1/4" cells and calibration curve number 1 for those cells. Each

-93of these calibration curves was checked periodically during the course of taking a set of points, and was found to be reproducible in every case. In the presentation of the experimental data, after some deliberation it was decided to use the metric system for any measurements made in the cryostat, because of its more frequent use in work involving equations of state, cryogenics, and phase equilibria. The remaining readings, those checks at approximately room temperature and pressure, are given in the English system, largely because of its familiarity in this country and also because the instruments used were calibrated in those units. It is sincerely hoped that no one will be confused or offended by this mixing of systems of units. B. Results and Discussion The Series A points, designating 5-1/4" analyzer cells, included points in which a gas-phase carbon dioxide mole fraction of less than 0.004 was expected. Most of the pressures at temperatures of 140, 150, and 160~K were covered in this series. At 140~K, points were run from 5 through 90 atmospheres. The 100 atmosphere point was not included because the gas density would be greater than 1.5 times the critical, which was not of particular interest in this study. Since it should be possible to predict the carbon dioxide concentrations quite accurately at low pressures, it would be expected that the mole fraction of carbon dioxide would fall quite rapidly from unity at the vapor pressure, 0.00184 atmospheres, as the total pressure is increased. As seen in Figure 17, there is a minimum

-94at about 25 atmospheres, after which the concentration begins to increase with pressure. This minimal pressure, as it is often termed, is of particular interest in the purification of gases by selective freezing. If the behavior were ideal, and all three assumptions discussed in Section II-F were valid, then the mole fraction Y1, would be given by the ratio P1/P and would show no such minimum point, but would continue to decrease with increasing pressure. The assumption of ideal behavior will hold reasonably well for low pressures, but is seen to be completely out of the question here. At about 60 atmospheres, there is an inflection in the isotherm, above which pressure the slope of the curve begins to decrease, this effect being much more apparent on a large scale ploto This behavior is consistent with that observed by Webster(l4) and others with this type of system. It is only reasonable to expect a limit to the increase of solubility of the condensed phase in the gasO At this temperature, check points were run at 70 and 1.0 atmospheres, with excellent agreement. The experimental accuracy of the carbon dioxide mole fraction is given as + 0.000001 except for the 10 atmosphere check, which was run on a calibration curve at lower gain, with a corresponding estimated accuracy of + 0.00002. At 1500K, similar behavior of the isotherm is observed. At the lower end, the carbon dioxide mole fraction would go to unity at 0.00841 atmospheres, the vapor pressure of pure carbon dioxide. The minimal pressure at this temperature occurs at 30 atmospheres having increased from 25 atmospheres at 140~o Again, an inflection is observed at higher pressure, this time around 75 atmospheres, as compared with

-95-.007 - -------.006.005.004 z 0 U,.003 w -i 160 1 K.002.001 0o 140 I 20 40 60 80 100 PRESSURE, ATM. Figure 17. Mole Fraction Yl vs. Pressure at 140~, 150~, 160~K

-9660 atmospheres at the lower temperature. For this isotherm, check points were run at 80, 50, and 10 atmospheres, the 50 atmosphere point being run with a different pressure gage and. calibration curve, as well as at a different flow rate. As before, there were no discrepancies between the check points and the originals. The gas-phase analysis was accurate to + 0.00002 at 60 atmospheres and above, and + 0.00001 at the lower pressures. The final isotherm of Series A is 160~K, also plotted in Figure 17, although the points at 5, 80-100 atmospheres were actually run in Series B because of higher carbon dioxide concentration. At this temperature, the mole fraction can be extrapolated to unity at 0.0314 atmospheres, and the minimal pressure has now increased to 35 atmospheres. At higher pressure, no inflection is observed for 1600K, although it appears that such a point may occur very close to 100 atmospheres, the maximum pressure studied. Four check runs were made at this temperature at 90, 70, 50 and 10 atmospheres. The 70 atmosphere point was run in both Series A and B, i.e., with the 3/4" as well as the 5-1/4" cells, as a special check of the correlation between results obtained with different cell lengths. The agreement is not exact, but the average, lying within the limits of error of both points, gives a very smooth curve. The other three check runs showed closer agreement. Accuracy of the mole fraction determination is + 0,00003 from 80-100 atmospheres, + 0.00002 from 10-70 atmospheres, and + 0.00005 at 5 atmospheres, due to the different ranges in calibration. Series B, using the 3/4" analyzer cells, includes all of 170 and 1800K, with the exception of the lowest pressures, as well as well as the four points at 1600~K discussed above.

-97At 170 K, Figure 18, the mole fraction of carbon dioxide would tend toward unity at 0.0993 atmospheres. The minimum point at this temperature is seen to occur at about 43 atmospheres, and although no inflection is observed at higher pressure, the isotherm becomes nearly linear around 100 atmospheres. The points at 50 and 10 atmospheres were rerun, and agreed very well. The accuracy is considered to be + 0.00007 at 90 and 100 atmospheres, + 0.00005 from 20-80 atmospheres, + 0.0001 at 10 atmospheres, and + 0.0002 at 5 atmospheres. The 5 atmosphere point was actually run in Series C, due to the relatively higher mole fraction of carbon dioxide. Experimental results for 180~K are also shown in Figure 18. At the low pressure end, the carbon dioxide mole fraction could be extrapolated to unity at 0.274 atmospheres. The pressure at which the minimum mole fraction occurs has now increased to just over 50 atmospheres, and the slope of the isotherm is still increasing at the maximum pressure covered, 100 atmospheres. The check runs, at 50 and 20 atmospheres, showed excellent correlation with the original points, the latter one being checked in Series C with the 5/16" cells, which makes the agreement particularly satisfactoryo The estimated gas-phase accuracy is + 0.0001 at pressures from 20-100 atmospheres, + 0.0002 at 10 atmospheres and + 0.0004 at 5 atmospheres, these latter two points actually being run in Series C. Series C, run with the set of 3/16" analyzer cells, included all the 190~K points, and also the three at 170 and 180~ mentioned previously. The vapor pressure at 190~ is 0.679 atmospheres, and the mole

-98-.08 -........07.06.05 i \zi 0 I-.04 4c 0 C 190 OK 5.03.02 180.01 O -~I 20 40 60 80 100 PRESSURE, ATM. Figure 18. Mole Fraction yl vs. Pressure at 170~, 180~, 190~K

-99fraction falls to a minimum at 60 atmospheres. As at 1800, no tendency for an inflection to occur is observed at up to 100 atmospheres pressure. The 5 atmosphere point was not run at this temperature, due to the limiations in the range of the analyzer cells. The runs checked at 190~ were at 90, 50, and 30 atmospheres. There was some difference at 90 atmospheres, but the original and check points were each within the other's specified limits of error. The accuracy of the points at this temperature is considered to be + 0.0002, except for 10 and 20 atmospheres, which is + 0.0004. The six isotherms have been plotted on two separate graphs because of the wide variance in mole fraction over the temperature range covered. The check points have also been included in these plots, but in all but a few cases appear identical to the original, especially on the scale used in Figures 17 and 18. From larger scale graphs, the best values have been determined, and these are tabulated in Table VIII. The only alterations made to the original data were to average the original and check point values at 1600K, 70 atmospheres, giving 0.00332 (from 0.00334 + 0.00002 and 0.005530 + 0.00003), and also to average those at 150~K, 80 atmospheres, to give 0.00222 (from 0.00223 + 0.00002 and 0.00221 + 0.00002). The only other point and check which could have been averaged without overspecification in the number of significant figures was that at 190~K, 90 atmospheres, and this was not done because the original point gave a smoother curve.

-100TABLE VIII MOLE FRACTION OF CARBON DIOXIDE AS A FUNCTION OF TEMPERATURE AND PRESSURE Pressure, Temperature ~K atmo 1.40 1 50 160 170 180 1.90 5 o00042 00192 o00699.0220 0o595 10 00024 o 00107 00395.0121 o.0325.0778 20 o00018 o00069.00251.00735 o0190 o0455 30 00019 o000oo64.00217 oo6o8.0151.03545 40 o 00025 00071.00217 00572 0137 o.0311 50 o 00043 0oo.00243.00 o 580.01,33 o0293 6o ooo0078 o00128 o00280 o00o68.0135 0290 70.00112 2. 2 332*.00685.0141 o 0293 80 00139 o00222**.003293 o00760 o01.50 0303 90.00158.00270 oo0046 o00860 o0o61 0320 10o o 00304 oo00536 oo00968 o01.76 o034o *Average of o00334 +.00002 and,.00330 +.00003 **Average of.00223 + o00002 and..00221. + o00002 As a point of particular interest, the 1.6 points that were checked, are listed below in Table IX, along with their respective limits of error based on the gas analyzer tests. Values of the gas velocity in the equilibrium vessel for each of these runs are also given. these being important in evaluating the effectiveness of the solid trapo Examination of Table IX with regard, to accuracy of gas-phase analysis shows that for fifteen of the sixteen points, both the original and check fall inside each. other's limits of errors The only exception is the point at 1600K, 70 atmospheres, for which a mole fraction 0,00332 seems to be the best value, this lying within the limits of both the experimental readingso With respect to effectiveness of the solid.

