Article

pubs.acs.org/jced

Correlation for the Viscosity of Normal Hydrogen Obtained from

Symbolic Regression
Chris D. Muzny,* Marcia L. Huber, and Andrei F. Kazakov
Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305,
United States

ABSTRACT: We report the results of a symbolic-regression methodology to

obtain both the functional form and the coefficients for a wide-ranging
correlation for the viscosity of normal hydrogen. The correlation covers the
temperature range from the triple-point temperature to 1000 K and pressures
up to 200 MPa and extrapolates in a physically reasonable manner to 2000 K.
The estimated uncertainty is 4 % for the saturated liquid from the triple point
to 31 K, with larger deviations as the critical region is approached. The
estimated uncertainty is 4 % for the supercritical fluid phase at pressures to
200 MPa. For the limited range of 200 K to 400 K at pressures up to 0.1 MPa,
the uncertainty is 0.1 %.

■ INTRODUCTION
Recent interest in reducing petroleum usage, lowering green-
critical point that contribute to a weak divergence of the
viscosity at the critical point.6
house gas emissions, improving air quality, and developing a The identification of these distinct contributions to the
more diverse energy infrastructure has led to renewed interest viscosity is useful because it is possible, to some extent, to treat
in hydrogen as a fuel and as an energy carrier. To advance new ηο(T), η1(T), and Δηc(ρ,T) theoretically. Unlike the critical
hydrogen technologies, it is important to have accurate contribution to the thermal conductivity that impacts a
thermophysical property information. Deficiencies identified1 relatively large region of temperatures and densities around
in currently available formulations for the viscosity surface2,3 the critical point, the critical contribution to the viscosity is
present an opportunity to provide an improved correlation. In limited to a very small region in close proximity of the critical
addition, a powerful new technique, symbolic regression,4 has point.6 For several fluids where there are sufficient data in the
become available to identify underlying trends in experimental critical region, it has been shown7,8 that the ratio of Δηc/η is
data. The purpose of this work is to demonstrate the use of greater than 0.01 only within 1 % of the critical temperature. In
symbolic regression to develop correlations of material this work, we will omit the critical contribution and consider
properties. In particular, we developed an improved, wide- only data outside of 1 % of the critical temperature and not
ranging correlation for the viscosity of hydrogen. near the critical pressure. Kinetic theory can be used to
The viscosity η(ρ,T) of a fluid can be expressed as the sum of
calculate the zero-density viscosity.9 In particular, advances in
independent contributions, as5
theory have led to accurate calculations of the dilute-gas
η(ρ , T ) = ηo(T ) + Δηexcess(T , ρ) + Δηc (ρ , T ) (1)
viscosity that can be used to supplement experimental values.
Mehl et al.10 presented quantum mechanical ab initio
calculations of the dilute-gas viscosity and thermal conductivity
Δηexcess(ρ , T ) = η1(T )ρ + Δηh(ρ , T ) (2) of normal hydrogen and parahydrogen. Rainwater and Friend11
presented a theory for the calculation of η1(T). However, there
Here, the first term, ηο(T) = η(0,T), is the contribution to the is almost no theoretical guidance concerning the contribution,
viscosity in the limit of zero density, where only two-body Δηh(ρ,T), so that its evaluation is based entirely on
molecular interactions occur. The term Δηexcess (T,ρ) accounts
experimentally obtained data. The goal of this work is to
for the increase in viscosity above the zero-density value at
elevated density, and it can be expressed in terms of an initial- apply a symbolic-regression methodology to this term to
density viscosity coefficient η1(T) and a term Δηh(ρ,T) identify a functional form that best represents the experimental
representing the contribution of all other higher-order effects data.
to the viscosity of the fluid at elevated densities including many-
body collisions, molecular-velocity correlations, and collisional Received: November 30, 2012
transfer. The term, Δηc(ρ,T), the critical enhancement, arises Accepted: March 7, 2013
from the long-range fluctuations that occur in a fluid near its Published: March 27, 2013
This article not subject to U.S. Copyright.
Published 2013 by the American Chemical 969 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979
Society
Journal of Chemical & Engineering Data Article

