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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
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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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