Paul K. Johnson

Analex Corporation, Brook Park, Ohio

A Method for Calculating Viscosity and Thermal

Conductivity of a Helium-Xenon Gas Mixture

NASA/CR—2006-214394

November 2006
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the

advancement of aeronautics and space science. The

NASA Scientific and Technical Information (STI)

program plays a key part in helping NASA maintain

this important role.

The NASA STI Program operates under the auspices

of the Agency Chief Information Officer. It collects,

organizes, provides for archiving, and disseminates

NASA’s STI. The NASA STI program provides access

to the NASA Aeronautics and Space Database and its

public interface, the NASA Technical Reports Server,

thus providing one of the largest collections of

aeronautical and space science STI in the world.

Results are published in both non-NASA channels and

by NASA in the NASA STI Report Series, which

includes the following report types:

• TECHNICAL PUBLICATION. Reports of

completed research or a major significant phase

of research that present the results of NASA

programs and include extensive data or theoretical

analysis. Includes compilations of significant

scientific and technical data and information

deemed to be of continuing reference value.

NASA counterpart of peer-reviewed formal

professional papers but has less stringent

limitations on manuscript length and extent of

graphic presentations.

• TECHNICAL MEMORANDUM. Scientific

and technical findings that are preliminary or

of specialized interest, e.g., quick release

reports, working papers, and bibliographies that

contain minimal annotation. Does not contain

extensive analysis.

• CONTRACTOR REPORT. Scientific and

technical findings by NASA-sponsored

contractors and grantees.

• CONFERENCE PUBLICATION. Collected

papers from scientific and technical

conferences, symposia, seminars, or other

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientific,

technical, or historical information from

NASA programs, projects, and missions, often

concerned with subjects having substantial

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and

technical material pertinent to NASA’s mission.

Specialized services also include creating custom

thesauri, building customized databases, organizing

and publishing research results.

For more information about the NASA STI

program, see the following:

• Access the NASA STI program home page at

http://www.sti.nasa.gov

• E-mail your question via the Internet to

help@sti.nasa.gov

• Fax your question to the NASA STI Help Desk

at 301–621–0134

• Telephone the NASA STI Help Desk at

301–621–0390

• Write to:

 NASA STI Help Desk

 NASA Center for AeroSpace Information

 7121 Standard Drive

 Hanover, MD 21076–1320
National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Contract NAS3–00145

Paul K. Johnson

Analex Corporation, Brook Park, Ohio

A Method for Calculating Viscosity and Thermal

Conductivity of a Helium-Xenon Gas Mixture

NASA/CR—2006-214394

November 2006
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by expert reviewer(s).
NASA/CR—2006-214394 1

A Method for Calculating Viscosity and Thermal Conductivity 

of a Helium-Xenon Gas Mixture 

Paul K. Johnson 

Analex Corporation 

Brook Park, Ohio 44142 

Abstract

A method for calculating viscosity and thermal conductivity of a helium-xenon (He-Xe) gas mixture 

was employed, and results were compared to AiResearch analytical data. The method of choice was that 

presented by Hirschfelder with Singh’s third-order correction factor applied to thermal conductivity. 

Values for viscosity and thermal conductivity were calculated over a temperature range of 400 to 1200 K 

for He-Xe gas mixture molecular weights of 20.183, 39.94 and 83.8 kg/kmol. First-order values for both 

transport properties were in good agreement with AiResearch analytical data. Third-order-corrected 

thermal conductivity values were all greater than AiResearch data, but were considered to be a better 

approximation of thermal conductivity because higher-order effects of mass and temperature were taken 

into consideration. Viscosity, conductivity, and Prandtl number were then compared to experimental data 

presented by Taylor. 

Introduction

In an effort to generate transport properties, namely viscosity and thermal conductivity, of a heliumxenon (He-Xe) gas mixture a method described by Hirschfelder was employed and compared to analytical 

data generated by AiResearcha

 (Coombs, 1970, pp. 274–373). The goal was to produce data similar to 

that of AiResearch’s with the added value of knowing the method with which the data was generated. 

This document will describe in detail the implementation of Hirschfelder’s method and compare the 

calculated properties to AiResearch data and a sampling of experimental data (Taylor, 1988). All tables 

and figures are located at the end of the document due to their size. 

The text Molecular Theory of Gases and Liquids by Hirschfelder, Curtiss, and Bird (1964) presents 

equations for the first-order viscosity and first-order thermal conductivity of dilute, monatomic, binary 

gas mixtures. A dilute gas assumes the effect of collisions involving more than two molecules is 

negligible. The equations are based on the Chapman-Enskog theory that models the molecules as spheres, 

and therefore the molecules have no internal degrees of freedom (Hirschfelder, 1964, p. 20). Calculations 

for the pure-gas properties and gas-mixture properties require temperature, molecular weights, mole 

fractions, and force constants of the species. 

Method

To begin, one must obtain the two force constants for each gas: collision diameter, σ, and potential 

parameter, ε/k. For helium σ = 2.576 Ǻ and ε/k = 10.22 K; for xenon σ = 4.055 Ǻ and ε/k = 229 K 

(Hirschfelder, 1964, app. table I–A). These values are based on the Lennard-Jones (6-12) potential. 

