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## Material Information- Title:
- Corresponding states relationships for transport properties of pure dense fluids
- Creator:
- Tham, Min Jack, 1935- (
*Dissertant*) Gubbins, K. E. (*Thesis advisor*) Reed, T. M. (*Reviewer*) Walker, R. D. (*Reviewer*) Blake, R. G. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1968
- Copyright Date:
- 1968
- Language:
- English
- Physical Description:
- xxii, 164 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Argon ( jstor )
Correlations ( jstor ) Correspondence principle ( jstor ) Fluids ( jstor ) Gases ( jstor ) Liquids ( jstor ) Molecules ( jstor ) Self diffusion ( jstor ) Transport phenomena ( jstor ) Viscosity ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic -- Chemical Engineering -- UF Fluid dynamics ( lcsh ) Fluids ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- An important consideration for the chemical engineer is his ability to predict reliable values for various properties of chemical substances that are needed for design equations. Because of the large number of combinations of substances, compositions, pressures, temperature, etc., that may be encountered, it is often neither feasible nor desirable to make experimental measurements of such properties. The need for reliable correlations has become more acute with the increasing use of high-speed electronic computers in designing chemical plants. Such design requires suitable mathematical expressions for properties as a function of operating conditions. Even when experimental data in tabulated form are available it is most conveniently introduced into the computer in the form of soundly based theoretical equations. In the long term the only satisfactory approach is one that is firmly based on molecular considerations. Theories for dilute fluids are well established. The kinetic theory of gases is capable of describing the thermodynamic and transport properties accurately. However, for dense fluids there is still no theory which can parallel the success of the dilute gas theory. Although there are several rigorous molecular theories for liquids, none have yet been developed to a stage that would yield numerical results. Theoretical study of transport properties is more difficult than that of thermodynamic properties. In thermodynamics, all equilibrium properties can be obtained once the partition function of the system is known. However, for transport processes each of the transport properties has to be formulated separately. This situation occurs because the thermodynamic equilibrium state is unique, whereas there are many types of non-equilibrium state. This dissertation considers corresponding states correlations of transport properties of fluids, particularly liquids. This principle has previously proved of great value to engineers in predicting thermodynamic properties. Chapter 1 provides a brief survey of the present status of theories of transport properties of dense fluids. Chapters 2-4 contain a detailed study of the free volume theory of viscosity and self-diffusion coefficient, with particular emphasis on developing corresponding states relations for the parameters involved. The next two chapters discuss a more direct corresponding states treatment, and a new theory is proposed for polyatomic molecules.
- Thesis:
- Thesis--University of Florida, 1968.
- Bibliography:
- Bibliography: leaves 155-163.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
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- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 000955748 ( AlephBibNum )
16996953 ( OCLC ) AER8377 ( NOTIS )
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CORRESPONDING STATES RELATIONSHIPS FOR TRANSPORT PROPERTIES OF PURE DENSE FLUIDS By MIN JACK THAM A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1968 PREFACE An important consideration for the chemical engineer is his ability to predict reliable values for various properties of chemical substances that are needed for design equations. Because of the large number of combinations of substances, compositions, pressures, temperature, etc.,that may be encountered, it is often neither feasible nor desirable to make experimental measurements of such properties. The need for reliable correlations has become more acute with the increasing use of high-speed electronic computers in designing chemical plants. Such design requires suitable mathe- matical expressions for properties as a function of operating conditions. Even when experimental data in tabulated form are available it is most conveniently introduced into the computer in the form of soundly based theoretical equations. In the long term the only satisfactory approach is one that is firmly based on molecular considerations. Theories for dilute fluids are well established. The kinetic theory of gases is capable of describing the thermodynamic and transport properties accurately. However, for dense fluids there is still no theory which can parallel the success of the dilute gas theory. Although there are several rigorous molecular theories for liquids, none have yet been developed to a stage that would yield numerical results. Theoretical study of transport properties is more difficult than that of thermodynamic properties. In thermodynamics, all equilibrium properties can be obtained once the partition function of the system is known. However, for transport processes each of the transport properties has to be formulated separately. This situation occurs because the thermodynamic equilibrium state is unique, whereas there are many types of non-equilibrium state. This dissertation considers corresponding states correla- tions of transport properties of fluids, particularly liquids. This principle has previously proved of great value to engineers in predicting thermodynamic properties. Chapter 1 provides a brief survey of the present status of theories of transport properties of dense fluids. Chapters 2-4 contain a detailed study of the free volume theory of viscosity and self-diffusion coefficient, with particular emphasis on developing corresponding states relations for the parameters involved. The next two chapters discuss a more direct corresponding states treatment, and a new theory is proposed for polyatomic molecules. The author is greatly indebted to Dr. K. E. Gubbins, chairman of his supervisory committee, for his interest, stimulation, advice and encouragement during the course of this research. He is also grateful to Dr. T. M. Reed for helpful discussions. Finally,he expresses his sincere appreciation to Dr. T. M. Reed, Professor R. D. Walker and Dr. R. G. Blake for serving on the committee. TABLE OF CONTENTS PREFACE......................................................... LIST OF TABLES................................................. LIST OF FIGURES............................................... .. LIST OF SYMBOLS................................................ ABSTRACT........................................................ CHAPTERS: 1. INTRODUCTION......................................... 1.1 The Time Correlation Function Theory............. 1.2 The Kinetic Theory of Liquids................... 1.3 The Enskog Theory................................ 1.4 Activation and Free Volume Theories.............. 1.5 The Correspondence Principle..................... 1.6 Empirical Correlations ........................... 1.7 Summary.......................................... 2. PREVIOUS WORK ON ACTIVATION AND FREE VOLUME THEORIES. 3. IMPROVED FREE VOLUME THEORY .......................... 3.1 Temperature Dependence of V .................... 3.2 Volume Dependence of E ......................... 4. TEST OF IMPROVED FREE VOLUME THEORY .................. 4.1 Corresponding States Relationships.............. 4.2 Test of Proposed Correlations ................... 4.3 Summary ......................................... Pa ge ii vii ix xii xx 1 1 3 8 14 16 16 20 22 27 28 35 41 42 44 52 TABLE OF CONTENTS (Continued) Page 5. PRINCIPLE OF CORRESPONDING STATES FOR MONATOMIC FLUIDS............................................... 55 5.1 Molecular Basis of the Correspondence Principle. 55 5.2 Previous Work.................................... 63 5.3 Test of Correspondence Principle for Inert Gases 65 5.4 Summary......................................... 84 6. PRINCIPLE OF CORRESPONDING STATES FOR POLYATOMIC NON- POLAR FLUIDS.. ....... ............................... 86 6.1 Problems in Polyatomic Fluids................... 86 6.2 Derivation of Corresponding States Principle for Polyatomic Molecules.............................. 90 6.3 Test of Correspondence Principle for Polyatomic Molecules.............................................. 103 6.4 Comparison with Theory for Thermodynamic Pro- perties......................................... 124 7. CONCLUSIONS.......................................... 132 7.1 Free Volume Theory............................... 133 7.1.1 Extension to Mixtures.................... 133 7.1.2 Electrolyte Solutions..................... 133 7.2 Corresponding States Principle.................. 134 7.2.1 Mixtures................................. 134 7.2.2 Polar Substances......................... 134 7.2.3 Fused Salts.............................. 134 7.2.4 Thermodynamic Properties................. 134 7.2.5 Prediction of c.......................... 135 TABLE OF CONTENTS (Continued) Page APPENDICES..................................................... 136 1. Chung's Derivation of the Free Volume Equations...... 137 2. Further Test of Proposed Free Volume Theory........... 144 3. Solution of Equations of Motion for Oscillator....... 148 4. Hamiltonian for the Three-Dimensional Oscillator..... 151 LITERATURE CITED...................... ...... .... ........ ......... 155 BIOGRAPHICAL SKETCH .................... ..... ................... 164 LIST OF TABLES Table Page 1.1 Test of Rice-Allnatt Theory for Shear Viscosity of Argon................................. ............. 6 1.2 Test of Rice-Allnatt Theory for Thermal Conductivity of Argon.............................................. 7 4.1 Viscosity Data Sources and Range of Conditions....... 46 4.2 Parameters for Viscosity Prediction.................. 47 4.3 Self-Diffusivity Data Sources and Range of Conditions 48 4.4 Parameters for Self-Diffusivity Prediction........... 49 5.1 List of Parameters of Monatomic Molecules............. 78 5.2 Coefficients of Saturated Liquid Viscosity Equation for Monatomic Molecules .............................. 80 5.3 Coefficients of Saturated Liquid Thermal Conducti- vity Equation for Monatomic Molecules ................ 81 5.4 Coefficients of Saturated Liquid Self-Diffusivity Equation for Monatomic Molecules..................... 82 5.5 Coefficients of High Pressure Viscosity Equations for Monatomic Molecules................................ 83 5.6 Coefficients of High Pressure Thermal Conductivity Equations for Monatomic Molecules .................... 85 6.1 Parameters for Correspondence Correlation of Poly- atomic Molecules. ................................... 105 6.2 Sources of Transport Property Data.................... 107 6.3 Coefficients of Saturated Liquid Viscosity Equation for Polyatomic Molecules............................... 117 6.4 Coefficients of Saturated Liquid Thermal Conductivity Equation for Polyatomic Molecules.................... 119 6.5 Coefficients of Saturated Liquid Self-Diffusivity Equation for Polyatomic Molecules.................... 