Towards An Equation of State for Gas Viscosities...

32nd Annual Symposium & Workshop IEA Collaborative Project on Enhanced Oil Recovery

- 1 -

Towards An Equation of State for Gas Viscosities

Klaus POTSCH1, T. GUMPENBERGER2 1OMV E&P, 2OMV E&P

Protteser Strasse 40, 2230 Gaenserndorf, Austria [email protected], [email protected]

Abstract

For many years it has been customary to calculate gas viscosities from their composition. Correlations exist and some attempts have been made to find an equation of state. This idea has been triggered by the fact that there exists an equation of state for the volume (or density). But for the transport property viscosity an EOS is not widely in use. The paper discusses firstly the data base for developing an EOS. Measurements in the past though were conducted with the light ends of the hydrocarbon spectrum. The theory suggests that the principle of corresponding states is valid which is acceptable if the components are not interacting with each other. This assumption may be not fulfilled for rich gas condensates, associated gases from light oils, gases from HPHT wells and gases with large amounts of sour components. This work investigates the principle of corresponding states and the calculation of the viscosity with an equation of state using literature data of the dynamic viscosity and the gas density. In conjunction with the experimentally determined density, one can easily calculate the kinematic viscosity. This procedure can yield some insight into how the transport property, the viscosity, can be better understood. The data analysis reveals that the kinematic viscosity follows different laws in the gas and the liquid phase. The results are compared with the tabulated values found in the literature and our own measurements with a falling body apparatus. Secondly p-T areas are identified where it is easy to find a formula for the viscosity.

Introduction Gas gains importance as an energy source, both for the industry and for private consumption. Therefore accurate knowledge of its properties is in demand. Especially the flow in the reservoir and in the pipeline is governed by its transport property, the dynamic viscosity. Gas viscositites in lab reports are usually calculated values. This paper will explore the question whether it suffices to calculate the viscosity or if we have to determine it experimentally. The central questions are therefore: � Are there enough data on which the correlations rest? � Are the data accurate enough? � Are the experimental methods comparable? � Do the correlations represent the experimental data with sufficient accuracy? � Do the correlations make sense physically? The literature on this topic is massive, therefore only a few articles were selected.


- 2 -

Amount of experimental data Dranchuk (1986) mentions in his paper more than 2000 data points. Chen (1993) talkes about a great number (how much is that?). Londono (2005) speaks of a database of 4,909 points. Elsharkawi (2006) derives his model from 2400 measurements. Gonzales (1970) used around 300 data points for developing his correlation. Shokir (2008) makes use of 1460 measurements. In this paper the data of Gonzales (1970) were used.

Accuracy of the experimental/correlated data For atmospheric pressure Carr (1953) lists an absolute error of 3.17% between the measured points and the graph for gas mixtures at a temperature of 93degC. Eakin’s data (1959) are accurate to 0.7%, Jossi’s (1962) to 4%. Dranchuk (1986), reports errors of maximum 0.97%. Londono (2005) when correlating the measured data notes an error of 1.36% in comparison to the Jossi model of 3.05%. Elsharkawy points out that the average relative error is about 2% for his correlation. Ling (2009) mentions that the gas viscosities for HPHT conditions cannot be extrapolated from low pressure data. He gives the most extensive overview of existing experimental data in the literature. Also non hydrocarbon components, like nitrogen do not follow the suggested correlations. We saw this as well in our measurements. (Fig. 8)

Experimental methods in use Diehl (1970) gives a thorough overview over all types of measurements. Presently four main methods are in use: � Oscillating cylinder viscometer – it is applied mainly for liquids. Gases can still be

measured with this method. � Rolling Ball Viscometer – here a ball rolls within a tube and the time is recorded

between two points. The viscosity is then correlated/calibrated with test substances. The application is mainly for liquids.

� Falling Body Viscometer – It is based on the same principle as the Rolling Ball method. Here the distance between two points can be made larger. The fall times can be made longer by making the annulus tight.