-101TABLE IX COMPARISON OF THE ORIGINAL AND CHECK RUNS T(~K) P(atm) V(cm/sec) y, 140 70 2.08 0.00112 + 0.00001 0.98 0.00112 + 0.00001 140 10 3.20 0.0Q024 + 0.00001 4.56 0.00023 + 0.00002 150 80 1.98 0.00223 + 0.00002 2.20 0.00221 + 0.00002 150 50 1.13 0.00094 + 0.00001 3.17 0.00094 + 0.00002 150 10 3.20 0.00107 + 0.00001 4.56 0.00107 + 0.00002 16o 90 2.26 o.oo00461 + 0.00003 1.4 0.0oo460 + 0.00007 16o 70 2.75 0.00334 + 0.00002 1.74 0.00330 + 0.00003 160 50 2.66 0.00243 + 0.00002 5.75 0.00243 + 0.00002 160 10 6.40 0.00395 + 0.00002 3.29 0.00394 + 0.00002 170 50 3.11 0.00580 + 0.00005 2o07 0.00580 + 0.00005 170 10 3.08 0.0121 + 0.0001 6.10 0.0120 + 0.0001 180 50 2.86 0.0133 + 0.0001 1.16 0o0132 + 0.0001 180 20 3.07 0.0190 + 0.0001 2,20 0.0189 + 0.0002 190 90 3.44 0.0320 + 0.0002 2.75 0.0322 + 0.0002 190 50 3.07 0.0293 + 0.0002 2.53 0.0292 + 0.0002 19 0 50 5.29 0.0554 + 0.0002 4.30 o0.0o554 + 0.0002..... I..,, I~~~~~~~~~~~m

-102-trap, it is seen from the table that the lower flow rate resulted in a lower mole fraction six times, a higher mole fraction four times, and the identical value the other six times. These random differences certainly do not give any indication that solid carbon dioxide passed through the filter and sublimed as the outlet gas was warmed. to room temperature. The experimental, mole fractions have previously been plotted as functions of pressure at constant temperature in Figure 17 and l8o It is also of interest to present these results as functions of temperature at constant pressure, this being given in Figure 19~ In this graph the 50, 50, 70, and 90 atmosphere isobars have been omitted, for the sake of clarity. Designation of the experimental points has also been neglected., for the same reasono As defined, and discussed in Section II-F, the enhancement factor e is a dimensionless parameter equal to the ratio of the carbon dioxide mole fraction in the gas phase to that which would exist for ideal behavior of the system. In this sense? the parameter e-. gives an indication of the departure of the system from the ideal behavior defined in II-F. The values for c tabulated in Table X have been computed from the experimental mole fractions given in Table VIII and the vapor pressures listed in Table I. The enhancement factor is also plotted in Figure 20 as a function of pressure at constant temperature. Examination of the enhancement factors of Table X shows several apparent discrepancies at low pressure, since plots of enchancement factor vs. temperature at constant pressure should yield smooth curves. From such plots, it appears that the 5 atmosphere points at

-1035-.040.035.030.025 z / I g/ 0;-.020 C.I I, _____X _ — w! ~-..015 140 150 160 170 180 190 TEMPERATURE, ~K Figure 19. Mole Fraction y/ vs. Temperature at Constant Pressure

P. ENHANCEMENT FACTOR E y P(Dl ct CD a), * H - (D 0 Cli i i) c-f —---- ---- ^ ^ -- ^ --- - --- - l, -- -- - - -- ^ "^ ^ ---- --- --- ---- --- --- -- -- -- — _____ \\\^ ^- S 0 p) \ ^ o^ c-f-\ \^/~ ~ s's g -— V ---:^0 —^ -- - (Cl'^^ ^ \^ (Cl — _ NA_.- _^ ^_ _ i \, \ f-":

-105140 and 160~ should be at or near their upper limits, respectively, in order to give a smooth curve. It also seems that the 10 atmosphere points at 140 and 150~ should be somewhat larger than the values given in Table VIII, although still within the limits given in the data. The computed values of enhancement factor are very sensitive to small changes in the mole fraction and/or the vapor pressure of carbon dioxid.e, and the scatter may be due in part to the values used here for the latter. TABLE X ENHANCEMENT FACTOR AS A FUNCTION OF TEMPERATURE AND PRESSURE!p P = y1 _ 1 Pressure Temperature, ~K atm 140 150 160 170 180 190 5 1.1414 1.1415 1o1131 1.1078 1,0858 10 1,3043 1.2723 1.2580 1.2185 1o1861 1.1458 20 1.9565 1.6409 1.5987 1o4804 1,3869 1.3402 50 350978 2.2830 2.0732 1.8369 1o6533 1.5641 40 5.4348 3.3769 2.7643 2,3041 2.0000 1.8321 50 11.685 5o5886 3.8694 2.9204 2 4270 2.1576 60 25.435 9.1320 5o3503 3.7341 2,9562 2.5626 70 42.609 14o316 7.4013 4o8288 3.6022 3.0206 80 60.435 21.118 10.013 6.1229 4.3796 3.5700 90 77.283 28.894 13 213 7.7946 5 2883 4.2415 100 - 56.147 17o070 9.7482 6.4234 5.0074 As mentioned earlier in presenting the results, the minimal pressure, that at which the mole fraction of carbon dioxide is a minimum for the given isotherm, is of particular use in the purification of a gas mixture by selective freezingo It is of interest to note

-106that a plot of this minimal. pressure as a function of temperature (Figure 21) shows a fairly smooth curve. The minimal pressure can itself be construed as some function of non-ideal behavior, as no such minimum would exist for ideal behavior. This statement is not meant to imply that the minimal pressure is that at which the system departs from ideal behavior; in fact, it occurs at a much lower pressure. The point to be made is that above the minimal. pressure the system is behaving non-ideally to such an extreme degree that the carbon dioxide concentration actually changes in the opposite direction to that predicted for ideal behavioro The term function is referred to above only in the sense that it would be expected that the system should behave more ideally as the temperature increases and becomes progressively farther removed from the critical point of nitrogen. The regular behavior of minimal pressure with temperature confirms this expectation~ The matter of phase determination should. also be discussed during the presentation and discussion of the experimental, results of this investigation. Inasmuch as it was not possible to view the mixture at the equilibrium point, there may be some conjecture as to a possible phase transformation, especially at high pressureo It would be expected that, for reasonably large initial percentages of carbon dioxide, there would be some pressure at which a transformation from the carbon dioxide solid-gas equilibrium in the presence of gaseous nitrogen would occur, resulting in the introduction of a liquid phase, as mentioned particularly in References 14, 21.

-10780 70 60 / / / 50 /',' / 140 150 160 170 180 190, — 20 140 150 160 170 180 190 TEMPERATURE, ~K Figure 21. Minimal Pressure vs. Temperature

-108This transformation never occurred in the present tests, because of the very low initial concentrations of carbon dioxide, which should. result in mixture critical temperatures below the range of this investigation, Further, there were no discontinuities of the concentrations as functions of pressure, as might be expected as a result of a change in solubility occurring with such a phase transformation. Final.y, there was no indication in the operating characteristics of the system to indicate in any way the presence of liquid in the equilibrium vessel, A further investigation might be worthwhile, in order to determine a more complete phase liagram for different initial concentrations of these componentso As a final point to be made, the data in Appendix D shows that the heat exchanger performance was very disappointing, the equilibrium vessel gas inlet temperature running much higher than had been the intention when the heat exchanger was designed.o This can be readily explained in that the flow rates used in the experimental runs were smaller than originally planned by a factor of 5, so that gas Reynolds numbers in the heat exchanger sometimes were as low as 2000 In addition, latent heat from the condensing carbon dioxide also caused performance to deteriorate. If the heat exchanger had been more effective in cooling the incoming gas, the temperature gradients in the block would have been almost nonexistent, but since these were sufficiently small even with the relatively warm gas, it was not found necessary to redesign this part of the apparatus.

