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Simple methods for the calculation of thermodynamic properties for metering
Conference Paper · May 2020
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Simple methods for the calculation of thermodynamic
properties for metering
Andrew Laughton
laughtonwork@talktalk.net
1
INTRODUCTION
This paper specifies a method to calculate viscosity and other properties, excluding
density, for use in the metering of natural gas flow.
This paper gives simplified methods for the calculation of (dynamic) viscosity,
Joule-Thomson
coefficient, and isentropic exponent for use in natural gas
calculations in the temperature range –20 to 40 oC (-4 to 104 oF) and pressures up
to 100 bar(abs) (1450 psia) in the gas phase.
For Joule-Thomson and isentropic exponent, the uncertainty of the equations
provided is greater than that obtained from a complete equation of state such as
GERG (reference [1]), ISO-20765:2 [2] or AGA-8 [3]; but is considered to be fit
for purpose. The equations are much simpler.
The motivation for providing simplified methods is mainly for the calculations
required, according to ISO-5167 or AGA-3, to measure flow of high pressure natural
gas with an orifice plate meter (references [4], [5], [6], [7], [8] and [9])
The basic mass flowrate equation is :𝑞=
𝐶
√1−𝛽 4
𝜋
(1)
𝜀 𝑑 2 √2.Δp.ρ
4
where C is function of  and Re, and of the type of orifice pressure tappings, and 
is a function of , P, p, and . The above standards differ in the functions for C
and . Although q is given by equation (1), iteration is required since C is a function
of Re and Re is a function of q. Similarly, given q, equation (1) does not directly
give p since  is a function of p. (See Appendix 6 for more details.)
The use of the equations in ISO-5167 (2003) [7] for calculating flow (q) for an
orifice plate meter, over a typical input range of temperature, pressure, differential
pressure, and gas composition, gives the following uncertainty equation (when the
only source of uncertainties is considered to be in the calculation of the required
gas thermophysical properties):
[u(q)/q]2
=
+
+
+
[
[
[
[
0.5
0.0006
0.002
-.0004
 0.0002 ]2 .[u()/]2
 0.0002 ]2 .[u()/]2
 0.0012 ]2 .[u()/]2
 0.0002 ]2 .[u()/]2
mass density
viscosity
isentropic exponent
Joule-Thomson coefficient
(2)
This equation can be used to estimate the required uncertainty for the calculation
of the properties.
For the contribution to the expanded uncertainty (U) (coverage factor k=2, 95%
confidence interval) in the flow to be less than 0.1 %, then
U()/ < 0.1 %
U()/ < 85 %
U()/ < 25 %
For the uncertainty contribution to be less than 0.02 %, then
1
U()/ < 125 %
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
U()/ < 0.02 %
U()/ < 17 %
U()/ < 5 %
U()/ < 25 %
Thus, density needs to be calculated as accurately as possible, while the calculation
does not need to be very accurate for the other properties. Their target uncertainty
is no better than about 25%.
2
VISCOSITY ()
2.1
Introduction
This section is concerned with gas phase viscosity (dynamic viscosity; S.I. units of
kg m-1 s-1 = Pa.s).
Gas phase viscosity typically increases with increasing temperature, as opposed to
liquid viscosity which decreases with increasing temperature. Both, gas and liquid,
tend to increase with increasing density (effectively pressure).
Gas phase viscosity is typically about 100 µP (micro-Poise) ( = 0.01 cP = 0.01
mPa.s) (liquid viscosity is typically about 0.5 mPa.s (cP) ).
2.2
Composition method
There are many methods for the calculation of gas phase (dynamic) viscosity.
Some, based in theory, are quite complicated. Of all the methods, the LohrenzBray-Clark (LBC) method is relatively simple, requires minimal component data,
and is a method that is widely implemented (reference [10]). It is the method
recommended here. One disadvantage is that it is sensitive to the input density;
but for the application considered here, accurate densities will be available, so this
is not a problem.
Below outlines the required parameters and equations to implement this method.
Table 1 – Component Parameters
CH4
N2
CO2
C2H6
C3H8
nC4
iC4
nC5
iC5
nC6
nC7
nC8
nC9
nC10
H2
O2
CO
H2O
H2S
He
Ar
neoC5
M (g/mol)
Tc (K)
Pc (bar)
ρc (mol/dm3)
16.04246
28.0134
44.0095
30.06904
44.09562
58.1222
58.1222
72.14878
72.14878
86.17536
100.20194
114.22852
128.2551
142.28168
2.01588
31.9988
28.0101
18.01528
34.08088
4.002602
39.948
72.14878
190.564
126.192
304.1282
305.322
369.825
425.125
407.817
469.7
460.35
507.82
540.13
569.32
594.55
617.7
33.19
154.595
132.86
647.096
373.1
5.1953
150.687
433.75
45.9920
33.9580
73.7730
48.7180
42.4661
37.9053
36.3729
33.7098
33.7823
30.4293
27.3107
24.9781
22.8198
21.0137
13.1500
50.3895
34.9821
220.640
89.9873
2.2746
48.5963
31.94
10.139342719
11.1839
10.624978698
6.870854540
5.000043088
3.920016792
3.860142940
3.215577588
3.271
2.705877875
2.315324434
2.056404127
1.81
1.64
14.94
13.63
10.85
17.873716090
10.19
17.399
13.407429659
3.24397
2
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
The above parameters are from ref.[1] (Table A5, p.3075); with Pc calculated as
the pressure from the GERG-2008 equation of state at T=Tc and ρ=ρc (to 4 decimal
places).