Because no measurements of σ and ε/k exist for gas mixtures, an empirical “combining law” for force 

constants between unlike molecules is recommended (Hirschfelder, 1964, p. 567). 

( ) 1 2 2

1 σ12 = σ + σ

 

a

AiResearch subsequently became part of AlliedSignal, and has since become part of Honeywell. 
NASA/CR—2006-214394 2

ε/ k12 = ε/ k1 ⋅ε/ k2

For the remainder of the report, subscript 1 will represent values for helium, subscript 2 will represent 

values for xenon, and subscript 12 will represent values for combined helium and xenon. 

Using the potential parameters, one can calculate the reduced temperatures, T*

, at any given 

temperature. 

1

1 k

T T

ε =∗ 2

2 k

T T

ε =∗ 12

12 k

T T

ε =∗

The reduced temperature allows one to look up table values for the integral Ω(2,2)*, and quantities A*, 

and B*

. Since all three values are functions of T*, table lookups will have to be performed at each desired 

temperature. The method requires values for Ω1

(2,2)*, Ω2(2,2)*, Ω12(2,2)*, A*

12, and B*12, listed in tables 1 and 

2. It should be noted that tables for Ω(2,2)*, A*

, and B* exist for both the Lennard-Jones potential and the 

modified Buckingham (6-EXP) potential. This report uses values from the Lennard-Jones table because 

the force constants are of the Lennard-Jones potential. 

Next, one calculates the first-order transport properties of the pure gases and of the hypothetical pure 

substance (Hirschfelder, 1964, pp. 528–529, 534–535). 

[ ] ( )∗ σ Ω

μ × = 2,2

1

2

1

7 1

1 1 10 266.93 M T [ ] ( )∗ σ Ω

λ × = 2,2

1

2

1

1 7 1 1 10 1989.1

T M

[ ] ( )∗ σ Ω

μ × = 2,2

2

2

2

7 2

2 1 10 266.93 M T [ ] ( )∗ σ Ω

λ × = 2,2

2

2

2

2 7 2 1 10 1989.1

T M

[ ] ( )

( )∗ σ Ω

+

μ × = 2,2

12

2

12

1 2 1 2 7 12 1

2

10 266.93

M M T M M

[ ] ( ) ( )

( )∗ σ Ω

+ λ × = 2,2

12

2

12

7 1 2 1 2

12 1

2

10 1989.1

T M M M M

where [μ]1 = first-order viscosity (g/cm-s) 

[λ]1 = first-order thermal conductivity (cal/cm-s-K) 

 T = temperature (K) 

 M = molecular weight (g/mol) 

 σ = collision diameter (Ǻ) 

 Ω(2,2)* = integral value that is a function of reduced temperature, T*

 Note: 1 g/cm-s = 0.1 kg/m-s 

 1 cal/cm-s-K = 418.4 W/m-K 

Using the values calculated above, the first-order viscosity and first-order thermal conductivity can be 

calculated (Hirschfelder, 1964, pp. 530 and 535). 
NASA/CR—2006-214394 3

[ ]

μ μ

μ

+

+

μ = X Y

Z

mix

1

1

[ ] [ ] [ ] 2 1

2

2

12 1

1 2

1 1

2

1 2

μ + μ + μ μ = x x x x X

[ ] [ ]

( ) [ ]

[ ][ ] [ ] ⎪⎭

⎪

⎬

⎫

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

μ

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

μ μ

μ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ +

μ

+⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

⎪⎩

⎪

⎨

⎧

μ = ∗ μ

1

2

2 1

2

2

1 1 2 1

2

12 1

1 2

2

1 2

12 1

1 2

2

1

1 1

2

1

12 4

2

5

3

M

x M

M M

x x M M

M

x M Y A

( ) [ ]

[ ]

[ ]

[ ] ⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ −⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

μ

μ +

μ

μ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ +

+⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ = ∗ μ

1

2 2

2

2 1

12 1

1 1

12 1

1 2

2

1 2

1 2

2

2 1

12 1 1

4

2

5

3

M

M

x

M M

M M

x x

M

M Z A x

[ ]

λ λ

λ

+

+ λ = X Y

Z

mix

1

1

[ ] [ ] [ ] 2 1

2

2

12 1

1 2

1 1

2

1 2

λ

+

λ

+

λ λ = x x x x

X

[ ] ( ) [ ] ( ) [ ] ( ) 2

2 1

2

2

12 1

1 1 2

1 1

2

1 2 U

x

U x x U

x

Y Y

λ

+

λ

+

λ λ =

() ( ) 2 (2) 1 2 2

2 1

1 Zλ = x U + 2x x U Y + x U

( ) ( )

1 2

2

1 2

2

1

12 12

1

2

1 1

5

12

12

1

15

4

M M

M M

M

M U A B − ⎟ + ⎠

⎞ ⎜

⎝

⎛ = − + ∗ ∗

( ) ( )

1 2

2

2 1

1

2

12 12

2

2

1 1

5

12

12

1

15

4

M M

M M

M

M U A B − ⎟ + ⎠

⎞ ⎜

⎝

⎛ = − + ∗ ∗

( ) ( ) [ ]

[ ][ ]