IX. COMPARISON OF PREDICTED AND EXPERIMENTAL RESULTS A. Numerical Comparisons In this final section, the experimental results given in Section VIII will be compared. with those predicted by the three methods used to solve the theoretical equilibrium equation. As discussed previously in Section IV, there will be two sets of solutions for each of these methods, one arising from the assumptions made to simplify solution of the equations, and the other from solution of the general equations. It is recalled, however, that both solutions are made under the assumption that the solid is pure carbon dioxide and is incompressible. The first method of solution is that making use of the Beattie-Bridgeman equation of state, an equation fitted empirically to a large set of P-V-T data for a pure substance. The combinations of coefficients to represent a mixture are also empirical, and have been decided upon by comparison with mixture P-V-T data. In applying this equation of state to the system under investigation, there is naturally some question as to how accurately the equation can be expected to represent the mixture, inasmuch as the pure gas constants and mixture combinations have been fitted, at much higher temperatures. A more severe problem is that the equation is intended for application below 0.8 times critical density, and the maximum densities reached at 140 and 1500K are of the order of 1.5 times critical, well beyond the range of the equation. -109

-110The second method, using a virial equation of state with the Lennard-Jones (6-12) potential and Ewald's revised method of force constant combination for interaction coefficients, is commonly termed theoretical, although the Lennard-Jones potential is itself an empirical representation of the intermolecular force, and the pure-gas force constants have been found from experimental virial coefficients. Some inaccuracy is being introduced in using the Lennard-Jones potential, as carbon dioxide molecules are non-spherical. Also, the force constants have been evaluated from data at temperatures greater than the range being studied hereo As before, the most severe problem will be that of the failure of the equation of state at high density, inasmuch as virial coefficients beyond the third could not be included.. The third method of solution, that of Prausnitz, can be classed as that of generalized behavior in that it incorporates the three-parameter theory of corresponding states to predict interaction coefficients. The pure-gas virial constants necessary for solution are those used with the preceding method, having been calculated from the Lennard-Jones potential. In this respect, this method will suffer the same equation of state failure as Ewald's method, since the equation for the mixture reduces to that for pure nitrogen in the case of the simplified solution. Thus, the three methods of solving the theoretical equations to be compared with the experimental results might properly be termed empirical, theoretical, and..-generalized behavior, respectively, even

-111though all are empirical in parto In the discussion to follow, all three will be referred to as predicted, simply as a distinction from results determined directly by experiment. The results and comparisons are presented in this section in the form of enhancement factors, in order to show a greater numerical contrast between different methods than is seen in the mole fractions. For example, at 140~K and 30 atmospheres: Yl Experimental Oo00019 3.0978 General Beattie-Bridgeman 0.00020 3.3284 General Ewald 0 00019 351032 General Prausnitz 0.00023 3.7663 Since the simplified and general solutions for each method are very close at the lower temperatures, it has been decided to plot only the three general solutions, and present both the simplified and general solutions in tabular formo The results listed in Table XI and plotted in Figures 22-27 have been calculated on the IBM 704 computer using the procedures given in Appendices B and CO The same program was used for the Ewald and Prausnitz solutions, since the equations to be solved are identical. The differences between the two lie in numerical values of the interaction coefficients, which were read as input data to the machine.

-112100 - - 60 40 30 -t/ 4 l -4 t LiI /I _ I __ — I I 0 1 5 - -------------,XBEATTIE- BRIDGEMAN -- -- EWALD 15" --- --- PRAUSNITZ 1 20 40 60 80 100 PRESSURE, ATM. Figure 22. General Solutions at 140~K

ENHANCEMENT FACTOR, E T - - f\)N (A CD 5 0C _ _ O_ ___0 0 0 0 0 0 otq 0 P0 (A CD 0 1\l IH1^^ TT HCm 0. O ^ _ _ _ _ _ VI 0 0 Co ___ ___ ___ "y^, __ __ _ _ __ _____ _ __ __ _______ 0' -~-4

ENHANCEMENT FACTOR,E - -, N 0 N. ______ o @ w P (s ax o un O 0 0 H OI; i i i 0 0 SB5Q^^l t ->0 c ______________ _sn^ _______ o^ oo&.

ENHANCEMENT FACTOR, e 0 0i No (. 01 ) -.4 0x o IN C I I~ (D VI~J~^1 Cl 0 o _ _ & m " -I ci0 ii i i i ii? m ^^ ^ ~, o> __________________________^^ ^_______ ___ __ __ ____~~i~c~as~

ENHANCEMENT FACTOR, E -' Gin N cQJ. 1~ c o'( -,, N~~~~~~~ (D P I 0 c-O H o g - c-f H. 0 OD 0 0 —._ ____I 0 ^r CD ^^&S.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~S

0 C N~~~~~~~~~~~~~~~~~~ _ _ _ _ _ _ _ _ _ ______0 0\ 0 Q o ___________________ <S________________________ o < W Z m ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ s m. 0 _______________ _ _ ___ _ _ _ _ _ o_ __ - r ~1) ----— ^ 10 ~ ~~~~~~ ~ ~ ~~~~~~~~~~~~~~~~ i i 0 imI)m * U013VJ 1N3W33NVH0 ~ IO.-I. 1N31N33NVHN3