� Capillary Tube Viscometer – This method offers the highest accuracy provided the capillary has a constant diameter, since the Hagen-Poiseuille flow is sensitive to the 4th power of the radius. It is used by Comings (1944) and in the API research Project 65 (Gonzales 1970).

Type of existing correlations and their physical approach The first and simplest approach to formulate the gas viscosity was in the dilute gas limit. Later attempts were made to include the dense gas region and liquids. To cover both phases is a challenge. In the gas phase the viscosity results from collisions of molecules, while in the liquid phase the viscosity is determined by the intermolecular forces. Near the critical point, both effects play a role. In the following the different correlations will be discussed. The aim of correlating the viscosity is to find a relationship between the dynamic viscosity µ and temperature T, pressure p and the composition zk in the following form


- 3 -

( ) ( )krrref

k zTpforzTpf ,,,, ==µµµ (1)

The reference point µref in several correlations is the atmospheric viscosity µa. Elementary kinetic theory This approach is connected with the names of Chapman and Enskog (1990)

2669.2

σµ

µµ c

ccc

MTwith

T

T == (2)

Correlation at atmospheric pressure For light hydrocarbons one can take the formula from Sutherland (1893) together with the mixing rules of Herning-Zipperer (1936)

kkkk

kkkm

kkk

kkkk

mm

mm

MSMB

SzSMz

MBz

BST

TBµ

9.134.75,0133.043.7

,,2/3

+=+=

==+

= ∑∑

∑ (3)

Correlations including pressure dependence As stated above, the viscosity is split into an atmospheric part depending on temperature only and a part that is depending on the gas density. The gas law makes the second term actually a general function of temperature and pressure. The difference between the viscosity and its value at atmospheric pressure is equal to the second term and is called residual viscosity.

( ) )()(, ρµµµ resa TTp += (4)

Enskog dense-gas kinetic theory This theory uses a rigid spherical molecule approach and can be expressed as

( ) ( )M

NbYbXXX

Y

TTp a

32

3

2,,7614.08.01

)(,

σπρµµ ==++= (5)

Y is a collision parameter. Carr et al. correlation (CKB, 1953) This is probably the most well know correlation. They use the Herning-Zipperer mixing rule (1) for the atmospheric viscosity plus a correction for the non-hydrocarbon components. This part of the viscosity is determined by a chart (Fig. 1-Fig. 3). Dempsey (1965) provided later formulae for an easier calculation.

( )[ ]( ) [ ]( )

( )( )( )gSHSH

gCOCO

gNN

gga

SHCONaa

z

z

z

FT

PasT

γγ

γγγµ

µµµ

log49.873.3

log08.924.6

log48.859.9

002026.001709.0log15.6188.8

22

22

22

222

0

0

+=∆

+=∆

+=∆

−+−=

∆+∆+∆+=

(6)

Jossi et al. (JST, 1962) and the Lohrenz et al. (LBC, 1964) correlations


- 4 -

The JST correlation was first developed for pure components and has the form

( ) ( )( )( )0093324.0,040758.0,058533.0,023364.0,1023.0

,10,

43210

3/22/1

6/14

0

4/14

=====

==+− ∑=

−

aaaaa

pM

TaTTp

c

c

i

iia r

ξρξµµ (7)

The LBC correlation is meant to be applied also for mixtures and finds application in most EOS software packages for both phases, gas and liquid. The viscosity parameter needs to be evaluated in scientific units. Gonzales et al (GEL, 1970) This correlation of the residual viscosity includes temperature, density and molecular mass of the gas. It has the general form

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

( )( ) ( )MTXMTY

MT

MTX

TM

TMMTK

MTXMTKPasTp MTY

,04.011.1,

0095.05.1914

57.2,

9.124.122

103.6777.0,

,exp,,2/33

,

+=

++=

++⋅+=

=−

ρµµ

(8)