120 6.6 Coefficients of High Pressure Viscosity Equations for Polyatomic Molecules............................. 122 LIST OF TABLES (Continued) Table Page 6.7 Coefficients of Gas Thermal Conductivity Equation for Polyatomic Molecules .............................. 125 6.8 Coefficients of High Pressure Thermal Conductivity Equations for Polyatomic Molecules.................... 126 6.9 Comparison of c with Corresponding Parameter of Hermsen and Prausnitz................................. 127 viii LIST OF FIGURES Figure Page 1.1 Theoretical and Experimental Saturated Liquid Viscosity of Argon Square-Well Model............... 9 1.2 Theoretical and Experimental Saturated Liquid Self- Diffusion Coefficient of Argon Square-Well Model... 10 1.3 Theoretical and Experimental Viscosity of Argon - Enskog Theory........................................ 13 1.4 Theoretical and Experimental Viscosity of Argon - Modified Enskog Theory ............................... 15 1.5 Theoretical and Experimental Thermal Conductivity - Horrocks and McLaughlin Theory....................... 17 1.6 Viscosity Isotherms of Krypton as Functions of Density............................................... 19 3.1 Viscosity of Argon.................................... 29 3.2 Viscosity of Nitrogen................................. 30 3.3 Variation of v* with T* .............................. 33 3.4 Activation Energy as a Function of Volume............ 36 3.5 Model of a Molecular Jump............................ 37 3.6 Corresponding States Correlation of Activation Energy vs. Volume ....................................... 39 4.1 Test of Free Volume Theory for Liquid Argon Viscosity 43 4.2 Test of Free Volume Theory for Liquid Xenon Viscosity 50 4.3 Test of Free Volume Theory for Saturated Liquid Methane Viscosity........................ ............... 51 4.4 Test of Free Volume Theory for Saturated Liquid Methane Self-Diffusivity ............................. 53 5.1 Smoothing Lennard-Jones [6,12] Parameters,-- vs. cu.. 68 kT c P 1/3 5.2 Smoothing Lennard-Jones [6,12] Parameters,(-- vs. C ......................................Tc./ ..... 69 5.3 Correspondence Principle for Saturated Liquid Vis- cosity Monatomic Molecules ......................... 70 LIST OF FIGURES (Continued) Figure Page 5.4 Correspondence Principle for Saturated Liquid Thermal Conductivity Monatomic Molecules ..................... 71 5.5 Correspondence Principle for Saturated Liquid Self- Diffusivity Monatomic Molecules.................... 72 5.6 Reduced Bulk Viscosity of Saturated Liquid Argon........ 74 5.7 Reduced Viscosity Isobars as Functions of Reduced Temperature Monatomic Molecules...................... 76 5.8 Reduced Thermal Conductivity'Isobars as Functions of Reduced Temperature Monatomic Molecules............... 77 6.1 Simple Correspondence Principle Polyatomic Molecules. 89 6.2 Simple Corresponderm Principle for Saturated Liquid Viscosity Polyatomic Molecules....................... 91 6.3 A One-Dimensional Oscillator Model..................... 93 6.4 Superimposed Rotational Motions of a Molecule.......... 94 6.5 Improved Correspondence Principle for Saturated Liquid Viscosity Polyatomic Molecules.......................110 6.6 Improved Correspondence Principle for Saturated Liquid Thermal Conductivity Polyatomic Molecules............ 111 6.7 Improved Correspondence Principle for Saturated Liquid Self-Diffusivity Polyatomic Molecules ................112 6.8 Test of Improved Correspondence Principle for High Pressure Viscosity......................................113 6.9 Test of Improved Correspondence Principle for Gas Thermal Conductivity Polyatomic Molecules............ 115 6.10 Test of Improved Correspondence Principle for High Pressure Thermal Conductivity..........................116 6.11 Simple Correspondence Principle Vapor Pressure....... 130 6.12 Improved Correspondence Principle Vapor Pressure.....131 A2.1 Test of Free Volume Theory for Liquid Nitrogen Viscosity...............................................145 x LIST OF FIGURES (Continued) Figure Page A2.2 Test of Free Volume Theory for Liquid Krypton Viscosity............................................ 146 A2.3 Test of Free Volume Theory for Liquid Neon Viscosity. 147 A4.1 A Three-Dimensional Oscillator Model................. 151 A A o AiA2,A3 ,A A23 A' A" al'a2'a3,a ,a5 a B B o o B" B1,B2 ,B3 ,B4 B' B" B a b o C1 a c c 0 c 1 c2 c3 c4 LIST OF SYMBOLS = pre-exponential factor of free volume equation for viscosity = A/T1/2 = arbitrary constants = constants = constants = coefficients of viscosity equation = molecular diameter = pre-exponential factor of free volume equation for self-diffusivity = B/T1/2 1/2 1/2 = (B 12)/(ok ) = arbitrary constants = constants = arbitrary constant = the rigid sphere second virial coefficient = coefficients of thermal conductivity equation = empirical constant = arbitrary constant = a characteristic factor defined by equation (6.37) = total concentration 3 = v/r = constant = (r2/rl ) = constant = (c 03/v-.) = constant 1 m = 6/kT- = constant m D = self-diffusion coefficient DAB = binary diffusion coefficient '. 1/2 1/2 D" = Dm l2/GI = reduced self-diffusivity by simple correspondence principle '1/2 1/2 D = Dm /co- = reduced self-diffusivity by proposed correspondence principle DAB = reduced mutual diffusion coefficient defined by equation (5.33) d = distance of closest approach of 2 molecules dl,d2 = coefficients of self-diffusivity equation 1 E = activation energy at constant volume v E = E /RT = reduced activation energy v v m e = varying activation energy per molecule e. = energy level e = height of potential barrier in equilibrium liquid e = E /N v v F. = y-component of intermolecular force on molecule i ly () time smoothed singlet distribution function f = time smoothed singlet distribution function -(2) f = time smoothed pair distribution function f = a function of density given by equation (1.23) g = universal function go(r) = equilibrium radial distribution function g = geometric factor H = Hamiltonian xiii HCM = Hamiltonian of center of mass coordinates h = Planck constant h = h/(oam) = reduced Planck constant I = moment of inertia I = viscosity constant used in equation (1.22) Jxx = defined by equation (1.6) J = defined by equation (1.5) K = kinetic energy KCM = kinetic energy of center of mass coordinates Krot = kinetic energy of rotational motion K1 = (n/n-6)(n/6)6/(n-6) k = Boltzmann constant k11 = defined by equation (6.8) k12 = defined by equation (6.9) k22 = defined by equation (6.10) L = Lagrangian 1 = jump distance M = molecular weight MA = molecular weight of component A MB = molecular weight of component B m = mass of molecule N = number of molecules N = Avogadro number n = repulsive exponent in [6,n] potential law n1 = singlet number density xiv P = pressure P = critical pressure c P = critical pressure of a mixture A-B cAB 3 P- = Pa /c = reduced pressure by simple correspondence principle = Pa /ce = reduced pressure by improved correspondence principle p = momentum ix = momentum in x-direction of molecule i (i) p = probability of a molecule having activation energy Se. and free volume v 1 p 1/2 p = p/(m) = reduced momentum Q = partition function Q = defined by equation (5.35) q = thermodynamic free volume r = position vector r.. = r r. --Lj -j -- r = r/C = reduced distance (s/c) = a constant factor used by Hermsen and Prausnitz k (166) s = defined by equation (1.7) T = temperature T = critical temperature c T = critical temperature of a mixture A-B cAB T = melting temperature m T = kT/E = reduced temperature by simple correspondence principle xv T t t U(r) U(O,O) U(x,Yx) x U(x1....xN) U(0) V V Vf V x V" Vk V 0 V o 1 V o v1* V o =kT/ce = reduced temperature by improved correspondence principle = kT/[e(s/c)k] = reduced temperature used by Hermsen and Prausnitz (166) = time 1/2 1/2 = tE /a m= dimensionless time N S u(r i) = pair potential iij i = potential energy of.a molecule at the center of a cell = potential energy of a molecule in a cell = potential energy of N molecules = potential energy of N molecules at the centers of their cells = molal volume = velocity vector = molal free volume = velocity in the x-direction = (V/3 ) = reduced molal volume = characteristic volume used by Hermsen and Prausnitz (166) = V/V m = V/V = reduced volume of Hermsen and Prausnitz (166) = minimum free volume per mole needed for molecular flow to occur = V /o = reduced minimum free volume o = minimum free volume when T* = 1.0 = V /V = reduced minimum free volume when T = 1.0 o m v = V/N = average volume per molecule vf = v v = free volume f o v = V /N m m v = V /N v = varying free volume v0 = (d/o)3 v =V /N = minimum free volume per molecule needed o o for molecule flow to occur WIW2 = number of configurations for distribution of molecules x. = x-component of position vector r of molecule i Y = average percent error(see page 80) y = b/V Z = configurational partition function z = number of nearest neighbors in a ring a = Lagrange multiplier = 1/kT = Lagrange multiplier 7 = free volume overlap factor = characteristic energy parameter in [6,n] potential = molecular friction constant S= shear viscosity S= dilute gas viscosity *o " = 2/(me)/2 = reduced shear viscosity by simple correspondence principle 2 1/2 r = =r /c(me) reduced shear viscosity by improved correspondence principle o = defined by equation (5.36) S= angle vector (in terms of Eulerian angles e,0,/) xvii e = x-component of an angular rotation x = bulk viscosity K= Kc2/(m)/2 = reduced bulk viscosity by simple correspondence principle K2 1/2 C = IKc /c(rn) = reduced bulk viscosity by improved correspondence principle A = Lagrange multiplier S= thermal conductivity X = dilute gas thermal conductivity o 2 1/2 1/2 S= XG m /kE = reduced thermal conductivity by simple correspondence principle 2 1/2 1/2 S= X m /cke = reduced thermal conductivity by improved correspondence principle = B'/A' = constant = constant T = a constant which has a value of 22/7 p = density (pD) = value of the product of density and self-diffusivity of a dilute gas p" = reduced density a = characteristic distance parameter in [6,n.] potential Gl ,2 = characteristic distance parameter in square-well potential 0 = a universal potential energy function 0x = angle associated with constant angular velocity = a universal function X = a factor defined by equation (1.13) Xh = probability of hole formation xviii x = angle associated with superimposed angular velocity x lo I,2 = the radial functions in the spherical harmonic expansion of the nonequilibrium distribution function g(r) (1, 1)- (2,2)* S (2 *= dimensionless collision integrals o = Pitzer factor O = angular velocity O, = angular velocity in a harmonic motion xix Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CORRESPONDING STATES RELATIONSHIPS FOR TRANSPORT PROPERTIES OF PURE DENSE FLUIDS By Min Jack Tham June, 1968 Chairman: K. E. Gubbins Major Department: Chemical Engineering Two approaches have been considered for developing corres- ponding states correlations for dense fluid transport properties. The first is based on the free volume theory for viscosity and self- diffusion coefficient. This theory has been improved to account quantitatively for the nonlinearity of constant volume plots of In(r/T /2) vs. T-1 and In(D/T /2) vs. T-1 for simple liquids, by the introduction of the temperature dependence of V and the volume dependence of E Corresponding states relations are presented for these two