TABLE XI COMPARISON OF PREDICTED AND EXPERIMENTAL RESULTS Notation: Experimental Results EXP Simplified Beattie-Bridgeman SBB Simplified Ewald SEW Simplified Prausntiz SPR General Beattie-Bridgeman GBB General Ewald GEW General Prausnitz GPR A. Enhancement Factor at l4o~K D. Enhancement Factor at 170~K P(atm) EXP SBB SEW SPR GBB GEW GPR P(atm) EXP SBB SEW SPR GBB GEW GPR 5 1.1414 1.1798 1.1713 1.2015 1.1798 1.1713 1.2015 5 1.1078 1.1037 1.1025 1.1107 1.1001 1.0988 1.1068 10 1.3043 1.4060 1.3841 1.4591 1.4060 1.3841 1.4591 10 1.2185 1.2177 1.2149 1.2332 1.2143 1.2114 1.2294 20 1.9565 2.0777 2.0007 2.2450 2.0777 2.0007 2.2450 20 1.4804 1.4939 1.4856 -1.5309 1.4916 1.4827 1.5279 30 3.0978 3.3284 3.1032 3.7663 3.3284 3.1032 3.7663 30 1.8369 1.8536 1.8344 1.9196 1.8541 1.8335 1.9193 40 5.4348 6.2615 5.5577 7.5452 6.2615 5.5580 7.5452 40 2.3041 2.3275 2.2896 2.4329 2.3347 2.2937 2.4395 50 11.685 19.223 19.244 34.397 19.223 19.244 42.730 50 2.9204 2.9569 2.8899 3.1175 2.9791 2.9053 3.1405 60 25.435 76.621 82.591 208.71 88.209 92.737 - 60 3.7341 3.7938 3.6867 4.0350 3.8480 3.7267 4.0948 70 42.609 151.72 - * - * 184.73 - * - 70 4.8288 4.8956 4.7422 5.2592 5.0146 4.8336 5.3984 80 60.435 240.78 - - 310.02 - - 80 6.1229 6.3134 6.1190 6.8626 6.5540 6.3108 7.1615 90 77.283 345.41 - - 472.83 - - 90 7.7946 8.0778 7.8593 8.8896 8.5266 8.2282 9.4789 100 9.7482 10.193 9.9654 11.333 10.966 10.611 12.390 *Solution Failed to Converge B. Enhancement Factor at 150~K E. Enhancement Factor at l80~K H P(atm) EXP SBB SEW SPR GBB GEW GPR P(atm) EXP SBB SEW SPR GBB GEW GPR 0 5 1.1415 1.1459 1.1410 1.1602 1.1455 1.1406 1.1598 5 1.0858 1.0924 1.0915 1.0969 1.0838 1.0828 1.0876 10 1.2723 1.3208 1.3085 1.3541 1.3204 1.3081 1.3537 10 1.1861 1.1875 1.1859 1.1976 1.1792 1.1774 1.1885 20 1.6409 1.7940 1.7546 1.8862 1.7940 1.7546 1.8862 20 1.3869 1.4101o 1.4057 1.4337 1.4033 1.3980 1.4257 30 2.2830 2.5310 2.4328 2.7309 2.5310 2.4328 2.7309 30 1.6533 1.6853 1.6754 1.7257 1.6822 1.6703 1.7210 40 3.3769 3.7643 3.5388 4.1789 3.7643 3.5388 4.1789 4o 2.0000 2.0270 2.0078 2.0885 2.0318 2.0085 2.0914 50 5.5886 6.0254 5.5379 6.9574 6.0628 5.5379 7.0120 50 2.4270 2.4516 2.4187 2.5401 2.4720 2.4310 2.5585 6o 9.1320 10.512 9.6743 13.145 10.703 9.8270 13.532 6o 2.9562 2.9776 2.9260 3.1012 3.0266 2.9601 3.1496 70 14.316 18.985 18.379 27.292 19.831 19.211 29.996 70 3.6022 3.6232 3.5493 3.7941 3.7226 3.6224 3.8974 -80 21.118 31.940 31.581 50.275 34.447 33.914 59.724 80 4.3796 4.4042 4.3067 4.6392 4.5875 4.4457 4.8380 90 28.894 48.541 46.806 77.976 53.967 51.120 98.856 90 5.2883 5.3305 5.2114 5.6505 5.6453 5.4552 6.0058 100 36.147 68.186 63.194 108.63 78.009 69.924 146.99 100 6.4234 6.4o50 6.2679 6.8320 6.9142 6.6671 7.4265 C. Enhancement Factor at l6o~K F. Enhancement Factor at 190~K P(atm) EXP SBB SEW SPR GBB GEW GPR P(atm) EXP SBB SEW SPR GBB GEW GPR 5 1.1131 1.1213 1.1180 1.1313 1.1199 1.1167 1.1298 10 1.1458 1.1688 1.1680 1.1763 1.1504 1.14 1 1.1564 10 1.2580 1.2607 1.2529 1.2833 1.2595 1.2517 1.2819 20 1.3402 1.3529 1.3514 1.3707 1.3367 1.3337 1.3522 20 1.5987 1.6148 1.5914 1.6718 1.6142 1.5906 1.6709 30 1.5641 1.5719 1.5685 1.6019 1.5605 1.5540 1.5875 30 2.0732 2.1102 2.0570 2.2199 2.1118 2.0576 2.2212 4o 1.8321 1.8325 1.8254 1.8771 1.8307 1.8178 1.8714 4o 2.7643 2.8235 2.7160 3.0166 2.8311 2.7209 3.0246 50 2.1576 2.1421 2.1293 2.2040 2.1575 2.1346 2.2150 50 3.8694 3.8787 3.6780 4.2120 3.9032 3.6950 4.2408 6o 2.5626 2.5080 2.4875 2.5909 2.5527 2.5150 2.6312 60 5.3503 5.4647 5.1226 6.0578 5.5362 5.1742 6.1504 70 3.0206 2.9371 2.9075 3.0459 3.0293 2.9712 3.1350 70 7.4013 7.8216 7.3107 8.9286 8.0162 7.4587 9.2140 80 3.5700 3.4350 3.3954 3.5758 3.6oo004 3.5156 3.7429 80 10.013 11.156 10.503 13.211 11.629 10.884 14.oo006 90 4.2415 4.0049 3.9557 4.1850 4.2788 4.1598 4.4710 90 13.213 15.536 14.755 19.005 16.535 15.576 20.846 100 5.0074 4.6477 4.5900 4.8749 5.0751 4.9126 5.3341 100 17.070 20.902 19.870 26.026 22.744 21.333 29.548

-119Upon examination of the results, some general trends are observed. Of the three methods used, Ewald's is generally the best, closely followed by that using the Beattie-Bridgeman equation. Values predicted by the Prausnitz method are nearly always high, and not nearly as good as the other two. The simplified solutions are seen to give as good or better results over much of the temperature range below 190~Ko For each of the methods, the general solutions lie somewhat below the simplified (in enhancement factor) at low pressures, and then become higher at pressures above the minimal pressure. The equations of state tend to fail at high density in the direction of predicting volumes too small, and the simplifying assumptions seem to exert a damping tendency on the equation, causing predicted mole fractions to be lower for those solutions than for the general case. In general, at low pressure the Beattie-Bridgeman equation predicts high and the Ewald either between the Beattie-Bri.dgeman and the experimental or else slightly low. At high pressures the curves normally line up downward from highest predicted e in the order GPR, SPR, GBB, SBB, GEW, SEW, although sometimes the GEW value is above that of the SBB. The notation used here is that of Table XI. At the lower temperatures, the entire set is above the experimental at high pressure so that for each method the simplified solution is better, since its prediction of C lies below that of the corresponding general solution in every case. As the temperature is increased, the entire set of curves decreases relative to the experimental, and the curves for the different solutions also become closer together. Throughout nearly all of the temperature range, the GEW

-120or SEW prediction is the closest to the experimental, followed by the SBB and SEW or GEW. At 1900, however, the predicted set has fallen relative to the experimental to such a degree that the GBB gives the best correlation with all other curves except the GPR lying below the experimental. At this temperature, even the SPR, which has always been high, crosses over at 90 atmospheres and dips below the experimental results. The correlation is in general very good except at 140~, this isotherm meriting special discussion. At this temperature, the Ewald and Beattie-Bridgeman equations prove to be very accurate at up to forty atmospheres, with the former being better throughout. However, the equations of state were not intended for densities greater than this, and consequently the solutions begin to show considerable deviations at fifty atmospheres. At seventy atmospheres, the Beattie-Bridgeman equation fails very badly, and the other two will not even converge to a solution. This presents the problem as to how much of this gross failure can be attributed to the high-density inaccuracy of the equations of state used, as opposed to failure of the interaction coefficient prediction and assumptions that the solid is pure carbon dioxide and also incompressible. This question can be resolved by eliminating the equation of state entirely as the second relation between the dependent variables y1 and v, as was done in the majority of the references cited earlier regarding this type of work. It is recalled that at low temperature the mixture equation of state reduces to essentially that for pure nitrogen, so it is possible to simply substitute experimental volumes directly into the integrated form of the equilibrium equation, and make a direct calculation of the

-121carbon dioxide mole fraction. This has been done with both the form derived using the Beattie-Bridgeman constants (4-86) and also that using the Ewald interaction coefficients (4-106). The resulting empirically calculated enhancement factors at 140~K are P(atm) Eq,.(4-86) Eq (4-106) 60 23.8 19.4 70 45 34.7 80 87 62 90 172 117.5 and the principal failure here is seen to be that due to the equation of state. This is not at all unexpected; in fact, the experimental range was chosen particularly to overextend the equations of state regarding density. It was hoped, however, that these equations would not fail as badly as they did at 140~. The results at 1.500, for example, are still reasonable at pressures considerably beyond the range for which the equations are considered accurate, that being roughly fifty atmospheres. Another prime interest, that of comparing simplified and genera: solutions, has led to the rather interesting conclusion that the simplifying assumptions do not impair the accuracy of the predictions until the carbon dioxide concentration reaches approximately 2 or 3 percent, which is somewhat higher than might be expectedo The preceding discussion gives some insight into the problems associated with the predictions being made in this investigation. Of the two principal factors, namely the inert-gas equation of state and the method of estimating virial-interaction coefficients, the former seems to be predominant. The Ewald and Prausnitz methods give a direct comparison of the latter, inasmuch as the same virial coefficients were used

-122for pure nitrogen and carbon dioxide in both cases. While the Prausnitz method is not of the caliber of the others as far as this system is concerned, it must be remembered that this procedure is an attempt to generalize gas mixture behavior, which in addition to being very convenient in calculations, can prove valuable in cases for which the fairly extensive information required by other methods is not known, The inaccuracies mentioned earlier in this section, such as those resulting from extrapolating the Beattie-Bridgeman constants range, considering carbon dioxide molecules to be spherical, etc., apparently had little effect on the overall accuracy of the results. B. Concluding Remarks The correlation of the theoretical results with those determined experimentally was, for the most part, very satisfactory except for 140~, which indicates that a more accurate high-density equation of state will be required to give accurate results in this density range. The author is presently conducting an investigation intended to apply the very accurate Martin-Hou equation to gas mixtures. Once a satisfactory means of coefficient combination has been determined, it is expected that this equation will enable accurate predication of composition over the entire pressure-temperature range of this investigation, including the 140~ isotherm.