Table 1. Summary of Available Dataa

first author year method purity/% est. unc./% no. pts. T range/K p range/MPa
Breitenbach32 1901 CAP na na 5 252 to 575 0.1
Markowski33 1904 CAP na na 3 287.6 to 457.3 0.1
Schmitt34 1909 TRANSP na na 10 78.3 to 458.4 0.1
Kamerlingh Onnes35 1913 CAP na 2 17 20 to 294 0.03 to 0.06
Vogel36 1914 OD na na 7 21 to 273.1 0.01 to 0.1
Gille37 1915 CAP na na 7 273 to 373 0.1
Verschaffelt38 1917 unknown na 1 1 20.35 0.1
Yen39 1919 CD na na 1 296.1 0.1
Gunther40 1920 SWP na na 8 15 to 273 0.01 to 0.1
Ishida41 1923 OIL na 0.3 1 321 0.1
Klemenc42 1923 TRANSP na na 1 273 0.1
Gunther43 1924 SWP na na 16 15 to 273 0.01 to 0.1
Trautz44 1929 TRANSP na na 12 195.2 to 523.2 0.1
Trautz45 1929 TRANSP na na 20 288.6 to 523.6 0.1
Trautz46 1929 TRANSP na na 15 192.4 to 524.9 0.1
Boyd47 1930 TRANSP na na 56 303 to 343 7.2 to 19.4
Trautz48 1930 TRANSP na na 6 298.2 to 523.2 0.1
Trautz49 1930 TRANSP na na 8 293.1 to 523.4 0.1
Trautz50 1930 TRANSP na na 8 289.8 to 1099.0 0.1
Trautz51 1931 TRANSP na na 12 295.2 to 523.4 0.1
Trautz52 1931 TRANSP na na 12 300.0 to 551.2 0.1
Trautz53 1931 TRANSP na na 12 292.6 to 523.2 0.1
Sutherland54 1932 OD na 0.4 11 74.8 to 293.8 0.1
Gibson55 1933 TRANSP 99.9 0.4 20 298 1.1 to 29.9
Trautz56 1934 TRANSP na na 4 293 to 523 0.1
Trautz57 1935 TRANSP na na 1 90 0.1
van Cleave58 1935 OD na 0.2 1 295 0.1
Adzumi59 1937 CAP na na 9 293 to 373 0.1
Keesom60 1938 OD na 2 14 14.5 to 20.3 0.01 to 0.1
van Itterbeek61,62 1938 OD na na 11 14 to 292.9 0 to 0.01
Johns27 1939 CAP na 1.1 42 14.3 to 20.7 0.15 to 0.28
Johnston63 1940 OD na 0.3 to 0.7 24 90 to 300 0.02 to 0.1
Keesom64 1940 OD na 2 21 13.9 to 20.5 0.01 to 0.1
van Itterbeek65 1940 OD na na 39 14.9 to 292.5 0 to 0.1
van Itterbeek66 1940 OD na na 6 14.9 to 291.8 0.01 to 0.1
van Itterbeek67 1941 OD na na 5 14.7 to 20.4 0.01 to 0.1
Wobser68 1941 HOP viscometer na na 5 293.1 to 371.2 0.1
Buddenberg69 1951 CAP 99.99 na 7 293 to 301 0.1
Kuss28 1952 CAP na 2 27 298−348 0.1 to 49
Kompaneets70 1953 CAP na na 7 284 to 873 0.1
Michels22 1953 CAP na 0.2b 95 298.1 to 398.1 2.6 to 186.3
Kestin71 1954 OD 99.99 0.2 1 294 0.1
Rietveld72 1957 OD na na 7 14.4 to 293.1 0 to 0.004
Coremans73 1958 OD na 2b 14 20 to 78 0.1
Kestin74 1958 OD 99.992 0.2 10 298 0.1 to 7.1
Kestin75 1959 OD 99.974 0.05 9 293 0.4 to 8.4
Rietveld76 1959 OD na 3 14 14.4 to 293.1 0.0005 to 0.005
Kestin77 1963 OD 99.999 0.2 13 293 to 303 0.1 to 0.6
Rudenko78 1963 CAP na 1.3 8 14.5 to 20.4 0.01 to 0.1
Barua79 1964 CAP 99.96 0.2 38 223 to 423 0.8 to 15.0
Diller17c 1965 TORC na 0.5 13 14 to 26 sat liquid
Diller17e (parahydrogen)17 1965 TORC crystal na 0.5 320 15 to 100 0.04 to 31.7
Menabde80 1965 OD na 2b 11 77.4 to 299.6 0.006
Tsederberg30 1965 CAP na 3 28 288.6 to 990.4 4.4 to 50.6
Golubev81 1966 CAP 99.99 1 96 77 to 273 0.9 to 49.1
Andreev82 1967 CAP 99.7 1.5 11 293 0.1 to 49
Konareeva83 1967 TOROC na 1.5 10 14 to 32 0 to 1
Kestin84 1968 OD 99.999 0.2 13 293 to 303 0 to 2.4
Rudenko85d 1968 FB na 1.5 91 33.2 to 300 0.1 to 217.1
Gracki86 1969 CAP 99.95 0.2 42 173 to 298 0.4 to 17.1
Guevara23 1969 CAP na 2b 23 1103 to 2152 0.1