( )

1 2

2

1 2

12

12

12

1 1 2 1

2

12 1

1 2

2

1 2

12 5

5

12

32

5 1

5

12

12

1

15 4

4

M M

M M B

A

B

M M

M M U Y A − ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − − ⎠⎞ ⎜

⎝

⎛ − +

λ λ

λ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = ∗ ∗ ∗ ∗

( ) ( ) [ ]

[ ]

[ ]

[ ] ⎟

⎠

⎞ ⎜

⎝

⎛ − +

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ −⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

λ

λ

+

λ

λ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = ∗ ∗ 1

5

12

12

1 1

15 4

4

12

2 1

12 1

1 1

12 1

1 2

2

1 2

12 B

M M

M M U Z A

where [μ]1 = first-order viscosity (g/cm-s) 

[λ]1 = first-order thermal conductivity (cal/cm-s-K) 

M = molecular weight (g/mol) 

x = mole fraction 

A*

12, B*12 = quantities for Lennard-Jones potential, functions of T*

First-order mixture properties calculated using the Hirschfelder equations above closely match the 

AiResearch data. The calculated Hirschfelder values compare well to Glenn Research Center (GRC) in-
NASA/CR—2006-214394 4

house curve fitsb that are representative of the AiResearch data. Over a temperature range of 400 to 1200 

K, calculated viscosity values differ from the GRC curve fits by no more than 0.19, 0.43 and 0.51 percent 

for He-Xe mixture molecular weights of 20.183, 39.94, and 83.8 kg/kmol, respectively; calculated 

thermal conductivity values differ by no more than 0.12, 0.20, and 0.37 percent. Tables 4 to 6 give 

calculated values and their comparison to the GRC curve fits. 

The Hirschfelder equations thus far have only produced first-order properties. Even though there are 

tables for the Lennard-Jones third-order correction factors for both viscosity and conductivity, they are 

only for pure substances (Hirschfelder, 1964, app. table I–P). Work has been done by Singh to calculate 

third-order correction factors for thermal conductivity. The correction factor, fλmix

(3), accounts for mass 

and concentration dependence and temperature dependence of higher-order contributions (Singh, 1992, 

p. 679). Singh presents a table of Lennard-Jones potential fλmix

(3) values, given in table 3, for He-Xe 

mixtures as a function of helium mole fraction for temperatures ranging from 400 to 2000 K. Corrections 

for the third-order viscosity of binary mixtures are unnecessary because viscosity is not as sensitive to 

unlike molecular interactions as is thermal conductivity (Mason, 1957, p. 80). Therefore, Hirschfelder’s 

first-order approximation of viscosity is considered adequate, but Hirschfelder’s first-order approximation 

of thermal conductivity is multiplied by Singh’s third-order correction factor to obtain the third-order 

approximation of thermal conductivity (Singh, 1992, p. 680). 

(3) (3) (1)

mix mix mix λ = f λ λ

As shown in tables 4 to 6, the calculated third-order thermal conductivity values are greater than the 

GRC curve fits by 2.13 to 2.31, 2.79 to 3.12, and 3.77 to 4.40 percent for He-Xe mixture molecular 

weights of 20.183, 39.94, and 83.8 kg/kmol, respectively, over a temperature range of 400 to 1200 K. 

This report has already shown good agreement between first-order calculations and the AiResearch data, 

which indicates that the AiResearch data reported in 1970 was most likely generated using first-order 

approximations. However, the third-order calculations are believed to be a better estimate of thermal 

conductivity because they account for mass and temperature effects. Given the large difference in 

molecular weights between helium and xenon, higher-order contributions apply especially to a He-Xe gas 

mixture (Singh, 1992, p. 681). 

Values calculated according to the method described in this report for first-order viscosity and thirdorder thermal conductivity are considered to be in good agreement with the AiResearch analytical data, 

noting Singh’s third-order correction factor. More importantly, this report documents the method used to 

calculate such values and ties together the supporting references. Using the equations given and tables 1 

to 3, one has all the information needed to calculate viscosity and thermal conductivity of a He-Xe gas 

mixture. Data values from the method described in this report were incorporated into NASA Glenn 

Research Center in-house design and analysis codes. 

Comparison to Experimental Data 

In an article entitled “Empirical Relationship for Higher Order Contributions to Thermal Conductivity 

of Gas Mixtures,” Taylor assembles experimental data for He-Xe transport properties from a host of 

references. Mixture molecular weights range from pure He to pure Xe, and temperatures range from 291 

to 793 K. Figures 1 and 2 compare Hirschfelder first-order viscosity and third-order thermal conductivity 

to the experimental data. However, the values shown have a high uncertainty because they were extracted 

from a graph in Taylor’s article. In general, the Hirschfelder viscosities match up well with the data and 

tend to over predict the data by no more than 3 percent; Hirschfelder conductivities match up well at the 

lower temperatures, but under predict the test data by as much as 8 percent at the higher temperature case. 