APPENDIX A ITERATIVE METHOD 1. Newton Method for One Equation in One Unknown Consider a function of one variable, F(v) = 0 (Al-l) In order to find the value of v satisfying (Al-l), assume some initial value, say v1. In general, Fl(v) will be unequal to zero, since vL is not the true solution. Then it will be necessary to correct Fl(v) by some amount AFj(v), in order to equate it to zeroo That is, Fl(v) 4 AF(v) = 0 (Al-2) The problem now reduces to that of specifying FiL(v) in terms of a correction Avl in v which will give a closer approximation to the solution of (Al-l) than did the first trial vio This relation can be given by Fl (v) [ v1 1 = F. (v) ~ Av (Al-3) dv 1 1A Now, equating (Al-2) and (Al-3), Av -F(v (Al -4) F1 (v) 1 or, the value v2 for the next trial is given by Fl(v) v FL(v- (Al-5) F'-(v) 1 -123

-124In general, for the i-th trial, i 1i i F(v) (A1-6) F (v) ^i~ - i - _ (A1-6) i The iterative process is repeated until the error Avi is less than the desired amount. 2. The Newton-Raphson Method for Two Equations in Two Unknowns Consider two non-linear functions in two variables, FA(Y, v) - 0 (A2-1) B(Y, v) = 0 (A2-2) In order to simultaneously solve the above set for y, v, assume some initial values Yl, vl1 Since these values will, in general not satisfy (A2-1), (A2-2), the functions FAl(y,v) and FBl(y,v) must be corrected by some amounts to equal zero, or FAl(yV) + ^FAl(yv) = 0 (A2-3) FBBi(y,) + AFB1(yv) = 0 (A2-4) Corrections to Yl and v1 to give improved values can be related to the function corrections in a manner analogous to that for one equation in one unknown, since the corrections can be expressed as: LFA1(Y,v) A [FA(Yv) Ay, + [FA(Yv (A2-5) FL dvy 1 av 1 AFB (y,v) 6 [FB(yv)] Ay + [FB(yv)] Av1 (A2-6) ~ d~y 1 v 1

-125Substituting (A2-3) into (A2-5) and (A2-4) into (A2-6) results in 6FA(Y.IV) [ FA(Yyv) [FA(Yv)] Ay, + [8] A v FA1(y,v) (A2-7) 6y 1 ov 1 [S ^ ] ^. i81F] A ~. p (y v) (A2y8) B' (yv) AY1 + [ (yv,V, FBl(yv) (A2-8) 6y 6v The resulting set of linear correction Equations (A2-7), (A2-8), can be solved simultaneously for the corrections Ay1, Avl, by matrix inversion. Therefore, Dy1 Dv Ay1=, Av -~ (A2-9) D1 DI where 1 y z L v 1 | [a (y]v) [FB(Yyv))] oy v 1 - F ( ydFA(yvv) FA1 (yv) [ a ]1 D = - FB1(yv) [FB(Yv)] (A2-1?v 1 F^(y,v) - F (yv) oy Dv1 = [aFB(Yv) ] FB(y,v) y 1 the values y2, v2 for the second trial are thus given by Y2 = y1 + DYl/Dl = vl + Dv1,/D1i(A2-11) V2 = v1 + DVl/D1

-126In general, for the i-th trial Yi+l Yi + DYi/Di (A (A2-12) vi+1 = Vi + Dvi/Di in which the D's represent the determinants of the i-th trial, as in (A2-10). The iterative procedure is repeated until both of the errors AYil Avi, are as small as desired. This method can be converted quite easily to a logarithmic form if it is found necessary to improve the convergence or preclude the possibility of y or v becoming negative during the course of the solution.

-127APPENDIX B FLOW DIAGRAM FOR BEATTIE-BRIDGEMAN EQUATION SOLUTIONS READ BB constants READ / ___- AI' BI, ai, bl, C, T, Pin P, START --. a, -ly in Ir — A2, B2. a2, b2, c2, Evaluate Evaluate Evaluate D1, D2, --—, D22fron Vl from P I's, Y's, W's (4-59), —, (4-70), (4-89) from (4-44), —,(4-57 (4-77), —, (4-85) (4-72), (4-75) (P =P. P A_ initial guess s/ = J=J-r _ for v from (4-89)\/ \/ 7fe"/!1 s Evaluate J: 50 — =v - v) F(v) BB eqn. F(V) (4-87) and F (v) c___< Evaluate PRINT F P(l |ooool ~ Y from (4-86) "Si mplified Solu then e from (2- T p v ( e ) ^y: 0.001 < J= 0!rC

-128Evaluate Evaluate Gen'l Set Partial Deriva( e )_< J=J+1 >_|FA (YiP) (4-58) tives of FA, FB FB (ypV) (4-76) l l PI N- 1 | Evaluate N oi Gti^na - P- SN o luti D PAppendix A2 ti<on J: 50 (yi 1:0.000005 P=P(At:0.00001 _ 5. T, P, i, v, L (2-43) PRINT / ^-. T,,!v, O~~~~~Ia.o.

-129APPENDIX C FLOW DIAGRAM FOR VIRIAL EQUATION SOLUTIONS READ PRINT T. P1~, V1S } Pin, T, PL, VIs Pin ^~START- Pmax- B1 B12 21 2 Pmax, B B12 I lC1, C112, C122, C2 C l Evaluate Evaluate -- initial guess -, J = \P =Pin or v from 4-log) L (4-lo8) I __Evaluate Evaluate F(v) J = J+l -- - J: 20 F(v) (4-107) v=v- t J and F'(v) PRINT / Evaluate Evaluate < <'^Simnplified Solu- e from Yl from:0( l >:.l tion" T, P, e, (^) (4-106) VI yl > ----- > / ~ Evaluate y: 0.001 J = FA ( v)(4-103)J'- -- I' \ / | FB Y V)(4-105),Yl __D'D Evaluate Evaluate Matrices D, Dyl Partial Deriva^P ~IIA- TDv Dvaccording to ivesofFF D Appendix A2

-130( p )-^ -^<J~J+u-' J:20 ) |'l:o.oooo0 viv+Av i i l + 20 III I. II II*l PRINT EVALUATE < "General Solution" from Av:0.00001 (2-43) MAAX P::10 P-P+0iO i} 0 \ PRINT / ( 5- )_ "No Solution" 5

APPENDIX D EXPERIMENTAL DATA The data following is given in the following notation and units: TIME: Minutes NOMTI Nominal Equilibrium Temperature, ~K NOMP: Nominal Equilibrium Pressure, ATM AMBT: Ambient Temperature, ~F AMBP: Ambient Pressure, in. HG. ANAL: Gas Analyzer Reading, Percent C02 ANACC: Analyzer error Limits, + Percent CO2 Ml: System Flow Rate, LBM/HR M2: Analyzer Flow Rate, SCFH P1: Equilibrium Pressure, PSIG P2: Pressure Downstream of Throttle, PSIG P3: Main Stream Flowmeter Pressure, PSIG P4: Analyzer Exit Pressure, PSIG TC-1: Equilibrium Gas Temperature, ~K TC-2: Vessel Inlet Gas Temperature, ~K TC-3: Block Side Top Temperature, ~K TC-4: Block Side Middle Temperature, ~K TC-5: Block Side Bottom Temperature, ~K TC-6: Block Bottom Corner Temperature, ~K TC-7: Block Core Temperature, ~K TC-8: Block Bottom Center Temperature, ~K TC-9: Inlet Tube Temperature, ~K TC-10: Outlet Tube Temperature, ~K TC-11: Outlet Upper Fitting Temperature, ~K TC-12: Outlet Upper Tube Temperature, ~K TC-15: Equilibrium Pressure Gage Temperature, ~F TC-14: Temperature Downstream of Throttle, ~F TC-15: Main Flowmeter Temperature, ~F TC-16: Analyzer Exit Temperature, ~F -131

