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The International Association for the Properties of Water and Steam

Berlin, Germany

September 2008

Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance

2008 International Association for the Properties of Water and SteamPublication in whole or in part is allowed in all countries provided that attribution is given to

the International Association for the Properties of Water and Steam

President:J. R. Cooper

School of Engineering and Materials ScienceQueen Mary, University of London

Mile End RoadLondon E1 4NS, England

Executive Secretary:Dr. R. B. Dooley

Structural Integrity Associates, Inc.2904 South Sheridan Way, Suite 303Oakville, Ontario, L6J 7L7, Canada

email: [email protected]

This release contains 9 pages, including this cover page.

This release has been authorized by the International Association for the Properties of Waterand Steam (IAPWS) at its meeting in Berlin, Germany, 7-12 September 2008, for issue by itsSecretariat. The members of IAPWS are: Argentina and Brazil, Britain and Ireland, Canada,the Czech Republic, Denmark, France, Germany, Greece, Italy, Japan, Russia, the United

States of America, and associate member Switzerland.This release replaces the Revised Release on the IAPS Formulation 1985 for the Viscosity ofOrdinary Water Substance, issued in 2003.

Further information concerning this release and other documents issued by IAPWS can beobtained from the Executive Secretary of IAPWS or from http://www.iapws.org.


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Contents

1. Introductory Remarks 2

2. Recommended Correlating Equation 2

2.1 Nomenclature 22.2 Reference constants 22.3 Dimensionless variables 32.4 Range of validity 32.5 Estimated uncertainty 42.6 Correlating equation 42.7 Critical enhancement 62.8 Simplified use outside the critical region 8

3. Recommendation for Industrial Application 8

4. Computer-Program Verification 8

5. References 9

1. Introductory Remarks

This release provides a correlating equation for the shear viscosity of pure watersubstance over an extensive range of fluid states. A discussion of the background,development, and validation of this formulation is presented in Ref. [1].

Section 2 of this release contains the correlating equation, necessary constants, rangeof validity of the equation, and estimates of the uncertainty of the correlation. Section 3

concerns the industrial application of the viscosity correlation, and Section 4 presents selectedvalues of the correlation at specific state points to enable computer verification of animplementation of the correlation.

2. Recommended Correlating Equation

2.1. NomenclatureTdenotes absolute temperature on the International Temperature Scale of 1990 denotes densityp denotes pressure denotes viscosity

2.2. Reference constantsThe reference constants used in this formulation for temperature, pressure, and density

agree with presently accepted values of the critical temperature, pressure, and density of waterrecommended by IAPWS [2], while the reference constant for viscosity has no physicalsignificance.


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reference temperature: T* = 647.096 K (1)reference density: * = 322.0 kgm3 (2)reference pressure: p* = 22.064 MPa (3)reference viscosity: * = 1.00 106 Pas (4)

2.3. Dimensionless variablestemperature: T = T/T* (5)density: = /* (6)pressure: p =p/p* (7)viscosity: = /* (8)

2.4. Range of validityEquation (10) below is recommended for computation of the viscosity for all

thermodynamically stable fluid states in the following ranges of pressure p and temperature T:

0 < p < pt and 273.16 KT 1173.15 K

pt p 300 MPa and Tm(p) T 1173.15 K300 MPa < p 350 MPa and Tm(p) T 873.15 K (9)350 MPa < p 500 MPa and Tm(p) T 433.15 K500 MPa < p 1000 MPa and Tm(p) T 373.15 K

In Eq. (9), Tm is the pressure-dependent melting temperature andpt is the triple-point pressureas given in Refs. [3,4]. For most applications, the IAPWS Formulation 1995 for theThermodynamic Properties of Ordinary Water Substance for General and Scientific Use [5,6]should be used to determine the densities used as input to Eq. (10), when the state point underconsideration is defined by pressure and temperature or by other thermodynamic variablesinstead of density and temperature.

In addition, IAPWS makes the following statements about extrapolation of Eq. (10) outsidethe range of validity given above:

For vapor states at temperatures below the triple-point temperature of 273.16 K and pressures less than or equal to the sublimation pressure, the viscosity calculation isdominated by the dilute-gas term, and this behaves in a physically reasonable mannerdown to at least 250 K.

For stable fluid states outside the range of validity of Eq. (10) but within the range ofvalidity of the IAPWS Formulation 1995 for the Thermodynamic Properties ofOrdinary Water Substance for General and Scientific Use [5,6], the extrapolationbehavior of Eq. (10) is physically reasonable.

At high temperatures, the extrapolation of the dilute-gas portion of Eq. (10) isphysically reasonable up to at least 2500 K.

