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Viscosity Equation and Data Analysis for Gases, Study notes of Engineering

The equation for calculating viscosity of gases based on temperature and pressure, along with data pairs and their deviations from reference values for nitrogen and argon. The data is fitted to a quadratic equation to enable calculation of viscosity at a specific pressure.

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NATL INST. OF STAND & TECH
AlllDS T7M3flS gg
I^EFEREIMCE Publi-
cations
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*NBS TECHNICAL NOTE 1186
U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards
Interpolation Formulas for
Viscosity of Six Gases:
Air, Nitrogen, Carbon Dioxide,
Helium, Argon, and Oxygen
103
.LI5753
Mo.llSC
1984
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NATL INST. OF^ STAND^ &^ TECH

AlllDS T7M3flS^ gg

I^EFEREIMCE Publi-

cations I/) Q

NBS TECHNICAL NOTE^1186

U.S. DEPARTMENT^ OF^ COMMERCE/National^

Bureau of Standards

Interpolation Formulas^

for Viscosity of^ Six^ Gases: Air, Nitrogen,^ Carbon^ Dioxide, Helium, Argon,^ and^ Oxygen

. LI

Mo.llSC

NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901.

The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (^) (1) a basis for the Nation's physical measurement system, (^) (2) scientific

and technological services for industry and government, (3) a technical basis tor equity in

trade, and (4) technical services to promote public safety. The Bureau's technical work is per-

formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with^ measurement

systems of other nations and furnishes essentia! services leading to accurate and uniform

physical and chemical measurement throughout the Nation's^ scientific^ community, industry,

and commerce; conducts materials research leading to improved^ methods of measurement,

standards, and data on the properties of materials needed by industry,^ commerce,^ educational

institutions, and Government; provides advisory and research^ services^ to other Government

agencies; develops, produces, and distributes Standard^ Reference^ Materials;^ and^ provides

calibration services. The Laboratory consists of the following^ centers:

Absolute Physical Quantities^ —^ Radiation Research —^ Chemical Physics —

Analytical Chemistry —^ Materials^ Science

THE NATIONAL^ ENGINEERING LABORATORY^ provides^ technology^ and^ technical^ ser-

vices to the public and private^ sectors to address national^ needs^ and^ to^ solve^ national

problems; conducts research (^) in engineering and applied science in support of these efforts;

builds and maintains competence in the necessary disciplines^ required^ to^ carry^ out^ this

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provides engineering measurement traceability^ services;^ develops^ test^ methods^ and^ proposes

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and develops^ and improves^ mechanisms^ to^ transfer^ results^ of^ its^ research^ to^ the^ ultimate^ user.

The Laboratory consists of the following centers:

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Engineering —^ Building Technology — (^) Fire Research — Chemical Engineering^

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection,

acquisiuon, application, and use of computer technology to improve effecfiveness^ and

economy in Government operations in accordance with Public Law 89-306^ (40 U.S.C. 759),

relevant Executive Orders, and other directives; carries out this mission^ by^ managing^ the

Federal Information Processing Standards Program, developing^ Federal^ ADP^ standards

guidelines, and managing^ Federal^ participation^ in^ ADP^ voluntary^ standardization^ activities;

provides scientific and technological advisory^ services^ and^ assistance^ to^ Federal^ agencies;^ and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology —^ Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. ^Some divisions within the center are located at Boulder, CO 80303.

National Bureau of Standards Technical Note 1 1 86

Natl. Bur. Stand. (U.S.), Tech. Note (^11) 86, 26 pages (Feb. (^) 1984)

CODEN: NBTNAE

U.S. (^) GOVERNMENT PRINTING OFRCE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

INTERPOLATION FORMULAS FOR VISCOSITY OF SIX GASES:

AIR, NITROGEN, CARBON DIOXIDE,^ HELIUM,^ ARGON,^ AND^ OXYGEN

Frank E. Jones

National Bureau of Standards

Washington, DC 20234

Equations for the calculation of viscosity for dry air,

nitrogen, carbon dioxide, helium, argon, and oxygen have been

developed as interpolation formulas fitted to experimental

data. The approximate ranges of strict application of the

equations are (^) the ranges of temperature (^) (20°C (^) £ t (^) _< SO^C) and

pressure (0.