It is mentioned by Ling (2009) that this correlation gives less accurate results if non-hydrocarbon components are present. Variations of this correlation were published by Sutton (2005) and Vishwanathan (2007). Dean and Stiehl The authors suggest two formulae for the atmospheric viscosity in dependance on the temperauture region

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

3/22/1

6/1

888.1

9/5

9/8

4402.5

11.1exp439.1exp108.0

5.10932.01338.0668.1

5.134.0

cmm

cmm

rrma

rra

ra

pM

T

PasT

TforTPasT

TforTPasTr

=

−−=−

>−=

≤=

ξ

ρρξµµµ

µµ

µµ

(9)

Friction Theory (Zeberg-Mikkelsen et al. 2002) As in the previous correlations the residual concept is applied. The calculation of the dilute limit of gas viscosity stays the same, the residual part of the viscosity contains a term with a quadratic in the repulsive pressure (from the EOS). patt and prep are the attractive and the repulsive pressure terms in the EOS.

( ) ( )

+++=

2

2

21,c

reprep

c

attatt

c

reprepa

p

pc

p

pc

p

pcCTTp µµ (10)

For light components the constant C, crep1, crep2, catt are known from matching the pure viscosity data. For gas mixtures linear mixing rules are applied.


- 5 -

New approach, a generalized chart In the correlations discussed so far, the corresponding state method (CSM) was combined with a residual viscosity approach. For parameters like pressure and temperature, reference quantities at the critical point are used, like in any other EOS calculation. In the literature for viscosity, its value at atmospheric pressure in dependence on temperature is taken. This is not consistent. One reference point should be taken for all variables. It would therefore be obvious to take the viscosity obtained at the critical point as well. Its determination is difficult, as it is with other critical parameters. In particular, the calculation of the density itself by an EOS has an accuracy of about 1%. Additionally, the critical density has to be known to arrive at the reduced density. The critical density follows from the gas law, provided Tc and pc are known. Zc is normally kept constant at 0.27. This is an additional error. The EOS employed for the density calculations are the Peng-Robinson (PR) and equations from the Benedict-Webb-Rubin (BWR) family.

Dimensional analysis (Bridgeman 1922, Buckingham 1914) states that all physical laws should be expressed as products. This makes it possible to introduce reference quantities and in such a way render the physical laws independent of the measurement system. That eliminates all correlations that work with the viscosity as the sum of atmospheric and residual viscosity. The general idea in this paper is to come up with a general chart (like the Standing- Katz Z-chart) for the viscosity. The functional dependence of the viscosity µ(p,T) resembles that of the volume V(p,T), provided that pressure and temperature trade places (Uyehara 1944) (Fig. 4). Attempts were made in the past to formulate an EOS for the viscosity (Guo 1997). With the viscosity displaying a volume-like behaviour, the idea arose to separate the volume effect by dividing the dynamic viscosity by the density and thus arriving at the kinematic viscosity. In order to start from experimental data, this requires viscosity and density measurements, like in Gonzales (1970) or in Diehl (1970). Unfortunately the kinematic viscosity inherits the experimental errors from measurements of two parameters, density and viscosity. On the other hand it is an excellent way to reveal the quality of the measurement. For this reason the data from Gonzales (1979) were taken. They reveal a higher accuracy than Diehl (1970). Data for pure substances like C1-C5 and C10 were available. In realizing this idea the following steps were undertaken:

1. Generate tables of the kinematic viscosity for each available pure component as a function of reduced quantities Tr=T/Tc, pr=p/pc. It appears that the plots from the table show no phase transition for the kinematic viscosity. That means that the separation of the mass and the volumetric components of the dynamic viscosity works (Fig. 6 and Fig. 7).

2. Determine the kinematic viscosity at the critical point by interpolation. 3. Seek a formula to calculate the critical kinematic viscosity for different components.