quantities. The improved free volume theory accurately describes the viscosity and self-diffusion coefficient behavior of liquids composed of simple, nonpolar molecules that may be expected to approximately obey a Lennard-Jones [6,121 potential law, for the density range of p > 2p c. It is in this range that most other theories fail. The only parameters required for the viscosity and self-diffusion coefficient prediction are melting and critical para- meters, together with one experimental value of viscosity (or self- diffusivity). Direct corresponding states correlations of transport pro- perties have also been studied, and provide a more general if slightly less accurate method of prediction. The simple corresponding states principle derived by Helfand and Rice from the time correlation func- tion expressions of statistical mechanics has been shown to apply closely for the transport properties of monatomic molecules over the entire range of temperature and density conditions for which measure- ments are available. In this simple form, however, it was found not to apply to polyatomic fluids in the dense phase. After careful re- examination of the four assumptions on which the derivation of the simple correspondence principle is based, it is shown that hindered rotation at high density is the most probable cause for the deviation of polyatomic fluids from the correspondence principle. A correspondence principle for polyatomic molecules has been derived, taking into account the effects of hindered rotation and making use of a simple harmonic oscillator model. The proposed equations have been tested exhaustively for the viscosity, thermal conductivity and self-diffusion coefficient of a large number of polyatomic fluids over a wide range of temperature and density condi- tions. The molecules tested include saturated hydrocarbons from methane to nonane, spherical molecules such as neopentane and carbon tetrachloride, flat molecules such as benzene, and simple diatomic molecules such as nitrogen. The improved correspondence principle predicts the transport properties of saturated liquids within experi- mental error. For high pressures the average percentage errors of the predicted viscosity and thermal conductivity are slightly higher. xxi It is also shown that the proposed model can account in a reasonable way for thermodynamic properties of polyatomic fluids. xxii CHAPTER 1 INTRODUCTION In this chapter a brief survey and evaluation is presented of the principal theories and empirical correlations that have been pro- posed to explain dense fluid transport properties. For pure fluids all transport properties are functions of temperature and density, namely, ) = T)(T,p), K = c(T,p), D = D(T,p) and X = X(T,p). None of the present theories provide a satisfactory means of predicting transport properties over the entire range of temperature and density conditions for even quite simple fluids. Also assumptions of dubious validity must be made in most cases in order to be able to make any comparison with experiment at all. In the following survey the more rigorous approaches are first described, followed by simple model theories and ending with the empirical correlations. 1.1 The Time Correlation Function Theory In the last decade a new theory, called the time correlation function theory (also known as the fluctuation-dissipation theory) has been developed to treat transport processes. This theory treats transport processes in terms of equilibrium ensembles whose properties are known. In a sense the time correlation function plays a similar role to the partition function in statistical thermodynamics. In statistical thermodynamics, all thermodynamic properties of any parti- cular system can be evaluated if the partition function is known. In the same manner the transport coefficients can be obtained if the appropriate time correlation functions can be calculated. However, in one respect the analogy breaks down. In statistical thermodynamics a single partition function determines all the thermodynamic properties, whereas in transport processes different time correlation functions are needed for different transport processes. A good review of the time correlation function approach to transport processes is given by Zwanzig (1). The time correlation function expressions for the transport coefficients may be derived by several methods. A simple derivation involves starting with expressions analogous to the Einstein equation for the self-diffusion of Brownian particle (2). The following expressions for transport coefficients are obtained. 0O Self-diffusion coefficient D = '0 SO Shear viscosity 7 = Vk o Bulk viscosity C = 'o Thermal conductivity X = -12 VkT where V (t) = velocity in the x-direction at time t N Jxy ~. Pixpiv 1 4 1 ly j xx ix ix jxx = + xF PV i im i iX i=l N N sx Pix ix 1> 2m i xx 2 i=l j=i + x..F.. 1 1] 1jX J N Pix _PV Pi m Nm Pix i=1 (1.3) (1.4) (1.5) (1.6) (ij xx (1.7) For dilute gases the time correlation functions may be evaluated and the expressions for the transport coefficients can be reduced to the forms obtained by solving the Boltzmann equation. However, for dense fluids no satisfactory expressions have as yet been obtained, although several approximate equations have been suggested (3). One approach has been to assume that the time correlation function decays exponen- tially. Alternatively one may perform molecular dynamics calculations (4-6). In this method a very large amount of computation is involved; the principal interest of such an approach is to provide data with which to compare various theoretical expressions for the time correla- tion functions. The time correlation theory gives a formal description of transport processes; however it is still not developed to a sufficiently advanced stage to be of practical value in the calculation of trans- port coefficients. To the chemical engineer, the most interesting feature of the theory in its present state of development is that it provides the basis for a rigorous development of the corresponding states principle for dense fluid transport properties. This is dis- cussed in a later section of this chapter. 1.2 The Kinetic Theory of Liquids The kinetic theory of dilute gases is already very well developed (7), and provides a method of accurately estimating their transport properties. It is therefore reasonable to attempt a similar development for dense gases and liquids. Such an attempt was first made by Kirkwood (8,9),whose approach was based on casting the Liouville equation for the distribution function into an equation of the Fokker-Planck type. The main disadvantage in this theory is that a large number of approximations have to be made in obtaining the -(1) -(2) Fokker-Planck equations for f and f the time smoothed singlet and pair distribution functions respectively. In addition to these deficiencies, Kirkwood's method has drawn much criticism for assuming the mean momentum change for a collision to be small. The equations derived by the method of Irving and Kirkwood (9) for transport coeffi- cients of liquids are as follows (10) Shear viscosity nkT +3 du n mkT S+ dr (r) )dr (1.8) Bulk viscosity 2 0 o____ 3 du(r) = 9kT r dr go(r)4o (r)dr (1.9) 9kT I dra0 0 Thermal conductivity 2 r 5k nTj n1 kT 1 C + C 3 u(r) u(r)) go(r) dr ~oTgg() r dr + u(r) r u(r) 0go(r) 2 r dr (1.10) kT Self-diffusion coefficient D = (1.11) where 2 2 = 3 p r 2g(r)7 u(r)dr (1.12) and o,'2 = the radial functions in the spherical harmonic expansion of the nonequilibrium distribution function g(r) A modification of the Kirkwood theory due to Rice and Allnatt (11,12) attempts to avoid the assumption that the mean collisional momentum change is small. In the Rice-Allnatt theory, a potential energy function is assumed in which the molecule has a hard core together with an outer shell which interacts with a soft attractive potential. A hard core collision leads to a large momentum transfer and tends to vitiate the Brownian motion approximation which is inherited in the Fokker-Planck equation. These authors therefore treated the rate of change of the distribution function f due to hard core colli- sions by means of an Enskog-type collision term, and treated the rate of change of the distribution function due to motion in the attractive field of surrounding molecules between collision by Kirkwood's approach. In order to calculate the transport coefficients from the Kirkwood or Rice-Allnatt theory one must have accurate values for the intermolecular potential, and also the equilibrium radial distribution function has to be known quantitatively. The latter requirement is especially difficult to meet since the experimental radial distribution function has not been determined accurately. Making use of the Kirkwood theory, Zwanzig, Kirkwood, Stripp and Oppenheim (13) were able to calculate the shear viscosity and bulk viscosity for liquid argon near the boiling point. Their calculated value for shear viscosity was O = 0.73x10-3 poise, while the experimen- tal value of shear viscosity for argon at the same temperature is -3 71= 2.39x103 poise. Thus the calculated value is in error by roughly a factor of three. For bulk viscosity, their computation yielded S= 0.36x103 poise. However, according to Naugle's (14) measurement of bulk viscosity for liquid argon at approximately the same temperature, -3 K = 1.7x10 poise. The discrepancy between calculated and experimen- tal values is more than a factor of four. Because of the non-trivial mathematics involved, the Kirkwood theory has not been tested for the temperature and density dependence even for very simple molecules such as argon. The Rice-Allnatt theory has been tested by Lowry, Rice and Gray (15) for viscosity. They found quite good agreement between calculated and experimental values of argon at densities slightly less than the normal liquid density (Table 1.1). To compare theory and experiment they assumed u(r) to be given by the Lennard-Jones [6,12] potential. The radial distribution functions, g(r) for different temperatures were those obtained theoretically by Kirkwood, Lewinson and Alder (16). The radial distribution functions g(r) and the derivative of u(r) were modified by introducing an empirical parameter cl, which was obtained from equation of state data. TABLE 1.1 TEST OF RICE-ALLNATT THEORY FOR SHEAR VISCOSITY OF ARGON 3-3 T)xlO poise Density, gcm 1.12 1.375 Temperature, OK 128 133.5 185.5 90 Pressure, atm. 50 100 500 1.3 c 0.9819 0.9827 0.9887 0.9705 r)(calc) 0.727 0.730 0.771 1.74 r(expt) 0.835 0.843 0.869 2.39 % error 13.0 13.4 11.3 27.2 Ikenberry and Rice (17) tested the Rice-Allnatt theory on the thermal conductivity of argon for the same temperature and density conditions as the viscosity values shown in Table 1.1. The calculated values were not entirely theoretical since they made use of experi- mental friction constants calculated from the self-diffusion coeffi- cient data of Naghizadeh and Rice (18). Close agreement was found between theory and experiment for the entire temperature and density range considered as shown in Table 1.2. -3 Density, gcm. Temperature, OK Pressure, atm. X(calc) X(expt) % error TABLE 1.2 TEST OF RICE-ALLNATT THEORY FOR THERMAL CONDUCTIVITY OF ARGON 4 -1 -1 XxlO4, cal.cm. sec.l C 1.12 128 133.5 185.5 50 100 500 2.83 1.85 1.77 2.96 1.89 1.86 4.4 2.1 4.8 Davis, Rice and Sengers (19) further modified the Rice-Allnatt theory by using the square-well potential energy function. Because of the simple form of this function the Rice-Allnatt theory is greatly simplified and the evaluation of the pair correlation function reduces to the determination of g(cq) and g(2 ). In order to calculate the transport coefficients from this theory, one must know the values of the parameters ,' c2' g(01)' g( 2) and have equation of state data. 1.375 90 1.3 1.84 1.87 1.6 In general o1, a and G for the square-well potential are available (20). In principle g(al) and g(2 ) can be calculated from equilibrium statistical mechanics. However there is still no satisfactory method of making such theoretical calculations accurately. Davis and Luks (21) estimated the values of g(al) and g(2) from equation of state and thermal conductivity data of argon by solving simultaneously the modified Rice-Allnatt expressions. They then calculated the viscosity coefficient and self-diffusion coeffi- cient of argon at various temperatures and pressures. The results are shown in Figures 1.1 and 1.2. The discrepancy between theory and experiment for viscosity is as much as 19% and for the self-diffusion coefficient is up to 34%. The experimental self-diffusivity data used by Davis et al. as shown in Figure 1.2 are apparently extrapolated values obtained from the self-diffusion coefficient data of Naghizadeh and Rice (18). Luks, Miller and Davis (22) have also made use of the modified Rice-Allnatt theory to calculate transport coefficients for argon, krypton and xenon. Discrepancies between theory and experiment become larger as the density increases. 