 

b

The source of the GRC in-house curve fits is undocumented. The only published source of data referenced by this report is from AiResearch 

(Coombs, 1970). GRC curve fits are representative of the AiResearch data and are used for their ease of comparison to the calculated values. 
NASA/CR—2006-214394 5

Taylor also presents his own Prandtl number experimental data for four mixture molecular weights. As 

shown in table 7, Hirschfelder over predicts Taylor’s Prandtl numbers by 2 to 4 percent. The data 

supports Singh’s application of a third-order thermal conductivity correction factor, because without it 

Hirschfelder would under predict thermal conductivity and over predict Prandtl number by even more. 

Conclusions 

The method employed for calculating transport properties of a He-Xe mixture was that presented by 

Hirschfelder with Singh’s third-order correction factor applied to thermal conductivity. For a temperature 

range of 400 to 1200 K, first-order viscosity values differed from the GRC curve fits by no more than 

0.51 percent for any of the He-Xe mixtures considered; third-order thermal conductivity values were 

greater than the GRC curve fits by as little as 2.13 percent and as much as 4.40 percent. First-order 

viscosity calculations were considered to be adequate because viscosity is not strongly affected by unlike 

molecular interactions, and third-order thermal conductivity calculations were considered to be an 

improvement upon first-order calculations because they account for higher-order effects of mass and 

temperature. 

Although the primary focus of this paper was not to compare calculated He-Xe properties to 

experimental data, the small amount of locatable data helped to substantiate Hirscfelder’s method as well 

as Singh’s third-order thermal conductivity correction factor. An extensive empirical database of He-Xe 

properties does not exist, so for the time being Hirschfelder’s method provides an acceptable alternative. 

References 

Coombs, M.G., Morse, C.J., and Richard, C.E. “Conceptual Design Study of Nuclear Brayton Cycle Heat 

Exchanger And Duct Assembly (HXDA).” NASA CR–72783, NASA Lewis Research Center, 

Cleveland, OH, 1970. 

Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B. Molecular Theory of Gases and Liquids. New York: John 

Wiley and Sons, Inc., 1964. 

Mason, E.A. “Higher Approximations for the Transport Properties of Binary Gas Mixtures. I. General 

Formulas.” Journal of Chemical Physics 27 (1957): 75–84. 

Singh, K., Dham, A.K., and Gupta, S.C. “Empirical Relationship for Higher Order Contributions to 

Thermal Conductivity of Gas Mixtures.” Journal of Physics B, Atomic, Molecular, and Optical 

Physics 25 (1992): 679–685. 

Taylor, M.F., Bauer, K.E., McEligot, D.M. “Internal Forced Convection to Low-Prandtl-Number Gas 

Mixtures.” International Journal of. Heat and Mass Transfer 31 (1988): 13–25. 
NASA/CR—2006-214394 6

TABLE 1.—THE INTEGRAL Ω(2,2)* FOR CALCULATING THE 

TRANSPORT COEFFICIENTS FOR THE 

LENNARD-JONES POTENTIALc

T* Ω(2,2) T* Ω(2,2)

0.30 2.785 2.7 1.069 

0.35 2.628 2.8 1.058 

0.40 2.492 2.9 1.048 

0.45 2.368 3.0 1.039 

0.50 2.257 3.1 1.030 

0.55 2.156 3.2 1.022 

0.60 2.065 3.3 1.014 

0.65 1.982 3.4 1.007 

0.70 1.908 3.5 0.9999 

0.75 1.841 3.6 0.9932 

0.80 1.780 3.7 0.9870 

0.85 1.725 3.8 0.9811 

0.90 1.675 3.9 0.9755 

0.95 1.629 4.0 0.9700 

1.00 1.587 4.1 0.9649 

1.05 1.549 4.2 0.9600 

1.10 1.514 4.3 0.9553 

1.15 1.482 4.4 0.9507 

1.20 1.452 4.5 0.9464 

1.25 1.424 4.6 0.9422 

1.30 1.399 4.7 0.9382 

1.35 1.375 4.8 0.9343 

1.40 1.353 4.9 0.9305 

1.45 1.333 5 0.9269 

1.50 1.314 6 0.8963 

1.55 1.296 7 0.8727 

1.60 1.279 8 0.8538 

1.65 1.264 9 0.8379 

1.70 1.248 10 0.8242 

1.75 1.234 20 0.7432 

1.80 1.221 30 0.7005 

1.85 1.209 40 0.6718 

1.90 1.197 50 0.6504 

1.95 1.186 60 0.6335 

2.00 1.175 70 0.6194 

2.10 1.156 80 0.6076 

2.20 1.138 90 0.5973 

2.30 1.122 100 0.5882 

2.40 1.107 200 0.532 

2.50 1.093 300 0.5016 

2.6 1.081 400 0.4811 

 

c

Table values taken from Hirschfelder, 1964, app. table I–M. 
NASA/CR—2006-214394 7

