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The Kinetic Theory of Dense Gases. the Calculation of the Viscosity and Thermal Conductivity Virial Coefficients " (1973)

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The Kinetic Theory of Dense Gases. the Calculation of the Viscosity and Thermal Conductivity Virial Coefficients Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1973 The kinetic theory of dense gases. The alcc ulation of the viscosity and thermal conductivity virial coefficients Mario Lorenzo Occelli Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Physical Chemistry Commons Recommended Citation Occelli, Mario Lorenzo, "The kinetic theory of dense gases. The calculation of the viscosity and thermal conductivity virial coefficients " (1973). Retrospective Theses and Dissertations. 6163. https://lib.dr.iastate.edu/rtd/6163 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with, a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. universiiy micfôîiîni» 300 North Zeeb Road Ann Arbor, Michigan 48106 A Xerox Education Company 73-16,970 I OCCELLI, Mario Lorenzo, 1942- I. THE KINETIC THEORY OF DENSE GASES. THE I CALCULATION OF THE VISCOSITY AND THERMAL CONDUCTIVITY VIRIAL COEFFICIENTS. Iowa State-University, Ph.D., 1973 Chemistry, physical I University Microfilms. A XEROX Company, Ann Arbor. Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. The kinetic theory of dense gases. The calculation of the viscosity and thermal conductivity virial coefficients by Mario Lorenzo Occelli A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Chemi s try Major: Physical Chemistry Approved Signature was redacted for privacy. arge of Maj york Signature was redacted for privacy. F the îla jor D p r tment Signature was redacted for privacy. For tth / ^duate College Iowa State University Of Science and Technology Ames, Iowa 1973 PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms, A Xerox Education Company ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. SELF CONSISTENT APPROXIMATIONS IN THE KINETIC THEORY 10 OF A MODERATELY DENSE GAS 2.1 Statistical Mechanical Theory of Transport 10 Phenomena 2.2 Self Consistent Approximations 17 2.3 Euler's Equations for an Imperfect Gas 35 2,1^. The Correlated Velocities and the Temperature 36 Tensors 2.5 A General Outline of a Method of Procedure to 1^.0 Evaluate Transport Coefficients for a Dense Gas 2.6 Correlated Parameters for Hard Spheres ij.3 3. THE VISCOSITY AND THERMAL CONDUCTIVITY SECOND VIRIAL 1^9 COEFFICIENTS 3.1 The Kinetic Equation of a Moderately Dense Gas, i|9 Enskog's Equation from the BBGKY Hierarchy 3.2 The Perturbation Solution of Boltzmann's Equation 56 3.3 The Viscosity Coefficient 62 3.1+ The Thermal Conductivity Coafficient 6tt- TRANSPORT COEFFICIENTS FOR THE HARD SPHERE AND 66 SQUARE WELL MODELS ii.l Transport Coefficients for Hard Spheres 66 1|.2 The Square Well Potential Model 69 4.3 Transport Coefficients for the Square Well 75 Potential iii 5. TRANSPORT COEFFICIENTS FOR THE LENNARÛ-JOKES MODEL 81; 5.1 'The Description of the Dynamics of Collision 8ij. 5.2 Reduction of the Virial Coefficients to Gomputa- 89 tional Forms 6. NUMERICAL RESULTS AND CONCLUSIONS 98 6.1 The Square Well Calculations 98 6.2 The Lennard-Jones Calculations 100 6.3 Numerical Results 103 7. LITERATURE CITED 121^. 6. ACKNOWLEDGMENTS 127 9. APPENDIX A 128 10. APPENDIX B 129 1 1. INTRODUCTION The purpose of the kinetic theory of gases is to explain the irreversible phenomena which are observed when­ ever a gas is not in a state of equilibrium. An isolated system is said to be in equilibrium when its macroscopic parameters are constant in time, Furthermore, if the system is not rotating and is free from external forces, the density n, the streaming velocity u and the tem­ perature T (the macroscopic parameters) of the gas are uni­ form. If a gas is not in equilibrium, fluxes will form with­ in the gas with the effect of establishing, very quickly, uniformity in the macroscopic parameters. In fact, after a time ^ seconds, (% is the ratio of the mean free path and some mean molecular speed ir), the irreversible phenomena cause the system to relax toward a state of local equilibrium characterized by a set of local macroscopic param­ eters (n,u,T) and by a velocity distribution function f^'~ fo^ ) of "Maxwellian form" it -(m/2kT)(v-u)^ fj"= n(m/27rkT)^ e 1.1 defined such that f^'Mr dv gives the average number of mole­ cules that can be found in a unit volume dr with velocities in the range v, v+dv during a time interval t, t+5t with St» It , In Eq, (1,1) the local macroscopic parameters are functions of position and time. The time dependence of 2 n, u and T is secular in the sense that on the To time scale these functions are constant, their change occurring only on the macroscopic time scale. By definition n = Jdv f^'' = J'dv fo' 1 .2 nu = J dv V f= Jdv V fo" 1.3 3nkT = fdv m(v-u)^f'" = fdv ra(v-u)^fo^'^ 1.4 2 2 " 2 It is easy to verify that fo^ as defined in Eq. (1.1) satis­ fies these equations. By a macroscopic parameter is meant any measurable property of the system whose value is averaged over a macro- scopically small time interval but large enough microscop­ ically not to be sensitive to the size of the interval chosen. An example will clarify this last statement. A Pitot tube (1) immersed into a moving fluid will record the time averaged of the force exerted by the fluid on the surface of the tube normal to the direction of the fluid motion. If this area would be of molecular size, and the time lag of the apparatus nonexistent, a fluctuation of the measured pressure would be observed. The pressure in this case would cease to be a macroscopic parameter in the sense a macroscopic parameter is used. Furthermore, if we let be the significant length of a region of in a gas. small enough macroscopically, to be in a state of local equilibrium, then = /o and 6? /^the mean free path of a molecule. This will imply that if a molecule is in the region at time t=0, at t= 2a it will still be in the region since it had not enough time to escape. Using this same argument we can reach the important conclusion that n, u and T are macroscopic paraPi- oters because they correspond to the average of collisionally conserved quantities. In fact, when two molecules of a dilute gas collide, their mass, momentum and kinetic energy are con­ served, Therefore, the average amount of mass, momentum and kinetic energy contained in the region dr, during a time inter­ val yZo t but macroscopically very small, will stay unchanged after the collisions that in time ,Jt have occurred in dr. It takes longer than time for n, u and T to change in a small region. Usually ^ is of the order 10"^ seconds. The time needed by a macroscopic parameter to change, is of the order of 10 sec. The above remark suggests the existence of another relax­ ation time T^~ L/C^si'i 0"^ see.,L is a macroscopic length and Cj,- the sound velocity. During this stage, the hydrodynamic stage, of the relaxation of the system toward a true equilibrium state, the time evolution of the system is determined by the rate of change of n, u and T as given by the three hydrodynamic equa­ tions. This set of equations is a direct consequence of the equations governing the conservation of mass, momentum and • - — -— — — — #3 «2 » k ^ fu = - ^.fuu.- 1.6 where P=nm is the mass density; U = +E is the total energy 2 per unit mass, E is the internal energy per unit mass defined in section 2.3; 3.
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