For the metastable subcooled liquid at atmospheric pressure, Eq. (10) is in fairagreement (within 5 %) with available data down to 250 K.


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2.5. Estimated uncertaintyThe uncertainties in this formulation are summarized in Figure 1; they can be

considered as estimates of a combined expanded uncertainty with a coverage factor of two.Thus the viscosity at any state point can be expressed as where is the applicable valuein Figure 1. The formulation reproduces the ISO recommended value of the viscosity at 20 oC

(293.15 K) and standard atmospheric pressure within the number of digits given in Ref. [7]; italso agrees with all values from 288.15 to 313.15 K at atmospheric pressure in Ref. [7] withinthe stated uncertainty of 0.17 % at 293.15 K.

Figure 1. Estimated uncertainty of the correlating equation.

2.6. Correlating equationThe viscosity is represented by the equation

),(),()( 210 TTT = . (10)

The first factor 0 of the product represents the viscosity in the dilute-gas limit and is given

by


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=

=3

0

0

100)(

ii

i

T

H

TT , (11)

with coefficientsHi given in Table 1.

Table 1. CoefficientsHifor )(0 T

i Hi

0 1.677521 2.204622 0.63665643 -0.241605

The second factor 1 represents the contribution to viscosity due to finite density:

=

==

6

0

5

01 )1(1

1exp),(

j

j

ij

i

i

HT

T , (12)

with coefficientsHijgiven in Table 2. The third factor 2 represents the critical enhancementof the viscosity.

Table 2. CoefficientsHij for ),(1 T i j Hij

0 0 5.200941011 0 8.508951022 0 -1.08374

3 0 -2.895551010 1 2.225311011 1 9.991151012 1 1.887973 1 1.266135 1 1.205731010 2 -2.813781011 2 -9.068511012 2 -7.724791013 2 -4.898371014 2 -2.570401010 3 1.61913101

1 3 2.57399101

0 4 -3.253721023 4 6.984521024 5 8.721021033 6 -4.356731035 6 -5.93264104

Note: CoefficientsHij omitted from Table 2 are identically equal to zero.


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2.7 Critical enhancementThe critical enhancement is only significant in a very small region in density and

temperature around the critical point. Although exactly at the critical point the viscosity isinfinite, the enhancement term contributes an amount greater than 2 % of the full viscosityonly within the following boundaries:

645.91 K < T< 650.77 K, 245.8 kgm3 < < 405.3 kgm3 . (13)

Thus, the critical enhancement is significant only within the boundaries specified by Eq. (13);outside this region, the enhancement is always less than the uncertainty in the formulation.This allows simplification for certain calculations (see Sections 2.8 and 3).

The function 2 is defined over the entire range of states by:

)Yx exp2 = , (14)

wherex is the critical exponent for viscosity and the function Yis defined for two ranges ofcorrelation length . For 0 0.3817016416 nm

( ) ( ) ( )

+= 2D

2CC

5DC 504

7651

5

1 qqqqqY , (15)

while for> 0.3817016416 nm

( ) ( )( )

( ) (

( )

)

( ) ( )

+=

)(12311

sin4

51

12sin

4

13sin

12

1

2/32CD

2C3

C

D2

C2C

DC

D

wLqqq

qqq

Y

, (16)

with

( )

+=

2

122

DD 1arccos q (17)

and with the functionL(w) given by

>

+=

1for,arctan2

1for,1

1ln

)(

C

C

qw

qw

w

wL . (18)

The variable w is defined by

+

=

2tan

1

1 D2

1

C

C

q

qw . (19)


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The critical enhancement of viscosity given by Eqs. (14)(19) is a function of the correlationlength :

=

00 , (20)

in terms of ( 0) which is defined by

( ) ( )

=

T

TTT RR,, (21)

Tp

=

. (21a)

When calculated by Eq. (21) is less than zero, it must be set to zero for calculations toproceed.1

The constants needed to compute the critical enhancement, 2 , are provided in Table 3.

Table 3. Critical-Region Constants

Constant Value

x 0.0681

Cq 1.9 nm

1D

q

1.1 nm 0.630

1.239

0 0.13 nm

0 0.06

RT 1.5

1 Due to the numerical implementation of the equation of state, the calculated singularity in the first derivative inEq. (21) may not occur exactly at Tc andc as it should. Therefore, calculated values of 2 may behave

unphysically at points extremely close to the critical point (approximately within 0.01 kgm3 ofc on the criticalisotherm). The formulation should be used with caution in this very small region.