TABLE 2.—THE QUANTITIES A* AND B* FOR CALCULATING THE TRANSPORT 

COEFFICIENTS OF MIXTURES FOR THE LENNARD-JONES POTENTIALd

T* A* B* T* A* B* 

0.30 1.046 1.289 2.7 1.094 1.103 

0.35 1.062 1.296 2.8 1.094 1.104 

0.40 1.075 1.296 2.9 1.095 1.102 

0.45 1.084 1.289 3.0 1.095 1.101 

0.50 1.093 1.284 3.1 1.095 1.100 

0.55 1.097 1.275 3.2 1.096 1.096 

0.60 1.101 1.263 3.3 1.096 1.099 

0.65 1.102 1.254 3.4 1.096 1.096 

0.70 1.104 1.242 3.5 1.097 1.096 

0.75 1.105 1.233 3.6 1.097 1.095 

0.80 1.105 1.223 3.7 1.097 1.097 

0.85 1.105 1.216 3.8 1.097 1.093 

0.90 1.104 1.206 3.9 1.097 1.093 

0.95 1.103 1.200 4.0 1.098 1.095 

1.00 1.103 1.192 4.1 1.098 1.093 

1.05 1.102 1.183 4.2 1.098 1.093 

1.10 1.102 1.179 4.3 1.099 1.094 

1.15 1.101 1.172 4.4 1.099 1.094 

1.20 1.100 1.169 4.5 1.099 1.092 

1.25 1.099 1.165 4.6 1.100 1.091 

1.30 1.099 1.159 4.7 1.100 1.093 

1.35 1.098 1.156 4.8 1.100 1.095 

1.40 1.097 1.154 4.9 1.101 1.091 

1.45 1.097 1.148 5 1.101 1.092 

1.50 1.097 1.143 6 1.103 1.090 

1.55 1.096 1.140 7 1.105 1.092 

1.60 1.096 1.137 8 1.107 1.090 

1.65 1.096 1.137 9 1.109 1.091 

1.70 1.095 1.133 10 1.110 1.094 

1.75 1.094 1.129 20 1.119 1.095 

1.80 1.094 1.127 30 1.124 1.095 

1.85 1.094 1.126 40 1.127 1.095 

1.90 1.094 1.124 50 1.130 1.095 

1.95 1.094 1.122 60 1.132 1.096 

2.00 1.094 1.119 70 1.134 1.095 

2.10 1.094 1.116 80 1.135 1.095 

2.20 1.094 1.113 90 1.137 1.096 

2.30 1.094 1.110 100 1.138 1.095 

2.40 1.094 1.108 200 1.146 1.095 

2.50 1.094 1.106 300 1.151 1.095 

2.60 1.094 1.104 400 1.154 1.095 

 

d

 Table values taken from Hirschfelder, 1964, app. table I–N. 
NASA/CR—2006-214394 8

TABLE 3.—VALUES OF fλmix(3) FOR VARIOUS TEMPERATURES 

AND COMPOSITIONS OF He-Xe MIXTURE FOR THE 

LENNARD-JONES POTENTIAL (x1 IS THE 

CONCENTRATION OF THE LIGHTER 

COMPONENTe

 OF THE MIXTURE)f

T (K) x1 fλmix(3)

400 0.01 1.0056 

 0.10 1.0272 

 0.30 1.0388 

 0.50 1.0374 

 0.70 1.0304 

 0.90 1.0192 

 0.99 1.0130 

 

800 0.01 1.0101 

 0.10 1.0289 

 0.30 1.0394 

 0.50 1.0377 

 0.70 1.0306 

 0.90 1.0191 

 0.99 1.0127 

 

1500 0.01 1.0137 

 0.10 1.0304 

 0.30 1.0396 

 0.50 1.0375 

 0.70 1.0302 

 0.90 1.0187 

 0.99 1.0125 

 

2000 0.01 1.0146 

 0.10 1.0310 

 0.30 1.0399 

 0.50 1.0377 

 0.70 1.0303 

 0.90 1.0188 

 0.99 1.0124 

 

e

In this case concentration of the lighter component is defined as mole fraction of helium. f

Table values taken from Singh, 1992 table 1. 
NASA/CR—2006-214394 9

TABLE 4.—VISCOSITY AND CONDUCTIVITY CALCULATED VALUES AND COMPARISON TO 

NASA GRC IN-HOUSE CURVE FITS FOR He-Xe MOLECULAR WEIGHT OF 20.183 kg/kmol 

Viscosity (10–6 kg/m-s) Thermal conductivity (W/m-K) 