-132Run Al A1 Al Al Al Al Al A2 TIME 30 30 30 30 30 30 30 40 NOMT 150 150 150 150 160 160 160 0 NOMP 80 760 0 50 0 60 70 0 A7BT0 5 70 70 AMBT 80 80 80 8081 78 AMBP 29.03 29.03 29. 03 29.03 29.03 90 29.03 i28.89 ANAL 0.223 0.172 0.128 0.094 0.243 0.280 0.334 0.112 ANACC 0.02 0.002 0.002. 0002 0 002 0.0012 Ml 0.95 0.900 0.85 0.85 0.85 0.85.95. M2 1.0 1.0 0 1.o 1.0o 1.0 1.0 1.0 P1 1160 1015 I870 1720 720 i865 1015 1015 P2 1 1 i 1 11 1 1 1 1 P3 j i 1 1 1 j 1 P4 0.6 0.6 0.6 0..6 0.6 0.6 TC-1 150.0 150.0 150.0 150.1 160.0 160.01 160.0 140.* TC-2 212.1 214.7 216.1 220.2 220.5w 222.9J 226.0i 211.1 T-3 150.2 150.2150.1 150.21 160.1 160.2j 160.2 140.1 TC-4 150.1 1501 50.1 150.1.11 160.2i 160.2 1 140.1 TC-5 150.1 0.1 150.1 150.1 160.0 160.19 160.O0 140.0 TC-6 1 0.2.2 150.2 150.1 150.2 160.1o 160.21 160.2j 140.2 Tc-?:' 15001 2 T-7 i 150.i 150.1 150.1 150.1 160.1 60. 1 60.1 16. 140.2 TC-8 i 150.0 150.0 150.01 15 0.0! 60.0 1 140.0 1o1 160.0 140 o i TC-9 202.6 203.3 204.31 207.6 208.9 212.31 215.0 198.31 TC-10 151.2 151.01 1 51 151.4 160.9 161.2 161.5 141.01 TOI11 162.2 151.3,10 15168. 141.01 TC-ll 162.2 163.0 160.8 163.3 168.5 171.6 171.61 161.51 TC-12 168.0 169.1 166.2 167.8 171.6 176.3 176.0 171.21 TC13 81.5 81.5 81.o0 81.5 81.5! 82.02 82.0 77.55 TC-14 81 81.5 1. 82.0 82.0 82.0i 82.0 82.0 79.0o TC-15 82.0 82.0 82.0 82.5 82.5 82.5 82.51 81.01 TC-16 81.5 81.5 8 821 82.0 82.01 82.0o 82.0o 79.0! 1I I i

-133RUN A2 A2 A2 A2 A2 A2 A2 A2 1W,.. i i TIME 30 3 30 30 30 30 30 30 NOMT 140 140 2 40 140 140 140 140 140 NONMP70 60 50 40 30 20 10 5 AMBT 78 7 78 7 79 79 79 79 79 AMBP 28.89 28.8992.9 28.89 2889 1 28.89 28.89 I828.89 ANAL 0.1121 0.0781 0.043 0.025 0.019 0.018 0.024! 0.0426 ANACC I0.001 0.001! O.l001 0.001 0.001 0.001 0.001 0.001 F~o I F~o ~Olool Ml 0.50 0.80 0.72 0.5 0.4 0.2 0.14 0.14 M2 1.0 J1.0 1.0 1.0 1.0 1.0 1.0 1.0 P1 11015 870o 720 572 426 1280 132 59 P2 I 1 I 1 1 1 1 1 f F i 1 i 8 P3 Jl 11 1 1 l jj 1 1 1 I P4 0.6 0.6 0O.6 | 06 O.0.6 0.6 0.6 0.6 0.6 TC-I S 140.0i 140.1j 140.o| 14-t 140.0; 10oOo 140.o 140.0;ioo TC-2 215.0| 211.3! 208.7 209.8 214.0 213.41 215.2 215.7 TC- 140. 140. 140.i. 140.0 140.1 140.0. TC-3 140. 1 40.1 140.1 140.0 140. 140.0 140.0 TCG-500140.0 o T-4 140.21 140.1 140.1 140. 140. 140.0j 140.l| 140. T- 140.1 140. 140. 140. 14. 1400.1 40 140 1 40.10 140. 0.0 TC-6 4.2 240.1 240.4 140.2 l 240.1 40.0 140 140. 140 140.3 IL40 ~4~i 10. 10.5 40i S A TC-8 140.0 140.0| 140.0! 140.0j 1400 4 | 140.0 140.0 TC-8 40.01 20.0! TC.-6 240.01 2400 240.2140 240.1! 14.01 1401 140.0.1 TC-9 o196.8o 200.6| 195.8 1u92.6 194.0 186.3 13.9 179.4 i Iqo~~~~~~~~1.0i~q.0 j 240.01 1g6.8~ ~~~958 19.6 194..0 TC-10 141.1 141.4 1.o 140.8 141.3 143.3 146.4 150.6 TO-b 1 14.1.11 14.1.4' 141O 14.0.8. TC-il 159.9 155.2 153.81 156.0 159.4 168.8 177.8 184.8 TC-12 170.0 164.5 157.3 163.0 164.8 177.0 185.5 193.8 TC-13 78.0 78.0 785 8.5 78.5 79.0 79.0 79.0 ~1 o s o.oB 8o.Q TC-14 79.0 79.o01 79.579..5 79.5 80.0 8. 8. TC-15 81.0 81.0 81.5 82.0 82.0 82.0 82.0| 82.0 TG-16 79.0 79.5 80.0o 80.5j 80.5 80 80.5 80.5 TC-16 79.01 79.5Sf5.5 8.Sj*

-134RUN A3 A3 A3 A3 A3 A3 A4 A4 I f,I TIME 35 L 40 30 30 30 35 30 30 NOMT 150 50 150 1 150 150 150 160 160 NOMP 430 5 40 30 AMBT 78 78 78 78 78 78 80 80 AMBP 28.952 28.95 28.95 28.95 28.95 28.95 28.95 28.95 ANAL 0.071 0.0641 0.069 0.107 0.192 0.094i 0.217 0.217 ANA C C,oi OOO. O.OO1 O. OO1 0.06'01 9' ANACC. 0.001 0.001: 0.001 0.001, 0.001 |0.001 0.002 0.002 I 0 00 5 0.3 M1 04.2 0.1 02.1 0. 0.5 I 0.3 M2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 P1 573 426 280 132 59 720 572 j 426 P2 | 1 1 1 1 1 1 P3 1 1 1 11 1 P4k 066.6 0.6 0.6 0.6 0.6 0 6 606 TC- 1 150. 150.0 o.0 150 o.1i 150.0 150. 1 160.0 o 160.0 TC-2 219.7 t 220.2 215.6 211.5 208.1 220 4 223.5 223.7 TC-3 1500 150 1501500.0 150.0 160. 160.0 TC-3 1 150 0 150. 0 15 150.01 155 0.150 160.0 160.0 TC-4 150.1i 150.0i 150.0 150.0. 150.0 150.0 160.0 160.0 TC-6 150. 1 150.10 150.2 150.1 150.2g 160.0l 160.10 TC-7 150.lJ 150.0 150.015150.10. 150.20 160.01 160. 1 Tc-5 iS0.015o.o~ 150.oo 150. o. o 150o.oi 160.0i 16o.o! TC-8 150.; 17.4 6 TC-~8 105 0.0'0 150 0 50.0; 150.0 150.0 600! 160 Is TC-9 212.1 203.9' 187.3 1821 1779201.2 204 TC-10, 151.81 151.2j 151.31 152.0 1525 150.5 160.3 161.4 TC-11 1179.6 172.7 176.5 177.0i 177 7 170.0 170.2 175.1 TC-12 s187.3 17653 183.3t 182.5j 183.3 1787 179.3 179.1 TC-13 78.0! 78.5 78.51 78.5 79.0 79.0 80*.0 80.0i TC1'9 7. 7 8o5.o5 8o. 0o 8 1. 0o 81.0 TC-14 79.01 79.5 79.5 7 TC-15 8 8282.0 82 8282.01 82.01 82.0o 82.01 TC-16 179.0 79.5 1 79.52 79.5 1 79.54 79.5 81.0o 810o I r fj I Sg I I