970 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979

Journal of Chemical & Engineering Data Article

Table 1. continued
first author year method purity/% est. unc./% no. pts. T range/K p range/MPa
Golubev87 1970 CAP na na 58 298 to 523 0.1 to 81.6
Kestin88 1971 OD 99.9995 0.1 3 295 to 308 0.1
Carey89 1974 ACST na 0.1 18 291 to 299 0.15 to 11
Chuang90 1976 CAP 99.999 0.5 37 173 to 273 0.4 to 50.6
Clifford21 1981 OD 99.9995 0.2 2 298 to 308 0.1
Lukin91 1983 CAP na 0.3 23 76.5 to 293.2 0.1
Nabizadeh12 1999 OD na 0.5 to 1 76 295.6 to 399.2 0.1 to 5.8
Maltsev13 2004 CAP 99.9 3 3 500 to 1100 0.3
May14 2007 2CAP 99.9999 0.08 32 213.6 to 394.2 0 to 0.11
Mehl10 2010 ab initio calculations na 0.08 to 10 20 20 to 2000 0
Hurly15 2011 GRN 99.9999 0.5 111 225 to 400 0.3 to 3.4
Yusibani16 2011 CAP 99.999 2 17 294 to 400 4.6 to 99.3
a
Abbreviations: est. unc., estimated uncertainty; no. pts., number of points; na, not available; 2CAP, twin capillary; ACST, acoustic resonance; CAP,
capillary; CD, constant deflection; FB, falling body; GRN, Greenspan viscometer; HOP, Hoppler viscometer; OIL, oil drop; OD, oscillating disc;
SWP, swinging plate; TOROC, torsional oscillating cylinder; TORC, torsional crystal; TRANSP, transpiration. Values in bold type are considered
primary data. bUncertainty ascribed by Assael et al.20 cIsotherms (32 K) close to critical excluded from primary data set. dOnly points above 150 K
considered in primary data set. eParahydrogen; data adjusted as described in text.

■ EXPERIMENTAL DATA
Previously, Leachman et al.1 reviewed the experimental
Table 2. Coefficients of Equation 4
i ai
viscosity data for normal and parahydrogen current to 2006. 0 2.09630·10−1
1 −4.55274·10−1
2 1.43602·10−1
3 −3.35325·10−2
4 2.76981·10−3

Figure 1. Temperature and density ranges of the experimental data.

Figure 3. Deviations of eqs 3 and 4 from the primary data and selected
secondary data for dilute gas.

Table 1 summarizes the presently available data for the viscosity

of normal hydrogen, including the measurement method, range
of experimental conditions, and an estimate of the experimental
uncertainty. The uncertainties are those supplied by the original
authors, except where noted. In many cases these estimates are
highly optimistic. There are few recent measurements; since
1990, only five new data sets have appeared in the
literature.12−16 Owing to the scarcity of reliable low-temper-
Figure 2. Temperature and pressure ranges of the experimental data. ature data, we supplemented the normal hydrogen data with
the parahydrogen measurements of Diller17 that were adjusted
by first calculating their densities with the Leachman equation
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Journal of Chemical & Engineering Data Article

with a primary experimental apparatus, that is, one for which a

complete working equation is available; the form of the working
equation should be such that sensitivity of the property
measured to the principal variables does not magnify the
random errors of measurement; all principal variables should be
measurable to a high degree of precision; there should be
information on sample purity or purification methods; and
explicit quantitative estimates of the uncertainty of reported
values should be given, taking into account the precision of
experimental measurements and possible systematic errors.
Unfortunately, very few data in Table 1 meet these standards.
Consequently, within the primary data set it is also necessary to
include results that extend over a wide range of conditions,
albeit with a poorer accuracy, provided that they cannot be
demonstrated to be inconsistent with other more accurate data
or with theory. In all cases, the accuracy claimed for the final
recommended data must reflect the estimated uncertainty in
the primary information.
Assael et al.20 reviewed the experimental data available for the
development of a dilute-gas correlation for hydrogen. Since the
time of that publication, May et al.14 provided very high
accuracy experimental data for the zero-density viscosity of
Figure 4. Deviations of eqs 3 and 4 from literature correlations for hydrogen over the temperature range 200 K to 400 K. We
zero-density viscosity.
selected these data for the primary data set for this temperature
range since it has a much lower uncertainty (0.08 %) than the
Table 3. Coefficients of Equation 792
data previously considered21,22 by Assael et al.20 For the highest
i bi temperatures (above 1100 K), we selected the measurements of
0 −0.1870 Guevara et al.23 In addition, to supplement the experimental
1 2.4871 data, we included as primary data the theoretical values from
2 3.7151 Mehl et al.10 who employed the spherical version of the
3 −11.0972 hydrogen intermolecular potential determined in ab initio
4 9.0965 calculations by Patkowski et al.24 to calculate the viscosity of
5 −3.8292 normal and parahydrogen using a full quantum-mechanical
6 0.5166 formalism. Comparisons of the ab initio values with the
experimental data of May et al.14 showed agreement to within
Table 4. Coefficients of Equation 9 the experimental uncertainty for temperatures from 298 K to
394 K. However, at higher temperatures, it was suggested10 that
i ci the use of a ground-state potential introduced a positive bias in
1 6.43449673 the calculated viscosities at high temperatures, so the
2 4.56334068·10−02 experimental values of Guevara were weighted more heavily
3 2.32797868·10−01 than the ab initio calculations for temperatures above 1100 K.
4 9.58326120·10−01 Finally, very recently Berg and Moldover25 critically reviewed
5 1.27941189·10−01 all measurements of the viscosity of 11 gases near 25 °C and
6 3.63576595·10−01 zero density and provided a recommended value for calibration
purposes for normal hydrogen that we have incorporated in the
of state for parahydrogen,18 and then recalculating an zero-density correlation.
equivalent normal-hydrogen pressure with the Leachman For primary data for the development of the excess
equation of state for normal hydrogen.18 This has the effect contribution, we selected the data of Diller,17 Golubev and
of assuming that the parahydrogen viscosity is essentially Petrov,26 Hurly,15 Johns,27 Kuss,28 May et al.,14 Mehl et al.,10
equivalent to normal hydrogen viscosity provided that the Michels et al.,22 Nabizadeh et al.,12 Rudenko and Slyusar,29 and
density is the same. Figures 1 and 2 display the data sets and Tsederberg,30 and Yusibani et al.16 Only data at temperatures
illustrate the range of data coverage in the T,ρ and T,p planes, greater than 150 K were selected from Rudenko and Slyusar.29
respectively. The data of Rudenko and Slyusar29 display considerable scatter,
We evaluated the data and assigned data to either a primary but it was necessary to include at least some of this data set
or secondary data set. Data considered as primary (indicated by since they provide the only high-pressure data at temperatures
bold type in Table 1) were used in the development of the between 150 K and 298 K. Preliminary work indicated that it
correlation, while secondary data were used only for was not possible to fit both the Rudenko and Slyusar data over
comparison purposes. The Subcommittee on Transport the temperature range 33 K to 150 K at pressures to 70 MPa
Properties (now known as The International Association for and the data of Diller over the temperature range 33 K to 100 K
Transport Properties) of the International Union of Pure and at pressures to 35 MPa to within their estimated uncertainties,
Applied Chemistry made recommendations for the selection of so we selected only the Diller data for this temperature and
primary data for fluid transport properties. These recommen- pressure range. We also excluded from consideration as primary
dations include19 that the measurements must have been made any data of Diller at isotherms near critical. We then subtracted
972 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979
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Table 5. Summary of Resultsa