T (K) Hirschfelder 

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Hirschfelder 

(third-order)g

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Conductivity 

third-order 

correction 

factor 

400 31.6340 31.5844 0.16 0.124434 0.121842 2.13 1.0220 

420 32.7501 32.6887 0.19 0.128500 0.125798 2.15 1.0220 

440 33.8311 33.7740 0.17 0.132511 0.129688 2.18 1.0220 

460 34.8957 34.8412 0.16 0.136472 0.133516 2.21 1.0219 

480 35.9283 35.8911 0.10 0.140381 0.137285 2.25 1.0219 

500 36.9556 36.9246 0.08 0.144259 0.141000 2.31 1.0219 

520 37.9641 37.9424 0.06 0.148001 0.144662 2.31 1.0219 

540 38.9587 38.9451 0.04 0.151694 0.148275 2.31 1.0219 

560 39.9383 39.9333 0.01 0.155329 0.151840 2.30 1.0219 

580 40.9011 40.9078 –0.02 0.158893 0.155361 2.27 1.0219 

600 41.8444 41.8689 –0.06 0.162369 0.158838 2.22 1.0219 

620 42.7750 42.8173 –0.10 0.165798 0.162274 2.17 1.0219 

640 43.7057 43.7534 –0.11 0.169264 0.165671 2.17 1.0218 

660 44.6249 44.6778 –0.12 0.172692 0.169029 2.17 1.0218 

680 45.5328 45.5908 –0.13 0.176083 0.172352 2.16 1.0218 

700 46.4371 46.4929 –0.12 0.179441 0.175640 2.16 1.0218 

720 47.3225 47.3845 –0.13 0.182764 0.178893 2.16 1.0218 

740 48.1999 48.2660 –0.14 0.186056 0.182115 2.16 1.0218 

760 49.0694 49.1377 –0.14 0.189318 0.185305 2.17 1.0218 

780 49.9314 50.0001 –0.14 0.192551 0.188465 2.17 1.0218 

800 50.7860 50.8534 –0.13 0.195756 0.191596 2.17 1.0218 

820 51.6334 51.6980 –0.13 0.198934 0.194698 2.18 1.0217 

840 52.4737 52.5343 –0.12 0.202087 0.197773 2.18 1.0217 

860 53.3072 53.3625 –0.10 0.205214 0.200821 2.19 1.0217 

880 54.1339 54.1829 –0.09 0.208318 0.203844 2.19 1.0217 

900 54.9539 54.9959 –0.08 0.211399 0.206842 2.20 1.0217 

920 55.7675 55.8017 –0.06 0.214457 0.209815 2.21 1.0217 

940 56.5745 56.6006 –0.05 0.217491 0.212765 2.22 1.0217 

960 57.3751 57.3929 –0.03 0.220504 0.215691 2.23 1.0216 

980 58.1693 58.1788 –0.02 0.223493 0.218596 2.24 1.0216 

1000 58.9570 58.9586 0.00 0.226460 0.221478 2.25 1.0216 

1020 59.7383 59.7326 0.01 0.229403 0.224340 2.26 1.0216 

1040 60.5056 60.5009 0.01 0.232274 0.227180 2.24 1.0216 

1060 61.2675 61.2639 0.01 0.235128 0.230001 2.23 1.0216 

1080 62.0242 62.0217 0.00 0.237965 0.232802 2.22 1.0216 

1100 62.7757 62.7746 0.00 0.240787 0.235583 2.21 1.0215 

1120 63.5222 63.5228 0.00 0.243592 0.238346 2.20 1.0215 

1140 64.2637 64.2665 0.00 0.246382 0.241091 2.19 1.0215 

1160 65.0004 65.0060 –0.01 0.249156 0.243817 2.19 1.0215 

1180 65.7323 65.7415 –0.01 0.251915 0.246526 2.19 1.0215 

1200 66.4595 66.4731 –0.02 0.254660 0.249218 2.18 1.0215 

 

g

 First-order thermal conductivity can be calculated by dividing the third-order conductivity by the third-order correction factor. 
NASA/CR—2006-214394 10

TABLE 5.—VISCOSITY AND CONDUCTIVITY CALCULATED VALUES AND COMPARISON TO 

NASA GRC IN-HOUSE CURVE FITS FOR He-Xe MOLECULAR WEIGHT OF 39.94 kg/kmol 

Viscosity (10–6 kg/m–s) Thermal conductivity (W/m-K) 