1.3 The Enskog Theory (7) Enskog extended the dilute gas kinetic theory to dense fluids composed of hard spheres. As a gas is compressed the mean free path, which for dilute gases is infinitely large with respect to the molecular diameter, gradually decreases until at high density it is of the same order of magnitude as the diameter of a molecule. During this process the intermolecular collision frequency tends to get larger due to the decrease in the mean free path; at the same time it decreases because -4 C" C) -r ,-- rJ -I C) I 1-4 U F-4 (asTod) O c : x c..J 0 u r c ** E- 4- r0 E-a 3 ,-1 i-1 -. J c c o 1-1 0 1 ) 1 I I I I I I I I I I I 10 I0 ,-44 II .1- C) C- c Gu m o / / 0 S/ a u cu // 0 O *t S i H O N // / / < I I I- / / -" I f / 0 *0 o - So i I I y. I o / -0 0D 0 C) 0 (i"a/m) /O x H" at close separations the molecules tend to shield one another from collisions with more distant neighbors. The net change in collision frequency was found to differ from that of a dilute gas by a factor S, which is given by b \b 02 /b 3 X = 1.0 + 0.625 ) b + 0.2869 2 + 0.115 ) (1.13) where b 7 iNO3 = the rigid sphere second virial coefficient. o 3 Assuming the collisional transfer of'momentum and energy between colliding molecules to be instantaneous, Enskog modified the Boltzmann equation for higher densities and solved it to obtain the following equations for transport coefficients. Self-diffusion coefficient Shear viscosity no Bulk viscosity K = 7o Thermal conductivity (monatomic molecules) where The value of y may be 0 0 b OD o (pD) Vy b -- + 0.8 + 0.761y V y b o 1.002 -- y V 0 + 1.2 + 0.757y V y (1.14) (1.15) (1.16) (1.17) b o y X obtained from the equation of state by PV + RT 1 + y RT Enskog suggested as an empirical modification of the hard sphere (1.18) theory the use of thermal pressure in place of external pressure. Thus according to this modification y becomes y = V( 1 (1.19) Equations (1.15) and (1.17) predict that when ('rV) and Q~V) are plotted as functions of y, the curves will go through minima at y = 1.146 and y = 1.151 respectively, having [V]min = 2.54570ob at y = 1.146 (1.20) and [XV]. = 2.938X b at y = 1.151 (1.21) mn o o In the above transport coefficient expressions, b0 is the only unknown parameter. It may be obtained from equations (1.20) and (1.21) by using experimental values of [rV]min and [XV]min The Enskog dense gas theory gives a useful approximate description of transport coefficients above the critical temperature and at densities less than the critical value. The theory fails at temperature below the critical temperature and at densities higher than the critical density. Under these conditions the rigid sphere approxi- mation is in serious error. The Enskog theory has been tested for thermal conductivity and viscosity of argon at various temperatures and densities (23). Figure 1.3 shows the comparison of calculated and experimental vis- cosities of argon at 0C and 750C. The value of bo used in these calculations was obtained from equation (1.20) using data at 0C. The agreement between theory and experiment is moderately good at 0 C for 13 Co O C - 00 0 0r ittS oo \ 8 0 o \\ C \O \O a o 00 0 0 0 u Q- \o I co co 0 < G \ 0 0 ( (as\od) 0 x L. densities up to about 0.8 g/cc. At higher densities the theory fails to predict the correct density dependence. As can be seen in Figure 1.3, the Enskog theory does not predict the correct temperature dependence. Dymond and Alder (25) recently modified the Enskog theory by using temperature dependent rigid sphere diameters obtained from the Van der Waals equation of state. With this modification, the theoretical and experimental values of viscosity are shown in Figure 1.4. The predicted temperature dependence is found to improve considerably; however the theory still fails to predict the density dependence at densities above 0.8 g/cc. 1.4 Activation and Free Volume Theories An activation theory for liquid transport properties was first proposed by Eyring (26). This type of theory assumes that a molecule spends a large fraction of its time oscillating about an equilibrium position in a cell, and only occasionally does it leave one cell to take up position in a neighboring vacant cell. Transport of mass and momentum are assumed to occur during such molecular transitions. Later modifications to Eyring's original theory have included the introduction of the concept of the fluctuating free volume. These theories are discussed in detail in Chapter 2. The free volume theories apply only at densities above approximately twice the critical value. Horrocks and McLaughlin (27) applied the activation and free volume theory to the thermal conductivity. They assumed a face- centered-cubic lattice structure for the liquid and that transfer of thermal energy down the temperature gradient was due to two causes: 15 o C--- C)l 60 0 O o N X 0 '- < CO 0 0 C 0 Q- \ CN -4 0 C 4 .0 \ \ 0 -e 0 O 0 Ca 1 0 \0d \O x O O C \' \ ( CC 0 0 C4 r(a c o 01 xL \' \ the actual transit of a molecule from one lattice site to another, and that due to the collisions of an oscillating molecule with its neighbors. For liquids the contribution due to the former cause is negligible when compared with that of the latter. The frequency of oscillation is determined by the molecular mass and the intermolecular force. Their theory has been tested for a number of simple liquids and the agreement between theory and experiment is often within 20%. Calculated and experimental thermal conductivities of a number of liquids are compared in Figure 1.5. 1.5 The Correspondence Principle The principle of corresponding states has been found very useful in the calculation of equilibrium properties of dense gases and liquids (28). However very few studies have been made of its applica- tion to transport properties of dense gases and liquids. As the principle of corresponding states will be discussed in great detail in later chapters it will not be discussed further here. 1.6 Empirical Correlations A very good review of the purely empirical methods of estimating transport coefficients is given in a new book by Reid and Sherwood (29). These methods have neither a theory nor a model to describe them. The many expressions proposed usually contain one or more empirical para- meters or constants which are said to characterize the structure or properties of the molecules; in most cases these parameters have to be supplied by the authors of the correlation. A typical example of these correlations is Souder's method for estimating liquid viscosity, which is one of the few empirical methods recommended by Reid and Sherwood. 0 - wI on 0, 0 0 0 ~J ~ NJ ( I oas UI-,m 'I ) OT x y 1- 01- I- 7 Souder's equation is I log(log 10r) =- p 2.9 (1.22) where T = liquid viscosity, centipoise -3 p = liquid density, gcm. I = viscosity constant calculated from atomic and structural constants supplied by the author M = molecular weight The results yielded by such methods are usually not very satisfactory. In spite of the author's claim that viscosity predictions are within 20%, errors are often greater than this. Thus the equation predicts a viscosity for acetic acid at 400C that is 36% too low (29). Similar empirical expressions are also available for liquid thermal conducti- vity and self-diffusion coefficient; the discrepancies between these equations and experimental data are of a similar order to that of the viscosity correlations. Among the empirical correlations, those based on residual- viscosity concepts have received much attention (30-32). Figure 1.6 is a plot of viscosity isotherms for krypton as a function of density. At the lower densities the curves are almost parallel, and this has led a number of workers to correlate the residual viscosity r no with density. The general form of this correlation is S- = f1(p) (1.23) where fl is a function of density only and oT is the dilute gas viscosity. However, as can be seen in Figure 1.6,the isotherms 0- C. - 0-. S L) C.) 000 0 0 0 L C) V 0C CN r- un ,-< (asTod) o0 x L .CO c~c N O 0 c4J 0 %-, 0 l1 o cJn O O > I-, (-) 0 cJ^4- intersect at higher densities, so that such correlations are not valid over the whole density range. The intersection occurs at densities of about twice the critical value. Above this density value the free volume theory is found to work well. 1.7 Summary Of the different approaches discussed above, the time correla- tion function theory and the kinetic theory of liquids are the most rigorous and correct descriptions of transport processes in dense fluids. In their present stage of development, however, they offer little immediate prospect of yielding methods of predicting transport properties for fluids of interest to chemical engineers. The Enskog dense gas theory provides good results for dense gases, but fails at densities a little above the critical value. For the lower densities quite good results may also be obtained from empirical equations, such as those employing the residual viscosity concept. At present a particular need exists for satisfactory methods of predicting transport properties for fluids at densities well above the critical, that is in the normal liquid density region. Toward this end two approaches are examined in detail in this dissertation. The free volume theory is first studied, and examined as a framework for developing predictive corresponding states relations for simple fluids. Although such a model lacks the desirable rigor present in the formal theories, it has the considerable advantage of being solvable, and suggests ways in which expressions may be obtained for mixtures. Chapter 2 briefly reviews previous work on free volume theories, and Chapters 3 and 4 contain the new contributions to the theory. The second approach consists of the development of corresponding states relationships directly from the time correlation function theory. The correspondence principle is particularly valuable when based on the rigorous statistical mechanical approach, but