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The Viscosity and Thermal Conductivity Coefficients of Dilute Nitrogen and Oxygen

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The Viscosity and Thermal Conductivity Coefficients of Dilute Nitrogen and Oxygen jras : .<; : .:' '.: » l\l 1 ^rf' ' » * ^ 1 350 The Viscosity and Thermal Conductivity Coefficients of Dilute Nitrogen and Oxygen G. E. CHI LDS AND H. J. M. HANLEY F *** ° Cf »* U.S. DEPARTMENT OF COMMERCE ,&': Z (A National Bureau of Standards \ n ' *"*C A U O? — THE NATIONAL BUREAU OF STANDARDS The National Bureau of Standards 1 provides measurement and technical information services essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The Bureau serves also as a focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . provides the central basis within the United States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. This Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics —Heat—Atomic Physics—Physical Chemistry—Radiation Physics—Laboratory Astrophysics 2—Radio Standards Laboratory, 2 which includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . conducts materials research and provides associated materials services including mainly reference materials and data on the properties of ma- terials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cry- ogenics 2—Materials Evaluation Laboratory—Office of Standard Reference Materials. THE INSTITUTE FOR APPLIED TECHNOLOGY . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The principal elements of this Institute are: —Building Research—Electronic Instrumentation—Textile and Apparel Technology Center Technical Analysis—Center for Computer Sciences and Technology—Office of Weights and Meas- ures—Office of Engineering Standards Services—Office of Invention and Innovation—Clearing- house for Federal Scientific and Technical Information. 3 1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D. C, 20234. 2 Located at Boulder, Colorado, 80302. 3 Located at 5285 Port Royal Road, Springfield, Virginia, 22151. UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director NBS TECHNICAL NOTE 350 ISSUED OCTOBER 31, 1966 THE VI SCOSI TY AND THERMAL CONDUCTIVI TY COEFFI CI ENTS OF Dl LUTE Nl TROGEN AND OXYGEN G. E. CHI LDS AND H. J. M. HANLEY Cryogenics Division Institute for Materials Research National Bureau of Standards Boulder, Colorado NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be listed or referred to in the open literature. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402 Price: 25 cents CONTENTS 2„ Potential Functions -----——-—-—-—— - ——- — • _____ 2 3o Kinetic Theory Transport Expressions--------------------- 3 4 Experimental Data ---• _________<._..=, = ,=, = _________ = .,.._____ 4 5.1 Nitrogen ____________ _____________ 5 6. Thermal Conductivity ------------------------------------- 15 8 . Acknowledgment ______„.,< --,.—>_,.__.____—_>.__ __.,,.___ 19 9. References ,_______. ,_____„___. ,___„______________ 22 III LIST OF FIGURES 1. Variation of e /k versus T (both in degrees Kelvin) for selected values of a using the Lennard- Jones potential function for nitrogen. 2. Best values of o for various values of y for the Kihara potential function for nitrogen. 9 3. Best values of r m for various values of a for the Exp: 6 potential function for nitrogen. 10 4. (a) Best values of o for three values of c for the Morse potential function, (b) Variation of o for a given value of c. Nitrogen. 11 5. Nitrogen percent deviation curves ( —i-^ ——- )x 100 LV T] calc y J of experimental and calculated viscosity coefficients for three potential functions using the best values selected by the method explained in the text. 12 6. Nitrogen and oxygen deviation curves calculated from the Kihara functions (for nitrogen Y =0.2, for oxygen y = 0. 1). These functions were selected as the best. 13 7. Oxygen percent deviation curves for three functions 14 - ^•e A-c a 1 c Percent deviation curves x p ") x lOo] ^c a 1 c for the thermal conductivity of nitrogen and oxygen. 16 9. Percent deviation curves of selected experimental thermal conductivity data calculated from Eqs (8) and (9). 17 IV LIST OF TABLES I. Best values of the parameters obtained from each potential function for nitrogen. 6 II. Best values of the parameters obtained from each potential function for oxygen. 7 III. Viscosity and Thermal Conductivity of Gaseous Nitrogen. 20 IV. Viscosity and Thermal Conductivity of Gaseous Oxygen. 