£ p^ £^

4 MPa; 0.4 <_ p <^40 atm) for the experimental

data. The estimates of relative resTdual standard deviation for

the fits (0.05% for air, 0.03% for nitrogen, 0.02% for carbon

dioxide, 0.02% for helium, 0.03% for argon, and 0.03% for oxygen)

are in close agreement with estimates of precision for the

experimental data.

Key words: Air; argon; calculation; carbon dioxide; helium;

nitrogen; oxygen; viscosity.

1. INTRODUCTION

In the calibration and use of flow metering devices the precise,

accurate value of the viscosity of the gas being metered is often

required. The availability of simple equations relating viscosity, y,

to temperature, t, and pressure, p, would be a convenience to the

engineer in calculating y. In the present work, simple interpolation

formulas have been developed which enable the engineer to conveniently

make precise, accurate calculations of y for dry air, nitrogen, carbon

dioxide, helium, argon, and oxygen using small readily available

hand-held calculators. The formulas are fitted to experimental data.

The data were selected from the literature on the basis of claimed

accuracy and precision, and of internal consistency. The last of these

criteria is particularly important for the development of empirical

equations for volume flow with a dominant term of the Poiseuille form,

the subject of a later paper. The sets of data published by Kestin and

3. DEVELOPrCNT^ OF^ EQUATIONS

The equations for^ y (t,p)^ were^ developed^ using^ a^ procedure^ [6] in

which u(t,p) is expressed as

y(t,p) =^ Pq (t,0)^ +^ Ay(p). (^) (1)

where y (t,0) is the viscosity at "zero pressure" and Ay(p) is the

experimental value of viscosity, Vr^^p.^ > minus y (t,0). The procedure

will be outlined below.

Air

The data pairs, (20.00°C, 181.94 yg/cm^s), (23.44°C, 183.75 yg/cm-s),

(23.90°C, 184.21 yg/cm-s), (25.00°C, 184.62 yg/cm-s), and (50.00°C, 197.

yg/cm-s), taken from Table 1, were fitted by least squares to an equation

quadratic in t to enable calculation of y at 0.101325 MPa, y-,. The small

departures of p from this value of pressure for these pairs are not

significant. The resulting equation is

y-j

= 170.368 + 0.605434 t - 1.33200 x 10"^^ t^,

(2)

where y-, is in yg/cm-s and t is in °C. To reduce y-, to "zero pressure,"

y ,^ the^ value^ of^ the^ increase^ in^ y per^ 0.10325^ MPa,^ 0.11^ yg/cm-s,^ estimated

from the tabulated data was subtracted from y-. resulting in

y^

= 170.258 + 0.605434 t - 1.33200 x lO'"^ t^. (3)

Calculated values of y are listed in Table 1

The difference, Ay, between the experimental values of y and calculated

values of y is also listed in Table 1. The values of Ay were fitted by

least squares to an equation quadratic in p. The resulting equation is

Ay =^ -2.44358^ x^

10"^ +

1.17237 (^) p +^ 0.125541 p^.^ (4)

Equations (3) and (4) were added together to synthesize the final

equation for calculating y:

y ca 1^^

= y^

+ Ay =^ 170.256 + 0.605434 t -^ 1.33200 x 10

" t

+ 1.17237 p + 0.

/, (5)

where y ,, is in ug/cin«s, t is in °C, and p is in MPa. Values of y^^i^ and

caic. —^ Co I c

.

differences between y^^^^ and experimental values, y^^^. , are^ listed

ca I c

.

me as (^).

,

in Table 1

.

The estimate of residual standard deviation (RSD), that is, the

estimate of the standard deviation of y^^i^

-y^^oc »^

''s 0-^^^ yg/cm-s for

ca (^) I c. me a 5

.

the 19 differences (n=19). The estimate of the relative residual^ standard

deviation (RRSD), that is, the ratio of RSD to the mean y^^^^ , is^ 0.05^.

me as

.