T (K) Hirschfelder 

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Hirschfelder 

(third-order)h

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Conductivity 

third-order 

correction 

factor 

400 33.1088 32.9759 0.40 0.080589 0.078295 2.93 1.0291 

420 34.3748 34.2279 0.43 0.083284 0.080888 2.96 1.0291 

440 35.5839 35.4583 0.35 0.085936 0.083439 2.99 1.0291 

460 36.7834 36.6677 0.32 0.088554 0.085950 3.03 1.0292 

480 37.9473 37.8569 0.24 0.091132 0.088423 3.06 1.0292 

500 39.1022 39.0267 0.19 0.093695 0.090861 3.12 1.0292 

520 40.2355 40.1776 0.14 0.096132 0.093265 3.08 1.0292 

540 41.3528 41.3104 0.10 0.098541 0.095636 3.04 1.0292 

560 42.4536 42.4256 0.07 0.100917 0.097977 3.00 1.0292 

580 43.5374 43.5239 0.03 0.103256 0.100288 2.96 1.0292 

600 44.6032 44.6059 –0.01 0.105553 0.102572 2.91 1.0292 

620 45.6541 45.6720 –0.04 0.107822 0.104828 2.86 1.0293 

640 46.6959 46.7229 –0.06 0.110098 0.107059 2.84 1.0293 

660 47.7231 47.7591 –0.08 0.112350 0.109265 2.82 1.0293 

680 48.7356 48.7811 –0.09 0.114580 0.111446 2.81 1.0293 

700 49.7509 49.7894 –0.08 0.116791 0.113605 2.80 1.0293 

720 50.7328 50.7845 –0.10 0.118978 0.115742 2.80 1.0293 

740 51.7063 51.7669 –0.12 0.121146 0.117857 2.79 1.0293 

760 52.6712 52.7371 –0.12 0.123295 0.119952 2.79 1.0293 

780 53.6279 53.6956 –0.13 0.125427 0.122027 2.79 1.0293 

800 54.5763 54.6427 –0.12 0.127540 0.124082 2.79 1.0294 

820 55.5165 55.5790 –0.11 0.129634 0.126118 2.79 1.0293 

840 56.4487 56.5049 –0.10 0.131712 0.128137 2.79 1.0293 

860 57.3728 57.4208 –0.08 0.133773 0.130137 2.79 1.0293 

880 58.2891 58.3272 –0.07 0.135818 0.132120 2.80 1.0293 

900 59.1975 59.2244 –0.05 0.137848 0.134087 2.81 1.0293 

920 60.0981 60.1130 –0.02 0.139863 0.136037 2.81 1.0293 

940 60.9910 60.9933 0.00 0.141863 0.137972 2.82 1.0293 

960 61.8763 61.8656 0.02 0.143847 0.139891 2.83 1.0293 

980 62.7541 62.7305 0.04 0.145817 0.141795 2.84 1.0293 

1000 63.6242 63.5883 0.06 0.147771 0.143684 2.84 1.0292 

1020 64.4869 64.4394 0.07 0.149711 0.145560 2.85 1.0292 

1040 65.3388 65.2841 0.08 0.151616 0.147421 2.85 1.0292 

1060 66.1838 66.1229 0.09 0.153509 0.149268 2.84 1.0292 

1080 67.0221 66.9561 0.10 0.155390 0.151103 2.84 1.0292 

1100 67.8538 67.7842 0.10 0.157259 0.152924 2.83 1.0292 

1120 68.6790 68.6073 0.10 0.159116 0.154732 2.83 1.0292 

1140 69.4979 69.4261 0.10 0.160962 0.156528 2.83 1.0292 

1160 70.3106 70.2406 0.10 0.162796 0.158312 2.83 1.0292 

1180 71.1172 71.0515 0.09 0.164619 0.160084 2.83 1.0291 

1200 71.9178 71.8589 0.08 0.166432 0.161844 2.83 1.0291 

 

h

First-order thermal conductivity can be calculated by dividing the third-order conductivity by the third-order correction factor. 
NASA/CR—2006-214394 11

TABLE 6.—VISCOSITY AND CONDUCTIVITY CALCULATED VALUES AND COMPARISON TO 

NASA GRC IN-HOUSE CURVE FITS FOR He-Xe MOLECULAR WEIGHT OF 83.8 kg/kmol 

 Viscosity (10–6 kg/m–s) Thermal Conductivity (W/m-K) 