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2.8 Simplified use outside the critical regionBecause the critical enhancement is insignificant except in a small region around the

critical point (described approximately by Eq.(13)), complexity and computing time may bereduced by omitting the critical enhancement for applications outside this region.2 This canbe done by setting 2 =1.

3. Recommendation for Industrial Application

For industrial uses where greater computing speed is needed, Eq. (10) should be simplified bysetting 2 =1. In addition, the IAPWS Industrial Formulation 1997 for the ThermodynamicProperties of Water and Steam [8,9], known as IAPWS-IF97, should be used to determine thedensity for use in Eq. (10) when the state point is defined by the temperature and pressure orother state variables. When IAPWS-IF97 is used to calculate densities, the error introducedby the use of the different thermodynamic formulation is smaller than the uncertainty of theviscosity correlation, provided the point is within the range of validity of IAPWS-IF97, exceptfor points close to and inside the near-critical region described by Eq. (13).

4. Computer-Program Verification

The following tables are provided to assist the user in computer-program verification. Theviscosity calculations are based on the tabulated temperatures and densities.

Table 4. Sample points for computer-program verificationof the correlating equation, Eq. (10), with 2 =1.

T(K) (kgm3) (Pas)

298.15 998 889.735100

298.15 1200 1437.649467373.15 1000 307.883622433.15 1 14.538324433.15 1000 217.685358873.15 1 32.619287873.15 100 35.802262873.15 600 77.430195

1173.15 1 44.2172451173.15 100 47.6404331173.15 400 64.154608

2 If a calculation is performed that uses the critical enhancement, Eq. (14), near the critical point but omits it farfrom the critical point, some discontinuity is inevitable. This discontinuity is less than 0.0051 % for single-phasestates outside a region near the critical point bounded by the equation

( )ii

iaT )mkg1()K1( 3

3

0

=

= ,

where a0 = 457.95895935062, a1 = 1.68077273385305, a2 = 3.24405775203984103, and

a3 = 1.43032446173023106.


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Table 5. Sample points for computer-program verification of the correlating equation,Eq. (10), in the region near the critical point.

T(K) (kgm3) (nm) 2 (Pas)

647.35 122 0.309247 1.00000289* 25.520677647.35 222 1.571405 1.00375120 31.337589647.35 272 5.266522 1.03416789 36.228143647.35 322 16.590209 1.09190440 42.961579647.35 372 5.603768 1.03665871 45.688204647.35 422 1.876244 1.00596332 49.436256* Correlation length < 0.3817016416 nm so Yis evaluated with Eq. (15).

5. References

[1] Huber, M. L., R. A. Perkins, A. Laesecke, D. G. Friend, J. V. Sengers, M. J. Assael,I. M. Metaxa, E. Vogel, R. Mare, and K. Miyagawa, J. Phys. Chem. Ref. Data, in

preparation.[2] International Association for the Properties of Water and Steam, Release: Values of

Temperature, Pressure and Density of Ordinary and Heavy Water Substances at theirRespective Critical Points, available at http://www.iapws.org (September 1992).

[3] International Association for the Properties of Water and Steam, Revised Release onthe Pressure along the Melting and Sublimation Curves of Ordinary Water Substance ,available at http://www.iapws.org (September 2008).

[4] Wagner, W., R. Feistel, and T. Riethmann, J. Phys. Chem. Ref. Data, in preparation.

[5] Wagner, W., and A. Pru, J. Phys. Chem. Ref. Data 31, 387 (2002).

[6] International Association for the Properties of Water and Steam, Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water

Substance for General and Scientific Use, available at http://www.iapws.org(September 1996).

[7] International Organization for Standardization (ISO), Viscosity of Water, ISO/TRTechnical Report 3666: 1998(E), Geneva.

[8] Wagner, W., J. R. Cooper, A. Dittmann, J. Kijima, H.-J. Kretzschmar, A. Kruse, R.Mare, K. Oguchi, H. Sato, I. Stcker, O. ifner, Y. Takaishi, I. Tanishita, J.Trbenbach, and Th. Willkommen, J. Eng. Gas Turbines & Power122, 150 (2000).

[9] International Association for the Properties of Water and Steam, RevisedRelease onthe IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Waterand Steam, available at http://www.iapws.org (August 2007).
http://www.iapws.org/relguide/IAPWS95.pdfhttp://www.iapws.org/relguide/IAPWS95.pdfhttp://www.iapws.org/relguide/IAPWS95.pdfhttp://www.iapws.org/relguide/IAPWS95.pdfhttp://www.iapws.org/relguide/IAPWS95.pdfhttp://www.iapws.org/relguide/IAPWS95.pdf

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