no serious attempt to apply the principle to dense fluid transport properties seems to have been made previously. This approach is examined in detail in Chapters 5 and 6. CHAPTER 2 PREVIOUS WORK ON ACTIVATION AND FREE VOLUME THEORIES Because of the present difficulties associated with developing a rigorous kinetic theory of dense fluids analogous to that available for dilute gases, considerable attention has been given to simplified models of the liquid state. Assumptions are introduced in the initial stages of the development, an attempt being made to obtain a model which incorporates the essential features of real liquids, but which is still solvable. Such attempts are exemplified by the activation and free volume theories for liquid transport properties. Eyring (26) may be considered a pioneer in the development of activation and free volume theories for liquid transport properties. In Eyring's original approach the equilibrium positions of molecules in a liquid were considered to be on a regular lattice similar to the crystal lattice in a solid. The transport of mass and momentum occurs as a result of a molecule jumping from one equilibrium lattice site onto a vacant neighboring site. In order to make such a jump a molecule is required to have the necessary activation energy to over- come the minimum energy barrier separating two adjacent sites, and at the same time a vacant site must be available. Using this model Eyring obtained the following expressions for liquid viscosity and self- diffusivity by making use of the absolute reaction rate theory: 1 e /kT S= 1/2 3(e m) (2.1) pv 12kT -e /kT D = 12 kT /e (2.2) 1/2 1/3 e (27nkT) qf More recently Weymann (35,36) used a statistical mechanical approach, and arrived at expressions very similar to those of Eyring. In his derivation of the equations for transport properties, Weymann gave a clearer picture of the physical model used and of the signifi- cance of the activation energy. Moreover his approach suggests a possible extension to liquid mixtures. Weymann also derived an expres- sion in which the probability of hole formation is related to the volume and energy needed to form a hole. The equations obtained by Weymann for viscosity and self-diffusion coefficient are: l \ /2e /kT 3= 1 /3 (2kT) /2e (2.3) 4h ivl/3 2Xhl 2kT 1/2 -e /kT D =-- -m e o (2.4) where Xh is the probability of hole formation. Eyring's activation theory was modified by McLaughlin (37) by introducing the concept of the probability of hole formation, as was done by Weymann. In his treatment, he assumed a face-centered- cubic lattice for the liquid. The most important modification made by McLaughlin was the attempt to relate the activation energy and energy of hole formation to the intermolecular forces by means of the Lennard-Jones and Devonshire cell theory of liquids. A similar expression for liquid viscosity was also obtained by Majumdar (38) using the tunnel model of Barker (39) in place of the Lennard-Jones and Devonshire cell model. The Eyring and Weymann theories yield good results for the viscosity of Arrhenius liquids (ones whose viscosity varies with temperature according to an equation of the type Inr = Ca + Ba/T), but cannot account for non-Arrhenius liquid behavior. In order to overcome this deficiency Doolittle (40-42) proposed empirical expres- sions which related liquid transport properties to the free volume, defined by vf = v Vo (2.5) where v is the hard-core volume of the molecule. His free volume o equations were placed on more solid theoretical grounds by Cohen and Turnbull (43,44) who derived them by a statistical mechanical method. They assumed that a molecule moves about in a cell in a gas-like manner, while the free volume available to each molecule fluctuates with time. A molecule is able to jump into a neighboring cell only if a free volume of a size greater than some minimum value v is available. They obtained the following expressions for the viscosity and self- diffusion coefficient: r = 1 2 exp (2.6) 3-ra f D = gla )1/2 exp o (2.7) The term "free volume" in these theories has a different meaning from that implied in thermodynamic free volume theories. The theories of Doolittle and Cohen and Turnbull assumed the molecules to behave as hard spheres; the free volume referred to here is the space in the fluid unoccupied by the hard spheres themselves. The average free volume per molecule is defined as v vo, where v = V/N. The free volume theory of these authors describes the viscosity behavior at atmospheric pressure, but fails to predict the temperature depen- dence of viscosity at constant volume. Recently, Macedo and Litovitz (45) proposed expressions for viscosity and self-diffusion coefficient in which the hole probability in Weymann's equations was replaced by the free volume expression obtained by Cohen and Turnbull. Molecular transport is assumed to occur if a molecule has sufficient energy, e to overcome intermolecular forces with its neighbors, and at the same time it has a free volume greater than the minimum free volume v needed for a jump to occur. The equations proposed for viscosity and self-diffusion coefficient are S= Aexp V v exp (2.8) D Bexp exp (2.9) By treating V and E as empirical constants Macedo and Litovitz o V have shown that equation (2.8) describes the viscosity behavior of a number of liquids over a range of temperature. Chung (46) has pre- sented an elegant statistical mechanical derivation of the equation of 26 Macedo and Litovitz. His derivation is presented in Appendix 1. Macedo and Litovitz assume the pre-exponential factor A in equation (2.8) to be proportional to temperature T although most other 1/2 workers predict a temperature dependence of T/2. Both types of temperature dependence of the pre-exponential factor have been tested in this work, and experimental results seem to give better agreement with theory when a temperature dependence of T1/2 is used. Thus throughout this work the pre-exponential factors A and B are assumed to be proportional to T1/2. Thus equations (2.8) and (2.9) may be rewritten as V E S= A exp ) exp (2.10) T /2o V-RT" D p 0o v (2.11) B exp exp -- 1/2 0 V V RT T o where A = o 1/2 T and B o T1/2 Macedo and Litovitz original equations contain a constant factor y which was introduced to account for the overlapping of free volumes. In the early part of this work, the constant 7 was evaluated for a number of molecules and was found to be close to unity. This factor will therefore not be included in the equations of viscosity and diffusivity. CHAPTER 3 IMPROVED FREE VOLUME THEORY The viscosity equation (2.8) has been tested by Macedo and Litovitz (45) for a variety of liquids and by Kaelble (47) for poly- meric substances. They treated the pre-exponential factor Ao, the activation energy E and the minimum free volume V0 as adjustable parameters characteristic of the substance considered. Equation (2.8) was found to describe the viscosity behavior of the substances tested by these workers moderately well. On the other hand, Naghizadeh and Rice (18) tested the theory for the self-diffusivity of simple fluids (such as the inert gases) and found that the agreement between theory and experiment was poor, especially in the high density region. If V is a constant, as assumed by Macedo and Litovitz, differentiation of equation (2.10) with respect to 1/T at constant volume (i.e. constant density) yields 1n(]/T1/2 E L 3(1/T) R (3.1) E represents the minimum energy required by a molecule to overcome intermolecular forces in making a jump and was also assumed constant by Macedo and Litovitz. Therefore equation (3.1) predicts that a plot o" In('r/T1/2) versus reciprocal temperature at constant volume should give a series of parallel straight lines. For a variety of nonpolar and slightly polar liquids over a moderate density and temperature range such a plot produces straight lines, but the slopes of the lines vary with volume. Moreover, when the results are plotted over a wide temperature range nonlinearity becomes apparent. Such nonlinearity is particularly marked for fluids composed of simple molecules. This is illustrated in Figures 3.1 and 3.2 for argon and nitrogen, for which data are available at constant volume over wide ranges of temperature and density. From the above discussion it is apparent that the equations as used by Macedo and Litovitz do not correctly predict qualitatively the effect of temperature and volume on the viscosity, especially for simple fluids. In addition, extensive experimental viscosity data are needed for each fluid in order to fit the adjustable parameters Ao, E and V ; when used in this way their equation is no more than an empirical correlation. In this chapter the physical significance of the parameters E and Vo is examined in the light of the theory, and their dependence on molecular type, temperature and volume is discussed. Interpretation of these quantities on the molecular level suggests corresponding states relationships which may be used to predict the parameters. The improved theory also explains why the theory of Macedo and Litovitz fails for simple fluids. 3.1 Temperature Dependence of Vo The parameter Vo of equations (2.10) and (2.11) represents the minimum free volume that must be available before a jump may occur. If the molecules may be treated as rigid spheres, as in the smoothed potential cell model of Prigogine (50), the free volume is independent -7.5 o- Experimental data (48) -8.0 3 -1 29.14 cm.mole o-o -8.5 o 30.45 oz 0 32.75 -9.0 - 0vo 36.3 -9.5 0- 39.15 -P 0 V0VVV`'"/ -10.0 -1C.5 I I I l 2.0 4.0 6.0 8.0 10.0 12.0 14.0 1/T x 103 (K-1) Figure 3.1. Viscosity of Argon. -o- Experimental data (49) 3 -1 32.6 cm.mole 35.1 o 37.2 44.5 S40.6 o oo :^/^ /^ 56.0 o /o I I , I . I 6.0 8.0 10.0 12.0 1/T x 103 (K-) Figure 3.2. Viscosity of Nitrogen. I 14.0 16.0 -8.0 -8.5 H -9.0 CM1 E -9.5 -10.0 L- -10.5 - -11.0 2.0 I 4.0 I I of the temperature at constant volume, and V should therefore be a 0 constant for a particular molecule, independent of temperature and density. Macedo and Litovitz (45) assumed this to be the case for all molecules treated by them. For more realistic potential models one would expect V to decrease somewhat with increasing temperature, since as temperature rises the average kinetic energy of the molecules increases, and molecules are thus able to approach each other more closely. According to such a viewpoint Vo should not be affected by the density at constant temperature. The extent to which this parameter depends on temperature will be determined largely by the repulsive portion of the intermolecu- lar potential energy curve. Since this part of the curve rises less steeply for simple molecules than for more complex polyatomic molecules, one would expect the effect of varying V to be most evident for the simpler molecules. To obtain a general expression describing the temperature dependence of Vo, a (6,n] pair potential energy function is assumed. Thus u= EK1 2 j (3.2) where 6/(n-6) 1 n-6 6/ The parameter n indicates the steepness of the repulsive part of the curve; for small molecules such as the inert gases n is close to 12, whereas for more complex molecules the best values of n is 28 or higher. For a group of roughly spherical molecules V may be assumed 0 to be proportional to the cube of the molecular "diameter." An estimate of the variation of the molecular diameter (and hence V ) with temperature may be obtained by equating the average kinetic energy of a two-particle system to the potential energy of the system at the distance of closest approach, d: 3kT = EK ( 6]3.3) or, in reduced form 3T = K1 [ 3 (--)2 (3.4) where T= kT C v a d7 Equation (3.4) may be solved to obtain v" as a function of T* for a various n values. The temperature dependence of v for several n values is shown in Figure 3.3. Assuming that V0 is proportional to d3, one may write V = (v (3.5) o a where ( is constant for a particular molecule, V at any temperature can be calculated from V 1 V = V o (T (3.6) o o v (T = 1) 2.0 4.0 6.0 8.0 kT/e Figure 3.3. Variation of v with T . Variaion a 1.0 0.9 0.8 0.7 0.6 0.5 0.4 10.0 1 1 where V is the value of V at T= 1. Thus a knowledge of V , o o o together with n and E, suffices to calculate Vo at any temperature. Some support for the above procedure is provided by values of the hard- sphere diameter calculated at various temperatures from equation of state data for inert gases by Dymond and Alder (25). These values are included in Figure 3.3, and agree well with the curve for n = 12. With n = c, no temperature dependence should be observed for V Under these conditions, from equation (2.10) [ ~1n(/T/2) (3.7) L (1/T) J R 1/2 -1 When In(n/T /2) is plotted against T- at constant volume for such nonpolar molecules as decahydronaphthalene, benzene, etc., approximately linear behavior is observed; for these molecules n is large and the variation of V0 with temperature is small. For simple molecules, such as the inert gases, for which n 12, the temperature dependence is more marked, and the model predicts noticeable nonlinearity on such a plot. This is as observed experimentally, as shown in Figure 4.1 in the next chapter. 