first author no. pts. T range/K p range/MPa est. unc./% AAD bias RMS
Breitenbach32 5 252 to 575 0.1 na 2.1 −1.7̀ 1.4
Markowski33 3 287.6 to 457.3 0.1 na 1.0 −1.0 0.1
Schmitt34 10 78.3 to 458.4 0.1 na 1.6 −1.6 2.2
Kamerlingh Onnes35 17 20 to 294 0.03 to 0.06 2 4.3 −4.2 3.9
Vogel36 7 21 to 273.1 0.01 to 0.1 na 5.4 1.3 7.0
Gille37 7 273 to 373 0.1 na 0.7 −0.7 0.3
Verschaffelt38 1 20.35 0.1 1 17.5 17.5 na
Yen39 1 296.1 0.1 na 0.4 0.4 0.0
Gunther40 8 15 to 273 0.01 to 0.1 na 13.8 11.2 14.3
Ishida41 1 321 0.1 0.3 0.7 0.7 na
Klemenc42 1 273 0.1 na 1.4 −1.4 na
Gunther43 16 15 to 273 0.01 to 0.1 na 13.0 10.4 13.5
Trautz44 12 195.2 to 523.2 0.1 na 1.3 0.7 1.4
Trautz45 20 288.6 to 523.6 0.1 na 1.2 1.2 1.6
Trautz46 15 192.4 to 524.9 0.1 na 1.1 0.9 0.8
Boyd47 56 303 to 343 7.2 to 19.4 na 6.7 −6.1 5.3
Trautz48 6 298.2 to 523.2 0.1 na 0.8 0.8 0.4
Trautz49 8 293.1 to 523.4 0.1 na 0.9 0.6 0.8
Trautz50 8 289.8 to 1099.0 0.1 na 2.2 2.2 1.2
Trautz51 12 295.2 to 523.4 0.1 na 0.9 0.9 0.5
Trautz52 12 300.0 to 551.2 0.1 na 0.9 0.9 0.5
Trautz53 12 292.6 to 523.2 0.1 na 0.8 0.8 0.4
Sutherland54 11 74.8 to 293.8 0.1 0.4 0.7 −0.1 0.8
Gibson55 20 298 1.1 to 29.9 0.4 0.8 −0.8 0.4
Trautz56 4 293 to 523 0.1 na 0.9 0.9 0.4
Trautz57 1 90 0.1 na 2.7 2.7 na
van Cleave58 1 295 0.1 0.2 0.5 0.5 na
Adzumi59 9 293 to 373 0.1 na 4.7 −4.7 0.1
Keesom60 14 14.5 to 20.3 0.01 to 0.1 2 25.0 −25.0 2.5
van Itterbeek61,62 11 14 to 292.9 0 to 0.01 na 6.2 −6.2 4.1
Johns27 42 14.3 to 20.7 0.15 to 0.28 1.1 1.5 -1.1 1.4
Johnston63 24 90 to 300 0.02 to 0.1 0.3 to 0.7 0.8 −0.8 0.7
Keesom64 21 13.9 to 20.5 0.01 to 0.1 2 13.5 −13.5 1.8
van Itterbeek65 39 14.9 to 292.5 0 to 0.1 na 103.6 90.1 208.5
van Itterbeek66 6 14.9 to 291.8 0.01 to 0.1 na 7.5 −7.5 4.7
van Itterbeek67 5 14.7 to 20.4 0.01 to 0.1 na 5.8 4.1 4.5
Wobser68 5 293.1 to 371.2 0.1 na 0.3 0.3 0.2
Buddenberg69 7 293 to 301 0.1 na 0.4 0.3 0.3
Kuss28 27 298 to 348 0.1 to 49 2 0.5 0.4 0.3
Kompaneets70 7 284 to 873 0.1 na 0.9 0.3 1.1
Michels22 95 298.1 to 398.1 2.6 to 186.3 0.2 0.3 -0.2 0.3
Kestin71 1 294 0.1 0.2 0.3 −0.3 na
Rietveld72 7 14.4 to 293.1 0 to 0.004 na 4.5 −4.1 5.0
Coremans73 14 20 to 78 0.1 2 3.6 −3.6 1.5
Kestin74 10 298 0.1 to 7.1 0.2 0.5 −0.5 0.2
Kestin75 9 293 0.4 to 8.4 0.05 0.9 −0.9 0.1
Rietveld76 14 14.4 to 293.1 0.0005 to 0.005 3 5.5 −5.5 6.9
Kestin77 13 293 to 303 0.1 to 0.6 0.2 0.4 −0.4 0.0
Rudenko78 8 14.5 to 20.4 0.01 to 0.1 1.3 7.1 7.1 0.7
Barua79 38 223 to 423 0.8 to 15.0 0.2 0.3 −0.1 0.4
Diller17 13 14 to 26 sat liquid 0.5 0.7 -0.6 0.7
Diller17** 320 15 to 100 0.04 to 31.7 0.5 2.2 -0.8 6.0
Menabde80 11 77.4 to 299.6 0.006 2 1.7 −1.7 0.8
Tsederberg30 28 288.6 to 990.4 4.4 to 50.6 3 1.5 1.5 1.0
Golubev81 96 77 to 273 0.9 to 49.1 1 3.2 2.1 3.9
Andreev82 11 293 0.1 to 49 1.5 0.5 −0.4 0.4
Konareeva83 10 14 to 32 0 to 1 1.5 5.0 3.7 4.0
Kestin84 13 293 to 303 0 to 2.4 0.2 0.4 −0.4 0.1
Rudenko85 91 33.2 to 300 0.1 to 217.1 1.5 7.8 -7.1 9.2
Gracki86 42 173 to 298 0.4 to 17.1 0.2 0.3 −0.1 0.3
Guevara23 23 1103 to 2152 0.1 2† 1.1 1.1 0.3