Kestin and Leidenfrost [1] estimated that for their measurements^ "a

precision ranging from +0.01% to +^0.07%, depending on the gas, has^ been

achieved. The final accuracy of the measurements is^ estimated^ to^ be^ of

the order of +0.05%." DiPippo and Kestin [2] estimated the precision of

their data to be +0.05%; they reached the conclusion that "no meaningful

assessment of the accuracy of the present data can be^ given^ if^ by^ accuracy

we mean the irreducible discrepancy between the best measurements available

at any particular time."

Ay =^ -1.12860^ X

10'^ +

1.24165 p +^ 9.87206 x

10'^

/. (10)

Equations (9) and (10) were added together to synthesize the final

equation for calculating y:

Vr..^.

= 167.214 + 0.392728 t + 1.22474 x 10"^^ t^

ca I c

.

+ 1.24165 p + 9.87206 x 10'^^ p^, (11)

where (^) y-,-]- is in yg/cni'S (^) , t is in (^) ^, and (^) p is in MPa,

The RSD is 0.05 yg/cm-s for n =^ 50; the RRSD is 0.03%. Kestin et. al.

[3] estimated^ the^ accuracy^ of^ their^ experimental^ measurements^ to^ be^ of^ the

order of +0.2% and the relative precision to be 0.03%. The accuracy and

precision estimates quoted above for air from [1] and [2] apply also to

nitrogen.

For (^) y in yPa'S , t in (^) ^, and (^) p in MPa (^) , equation (11) becomes

-2 (^). ,. oo... .. .n-4. ^ II-* /xi/i -^ < \Jk y I y X V III M^ M^ It^ I^ I^ *. calc.

y„-,. =^ 16.7214^ +^ 3.92728^ x^10

"

t +^ 1.22474 x 10

" t

+ 0.124165 p + 9.87206 x

10""^ p^' (12)

For y in 10"^ lb ft" sec" , t in °C, and p in PSI_, equation (11) becomes

-2 (^) r, ^or.r.-7 Tn-5 x ^ II • <.v\ <^ --^ y tr^ <VJI II 'M III JfllJ^ /^ »^ I^ f^1 -f- calc.

y.,n. =^ 11.2363^ +^ 2.63901^ x^10

^

t +^ 8.22987 x 10

" (^) t

+ 5.75263 X 10"^^ p + 3.15351 x

10"^ (^) p^. (13)

Carbon Dioxide

The data pairs, (20.00°C, 146.63 yg/cm-s), (25.00°C, 149.09 yg/cm-s),

(30.40°C, 151.81 yg/cm-s), and (49.12°C, 161.75 yg/cm-s), taken from Table 3,

were fitted by least squares to an equation quadratic in t to enable

calculation of y-,. The resulting equation is

y, =^ 137.335^ +^ 0.441133^ t^ +^ 1.12987^ x^

10"^

t^. (14)

To reduce y-, to y , 0.13 yg/cm«s estimated from the tabulated data was

subtracted from y-, resulting in

u =^ 137.205 + 0.441133 t +^ 1.12987 x

10"^

t^. (^) (15)

The values of Ay were fitted by least squares to an equation quadratic

in p. The resulting equation is

Ay =^ 0.133827 +^ 6.28105 x

10"^

p +^ 0.562974^ p^.^ (16)

Equations (15) and (16) were added together to synthesize the final

equation for calculating y:

v^r.i.

= 137.339 + 0.441133 t + 1.12987 x 10"^^ t^

Cd I C

+ 6.28105 X 10"^^ p + 0.562974 p^, (17)

where y-^-i- is in yg/cm'S , t is in %^, and p is in MPa.

The RSD is 0.04 yg/cm-sfor^ n =^11 ; the RRSD is 0.02%. Kestin et. al

[4] estimated^ the^ accuracy^ of^ the^ 25.00°C^ point,^ "expressed^ as^ the^ maximum

Equations (20) and^ (21) were^ added^ together^ to^ synthesize^ the^ final

equation for calculating y:

y T

= 185.975 + 0.530773 t -^ 1.04982 x

10"^

t^ ^calc.