It was found that the function in Equ.(11), displayed in Fig. 5 serves the purpose

6315.00922.07.4ln cc N⋅+=ν (11)

4. Calculate the reduced kinematic viscosity νr=ν /νc. The results are given in tabular form in Tab.1 and in graphical form in Fig. 6 and Fig. 7.

The general chart has the same characteristics as the Standing- Katz Z-chart. There are only two free parameters, pc and Tc, for aligning the viscosity data of different gases into the


- 6 -

general chart. From EOS we know that a third parameter is necessary to better describe reality. The reverse way, namely how to come up with the dynamic viscosity for a given gas mixture, follows the sequence given below:

1. Calculate pr and Tr for the given situation. 2. Determine νc from 3. Determine the Z-factor from Standing-Katz or from an EOS (the BWR-type is

suggested) 4. Calculate the dynamic viscosity via the formula

( )r

crrcrc TRZ

ZMpTp

⋅⋅⋅⋅⋅=⋅⋅= ννννρµ , (12)

Zc can be adjusted so that the data fit a calibration gas.

Discussion of the generalized chart (Fig. 6 and Fig. 7) For high temperatures, this is the region where gas shows ideal behavior, the curves of ln(νr) exhibit a linear behaviour with ln(pr). That leads further to a dynamic viscosity that is only a function of temperature

( ) .lnln rrr pTa −=ν (13)

Looking at Fig. 7, one notices that the slope of the curves is almost constant towards high temperatures. Its value is 1.97.

( ) .ln97.1ln rrr Tpc ⋅+=ν (14)

Assuming for the sake of simplicity that the slope is 2, it is possible to derive a simple formula:

( ) [ ]r

crr

rc

TZ

ZpPas

Z

TZTp =≈ ρµµ using,43, (15)

Fig. 7 suggests that for pr<1 the kinematic viscosity follows a simple formula in the region Tr<1:

( ) ( ) ( ) ( )3ln2ln , −− == rr Tr

r

c

ccTrr T

Z

p

TR

MpTporTT

νµν (16)

Experimental work The series of viscosity measurements were performed using a falling body viscometer. The travelling times of the falling body were calibrated to data from the literature (Gonzales 1970). As a first test nitrogen was measured. The viscosities read from the µ-t chart did not coincide with tabulated data. Even if the CSM were applied there would be discrepancies at higher pressures.


- 7 -

From Gonzales (1970) a gas mixture was selected that had almost the same composition as a sample physically available in house. The viscosity was determined by entering the runtime in a calibrated µ-t graph. The agreement of the data could be improved by a better temperature control, but for a general check it is satisfactory ( Tab. 2 and Fig. 9). The evaluation via the CSM showed good reliability. Conclusions

1. By splitting the dynamic viscosity into a product of density and kinematic viscosity, the mass and the volume effect were successfully separated. The kinematic viscosity showed a jump at the phase boundary due to the different forces acting on the molecules in the gas and the liaquid state.

2. The CSM proved to be a sound method to estimate viscosities, physically compatible with dimensional analysis.

3. Gas mixtures with high nitrogen content should be measured. 4. Data in the critical region have to be taken with caution. 5. Regions in the p,T area are identified where the kinematic viscosity can be expressed

by analytical formulae. Acknowledgement The authors are grateful to OMV for the permission to publish this work.


- 8 -

Nomenclature k Boltzmann constant m mass of a molecule M molecular mass N Loschmidt number Nc carbon number p pressure R gas constant T temperature zk composition Z real gas factor

Greek µ dynamic viscosity ν kinematic viscosity ρ density σ collision diameter ζ viscosity parameter

Subscripts a atmospheric att attractive c critical r reduced rep repulsive res residual

References, literature Dean,D.E.; Stiel,L.I.;AIChE J.,11,526 (1965) Born, M.; Green, H.S.; “A General Theory of Liquids, I. Molecular Distribution Functions”