T (K) Hirschfelder 

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Hirschfelder 

(third-order)i

GRC 

curve fit 

% diff 

Hirschfelder 

from GRC 

Conductivity 

third-order 

correction 

factor 

400 31.8624 31.7122 0.47 0.032369 0.031045 4.26 1.0401 

420 33.1977 33.0301 0.51 0.033514 0.032130 4.31 1.0401 

440 34.4551 34.3261 0.38 0.034630 0.033197 4.32 1.0401 

460 35.7171 35.6008 0.33 0.035735 0.034248 4.34 1.0401 

480 36.9497 36.8545 0.26 0.036822 0.035284 4.36 1.0401 

500 38.1669 38.0876 0.21 0.037901 0.036305 4.40 1.0402 

520 39.3633 39.3007 0.16 0.038921 0.037312 4.31 1.0402 

540 40.5416 40.4942 0.12 0.039928 0.038305 4.24 1.0402 

560 41.7021 41.6686 0.08 0.040924 0.039286 4.17 1.0402 

580 42.8448 42.8244 0.05 0.041908 0.040254 4.11 1.0402 

600 43.9699 43.9621 0.02 0.042879 0.041211 4.05 1.0402 

620 45.0779 45.0822 –0.01 0.043840 0.042155 4.00 1.0402 

640 46.1693 46.1852 –0.03 0.044794 0.043089 3.96 1.0403 

660 47.2430 47.2715 –0.06 0.045737 0.044012 3.92 1.0403 

680 48.2983 48.3418 –0.09 0.046671 0.044925 3.89 1.0403 

700 49.3648 49.3965 –0.06 0.047601 0.045827 3.87 1.0403 

720 50.3808 50.4362 –0.11 0.048515 0.046720 3.84 1.0403 

740 51.3882 51.4614 –0.14 0.049421 0.047603 3.82 1.0403 

760 52.3870 52.4725 –0.16 0.050319 0.048477 3.80 1.0403 

780 53.3771 53.4702 –0.17 0.051211 0.049342 3.79 1.0404 

800 54.3585 54.4548 –0.18 0.052095 0.050199 3.78 1.0404 

820 55.3312 55.4271 –0.17 0.052971 0.051047 3.77 1.0404 

840 56.2952 56.3873 –0.16 0.053841 0.051886 3.77 1.0404 

860 57.2505 57.3362 –0.15 0.054703 0.052718 3.77 1.0404 

880 58.1972 58.2741 –0.13 0.055559 0.053542 3.77 1.0404 

900 59.1351 59.2017 –0.11 0.056408 0.054358 3.77 1.0404 

920 60.0645 60.1194 –0.09 0.057250 0.055166 3.78 1.0404 

940 60.9853 61.0277 –0.07 0.058085 0.055968 3.78 1.0404 

960 61.8976 61.9271 –0.05 0.058914 0.056762 3.79 1.0404 

980 62.8015 62.8181 –0.03 0.059737 0.057549 3.80 1.0404 

1000 63.6970 63.7013 –0.01 0.060553 0.058329 3.81 1.0404 

1020 64.5844 64.5772 0.01 0.061363 0.059102 3.83 1.0404 

1040 65.4631 65.4461 0.03 0.062164 0.059869 3.83 1.0404 

1060 66.3339 66.3087 0.04 0.062960 0.060629 3.84 1.0403 

1080 67.1969 67.1654 0.05 0.063749 0.061383 3.86 1.0403 

1100 68.0522 68.0167 0.05 0.064533 0.062130 3.87 1.0403 

1120 68.9001 68.8631 0.05 0.065311 0.062872 3.88 1.0403 

1140 69.7406 69.7051 0.05 0.066084 0.063607 3.89 1.0403 

1160 70.5739 70.5431 0.04 0.066851 0.064336 3.91 1.0403 

1180 71.4003 71.3777 0.03 0.067613 0.065060 3.92 1.0403 

1200 72.2199 72.2092 0.01 0.068370 0.065778 3.94 1.0403 

 

i

First-order thermal conductivity can be calculated by dividing the third-order conductivity by the third-order correction factor. 
NASA/CR—2006-214394 12

TABLE 7.—He-Xe CALCULATED AND EXPERIMENTALj

 PRANDTL NUMBER 

AT FOUR MIXTURE MOLECULAR WEIGHTS 

He-Xe mol. wt. Temperature Prandtl number % Diff Hirsch. 

(kg/kmol) (K) Taylor Hirsch. from Taylor 

14.5 972 0.301 0.308 2.33 

28.3 982 0.231 0.240 3.90 

40.0 941 0.214 0.224 4.67 

83.8 962 0.251 0.261 3.98 

 

j

Table values taken from (Taylor, 1988, table 2). 
NASA/CR—2006-214394 13

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 20 40 60 80 100 120 140

He-Xe Mixture Molecular Weight (kg / kmol)

* For temperatures below 400 K f λmix

(3) was linearly extrapolated

Thermal Conductivity (W / m-K) 

793 K

363 K

291 K

Hirsch.*

Figure 1.—He-Xe calculated and experimentalk thermal conductivity as a 

function of molecular weight. 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 20 40 60 80 100 120 140

He-Xe Mixture Molecular Weight (kg / kmol)

Viscosity (10 -5 kg / m-s) 

550 K

500 K

400 K

293 K

Hirsch.

Figure 2.—He-Xe calculated and experimentalj viscosity as a 

function of molecular weight. 

 

k

Experimental values from Taylor’s article were extracted from a graph. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

REPORT DOCUMENTATION PAGE

2. REPORT DATE

19. SECURITY CLASSIFICATION

 OF ABSTRACT

18. SECURITY CLASSIFICATION

 OF THIS PAGE

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Form Approved

OMB No. 0704-0188

12b. DISTRIBUTION CODE

8. PERFORMING ORGANIZATION

 REPORT NUMBER

5. FUNDING NUMBERS

3. REPORT TYPE AND DATES COVERED

4. TITLE AND SUBTITLE

6. AUTHOR(S)

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT

13. ABSTRACT (Maximum 200 words)

14. SUBJECT TERMS

17. SECURITY CLASSIFICATION

 OF REPORT

16. PRICE CODE

15. NUMBER OF PAGES

20. LIMITATION OF ABSTRACT

Unclassified Unclassified

Final Contractor Report

Unclassified

1. AGENCY USE ONLY (Leave blank)

10. SPONSORING/MONITORING

 AGENCY REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546–0001

Available electronically at http://gltrs.grc.nasa.gov

November 2006

NASA CR—2006-214394

E–15672

WBS 463169.04.03

NAS3–00145

19

A Method for Calculating Viscosity and Thermal Conductivity of a HeliumXenon Gas Mixture

Paul K. Johnson

Gas mixtures; Viscosity; Thermal conductivity; Helium; Xenon; Brayton cycle;

Closed cycle

Unclassified -Unlimited

Subject Category: 20

Analex Corporation

1100 Apollo Drive

Brook Park, Ohio 44142

A method for calculating viscosity and thermal conductivity of a helium-xenon (He-Xe) gas mixture was employed, and

results were compared to AiResearch (part of Honeywell) analytical data. The method of choice was that presented by

Hirschfelder with Singh’s third-order correction factor applied to thermal conductivity. Values for viscosity and thermal

conductivity were calculated over a temperature range of 400 to 1200 K for He-Xe gas mixture molecular weights of

20.183, 39.94, and 83.8 kg/kmol. First-order values for both transport properties were in good agreement with

AiResearch analytical data. Third-order-corrected thermal conductivity values were all greater than AiResearch data,

but were considered to be a better approximation of thermal conductivity because higher-order effects of mass and

temperature were taken into consideration. Viscosity, conductivity, and Prandtl number were then compared to experimental data presented by Taylor.

Project manager, Lee S. Mason, Power and Electric Propulsion Division, NASA Glenn Research Center, organization

code RPT, 216–977–7106.