21 THE VISCOSITY AND THERMAL CONDUCTIVITY COEFFICIENTS OF DILUTE NITROGEN AND OXYGEN G. E. Childs and H. J. M. Hartley The coefficients of viscosity and thermal conductivity for dilute nitrogen and oxygen were examined using a method proved suitable for argon. Given the kinetic theory expressions for the transport coeffi- cients, this method indicates a selection of a potential function and its parameters to correlate theory with experimental data. The potential functions chosen were the Lennard- Jones, Kihara, Exp: 6, and the Morse. It was found that the Kihara was most suitable and theoretical viscosity coefficients were computed with this function. The usual correction to the kinetic theory equation for thermal conductivity, the Eucken correction, was found not to be sufficient and it was decided to use an empirical polynomial equation to correlate the thermal con- ductivity coefficients. Tables of the transport coefficients for both gases are given between 100 and 1000°K. Key Words: Dilute gases, Nitrogen, Oxygen, Transport coeffi- cients, Lennard-Jones, Kihara, Exp: 6, Morse potential functions, Eucken correction, Correlation. 1. INTRODUCTION The viscosity and thermal conductivity coefficients of dilute nitro- gen and oxygen can be calculated from the Chapman-Enskog kinetic theory expressions [l] *. As these expressions are known to be sat- isfactory, the problem, as for all dilute gases, is that of choosing a suitable potential function and the best force constants for that function. The method of selection used here is the same as that proved successful for dilute argon [2, 3] . The kinetic theory thermal conductivity equation is not applicable for polyatomic gases and the usual procedure is to apply the Eucken correction [l] . However, it was found that this did not give satisfactory results and it was necessary to use a polynomial expression. ^Numbers in brackets refer to references. 2. THE POTENTIAL FUNCTIONS The discussion was restricted to the four most commonly used functions: the Lennard-Jones, the Kihara, the Exp: 6, and the Morse. The Kihara, in particular, has received much attention in the literature recently [4, 5, 6]. The Morse function has also been described by sev- eral authors [7, 8]. As the functions are well known and have been fully discussed, it is necessary only to outline them here. If U(r) is the interaction potential of two molecules separated by distance r, and £ is the maximum energy of attraction, or energy minimum, the potentials are written: Lennard-Jones r lrf b U(r) =4e (o7r) - (a/r) (i) |_ J , where cr is the value of rat U(r) =0. Kihara M° - a \l2\; { G - a U(r) = r> a (2) _ v. r - a J V r - a U(r) r ^ a, Here the finite size of the molecule is taken into consideration by including a core diameter, a. (For the Lennard-Jones, a = 0.) A reduced parameter Y is defined as a/(7. Exp: 6 a(1 - ,/ °<')-T^s[!- ^ .^/')']. (3) : where r is the value of r at the cc a ffl energy minimum and parameter which represents the steepness of the repulsive part of the function. - Morse U(r) =e{exp [-2 (r-r B -2 exp [- )(r-r.)] (4) (f ) )] (f } ^ where c is related to the curvature of the potential at r = r, . 3. KINETIC THEORY EXPRESSIONS FOR THE VISCOSITY AND THERMAL CONDUCTIVITY COEFFICIENTS The kinetic theory for a dilute gas is formally complete [1]; the Chapman- Enskog treatment of the Boltzmann equation gives the vis- cosity and thermal conductivity coefficients in terms of collision integrals which are functions of the gas dynamics and thus of the inter- molecular potential. It is the lack of knowledge of the latter -which restricts the applicability of the kinetic theory expressions. These expressions are: Viscosity (r\) V3 266.93 (MT) r _x -i rjlOmn? = J— f g cm sec , (5) 2 (3 ' 2) R fi *(T*) ^ Thermal Conductivity (A.) 1/; 7 8322.4 (T/M) _i _i _ a .,, A10lin = f, J cm sec deg , (6) 3):,; K 2 Q( 2 ' R ( T ^) where: M = molecular weight. (M = 28 . 134 for nitrogen M =31.9988 for oxygen) R = a distance parameter, i.e. , R - (J for the Lennard-Jones, Kihara, and Morse; and R = r m for the Exp: 6. T = the absolute temperature, °K. *'* (3 2 ^ Q (T#) = the reduced collision integrals (reduced by dividing by the integrals for the rigid sphere case) at the reduced temperature T*, where T* = T/(e /k) with k the Boltzmann constant The terms f and f. account for higher mathematical approxima- tions to v\ and X and are slowly varying functions of T* which seldom differ from unity by more than about 0.
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