Proc.Roy.Soc. (London) A188, 10-18, 1946. Bridgman, P. W.; “Dimensional Analysis”, Yale University Press, ISBN 0548910294 1922 Buckingham, E.; “On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis”,

Phys. Rev. 4: 345, doi:10.1103/PhysRev.4.3451914. Carr, N.L.; “Viscosities of Natural Gas Components and Mixtures”; Inst. Gas Technology

Res.Bull.No.23, June 1953. Carr, N.L.; Kobayashi, R.; Burrows, D.B.; “Viscosities of Hydrocarbon Gases under Pressure”, 1953,

SPE 297-G Chapman, S; Cowling, T.G.; “The Mathematical Theory of Non-uniform Gases: an Account of the

Kkinetic Theory of Viscosity, Thermal Conduction, and Diffusion in Gases”, Cambridge University Press, 1990

Chen, Z.A.; Ruth, D.W.; “On Viscosity Correlations of Natural Gas”, 1993, CIM 93-02. Dempsey, J.R.; “Computer Routine Treats Gas Viscosity as a Variable” , OGJ,Aug 1965, 141-143. Diehl, J.; et al.; Viscosity, Edition Technip 1970. Dranchuk, P.M.; Islam, M.R.; Bentsen, R.G.; “A Mathematical Representation of the Carr, Kobayashi

and Burrows Natural Gas Viscosity Correlations”, 1986, JCPT86-01-03. Eakin,B.E.; Ellington, R.T.; “Improved High Pressure Capillary Tube Viscometer”, 1959, SPE 1142-G. Eakin,B.E.; Ellington, R.T.; “Predicting the Viscosity of Light Hydrocarbons”, JPT,15,p210-214, 1963. Elsharkawi, A.M.; “Efficient Method for Calculations of Compressibility, Density, and Viscosity of

Natural Gases”, JCPT, June 2006, Vol. 45,No.6 Gawish, A.; Al-Homadhi, E.; State of the Art; Natural Gases Viscosity under Reservoir Conditions, 2005,

SPE 106326. Giddings, J.G.; Kobayashi, R.; Correlation of the Viscosity of Light Paraffin Hydrocarbons and Their

Mixtures in the liquid and Gaseous Regions, 1964, SPE 719. Gonzales, M.; Eakin, B.E.; Lee,A.L.; “Viscosity of Natural Gases, Monograph on API research Project

65” , 1970, p88ff. Guo, X.Q.;Wang, L.S.; Rong,S.X.; Guo, T.M.; “Viscosity Model Based on Equation of State for

Hydrocarbon Liquids and Gases”, Fluid Phase Equilibria, 139,405-421, 1997. Herning, F.; Zipperer, L.; “Calculation of the Viscosity of Technical Gas Mixtures from Viscosity of the

Induvidual Components”, Gas- und Wasserfach,79, 69-73, 1936. Jeje, O.; Mattar, L.; “Comparison of Correlations for Viscosity of Sour Natural Gas, JCPT, July 2006,

Vol.45, No.7. Jossi, J.A.; Stiel, L.I.; Thodod, G.; “The viscosity of Pure Substances in the Dense Gaseous and Liquid

Phases”, AIChE J, March 1962, 8, No.1, 59.


- 9 -

Ling, K.; Teodoriu, C.; Davani, E.; Falcone, G.; “Measurement of Gas Viscosity at High Pressures and High Temperature”s, 2009, ITPC 13528.

Lohrenz, J.; Bray, B.G.; Clark, C.R.; “Calculating Viscosities of Reservoir Fluids from Their Composition”, 1964, SPE 915.

Londono, F.E.; Archer, R.A.; Blasingame, T.A.; “Correlations for Hydrocarbon-Gas Viscosity and Gas Density – Validation and Correlation of Behavior Using a Large-Scale Database”, 2005, SPE 75721.

Reynolds, O.; “On the Theory of Lubrication and an Experimental Determination of the Viscosity of Olive Oil” ; Phil. Trans., 177, pp.157-234, 1886.