6.99813 X

10"^

p, (22)

where y , is in yg/crri'S, t is in °C, and p is in MPa.

calc. ^^^^ —

The RSD^ is^ 0.04^ yg/cm-s for n^ =^ 15; the^ RRSD^ is^ 0.02%.^ The^ accuracy

and precision estimates quoted above for air from [1] and [2] apply also to

helium. The 25.00°C °C value is estimated to be accurate within +^ 0.1% [5],

For (^) y in yPa^s (^) , t in (^) ^, and (^) p in MPa (^) , equation (^) (22) becomes

-2 (^).. r.nr.nn -,^-4 (^). ^ IX --l/»-» H- t-.^ <ll//< V III fi/iiii.#^ ..^ .*. calc.

y„,-,_ =^ 18.5975^ +^ 5.30773^ x^10

"

t -^ 1.04982 x 10

" t

  • (^) 6.99813 X 10"^^ p. (23)

For y in 10"^ lb ft" sec" , t in °C, and p in PS I , equation (22) becomes

y^.i.

= 12.4969 + 3.56663 x 10"^^ t - 7.05446 x 10"^^ t^

ca I c.

3.24228 X

10"^

p. (24)

Argon

The data pairs, (20.00°C, 222.86 yg/cm-s), (24.39°C, 225.79 yg/cm-s),

(25.00°C, 226.36 yg/cm.s), and (50.31°C, 243.43 yg/cm-s), taken from Table 5

were fitted by least squares to an equation quadratic in t to enable

calculation of y-.. The resulting equation is

y-,

= 208.940 -^^ 0.702190 t - 3.30712 x 10"^^ t^. (25)

To reduce y-, to y , 0.21 yg/cm-s estimated from the tabulated data was

subtracted from y-, resulting in

y^

= 208.730 + 0.702190 t - 3.30712 x 10 ^^ t^. (26)

The values of Ay were fitted by least squares to an equation quadratic

in p. The resulting equation is

Ay =^ 3.17420 X

10"^ +

1.73987 p +^ 0.152358 p^.^ (27)

Equations (26) and (27) were added together to synthesize the final

equation for calculating y:

y.=i.

= 208.762 + 0.702190 t - 3.30712 x 10"^^ }

ca (^1) c

.

+ 1.73987 p + 0.152358 p^, (28)

where (^) u^^-i- is in yg/cm^s (^) , t is in (^) ^, and (^) p is in MPa^.

The RSD is 0.07 yg/cm-s for n =^ 38; the RRSD is 0.03%. The estimates

of accuracy and precision of the experimental data for argon^ are^ those

quoted above from [1], [2], and [3].

For y in yPajs, t in °C, and p in MPa, equation (28) becomes

-2 (^).. .n^no in-5.

calc.

y__-,- =^ 20.8762 +^ 7.02190 x 10

^

t -^ 3.30712 x 10

" t

+ 0.173987 p + 1.52358 x 10'^^ p^.^ (29)

where locale. ^^ ^" Mg/cm-s , t is in

^, and (^) p is in atm.

The RSD^ is^ 0.06^ g/cm^ s for n =^ 16; the RRSD is 0.03%. The

estimates of accuracy and precision of the experimental data for oxygen

are those quoted above from [1], [2], and [3].

For (^) y in uPa-s (^) , t in (^) ^, and (^) p in MPa (^) , equation (34) becomes

^calc

" '•9.0395^ +^ 6.50043^ x

10"^

t -^ 8.97542 x

10"^

t^

+ 0.129185 p + 1.28975 x 10"^^ p^. (35)

— fi -1 1

For y in 10"^ lb ft" sec" , t in **C, and p in PSI, equation

(34) becomes

^ ,^^ =^ 12.7940^ +^ 4.36809^ x^

10"^

t -^ 6.03121^ x

10"^

t^ calc.

+ 5.98525 X 10"^^ p + 4.11587 x 10"^^ p^. (36)

4. RANGES OF APPLICATION OF THE EQUATIONS

The equations developed in this paper are interpolation formulas

fitted to experimental data. The range of strict application is

indicated by the range of t and p of the experimental data listed in

Tables 1-5. In the absence of measurements of y for temperatures below

20°C of comparable quality to that of the data in the tables, there are

two options in extending calculations below 20" C (0 <^ t <^ 20*0):^ 1)

apply the extended law of corresponding states [3,7-10]; 2) extrapolate

using the equations developed here, with probable loss in accuracy.

Either option might be followed until suitable experimental data at

temperatures below 20''C became available.