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Journal of Chemical & Engineering Data Article

Table 5. continued
first author no. pts. T range/K p range/MPa est. unc./% AAD bias RMS
Golubev87 58 298 to 523 0.1 to 81.6 na 0.7 0.3 0.8
Kestin88 3 295 to 308 0.1 0.1 0.1 −0.1 0.1
Carey89 18 291 to 299 0.15 to 11 0.1 0.5 0.1 0.5
Chuang90 37 173 to 273 0.4 to 50.6 0.5 0.7 0.7 0.8
Clifford21 2 298 to 308 0.1 0.2 0.2 −0.2 0.0
Lukin91 23 76.5 to 293.2 0.1 0.3 1.4 −1.4 1.0
Nabizadeh12 76 295.6 to 399.2 0.1 to 5.8 0.5 to 1 0.9 -0.9 0.7
Maltsev13 3 500 to 1100 0.3 3 1.4 −1.4 0.3
May14 32 213.6 to 394.2 0 to 0.11 0.08 0.1 0.0 0.1
Mehl10 20 20 to 2000 0 0.08 to 10 0.5 -0.5 0.7
Hurly15 111 225 to 400 0.3 to 3.4 0.5 0.3 -0.2 0.3
Yusibani16 17 294 to 400 4.6 to 99.3 2 1.6 0.3 1.7
a
Abbreviations: est. unc., estimated uncertainty; no. pts., number of points; na, not available; AAD, absolute average percent deviation; bias, average
percent deviation; RMS, root mean square. Values in bold type are considered primary data. Double asterisk (∗∗) = parahydrogen.

Figure 5. Comparison against the primary data set of the present

model as a function of temperature for pressures up to 200 MPa.

Figure 7. Comparison against the primary data set as a function of

temperature for three correlations.