1 In addition to the above comments, the parameter V should be related in some way to molecular size. If V may be made dimension- less with some suitable reducing parameter, the resulting reduced minimum free volume may be quite constant for a series of similar molecules. Since free volume theories apply best at high densities, the molal volume at the melting point seems a more appropriate reducing parameter than the critical volume. Thus V1 o V = v (3.8) m 3.2 Volume Dependence of E Macedo and Litovitz (45) treated the minimum activation energy E as an empirical constant, independent of density and temperature. However, as pointed out by Brummer (51), E may be expected to vary with the average intermolecular distance, and thus with density. The value of Ev becomes larger as the molal volume decreases because of the increase in repulsive intermolecular force between molecules at close separations. From equations (2.10), (3.4) and (3.6) one may obtain E 1/2 _v = Lim [ln(/T 1) (3.9) R T-+0 d(1/T) so that Ev may be calculated from experimental viscosity values. For more complex molecules where n is large,a plot of In(7/T1/2) versus 1/T is found to be approximately linear, and the requirement T 0 in equation (3.9) is less stringent. Figure 3.4 shows the volume depen- dence of E for several fluids as calculated from equation (3.9). The activation energy arises from the motion of the jumping molecule from its initial equilibrium position through a region of higher potential energy to its final position. A model of the situation is shown in Figure 3.5 in which a molecule jumps from A to C, and passes through a region B in which it has to squeeze through a ring of z molecules. The activation energy may be written r.. U! C-, 2-i U 1-i ,. , r..) u c\ C4 z D 0 0 0 3 0 0 0 0 ( aiom'lwo)Az 0 -4-4 o ii o e 00 0 C.i > > c* .,-4 C-) o c i- i Q) > i- & 0 !- E z Molecules r0 1 0r 0 B Figure 3.5. Model of a Molecular Jump. I-- e = z[u(r2) u(r) = KlZE 2 a n - 6" (3.10) c1 = 3 v/rl 1 c2 = (r2/r)3 c3 = C 1/Vm c4 = kT Then e = K zkT c 3 m 3 ) ( 3vm )2 v m4 c2V c2 (3.11) c v n/3 C The parameters cl, c2, c3, c4 and n, the repulsive eponent should be approximately the same for a group of similar molecules and thus equation (3.11) suggests a corresponding states relation of the form E = E (V*) V V where and (3.12) E V E - v RT m V, V V~ 7 Figure 3.6 shows the correlations of reduced activation energy as a function of reduced molal volume for several nonpolar molecules. Good agreement is obtained at high reduced volumes, but some scatter is Define o Ar (48) o N2 (49) CH4 (52) a CO, (53) 0 i-C4H10 (54) g CS2 (54) v n-C6H14 (55) . C6H5C1 (55) .C C6H6 (55) 3 CC4 (55) B A 1 \ -\ v" "\ oot- 1.0 V/V mn Figure 3.6. Corresponding States Correlation of Activation Energy vs. Volume. 7.0 6.0 1 5.0 4.0 ,- -3.0 2.0 1.0 0.0 0.8 I observed at lower values of V", where the curve rises steeply. This may be attributed to differences in n, the repulsive exponent, for the molecules within the group. Thus two curves A and B are plotted in Figure 3.6. Curve A is for simple molecules obeying a Lennard-Jones [6,12] potential. Curve B shows an approximate relationship for more complex molecules which obey a potential law in which n is larger than 12. Sources of viscosity data from which values of Ev are cal- culated are given in Figure 3.6. Taking into consideration the temperature dependence of V and the volume dependence of Ev, the modified viscosity and self- diffusion coefficient equations become: -2 A exp -- ) exp E ) (3.13) T' V V (T) RT o and D/ V (T) / E (V) 2 -B exp ( ) exp ) (3.14) 1/2 V V (T) x RT T o CHAPTER 4 TEST OF IMPROVED FREE VOLUME THEORY In order to perform a rigorous test of the theory, viscosity and self-diffusion coefficient data over a wide range of temperature and density conditions are most desirable. In this work only fluids composed of simple molecules which approximately obey the Lennard-Jones (6,12) potential law will be considered. These fluids include the inert gases and most diatomic fluids-and perhaps methane, fluoro- methane and tetradeuteromethane. The only monatomic and diatomic fluids for which viscosity data over a wide range of temperature and density conditions are avail- able seem to be argon and nitrogen. Zhadanova (48,49) has reported viscosity measurements for these fluids for experimental conditions corresponding to pressures up to several thousand atmospheres. The accuracy of the experimental argon data appears to be of the order 6% over the entire range of conditions. However, the nitrogen data are in poorer agreement with measurements of other workers, and the accuracy appears to be of the order 12-15%. The only high pressure self-diffusivity measurements for simple liquids seem to be those of Naghizadeh and Rice (18), who reported measurements for argon, krypton, xenon and methane over a reasonable range of temperature and at pressures up to a more than one hundred atmospheres. Naghizadeh and Rice claimed that the uncertainty of their experimental data was less than 5%. However the disagreement between their argon data and those of Corbett and Wang (56) is about 12%, while their xenon data differ from those of Yen and Norberg (57) by as much as 86% and their reported methane data are about 11% higher than those of Gaven, Waugh and Stockmayer (58). 4.1 Corresponding States Relationships The following procedures were employed to test the theory for viscosity and to establish the corresponding states relationships in equations (3.8) and (3.12). (1) The experimental data were plotted as ln(]/T1/2) against -1 T at constant volume. Values of E were estimated at various volumes from the lines extrapolated to low temperature, since from equation (3.9) v Lim ln('yn/T1/2) 14. R T->0 (1/T) (2) Values of the parameters A and V were obtained from 0 0 the best fit to the viscosity data using the computer, equations (3.4) and (3.6) being used to obtain the temperature dependence of V . Values of Ev for argon and nitrogen estimated by procedure (1) are included in Figure 3.6. Theoretical and experimental liquid argon viscosity values are shown in Figure 4.1. Agreement between theory and experiment is found to be within 5%. Similar agreement between theory and experiment is found for nitrogen, with V = 17.0 cc/mole and A = 2.53x105 poise (K)-/2, and is shown in Appendix 2. The theory provides a satisfactory fit to the data for densities above twice the critical value. Thus as can be seen in Figure 4.1 the theory fails for molal volumes of 39.15 cc/mole and above for argon. Similar behavior is also observed for the nitrogen viscosity (see Figure A2.1 3 -1 29.14 cm.mole 1, -, 30.45 34.4 36.3 ^ 39.15 (theory) S39.15 (expt.) o- Free Volume Theory -- o--- 9hadanova (48) o Lowry et al. (15) r De Bock et al. (59) 0 Saturated Liquid (60) 8.0 10.0 1/T x 103(K-1) 12.0 14.0 Figure 4.1. Test of Free Volume Theory for Liquid Argon Viscosity. -7.5 -8.0 - -8.5 -9.0 -9.5 -10.0 -10.5 -11.0 2.0 4.0 in Appendix 2), where the theory fails for molal volumes of 56.0 cc/mole and above. The density range in which the theory applies is sharply defined and is similar to that in which similar models for thermo- dynamic properties, such as the Prigogine smoothed potential model (50), apply. The theory appears to work well for temperatures below about 1.5 Tc, although the temperature range in which the theory applies is less clearly defined. The values of the parameters E and V found for argon and v o nitrogen may be used to form the basis of corresponding states cor- relations of these parameters as described in Chapter 3, from which viscosity and self-diffusivity values of other molecules may be pre- dicted. The average value of the reduced minimum free volume for argon and nitrogen is 0.554, so that V1 V1 =_V = 0.554 (4.2) o V m 4.2 Test of Proposed Correlations To use equations (2.10) and (2.11) to estimate the viscosity and self-diffusion coefficient for simple molecules, values of V were o calculated from equations (3.4), (3.6) and (4.2), assuming n = 12 and taking values of e/k determined from gas viscosity data (20). Values of E were found using curve A of Figure 3.6. The best values of A v 0 were obtained by substituting experimental viscosity data in equation (2.10) for each molecule. In a similar manner, by fitting experimental self-diffusion coefficient data of each molecule into equation (2.11) the best values of Bo were obtained. Other than the viscosity data for argon and nitrogen ofZhadanova (48, 49) and self-diffusion coefficient data of Naghizadeh and Rice (18), very few high pressure viscosity and self-diffusivity data are available. Thus most of the viscosity and self-diffusivity data used are for saturated liquids. Table 4.1 shows the viscosity data sources and ranges of temperature and density for each of the molecules studied. The different para- meters which are required for the estimation of viscosity with equation (2.10) and the average percent deviation between theory and experiment are shown in Table 4.2. Tables 4.3 and 4.4 provide similar information for the self-diffusion coefficient of the molecules studied. The average percent deviation between theory and experiment is about 6% or less for viscosity. For self-diffusion coefficient the discrepancy between experiment and theory is found to be much larger, being about 15% in most cases. The discrepancies between theory and experiment for both viscosity and self-diffusion coefficient seem to be of the same order as the accuracy of the experimental data for the liquids studied. The viscosity data of Zhdanova for nitrogen at high pressures seem to be in poor agreement with those of other workers (66), and saturated liquid viscosity data (66) were used in obtaining che A value given in Table 4.2 for this fluid. Experimental and predicted viscosities for xenon and methane are compared in Figures 4.2 and 4.3 (similar plots for Ne, Kr and N2 are shown in Appendix 2). The predicted values become less reliable as density falls to values approaching 2pc (for methane this corresponds O-1 O N' (U*z ^ v 0 r C4 0 11o c0 Lt <-\ 1-1CO ~ >~ a o cr> U0 0 co CNJ L 0n 'I cO r - Q) LC ) I' 1.0 10 I' D c > Pc d) ci c C '. cn 0 ) r- 4-4 ( a) o co p Pc; ID It 0 oD CO c S-J 3 0 Q) olc HO CO ?