12

Hanley et al. [11] have developed a functional form to represent

critically evaluated viscosity and thermal conductivity coefficient

data, and have generated tables. The gases treated by Hanley et al

include nitrogen and argon. Values of y calculated using the formulas

in the present work have been compared with interpolated values from

the tables in [11] at 0°C, S^C, 10"*C, IS^'C, 20''C, and 25°C, and 0.

MPa, for nitrogen and argon. The deviation of the values in the

present work from^ the values^ from^ reference^ [11], expressed^ in^ percent

are: for nitrogen, +0.26%^ at O^C, +0.02%^ at S^C, -0.12%^ at 10°C,

-0.28% at IS^'C, -0.32% at 20*0, and -0.38% at 25''C; for argon, -1.16%

at 0°C, -1.10% at S^C, -0.99%^ at 10*^0,^ -0.86%^ at 15°C, -0.78%^ at 20''C,

and -0.66% at 25°C. These deviations are all well within the

uncertainty, j^ 2%, estimated by Hanley et al. for their tables.

5. CONCLUSIONS

Equations (interpolation formulas fitted to experimental data) for

the calculation of y for dry air, nitrogen, carbon dioxide, helium,

argon, and oxygen have been developed. The estimates of relative

residual standard deviation for the fits are in close agreement with

the estimates of precision for the experimental data in the above

stated ranges.

6. ACKNOWLEDGMENTS

The author is deeply grateful to J. Kestin for his helpful

suggestions and to him and his collaborators for the experimental data

on which the present work is based. The typing of the manuscript by

Susan Johnson is gratefully acknowledged.

References

1. Kestin, J., and Leidenfrost, W., "An Absolute Determination of the

Viscosity of Eleven Gases," Physica, Vol. 25, 1959, pp. 1033-1062.

2. DiPippo, R., and Kestin, J., "The Viscosity of Seven Gases up to

500°C and Its Statistical Interpretation," Proceedings Fourth

Symposium on Thermophysical Properties , American Society of

Mechanical Engineers, College Park, Md., 1958, pp. 304-313.

3. Kestin, J., Paykoc, E., and Sengers, J.V., "On the Density Expansion

for Viscosity^ in^ Gases,"^ Physica, Vol. 54, 1971, pp. 1-19.

4. Kestin, J., Ro, S.T. , and Wakeham, W.A., "Viscosity of Carbon

Dioxide in the Temperature Range 25-700°C," J. Chem. Phys., Vol. 58,

No. 8, 1972, pp. 4114-4118.

5. Kestin, J., Khalifa,^ H.E.,^ Ro, S.T., and Wakeham, W. A., "The

Viscosity and Diffusion Coefficients of Eighteen Binary Gaseous

Systems," Physica, Vol. 88A, 1977, pp. 242-260.

6. Kestin, J., and Whitelaw, J.H., "The Viscosity of Dry and Humid Air,"

Int. J. Heat Mass Transfer, Vol. 7, 1964, pp. 1245-1255.

7. Kestin, J., Ro, S.T., and Wakeham, W.A., "An Extended Law of Corres-

ponding States for the Equilibrium and Transport Properties of Noble

Gases," Physica, Vol. 58, 1972, pp. 165-211.

8. Kestin, J., Ro, S.T., and Wakeham, W.A., "Viscosity of the Noble

Gases in the Temperature Range 25-700°C," J. Chem. Phys., Vol. 56,

No. 8, 1970, pp. 4119-4124.

9. Kestin, J., and Mason, E.A. , "Transport Properties in Gases

(Comparison Between Theory and Experiment)," AIP Conf. Proc. No. 11,

1973, Edited by J. Kestin, pp. 137-192.

10. Hellemans, J.M., Kestin, J., and Ro, S.T., "The Viscosity of Oxygen

and Some of Its Mixtures With Other Gases," Physica 65_, 1973, pp. 362-375,

11. Hanley, H.J.M., McCarty, R.D., and Haynes, W.M., "The Viscosity and

Thermal Conductivity Coefficients for Dense Gaseous and Liquid Argon,

Krypton, Xenon, Nitrogen, and Oxygen," J. Phys. Chem. Ref. Data 3^,

1974, (^) pp. 979-1018.

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