MPa. Finally, we note that temperatures for all data were

converted to the ITS-90 temperature scale31 and the equation
Figure 6. Comparison against the primary data set of the present of state of Leachman et al.18 was used to provide the density for
model as a function of pressure for temperatures between 200 K and each experimental state point. The uncertainty in density
400 K. calculated from this equation of state is estimated to be 0.1 % at
temperatures from the triple point to 250 K and at pressures up
off the dilute contribution to obtain the excess viscosity. Any to 40 MPa, except in the critical region, where uncertainties are
data points that resulted in a negative value of the excess larger. In the region between 250 K and 450 K and at pressures
viscosity were also excluded from the primary data set. The to 300 MPa, the uncertainty in density is 0.04 %, while at
resulting primary set for the excess viscosity covers the temperatures between 450 K and 1000 K, the uncertainty in
temperature range 14 K to 2000 K at pressures up to 217 density increases to 1 %.
974 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979
Journal of Chemical & Engineering Data Article

May et al.14 and Guevara et al.23 to within their ascribed

experimental uncertainty, and the theoretical values of Mehl et
al.10 to within 1 % for temperatures less than 1000 K. The
deviations from Mehl et al.10 increase at higher temperatures to
3 % at 2000 K. As mentioned earlier, we have chosen the
experimental values of Guevara et al.23 over the ab initio
calculations for the highest temperatures.
There are several correlations for the zero-density viscosity of
hydrogen3,16,20,94 with which comparisons can be made; these
are shown in Figure 4. We include the correlation of McCarty94
that, although unpublished in the literature, has been adopted
as the default model in the REFPROP95 software package. The
high-temperature values of the zero-density correlation of
McCarty94 are based on a modified Enskog model of Hanley et
al.96 that incorporates a Lennard-Jones 12−6 potential with
quantum-mechanical collision integrals for parahydrogen for
the dilute gas; we were unable to ascertain the basis of the high-
temperature behavior of the Vargaftik et al.3 correlation. At high
temperatures, the present correlation agrees well with the
correlation of Assael et al.20 and that of Yusibani et al.,16 due in
part to the selection of the Guevara et al.23 data set as primary.
The low-temperature behavior of the present correlation is
based on the ab initio calculations of Mehl et al.10 and the
highly accurate recent data of May et al.14 that were unavailable
to some of the earlier researchers. The Yusibani et al.16
correlation is valid for 40 K < T < 2130 K, and is in good
agreement with our results.
The Excess Contribution. The excess contribution, eq 2,
represents the behavior of the viscosity outside of the critical
Figure 8. Comparison against the primary data set as a function of region as a function of both density and temperature. The
pressure for three correlations. initial-density coefficient of the viscosity η1(T) may be written5
η1(T ) = Bη(T )η0(T ) (5)
Zero-Density Limit. The zero-density limit of the viscosity
η0 (T) may be approximated by the expression5 where Bη(T) is the second viscosity virial coefficient and η0(T)
0.021357(MT ) 0.5 is the zero-density contribution from eqs 3 and 4. When
η0(T ) = sufficient high-quality, low-density data are available, such as for
σ 2 :*(T *) (3) water,97 the initial density dependence may be obtained directly
where : * is a reduced effective cross section, M = 2.01588 is from experimental data. Rainwater and Friend11,98 developed a
the molar mass in g·mol−1, σ is a length scaling parameter in theoretical representation of Bη that later was expanded upon
nm, η0 is in μPa·s, and the temperature T is in K. It is common by Vogel and co-workers.5,99 More recently, Behnejad and
to express the effective cross section with the functional form5 Miralinaghi92 used the Rainwater−Friend theory to develop an
4
expression for the second viscosity virial coefficient of hydrogen
that we adopt in this work:
ln(:*(T *)) = ∑ ai(ln(T *))i
i=0 (4) Bη*(T *) = Bη /σ 3 (6)
where the reduced temperature is T* = kBT/ε and ε/kB is an 6
energy scaling parameter in K. For the scaling parameters σ and
ε/kB we adopt the same values used by Behnejad and
Bη*(T *) = ∑ bi(T *)−1
(7)
Miralinaghi,92 namely σ = 0.297 nm and ε/kB = 30.41 K. We i=0

then fit the primary experimental data for the dilute gas10,14,23 where T*, σ are as defined earlier and the coefficients are
with the ODRPACK orthogonal distance regression package93 presented in Table 3.
and weighting factors of the square of the inverse of the Little theoretical guidance exists for the determination of the
experimental uncertainty for all points except the low- term Δηh(ρ,T) in eq 2, so we explore the use of symbolic
temperature (T < 100 K) values of Mehl et al.10 that were regression to determine both the coefficients and the functional
given increased weights in order to have the final representation form of this contribution.
be within 1 %. In addition, we constrained the fit to agree with Symbolic Regression. We employed the technique of
the recommended zero-density value of Berg and Moldover25 symbolic regression (SR)4,100,101 in our effort to find a new
at 25 °C, 8.8997 μPa·s. The final values of the coefficients are viscosity correlation for normal hydrogen. Shokir and
given in Table 2. Deviations of the primary data and the Dmour102 demonstrated the use of a form of symbolic
correlation given by eq 3 and 4 are shown in Figure 3. Also regression to obtain a correlation for the viscosity of pure
shown are deviations from secondary data sets measured after hydrocarbons and hydrocarbon gas mixtures. Symbolic
1950. The dilute-gas expression given by eq 3 and eq 4 with the regression is a specific application of genetic programming
coefficients in Table 2 represents the experimental data sets of (GP) that allows one to explore arbitrary functional forms in
975 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979
Journal of Chemical & Engineering Data Article