Shokir, E.M.El-M.; “Novel Density and Viscosity Correlations For Gases and Gas Mixtures Containing Hydrocarbon and Non-Hydrocarbon Components”, JCPT, Oct. 2008, Vol.47, No.10.

Sutherland, W; “Viscosity of Gases and Molecular Force”, Phil.Mag.,36, series 5, pp.507-531, 1893. Sutton, R.P.; “Fundamental PVT calculations for Associated and Gas/Condensate Natural Gas Systems”,

SPE97099, 2005. Uyehara, O.A.;Watson, K.M.; Natl.Pet.News, 36, R-714, 1944. Vishwanathan, A.; “Viscosities of Natural Gases at High Pressures and Temperatures”, MS thesis,

Texas A&M University, College Station, 2007. Zeberg-Mikkelsen, C.K; Quinones-Cisneros, S.E.; Stenby, E.H.; “Viscosity Prediction of Natural Gases

Using the Friction Theory”, Int.J. of Thermophysics, 23(2), 437-454, 2002.


- 10 -

Tables and Figures lnTr\

lnpr -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-0.598 1.523 1.539 1.522 1.522 1.522 1.52 1.533 1.551 1.578 1.607 1.663 1.731 1.845 2.015

-0.511 1.252 1.245 1.232 1.233 1.237 1.237 1.244 1.272 1.291 1.319 1.366 1.44 1.554 1.719

-0.431 2.278 0.968 0.957 0.968 0.974 0.978 0.978 1.004 1.024 1.057 1.107 1.193 1.314 1.5

-0.357 2.62 2.061 0.768 0.765 0.772 0.771 0.782 0.793 0.813 0.847 0.897 0.963 1.123 1.321

-0.288 2.87 2.4 1.85 0.602 0.608 0.612 0.625 0.651 0.67 0.706 0.754 0.825 0.97 1.18

-0.223 3.1 2.6 2.091 1.521 0.48 0.471 0.479 0.517 0.543 0.582 0.642 0.734 0.866 1.033

-0.163 3.3 2.79 2.276 1.735 1.192 0.342 0.338 0.384 0.419 0.469 0.539 0.641 0.768 0.934

-0.105 3.45 2.951 2.445 1.911 1.38 0.792 0.222 0.256 0.3 0.36 0.442 0.551 0.682 0.842

-0.051 3.611 3.112 2.616 2.068 1.548 0.973 0.4 0.104 0.177 0.26 0.353 0.463 0.601 0.769

0.000 3.73 3.246 2.733 2.214 1.7 1.149 0.574 0 0.071 0.158 0.267 0.389 0.529 0.71

0.049 3.848 3.346 2.835 2.32 1.827 1.29 0.723 0.178 -0.12 0.048 0.181 0.314 0.465 0.639

0.095 3.951 3.447 2.943 2.437 1.93 1.405 0.864 0.306 -0.122 -0.022 0.114 0.248 0.408 0.591

0.140 4.041 3.538 3.034 2.529 2.022 1.507 0.978 0.439 -0.047 -0.083 0.058 0.196 0.36 0.544

0.182 4.123 3.621 3.118 2.615 2.111 1.602 1.083 0.56 0.064 -0.109 0.015 0.153 0.314 0.497

0.223 4.208 3.706 3.204 2.701 2.196 1.69 1.179 0.667 0.173 -0.082 -0.015 0.116 0.273 0.458

0.262 4.277 3.775 3.273 2.77 2.268 1.764 1.26 0.754 0.271 -0.035 -0.037 0.083 0.237 0.411

0.300 4.347 3.846 3.345 2.844 2.343 1.842 1.343 0.85 0.366 0.014 -0.043 0.058 0.2 0.387

0.336 4.408 3.909 3.41 2.911 2.415 1.913 1.419 0.932 0.447 0.076 -0.026 0.039 0.167 0.352