-135RUN A4 A4 A4 A5 A5 A5 B1 B3l i | 50, TIME 30 3 30 40 70 50 3 335 NOMT 160 160 160 160 150 140 18 180 NOMP 20 10 10 10 0 10 0 870 AMBT 80 L 80 80 78 78 78 78 78 AMBP 28.952 28.95 28.95 29. 0129 29.01 29.01 29. 29.31 ANAL 0.2251 0.394 0.2 43 0.395 0.107 0.023 1.50 1.41 ANACC 0.0020 0.002 0.002 0.002 0.00 0.002 0.01 M1 i 0.2 0.1 0.6 0.2 0.2 0.2 0.83 0.6 M2 i. 1.0.0 1.0 0.5 0.5 0.5 0.75 0.75 2 13 132.1160 Pi1 279 i 132 7 720' 132 i 132 132 1160 15 P21 1 1 1 1 1, 1 1 1 P3 1 1: 1 1 i 1 6, 1 1 P3I P4 0 0.6 0.6 0.6 0.6.6 0.6 0.6.6 TC-1 160.0 160.1 | 160.0 160.0 150.0 140.0O 180.0 180 0 TC-2 i 222.8 i 220.0 222.5 205.7 212.0 j 210.5 229.9 229 TC-3 160.0 160.0 160.1 160.1 i 10.1 180. TC - i160.0 160.0 160.0 o.o 0 150.0 140. 180.1 180.1 TC-5 160 160.0 1610.0 160.0 150.0 140.0 180.0 180.0 TC-6 160.11j 160.2 160.1 160.1j 150.1 140.1 180.1 r 180.1' TC-7 i 160.0. 160 60 160 TC -8 160.0 160.1 160.1 0 180. 10 180. 40 16o.lj 160~~: 16o.0 2'i l o80. TC-10 162.0 I 1 64. i 161.7 1968.. 159.0 146.8J 181.6 182L.2 4*4.71.7 159.0O 146 TC-12 186.3 194.1 180.61 182.5 177.36 176.6: 202.9 196.61 TC-13 8 0.1 80.0 80.0 7.5 77.5 77.5 77.0 77 TC-14 81.O| 8 1.0 81.5 7 7 8.0 78.5 78.o0 78.0 TC-15 82.5 82.5i 82.5 80.01 80.0 80.5 79.0 79.0 i 160.0 1 6 0f.0 i TC-16 81.0o 81.01 81.0 78.1 85 9.0 79.01 79.0 1 i i' 7 79 TC-10 162 164 1 161 1 1687t 15 O i 16 8~ 81 6~18i 2

-136RUN B1 B2 B2 B2 B3 B3 B3 B3 TIE 1 333 1 30 TIME30 30 30 30 35 30 30 30 NOMT 180 160 160 160 170 170 170 1170 NOMIP 60 90 80 870 80 0 60 AMBT 78 79 79 79 79 79 79 79 729 I 31 AMBP 29.31 29.312 29.31 29.311 29.31 29312931.31 29. ANAL o51 0618: o 58o:' ANAL |1.35 0.461 0.393 0.330i 0 7600 o.685' 0.618 0.580 ANACC | 0.01 0.0031 0.003 0.003j 0.005 O 0.005i O.OOj5 0.0 M1 0.6 1.05 0.95 0.6 0.83 0.83 0.6 0.6 M2 0.75 0.75 0.75 0.75 0.75 0.75 0.75 1075 P1 865 1310 1 1160 1015 1160 1015 870 1720 P23 1 |1 1 1 1 P3 1 1 1 1 1 1 1 1 1 P4 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 TC-1 180.0 160.0 160.1 160.0 170.0 170.0 170.0 170.0 TC-2 229.2 211.3 221.2 212.8 221.0 223.6 223.7 225.0 T 160.1 160.11 170.1 a TC-3 180.1160.1 160.1 160.1 170.11 1 70..2 170.2 TC-4 180.1 160.1 160.1 60.1701 170.1 170. 2 170. 1 -IC-5 | 180.01 11 60.0 160. 160.0 170.0 170.0 170.0 170.0 TC-6 180.1 160.2 160.21 16.0.2 170. 10. 170.2! 170.1i TC-7 i 180. 160.| 160.2 160.1 170.1 170. 1 170.1 170.1 TC- 180.0 160.0 16 0. 160 16 0.1 170. 1.01 7170.01 170.0 TC1-8 160o 160.2 176 TC-9 221..4 203.6 212.9 202.9 212.0 213.3 214.2 t 215.5i TC-10 181.3'161.31 161.61 160.81 170.7 171.05 171.1i 171.4 TC-11 188.4 166.3 172.3168.6 176.5 177.5S 177.7 178.2i TC-12 194.3 173.0 181.2 175.8 182.3 7.7 184.7 184 TC-13 77.0 77.0 77.5 79.0 78. 78.0 78 78.O TC-14 78.5 78.51 78.5 79.0 79.5 7 797.5 79.51 TC-15 79.5 79.09 79.5 80.0 80.0 80.0 80.0 80.0i TC-16 79.01 79.0 79.5 79.5 79.51 79.5 80.o 80.

-137RUN B3 B3 B3 B3 B3 B4 B4 B4 3o J j | TIME 30 30 30 30 40 335 NOMT 10 170 60180 iNOMT 170 1i 7 170 70 170 17 NOMP 5 0 40 0 5 10 0 50 AMBT 80 0'8 80 8 80 8080 8 AMBP7 37 29.37 29.37 29. 37 2937 29.37 ANAL j 0.580 I 0.572 0.608i 0.735 0.699 1.21 1.20 1.33 ANACC 005 0 005 005 0 0050 0.005 0.01 0.01 0.01 Ml 0.4 0.4 0.3 0 0.2 0.07.1.2 0.5 N2 I~'~~j i o7 1 0.75 M2 0.75 ~ 0.75.75 0. 0.75 75 j 775 Pi 720 573 2 P 1 |720 3 573 426 280 59 132 132 720 P2 1 t 1 1 I 1 1 1 s 1 1 P3 1' 1 1 1 1 P3 i''I 1' 1 P4 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 TC-1 170. 170.0 170.0 170.0 160.1 170.0 170.0 180.0 TC-2 224.8 224.1 226.6i 215.2j 216.0 205.8i 219.71 225.0 TC201-3 10 170.1 17 170. 170.0 160. 170.1 180.0! TC-4 170.1 1 70.11 1 70. 170.0 160.2 170.1 1 180.0 TC-5 1 0.0. ~0 170' i 180. o -_5 170.0 7 170. 0.0, 160.0 170.0 170 180.0 TC-6 i 170.2 170.1 170.1 170.2 160.21 1 70.1 180.2 17o.oI 170.o17 170.0 TC-7 17 17021 170 1701.11 160.11 170.0 170.0J 180.1 TC-8 170.00 170.0 170.0 170.0 160.0 170 17 0.0 TC-9 221.7 215.21 210.5 199.3 191.21 188.6| 200.8. 220.2 TC-10 171.51 171.4 171.7- 171.31 164.O0 172.31 172.71 181.2 TC-ll 1 84.7 180.0 179.6 180.81 176.3 180.8 185.3, 189.3i TC-12 186.2j 182.7i 183.3 181.2 2 195.3 193.7 1811 20 195-31 193 71 TC-13 79.0 79.0 79. 0 79. 79.0 79.0O 79.0 79.0 TC-14 80.51 80.5 81.o081.o 81.o 80.05 80.5 TC-14. 8. 0 TC-15 81.5j 81.5i 82.o 82.0' 82.0o 82.0o 82.0o 82.01 TC-16 80.0j 80.0o 80.5 80.5 80.5' 81.0o 81.0o 81.0 I 1I i

-138RUN B4 B4 B4 B4 B5 B5 B6 B6 TIME 30 0 30 3 350 30 40 4 NOMT 180 180 i 180 180 18800 170 170 NOMP | 50 | 4 Ij0 j 30 20 100 90 j AMBT | 80 | 80 81 81 81 81 AMBP 29.37 29.37 29.37 29.37 29.39'29.39 29.39 29.39 ANAL 1.32 1.37 1.51 1.90 1.76 1.61 0.968 0.860 ANACC 0.01 0.011 1 0.001 0.01 0.0.07 07 r Ml 1 0.2 0.4 0.3 0.2 0.95'0.83 1.05 0.95 M2 i 0.75 7 0..75 0.75 075 0.75 0 75 075 10.75 P1 720 I 573 426 279 ~ 1460 1310 1 1460 1310 P2 1 1 1 1 1 1 1 1 P3 0 1 77 1 I I l S i1 1 1 1 P4 0i 0 06 0 6 06 6 6 0.6 0.6 0.6 TC-1 180.0 180.1 180.0 180.0 180.1 180.0 170.0 170.0 TC-2 223.4 224.7 223.9 225.4 225.4 224.6 225.7 225.0 | TC-3 180.0 180.2 180.1 180.1 180.2, 180.1 170.2 170.1 TC-4 180.0 180.2 180.1 180.1 180.2 180.1 170.2 170.1 TC-5 180.0 180.0 180.0 180.0 180.0 180.0 170.0 170.10 TC-6 180.0 180.2 180.1 180.1 180.2 180.1 170.1 170.1 TC-7 180.0 180. 1 8 180|.0 180. 1801.11 170.1 170.1 TC-8 180.0 180. 180.0 180.0 180.0 180.01 170.0 170. TC-8 180.0 1800. 180.0 180.0 i. 1 TC-9 216.3 219.2 213.4 211.6 221.5 221.3 217.7 212.9 TC-10 180.8 181.3 180.7i 181.4 181.3 180.91 171.1 171.4 TC-11 190.0 189.2 186.8 188.0 187.8! 186.7 178.6 179.5 TC-12 197.7 193.3 188.0 195.6 194.7 193.4 187.0 189.2 TC-13 79.0 79.0 79.5 79.5 79.5 79.5 81.0 81.0 TC-14 80.5 81.0 81.0 810 81.0 81.0 81.5 81.5 TC-15 82.0 82.0! 82.0 81.5 82.0 82.0 82.5 83.0 TC-16 81.0 81.0 81.0 81.5 80.5 80.5 81.0 81.5 I i I!