order to fit data. These functional forms are constructed by behavior and the relative simplicity of numerous models.
using a set of operators, parameters, and variables as building Finally, the constant elements of the model were reoptimized
blocks for functions of arbitrary complexity. Using a classic GP using a differential evolution nonlinear fitting algorithm to
algorithm for this function search is very slow and, even for produce the final functional form:
relatively simple data fitting problems, requires many days of ⎡
computing on current computing hardware. An improved GP η(T , ρ) = η0(T ) + η1(T )ρ + c1ρr2 × exp⎢c 2Tr + c3/Tr
algorithm based on multiobjective optimization and preferential ⎢⎣
evolution of models near the Pareto front has recently been
developed103 that greatly improves the efficiency of the search c4ρr 2 ⎤
and brings it into the realm of current, desktop computer + + c6ρr 6 ⎥
c5 + Tr ⎥⎦ (9)
technology. This method addresses many of the numerous
problems associated with SR including the issues of runaway In this equation η0(T) is obtained from eqs 3 and 4, η1(T,ρ)
complexity, evolutionary lock-in, and slow model development. from eqs 5 to 7, the scaled temperature is Tr = T/Tc and the
There are two available software realizations of SR104,105 that scaled density ρr = ρ/ρsc where the temperature is expressed in
use similar overall architecture, but vary in the details of the Kelvin, densities in kg·m−3, and the viscosity η is expressed in
algorithm implementation. (Certain trade names and company μPa·s. The quantity ρsc is a compressed-state density used for
names are mentioned to specify adequately the materials used. scaling that the symbolic regression procedure identified as 90.5
In no case does such identification imply endorsement by kg·m−3.
NIST, nor does it imply that the materials are the best.) Comparison with Experimental Data and Previous
However, both packages produce a similar result after Correlations. Comparisons with the experimental data are
optimization, which is a set of functions that are ranked by a presented in Table 5, which gives the number of data points,
combination of complexity and quality of data fit. We have the original authors’ estimated uncertainty of the data, average
found that using both packages allows us to increase the percent deviation, average absolute percent deviation, and a
diversity of models suggested by SR and, therefore, gives us a root-mean-square error of each data source. We define the
better pool of results from which to choose. Our experience percent deviation as P = 100 × (ηexp − ηcalc)/ηexp, where ηexp is
indicates that some criteria necessary for a good viscosity data the experimental value of the viscosity and ηcalc is the value
correlation are very difficult to automatically enforce in SR and calculated from the present correlation. The average absolute
must be enforced during the postoptimization model selection percent deviation (AAD) is found with the expression AAD =
process. These criteria include reasonable extrapolation (∑|P|)/n, where the summation is over all n points; the average
behavior and exclusion of functions that produce nonphysical percent deviation (bias) is AVG = (∑P)/n, and the we use
results such as negative viscosities or infinite viscosities. RMS = ([n∑P2 − (∑P)2]/n2)1/2. Table 5 summarizes the
Our specific choice of model search for the viscosity of performance of the new model. Figures 5 and 6 show
hydrogen was the following. We initialized SR optimization comparisons with the primary data for normal hydrogen over
with the set of operators {+,-,*,/,Exp,∧} and the operands the temperature range 200 K to 400 K at pressures to 200 MPa;
{constant,T,ρ} to simultaneously optimize function complexity the agreement is to within about 4 %. For the region between
and mean absolute error. Initial weights based on the estimated 200 K and 400 K at pressures up to atmospheric, the present
uncertainty of the data were also utilized. The form optimized correlation reproduces the high-accuracy data of May et al.14 to
was within 0.1 %. In Figures 7 and 8 we compare the performance
f (T , ρ) = η(T , ρ) − η0(T ) − η1(T )ρ of the correlation given by eqs 2 and 3, eqs 5 to 7, and eq 9
(8) against that of the correlation of Vargaftik et al.,3 and that of the
where η0 is obtained from eqs 3 and 4 and η1 is from eqs 5 to 7. McCarty model, as implemented in the NIST Standard
Several hundred individual evolutions that each typically Reference Database 23 (REFPROP).95 As mentioned earlier,
involved ∼1011 function evaluations and 100 h of c.p.u. core the default correlation in v9.0 and earlier of REFPROP is based
time were performed to identify an optimal function that is on the unpublished work of McCarty;94 this model relied
both relatively simple and able to fit the data well. Some terms heavily on the data of Diller.17 The recently developed model
appeared repeatedly; one was a scaled density, in this case of Yusibani et al.16 is not included in the comparisons because it
0.011ρ (where ρ has units of kg·m−3), or equivalently ρ/90.5. is limited to temperatures above 40 K for the dilute gas and
Therefore, we made additional runs where the density and above 100 K for pressures up to 220 MPa. As shown in Figures
temperature were both scaled, with Tr = T/Tc and ρr = ρ/ρsc . 7 and 8, the SR correlation performs comparably to the
For temperature, we scaled with the critical temperature 33.145 Vargaftik3 and McCarty94,95 correlations. The largest deviations
K;18 for density we used the scaling factor ρsc = 90.5 kg·m−3. shown in Figures 7 and 8 for the SR correlation are for
This value is interesting in that it is a very compressed liquid parahydrogen data of Diller17 as the critical region is
state, close to the value of the density of the liquid at the triple approached. There are few liquid-phase data available, but
point (77.00 kg·m−3). In 1971, Hildebrand106 modified the comparisons with the data of Johns27 and the saturated liquid
earlier work of Batschinski107 and introduced the concept of an data of Diller17 indicate agreement with the data to within 4 %.
intrinsic volume where the fluidity is zero. Since then, various The McCarty model was not developed for pressures above
researchers5,108−110 have successfully used variations of this 100 MPa, and this is evident from Figure 8.
concept to model the viscosity of a range of fluids, and it is
interesting that symbolic regression suggests a scaling in terms
of a very compressed-state density. Future work will investigate
■ CONCLUSION
A new wide-ranging correlation for the viscosity of normal
if this same type of term arises for other fluids. hydrogen was developed by way of a symbolic regression
The final choice of model function was done in an ad hoc methodology. The correlation covers the temperature range
fashion after reviewing the overall fits to data, the extrapolation from the triple point to 1000 K and pressures up to 200 MPa
976 dx.doi.org/10.1021/je301273j | J. Chem. Eng. Data 2013, 58, 969−979
Journal of Chemical & Engineering Data Article