0.372 4.473 3.984 3.477 2.978 2.478 1.986 1.492 0.999 0.522 0.136 -0.013 0.028 0.132 0.332

0.405 4.56 4.064 3.537 3.04 2.545 2.053 1.561 1.072 0.599 0.196 0.009 0.02 0.108 0.305

0.438 4.62 4.133 3.6 3.103 2.607 2.12 1.626 1.137 0.666 0.251 0.036 0.024 0.089 0.281

0.470 4.698 4.196 3.694 3.192 2.67 2.183 1.683 1.192 0.714 0.313 0.053 0.002 0.082 0.259

0.531 4.808 4.307 3.806 3.305 2.804 2.301 1.803 1.302 0.832 0.415 0.124 0.031 0.075 0.218

0.615 4.963 4.464 3.964 3.464 2.964 2.463 1.967 1.462 0.995 0.56 0.231 0.082 0.075 0.163

0.693 5.108 4.609 4.109 3.61 3.11 2.609 2.113 1.615 1.14 0.695 0.341 0.141 0.09 0.125

0.788 5.279 4.782 4.284 3.786 3.285 2.786 2.288 1.792 1.323 0.871 0.493 0.243 0.14 0.115

0.875 5.442 4.942 4.443 3.944 3.446 2.946 2.451 1.952 1.487 1.036 0.641 0.348 0.2 0.136

0.956 5.59 5.089 4.589 4.09 3.593 3.094 2.599 2.095 1.635 1.184 0.777 0.453 0.257 0.164

0.993 5.663 5.162 4.661 4.16 3.661 3.163 2.669 2.159 1.705 1.251 0.84 0.503 0.291 0.188

Tab. 1 Generalized data of reduced kinematic viscosities as a function of reducted temperature and reduced pressure

C1 GEL #5 mixture F-1 exp CSM

p abs µ µ µ µ bar µPas µPas µPas µPas

55.2 12.18 12.8 12.62 12.6 137.9 15.75 17.5 18.91 18.1 275.8 24.17 29.1 30.5 30.3 482.7 33.78 40.8 43.85 42.5

Tab. 2 Comparison of experimental and tabulated data


- 11 -

Fig. 1 Viscosity ratio versus reduced pressure (Carr 1953)

Fig. 2 Viscosity ratio versus reduced temperature (Carr 1953)

Fig. 3 Viscosity of paraffin hydrocarbon gases at p=1atm (Carr 1953)


- 12 -

Fig. 4 Reduced viscosity versus Tr and pr

4.7

4.75

4.8

4.85

4.9

4.95

5

5.05

0 2 4 6 8 10 12

single carbon number Nc

ln (

nyc)

data

fit

Fig. 5 Critical kinematic viscosity as a function of the single carbon number Nc

ln(Tr)

-1

0

1

2

3

4

5

6

-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5

ln(pr)

ln(n

yr)

-0.598

-0.511

-0.357

-0.223

-0.105

0.000

0.223

0.438

0.615

0.788

0.993

Fig. 6 Reduced kinematic viscosity as a function of reduced pressure


- 13 -

ln(pr)

-1

0

1

2

3

4

5

6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

ln(Tr)

ln(n

yr)

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.5

2.5

3.0

Fig. 7 Reduced kinematic viscosity as a function of reduced temperature

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

reduced pressure

kine

mat

ic v

isco

sity

C1

N2

Fig. 8 Kinematic viscosity of methane and nitrogen

0

5

10

15

20

25

30

35

40

45

50

0 100 200 300 400 500

abs. pressure [bar]

dyna

mic

vis

cosi

ty [

µP

as]

GEL#5

mixture F-1, experiment

mixure F-1, CSM

Fig. 9 Dynamic viscosities from experiment and CSM for a gas mix

Towards An Equation of State for Gas Viscosities...

Documents

Transcript of Towards An Equation of State for Gas Viscosities...