- s 0) fr. I--( 0N I I ,d 04 0 'D '.0 uLt -d r. i ir cO cO I I I I I I I -"1 1* rC CNJ I D Z cc u f m fc 0 Ln t ) "n 00 cc in i 0 r- C 0 ,t4 Ci I CV C 11-- )- co 1- I I ' I I I I i I co0 o C4 0 o c r- cn 1 01% r- cc- I' O 0-1 Sz 0 z 4 C CM C4 0 C SXz ; u u u 4' . Nt Z cj! H cj3 0 U cj I- '' 47 mo OO 0 B~2 0-- > o )o ' Ci CO , o Si cI t- c ^ cC- H C-) E-4 M - C) c 0c 0 > Co 0 o-^- vN- c/L/ 0 00 0 0 J. CNI CO I OC C Ni CP) Ci CO O O c) -7 r^ ui Z < c in o 0 C- a-) 0- A^- Q 0 0 D'- C CI <-l O-i o o w Lr ,4 C : S C O0 0-I O0 CO O 0 O uC) UH 0 0 4 0 U) L ."- ( ) O 00 00 r-4 < < c' Q < 0) 0 Cl H o o a oo (, 3O 0 c -Z <-Z c< -Z LCi EH Oc' oo C3 Cn C LCn LC < > C'. c< <- ( ( <-Z u0 CI3 CC I Co Q) C c 0 Q -4 D O CC -Z CrZ E -4 i-I CM F- '-4 >-4 CCC L0 0 CD C0 \ CO 0l Cl E-4 Cl !-4 }H CL 0 C 49 M0C 00 0 0 Lu- r--. -. L"Th G' CM Z o > c GIN '-i ,i q 00 --1 0 0 0 i-4 co ON czt r- Cm cm c uii m c) r- C , 0 0 0 0 0 1-1 0 C N Co 0 0 F- ,, i-ii E I 1o co > 0 o 0 L L a-1 (N c-C -N m c -z O i-i -E N ( N (N rc- 0 1-4 4-i 0 E: V0 c0 'Z0 00 0 C C 0 (N f - ^^X U -7.0 Free Volume Theory o Experiment (64,65) -7.5 -3 44.5 cm.mole -8.0 - 50. -8.5 5 I: -9.0 -9.5 -10.0 I I I I 1.0 2.0 3.0 4.0 5.0 6.0 7.0 1/T x 103 K-1) Figure 4.2. Test of Free Volume Theory for Liquid Xenon Viscosity. a U J 0 oo E / o o C 0 D CC -_J -4 0 0O 0 o L- C 0 -~ o 0 O 0 4 0 > "-4 U a0 3 0 C ,- C- 0 0 (as-od) 01 x to V = 49.5 cc/mole). Figure 4.4 compares theory and experiment for the self-diffusion coefficient of saturated liquid methane. The expressions for the pre-exponential factors of the viscosity and self-diffusion coefficient equations of Eyring (26) 2/3 1/2 and Weymann (35) suggest that the groups (AoV //M ) and (B M1/2/Vm /3) may be approximately the same for different molecules. Values shown in Table 4.2 indicate that the quantity (AoVm2/3/M /2 is approximately constant for inert gas liquids, but for the other molecules a range of values is found. Similar behavior is observed for the group (B M1/2/Vm/3). Apparently the expressions given for the pre-exponential factors by Eyring and Weymann are not generally valid, and cannot form the basis of successful corresponding states relationships for Ao and B In general the value A (or B ) can be determined from a single experimental value of viscosity (or self- diffusivity) for a given fluid. This can then be used together with the proposed correlations for V and E to predict viscosity (or self- diffusivity) values at other temperatures and densities. 4.3 Summary By allowing for the variation of V with temperature it is possible to account quantitatively for the nonlinearity of constant volume plots of In(r/T1/2) vs. T-1 and In(D/T1/2) vs. T-1 for simple liquids. The physical model predicts a density dependence of E and thus explains the variation with density of the slope of constant volume plots of In(T1/2) vs. T- and In(D/T1/2) vs. T1. With these improvements the free volume theory is able to accurately describe the viscosity and self-diffusivity behavior of liquids composed of r- 0 CC *r-1 o 0. Cr) E-' O- C> 0 ao c 0 \ 0 \ 0 -i * \ o C 0 o -o 0 r r- \I 0 0 0 0 Lr 0 ( Doas-uI ) o0 Y z- Z simple, nonpolar molecules that may be expected to approximately obey a Lennard-Jones [6,12] potential law. The theory works well at densities p > 2pc, where other correlations fail. The fact that both V1 and E values vary in a corresponding 0 v states way indicates that the free volume theory provides a reasonably correct picture of viscosity and self-diffusivity behavior for these liquids, and shows that the equations' success does not arise only from the availability of three adjustable parameters. The simple expressions given for A and B by Eyring and others, however, do not seem to be generally applicable. CHAPTER 5 PRINCIPLE OF CORRESPONDING STATES FOR MONATOMIC FLUIDS The free volume equations discussed in Chapters 2-4 accurately predict viscosities and self-diffusion coefficients for nonpolar fluids in the density range p > 2p. However, the theory cannot be readily extended to include thermal conductivity, and is limited to the above density range. In this chapter and the next a general and direct application of the corresponding states principle is examined. The treatment for monatomic molecules does not assume any simple model, and should be applicable to all transport properties over the entire range of densities and temperatures. 5.1 Molecular Basis of the Correspondence Principle If correctly applied, the principle of corresponding states provides an accurate and very useful method for predicting physical properties of pure substances, under conditions for which no data are available, from the measured properties of one or more substances. The corresponding states principle has been widely applied to the thermodynamic properties of pure substances; thus the generalized charts of Hougen, Watson and Ragatz (28) and of Hirschfelder, Curtiss and Bird (20) have proven of great practical use in engineering calculations. The principle of corresponding states was first suggested by Van der Waals in 1873 as a result of his equation of state. It was not until 1939, however, that a rigorous and more general theoretical derivation of the principle was attempted by Pitzer (74) for thermodynamic properties using statistical mechanics. Pitzer presented his derivation based on the following assumptions: Assumption 1. Quantum effects are negligible. Assumption 2. The intramolecular degrees of freedom (rotational and vibrational) are indepen- dent of density. Assumption 3. The intermolecular potentials are pair- wise additive. Assumption 4. The potential energy for a pair of molecules has the form u = EO (5.1) where r = the intermolecular distance E = characteristic energy S= characteristic distance 0 = a universal function Using Assumptions 3 and 4 the configuration energy may be written as N U C (5.2) i The configurational partition function may then be put in reduced form (50) V V N Z exp dr .... dr o o i 03N Z = Z (T ,V*,N) (5.4) N. where N = number of molecules -A- kT T = reduced temperature V V = = reduced molal volume a Since all equilibrium configurational properties of interest may be evaluated from the configurational partition function Z and its temperature and volume derivatives, equation (5.4) represents a general proof of the principle of corresponding states for thermo- dynamic properties of pure substances, provided all the above assumptions are observed. Thus the reduced equation of state is SPc 03 T nZ P - kTT v )T,Ni = P (T ,V ) (5.5) where it is noted that intensive properties do not depend upon N. The principle of corresponding states for thermodynamic properties may also be derived by the method of dimensional analysis (75). However in this method the conditions under which a molecule may obey the corresponding states principle are not explicit. One way of deriving the principle for transport properties is by di.aensional analysis, with the reduction performed using either the critical constants or appropriate combinations of molecular parameters. A rigorous statistical mechanical derivation for trans- port properties similar to that mentioned above for thermodynamic properties, is possible,as has been shown by Helfand and Rice (76). The transport coefficients may be expressed in terms of equilibrium time correlation functions. The equations for the shear and bulk viscosity, thermal conductivity and self-diffusion coefficients are given in equations (1.1) to (1.4) of Chapter 1. Using the four assumptions made in deriving the principle for thermodynamic pro- perties, it is possible to put these equations in a reduced form. The derivation of the principle for the self-diffusion coefficient is given below as an example of the procedure. The self-diffusion coefficient expressed in terms of the time integrals of appropriate autocorrelation function has the form D = 0 Based on Assumptions 1 and 2 given above the time correlation function .V V V (0)V (t)e- d. ...d Ndr ...drN x V e. ePHdf1. .d2Ndrl...drN -co o Assumption 2 is implicit, since the integrals are performed only over center of mass coordinates. The Hamiltonian H for the center of mass coordinates is given by N 2 H = + U (5.8) i=2m i=l and if Assumptions 3 and 4 are satisfied, we may write iU i(j Now, if we define reduced quantities distance r = temperature T time t = momentum p = volume v = as r C- kT t 1/2/m 1/2 tc /m 0. p/(ml/2 1/2) v/o3 then N *2 SP (O)p (t")exp - m -. 2T* S x v- N *2 T2T -co o 1=1 N 1 Np -1\ L.. d-N ij = -- g(T ,V* ,t ) 1 ... (5.15) -T T (r ) d 1...dE dr .d i where g is a universal function of T V and t. From equation (5.6) 1/2 D 1/---- g(T*,V )dt* (5.16) m o (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) 60 or combining equations (5.15) and (5.16) D = D^(T*,V*) (5.17) where D* = D F (5.18) Similar relationships may be derived for shear and bulk viscosity and thermal conductivity, ]= = (T",V ) (5.19) S= *(T",V*) (5.20) = X'(T',V) (5.21) where 2 7O (5.22) K 02 = -= (5.23) >1 = x (5.24) k V The corresponding states principle may be derived under less restrictive conditions than those used above. Although it is then of general application it loses the attractive simplicity of the two- parameter laws expressed by equations (5.17), (5.19), (5.20) and (5.21). Thus, Assumption 1 is readily removed by using quantum mechanical expressions in place of integrals over phase space. This results in the introduction of a reduced Planck constant in the expressions for transport properties, D D'(T ,V",h ) (5.25) -' = r(T, V", h ) (5.26) = ic'(T', V, h ) (5.27) = (T V h ) (5.28) where h Donth (77) showed that the corresponding states principle could be proved for thermodynamic properties without assuming pairwise addi- tivity of potentials (Assumption 3). Thus in place of Assumption 3 he assumed that the total potential energy of the system might be expressed in the form rr r U = E a3, -3 -,... (5.29) where 0 is a universal function of the reduced position coordinates. Recently Hakala (75) derived a more general form of the corresponding states principle for the thermodynamic properties by means of dimen- sional analysis. The effect of his treatment is to remove Assumption 4. The corresponding states principle is shown to apply to substances which do not obey a pair potential of the form given by equation (5.1), provided the necessary additional reduced parameters are included in the correlation. It is of interest to note that the above corresponding states relations for D and n may be derived from the free volume equations. The assumptions implied are in this case less clearly defined. The expression for self-diffusion coefficient derived from free volume theory is given by equation (2.11) as D p V 0 S/2= Boexp _v exp (- V Vo (5.30) Equation (5.30) can be expressed in terms of reduced variables as E" V S = B exp exp (5.31) o or D* = D (T*,V) (5.32) where E S= E (V*) (see Chapter 3) v NE v V 3 V V* = V(T) (see Chapter 3) o 3 o 1/2 and it is assumed that B* o = constant. o 1/2 ok In this chapter we consider only fluids composed of monatomic molecules (excluding He) so that Assumptions 1 and 2 may be regarded as fully satisfied. The inert gas molecules have been found to obey the principle of corresponding states for thermodynamic properties (20,74) and transport properties at low density (78) within experi- mental accuracy. Pairwise additivity of potentials seems to be a good assumption up to reasonably high densities for the molecules considered in this chapter (79-81). Ross and Alder (82) have pre- sented experimental evidence that nonadditivity becomes a significant factor for these fluids only at very high densities, of about 4 times the critical value. Assumption 4 concerning the form of the pair potential energy is only approximately satisfied by monatomic mole- cules considered here, as discussed by Smith (83). 