and extrapolates in a physically reasonable manner to 2000 K. (15) Hurly, J. J. Acoustic relaxation and virial coefficients of H2 and
The dilute-gas viscosity agrees to within the experimental O2, and viscosity of H2. Int. J. Thermophys. 2011, submitted.
uncertainty of the most accurate gas phase data of May et al.14 (16) Yusibani, E.; Nagahama, Y.; Kohno, M.; Takata, Y.; Woodfield,
over the temperature range 213−394 K at pressures to 0.11 P. L.; Shinzato, K.; Fujii, M. A capillary tube viscometer designed for
measurements of hydrogen gas viscosity at high pressure and high
MPa, and also reproduces the recommended value of Berg and
temperature. Int. J. Thermophys. 2011, 32 (6), 1111−1124.
Moldover25 at 25 °C and zero density. Outside of that region, (17) Diller, D. E. Measurements of the viscosity of parahydrogen. J.
the estimated uncertainty is 4 % for the saturated liquid and Chem. Phys. 1965, 42 (6), 2089−100.
supercritical fluid phases, except along the saturated liquid (18) Leachman, J. W.; Jacobsen, R. T.; Penoncello, S. G.; Lemmon,
boundary above 31 K and in the near-critical region, where the E. W. Fundamental equations of state for parahydrogen, normal
uncertainty is larger. The simplicity of this new correlation hydrogen, and orthohydrogen. J. Phys. Chem. Ref. Data 2009, 38 (3),
makes it easy to implement and demonstrates the power of 721−748.
symbolic regression in finding relatively simple functional forms (19) Assael, M. J.; Ramires, M. L. V.; Nieto de Castro, C. A.;
for data correlation. Wakeham, W. A. BenzeneA further liquid thermal-conductivity

■
standard. J. Phys. Chem. Ref. Data 1990, 19 (1), 113−117.
AUTHOR INFORMATION (20) Assael, M. J.; Mixafendi, S.; Wakeham, W. A. The viscosity and
thermal conductivity of normal hydrogen in the limit of zero density. J.
Corresponding Author Phys. Chem. Ref. Data 1986, 15 (4), 1315−22.
*E-mail: chris.muzny@nist.gov. (21) Clifford, A. A.; Kestin, J.; Wakeham, W. A. The viscosity of
Notes mixtures of hydrogen with three noble gases. Ber. Bunsenges. Phys.
The authors declare no competing financial interest. Chem. 1981, 85 (5), 385−8.

■
(22) Michels, A.; Schipper, A. C. J.; Rintoul, W. H. The viscosity of
ACKNOWLEDGMENTS hydrogen and deuterium at pressures up to 2000 atm. Physica
(Amsterdam) 1953, 19, 1011−28.
We thank Mark Kotanchek and Katya Vladislavleva of Evolved (23) Guevara, F. A.; McInteer, B. B.; Wageman, W. E. High-
Analytics, LLC, for their encouragement and expertise in the temperature viscosity ratios for hydrogen, helium, argon, and nitrogen.
use of the symbolic regression method.

■
Phys. Fluids 1969, 12 (12), 2493−505.
(24) Patkowski, K.; Cencek, W.; Jankowski, P.; Szalewicz, K.; Mehl, J.
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