-139RUN B6 B7 A6 A6 A6 A7 A7 Cl TIME 50 S 45 50 30 30 45 I 30 30 NOMT 160 160 150 150 150 J 140 i 140 190 NOMP 90 100 100 90 80 90 100 90 809 AMBT 81 79 79 79 7 78 AMBP 29.39 28.97 28.97 28.97 28. 2897 28.97 28.65 ANAL 0.460 0.536 0.304 0.270580.139 j40'i''' ANACC 0.007i 0.003.002 0.002 0002 02 0. 0.001 0.001 0.02 Ml 0.6 1.05 [ 1.05 1.1. 0 1.05.15 105 0.83 M2 1075 0 0.75 5 M2 0.75 0 0.75 0.75 0.75 0.7 5.75 0.75 0.5 P1 1310, 1455 1455 1310 1160 1310 1160 1460 P2 1 1 1 1 1 1 1 P3 1 1 1 1 1 P3 ~. I. I. I I i.. P4 0.6 0.6 o 0.6 0.6 0.6 0.6 0.6 0.6! TC-1 160.0 160.1 150.1f 150.1 150.0 140.0 140.1 190.0 TC-3 160.2 160.2| 150.2! 150.1 150.2 140.1 140.2J 190.0 TC-4 160.2! 160.2 150.1 150.11 150.1 140.0 140.2j 190.0 TC-5 160.0 160.0 150.0, i0.01 150.0 140.0 140.0 190.0J TC-6 160.2 160.1 150.21 150.2 150.2 140.21 140.29 190.1s TC-7 160.1 160.2 150.2 150. 1 150. 1 140.1 140.1j 190. 0 TC-8 160.0 160.0 150.0 150.0 150.0 140.0 140.0; 190.0O TC-9 211.31 220.5 207.5j 205.81 205.5i 197.91 197.41 234.34 TC-10 161.4! 161.91 151.82 151.9 152.0j 143.20 142.7 191.Oi TC-11 173.31 173.0 165.21 162.2 162.56 154.7J 154.33 200.5' I I I I E TC-12 182.2 182.8| 174.9 170.5 169.8[ 163.3J 163.1i 210.3j TC-13 81.0 78.0 77.5 78.0 78.0 78.0 78.0 77.0 TC-14 82.0 79.0 78.0 78.0 78.5 78.0 78.5 78.5 TC-15 83.0 80.0 79.0 79.0 79.0 79.0 79.5 79.5 80.0 ~TC-1655 79.0 7.0 79. 79.0 79.0 0.0 80 TC-16 181.5 79.0 79.0 79.0 79.0 179.0 79.0 178.5 I~~\ i i i I

-140RUN C1 C1 C1 Cl1 C1 C1 1 1 TIME S 45 35 30 45 30 30 40 30 NOMT 190 190 190 190 190 4190 190 190 NOMP j 90 80 90 70 6 50 50 40 AMBT [ 78 78 79 79 79 79 77 77 AMBP 28.65 28.65 28.65 28.65 28.65 28.65 28.82 28.82 ANAL | 3.22 3.03 3.20 2. 2. 930 23 2.92 3.11 ANACC 0.02 0.02 0.02 02.02 0.2 0.02 0.02 Ml 0.83 0.83 1.05 0.6 0.6 0.5 0.4 0.4 M2 0.5 0.55 0.5 0.5 0.5 0 5 0.5 0.5 P1 1310 1160 1310 1015 865 720 i 720 573 P2 1 1 1 P3 1 1 1 1 P4 0.6 0.6 06 0 0.6 0.6 0.6 f0.6 TC- 1 190.0 190. 0 1.0 190.0 190.0 190.0 190.0 190.1 190.0 TC-2 245.2 242.6 245.3244.8 244.2 246.0 248.4 249.3 TC-3 0.2 1 190.2 190.1 190.2 TC-4 190.2 190.2 190.1 190.2' 190.0 190.001 190.1 190.1 TC-5 190.019.0190.0190.0 190..0 190.0 190.0 190..0 190.0 190.0 TC-6 190.1 190.1 190.1 190.0 190.0 190.0 190.2 190.1 TC-7 190.02 190.10 190.1 190. 190 TC-8. 190.0 90.01 190.0 190.01 190.0, 190.0O 190.0 190.0 TC-85 190.01 190.0~ 190.0, 190 0i0 190.0 190.0, TC-690 1 } 190 190li190 0 190 01900190236.6 9 TC-9 J 235.5 232.54 235.9 236.3 236.6 236.7z 233.2' 237.0: TC-11 1 202.0 199.7 199.4 202.0 202.4 204.0 20.3 209.2 TC-12 209.7 205.7 206.42 210. 210.5' 212.1 215.7 223.51 TC-13 78.0 78.0 78 7.5 78.0 78.0 76. 76.0 TC-14 79.0 79.0 78.5 78.5 79.0 78.5 77.0 77.0 | TC-15 80.0 i 80.0 80.0 80.0 r 79.5 79.5 j 77.0 77.5 TC-16, 79.0 79.0 79.0 79.0 79 0 79.0 i 77.0 77.0 i i i. i,,~

-lFlRUN C1 C1 C C1 C1 C2 C2 C2 TIME 1 35 340 35 0 4 30 NOMT 190 190 t0 180 700 180 9o 190 NOMP 30 30 20 10 5 5 2010 AMBT 77 77 77 77 78 7878 AMBP 28.82 28.82 i2882 2882 28.8728.87 28 87, ANAL 3.54 3.54 1.89 i3.25 2.20 i5.95 4,55 7.78 I ~ 3.25 20 5.95' 7 8 ANACC 0.02 0.02 j0.02!0.02 i0.02 0.04 0.04 0.04 M1 I 0.3 1 0.4 0.14 0.1 0 0.07 0 07 2 01 M2 00.55 0.5 0.5 |0.5 0.5 0 05 05 P1 426 426 279 132 59 159 j279 132 P2 i 1 i 1 1 P3 2 1 1 1 i 1 1 1 1 i 1 P3 1 P4 i0.6 06 0.6,0.6 0.6 0,6 0.6 I 4,6 O a86 0 19.0 I 0. TC-1 190.0 190.1 80.0 i180.0 i170.0 180.0 190.0 190.0 TC-2 1248.9 248.6 233.7 i227.8 i220.1 227.3 227.5 228.9r TC- 3 190.2 i190.2 180.1 180.2 170.1 i 180.1i 190.0 190.1 TC-4 190.1 190.1 180.1 180.1 170.1 180.l| 190.0 190.1| I~T~~ ~'OC-5 10 180. 0 190.0 190 0.0 TC-5 190.1 1 190.2 180.1 180.1 170.1 TC- 1908. 9 1 TC-6 190.1 190.1 180.1 180.1 170.1 180. 190.0 190.0, TC-,7 190.0 i 190.1 1800 1800 1170.0 180.01 190.0 i 190.0 TC-8 190.0 190.0 j180.0 180.0 170.0 [ 180. J 190.0 190.0 a 180.0 170.0 18 TC-9 f232.7 1233.5 3213.8 203.9 196.4 204. 3 213.2 214.3, TC-10 192 0 192 0 192.2 183.0 174.1 184.9 191,4 192.9' TC-11 209.2 205.5 i195.3 199 4 196.2 207.2 205.7 205.9 TC-12 215.9 208.4 199.6 209.0 202 2017.34 210.3 219.5 TC-13 760 760 77.0 77.5 77.5 77.0 770 77. 177.0i 18.77 77.5 17 TC-14 177.0 577.0 j78.0 78.5 77.5 77.55. 775 TC-15,77.5 78.0 79.0 79.0 79.0 78.5 78.5 78.57 5 TC-16 77.0 77.0 77.5 78.0 78.0 78.0780 78. TC-16 77.0 177 77 7.0 77.50 78.0 78

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