5.2 Previous Work Previous work on the application of the principle to trans- port coefficients of inert gases and other simple polyatomic molecules in the dense gas and liquid states has been reported by Cini-Castagnoli, Pizzella and Ricci (84), Kerrisk, Rogers and Hammel (85), Rogers and Brickwedde (86), Boon and Thomaes (87,88), Boon, Legros and Thomaes (64), Tee, Kuether, Robinson and Stewart (89), Trappeniers, Botzen, Van Den Berg and Van Oosten (90), and Trappeniers, Botzen, Ten Seldam, Van Den Berg and Van Oosten (91). Cini-Castagnoli, et al. applied the principle to the viscosity and thermal conductivity of saturated liquid Ar, CO, N2, CH4 and 02. Correla- tions appeared to be poor, especially for the reduced viscosities. Kerrisk et al. and Rogers and Brickwedde examined the principle of 3 4 corresponding states for the light molecules He He H2, D 2 T2 etc., where quantum effects are important. They tested the principle for the saturated liquid viscosity and thermal conductivity of these substances. In their correlation they introduced the reduced Planck * constant h as a third parameter as required by equations (5.26) and (5.28). Boon et al. tested the validity of the corresponding states principle for the saturated liquid viscosity of Ar, Kr, Xe, 02, N2, CO2, CH and CD They found that the inert gases obey the principle closely, but even relatively simple polyatomic molecules such as 02, C':. and CD, wcre found to deviate markedly. Self-diffusion coefficients and binary diffusion coefficients of more than 14 simple molecules in the dilute gas, dense gas and liquid states have been tested by Tee et al. They defined the reduced diffusion coefficient D as AB c T cAB DAB P cAB oAB AB T cAB AB / * AB M1 1 T cAB M M where c here represents the total concentration, and obtained the equation for the diffusion coefficient of a dilute gas the T T D = 4.05 x 10- cAB AB(1,1)* (5.33) from relation (5.34) They plotted their results as D" against P at constant T where AB R R P T P= P and TR However, the data they present seem to be too c c scattered to justify the test. Trappeniers et al. applied the corresponding states principle to the viscosity of inert gases up to high densities for temperatures above critical. They defined the reduced viscosity as Q = n where 5 T 0 16 (2,2)* 16 n R and 2 T ) -n (5.35) (5.36) (5.37) Q" was then related to p and T*. 5.3 Test of Correspondence Principle for Inert Gases No systematic and consistent test of the principle for all three transport properties of dense fluids over a wide range of tempera- tures and densities seems to have been made. In particular the trans- port properties of fluids at very high pressures (and thus high densi- ties) have not been examined. The behavior at high densities is of particular interest since it is under these conditions that Assumption 3 may become invalid. Also, errors arising from differences in the repulsive portion of the intermolecular potential for the molecules considered (Assumption 4) may become apparent at high density. The essential simplifying factor in the case of monatomic (as opposed to polyatomic) molecules is that Assumption 2 is fully satisfied. A comparison of the correspondence behavior for monatomic and polyatomic molecules may therefore throw light on the validity of this assumption. Calculations are presented in this chapter to test the correspondence principle for all three transport coefficients over the full range of temperatures and pressures for which data are available. The extension to polyatomic molecules is discussed in the following chapter. Reduction of the transport coefficients are performed using Lennard-Jones [6,12.] potential parameters rather than critical constants because the former have more theoretical significance. The use of critical constants is justified when the group of substances accurately obey the same potential function, Assumption 4 (50). In such a case it would in principle be immaterial whether potential function or critical parameters were used. However, in practice the group of molecules do not all accurately obey Assumption 4, and the potential parameters are force-fitted to some semi-empirical equation, in this case the Lennard-Jones (6,12] potential function. Under such condi- tions the use of potential parameters rather than critical constants may improve the correlation obtained. In this work, reduction with both critical constants and potential parameters (smoothed in the way described below) was tried. The use of potential parameters was found to give consistently better results. The potential parameters reported in the literature by differ- ent workers show substantial variations. Thus for krypton values of C, the characteristic energy, reported by different authors vary by as much as 14%. It is therefore desirable to find some means of averaging these parameters. Moreover, Reed and McKinley (92) have shown that polyatomic molecules do not all obey the simple two-para- meter Lennard-Jones 16,12] potential energy function. Since Assumption 4 requires that all molecules should obey the same potential energy function in order to arrive at the same corresponding states, an attempt was made to force the polyatomic molecules to fit a two- parameter potential energy function by adopting the method used by Tee, Gotoh and Stewart (93). According to the Pitzer compressibility correlation (94) the volumetric behavior of a fluid, and thus the intermolecular forces present, are characterized by P T and the c c acentric factorO Thus the intermolecular force constants E and a should depend on these three quantities, so that one obtains the dimensionless relations = f(() (5.38) kT c P p 1/3 SkT ) a = f'() (5.39) S c where f and f' are universal functions. In Figures 5.1 and 5.2 P 1/3 values of (c and for twenty different molecules are SkTC kT c c plotted against their corresponding Pitzer factor c. The best straight lines were drawn through them by the method of least squares. Poten- tial parameters used in these figures were values calculated from gas viscosity data, and were taken from Hirschfelder, Curtiss and Bird (20) and Tee, Gotoh and Stewart (93). Where several values for E and a were reported for a given molecule the mean values are plotted in Figures 5.1 and 5.2, and the maximum deviation is indicated. The calculations presented in this chapter and the next make use of smoothed potential parameters obtained from these two figures. Values of the potential parameters used are shown in Tables 5.1 and 6.1. Values for molecules other than those tabulated may be calculated from the equations for the best lines shown in Figures 5.1 and 5.2 which are = 0.7932 0.6723C (5.40) kT c / c \1/3 = 0.4583 + 0.1213C (5.41) kT Figures 5.3, 5.4 and 5.5 are reduced plots of saturated liquid viscosity, thermal conductivity and self-diffusion coefficient as 'D \ CO 0 CM N =C O = CM CM Q r- Ci \o CM M NO U U U u <> > C) c, x C) 0 ' - r -- 'i I, ,I-, - ~-----/- / - 0 0 0 3 0 o ulU F-- c C) 0 u 0 0 o Crm -o I-3 0 0 0 ':1 0 4- C, C)- I o ,--1 N r Noo co> > c ,c jO H 0 0- p ^ C C O U V o F te<^f Zo ( Q iU U tO 0 10 O0 2 o u"8 C7 -J C1) 04 r --' SP .C) C '0 C)- C O r( O O o o P,> 1 q6 \ o\ 0 0 t 6 ~~~Q 0MX S I I I I--I I I I -IIf-. I I I - Ul) 0 cn 0 00 C O *H -l 0 0 -a cC O ;-I o uC *- *-1 LQl 0 ':3 O CC) CO o -4 o .-4 0 C- - o o 0 o 0 0 0 C)' ou 00 03- CO I 1 I Ii I i i i i I , p. oij~ C ri f) C4 0 0 0 0O O O 4-J *-4 > '4-1 CO 4c U-4 c* 4-1 0 C/I C) 0 0 *rl -11 0 C)) t 'ci 0o Oi *r- i- functions of temperature for the inert gases (argon, krypton, xenon and neon). For pure saturated liquids there is only one degree of freedom, and the reduced transport coefficients depend only on reduced temperature. In the saturated liquid region, the inert gases appear to obey the corresponding states principle for the above transport properties reasonably well with the exception of the thermal conducti- vity data of neon. The discrepancy observed in Figure 5.4 for neon may be due to quantum effects, since neon is the second lightest molecule among the inert gases. Experimental data for bulk viscosity are scarce. The only bulk viscosity measurements made seem to be those reported by a few workers for argon (14,95,96). Figure 5.6 is a reduced plot of bulk viscosity of saturated liquid argon (extrapolated data) as a function of temperature. Great experimental difficulties are involved in bulk viscosity measurements, and the data are very scattered. Even among the data of the same worker the average percent deviation is as much as 15%. In the derivation of the correspondence principle for trans- port properties shown above, reduced viscosity is found to be a function of reduced temperature and reduced molal volume. Thus according to equation (5.19) = (T"',V*). However it is more convenient in practice to plot the transport properties as a function of temperature and pressure. Since reduced molal volume may be shown to be related to reduced pressure through the equation of state, one may e::press as r"(T",P'). Reduced viscosities of inert gases plotted as a function of reduced temperature at constant reduced pressures are shown in 0 ClC 4J 0 0O 0 r-- M 0 SO0 O 0 0 0 0 0 0 N ) Figure 5.7. The corresponding states principle appears to apply closely to viscosity of inert gases over the entire range of tempera- ture and pressure for which experimental data are available. Data plotted in Figure 5.7 cover densities from the dilute gas phase to the dense gas and liquid phase, and correspond to pressures from zero to over 2000 atmospheres. A similar plot for the thermal conductivity of inert gases is shown in Figure 5.8. In this plot the correspondence principle is found to apply well for argon, krypton and xenon over the entire temperature and pressure range for which data are avail- able. The neon data in Figure 5.8 obey the simple correspondence principle at high temperatures, but not at low temperatures. This behavior for neon suggests that quantum effects are important for this fluid at low temperatures. The reduced viscosity and reduced thermal conductivity data are found to fit equations of the form ln-1 *-2 a-3T In' 1 = al + a22T + aT2 + a T*3 + a5T -4 (5.42) and InX' = b1 + b2T-1 + b3T*-2 + b T*-3 + b 5T4 (5.43) at constant reduced pressure or along the saturation curve. The reduced self-diffusion coefficient, on the other hand, is found to fit an equation of the form InD = dl + d2 T (5.44) The principal parameters used to test the correspondence principle for monatomic molecules are shown in Table 5.1. The C) 0 0 C.-i -e u 0 U1. c, 0 O4 L) N- CO) , ^ E- o ( G N 0 0 C3 0 U 0 0 0 i- C; 0 U 0 01 C) C) 01 *r- *l- O- U, |