Mmix = ∑N
i=1 X i Mi
Mixture parameters:
(3)
Xi is component mole fraction
1
Vcmix = ∑𝑁
𝑖=1 𝑋𝑖 ρc
𝑖
Pc
Tcmix = ∑𝑁
𝑖=1 𝑋𝑖 Tc𝑖
Viscosity at low pressure:
𝑖
Pcmix = ∑𝑁
𝑖=1 𝑋𝑖 1.01325
Tr = T / Tci
(4)
(5)
0.00034Tr0.94
Tr  1.5
=
Tr > 1.5
 = 0.0001778(4.58Tr - 1.67)0.625
(7)
for H2 & He
 = 0.0001(7.08Tr + 2.26)
(8)
1
⁄
MW𝑖 2
Component viscosity:
𝜂𝑖 =
Mixture viscosity:
𝜂mix =
1
⁄
Tc𝑖 6
𝜉=
2⁄
3
(1.01325)
0.72
𝛼
(9)
∑𝑁
𝑖=1 𝑋𝑖 𝜂𝑖 √M𝑖
(10)
∑𝑁
𝑖=1 𝑋𝑖 √M𝑖
2⁄
1⁄
Viscosity at high density:
Pc𝑖
(6)
2 Pc 3
Mmix
mix
(11)
1⁄
6
Tcmix
where the units are :
Mmix [g/mol], Pcmix [atm],
r = Vcmix ρ
P [bar], T [K],
Tcmix [K]
ρ calculated by AGA8 [3]
δ = 0.1023 + 0.023364r + 0.058533r2 - 0.040758r3 + 0.0093324r4
(12)
Viscosity:
(13)
𝜂 = 𝜂mix + ξ (𝛿 4 − 0.0001)
This is the viscosity of the natural gas mixture (units mPa.s = cP)
(ξ is a group of parameters with, in principle, the same dimensions as viscosity and is
commonly used as a reducing parameter that brings viscosities to an equivalent,
dimensionless value. Actually ξ is (MW/NA)1/2(Pc)2/3/(RTc/NA)1/6, where NA =
6.0221367×1023 mol-1, R = 8.314510 J/mol.K and MW [kg/mol], Pc [Pa], Tc [K].
Although, as above, it is defined with MW, Pc & Tc having engineering units, which
means it isn’t dimensionless and thus care must be taken with the equations which have
units conversion factors ‘hidden’ in the coefficients.)
From the following experimental data, the estimated uncertainty of this method is
about 4% (95% confidence interval). (Bias=-0.31 %, RMS=1.59%; actually 95.1%
of errors are within 3.8%).
Note that the experimental data was not used in the development of the LBC
equation.
Total number of points
721
Temperature range
260 to 344 K (-13 to 71 oC)
Pressure range
1 to 127 bar
3
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Experimental data is from the following references :[11]
Carr
(1953)
3 mixtures
55 points
[12]
Golubev
(1959)
1 mixture
17 points
[13]
Gonzalez et al.
(1970)
8 mixtures
35 points
[14]
Nabizadeh et al.
(1999)
1 mixture
32 points
[15]
Assael et al.
(2001)
1 mixture
22 points
[16]
Schley et al.
(2004)
3 mixtures
521 points
[17]
Langelandsvik et al. (2007)
2 mixtures
39 points
The above mixture compositions are given in Appendix 3.
The following figures show the distribution of the errors.
Figure 1 LBC errors in viscosity as a function of temperature and of pressure
Figure 2 Histogram of LBC errors in viscosity
4
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
If a detailed composition is not available, but only bulk properties, e.g., calorific
value (CV), relative density (RD), and CO2 mol%, then this can be converted to an
equivalent N2/CO2/CH4/C3H8 mixture, and the above equation for viscosity can be
used for this equivalent 4 component mixture.
This 4 component mixture has 2 unknown mol% (N2-mol% and C3H8-mol%) (CO2mol% is given, and CH4-mol%=100-N2-CO2-C3H8). These 2 unknowns are
determined from the CV and RD. The procedure is to assume Z (e.g. 0.9975), solve
the linearized CV and RD equations, update Z, and repeat until converged.
Convergence is rapid since Z does not change much with natural gas composition.
2.3
Simple viscosity equation
When only temperature and (mass) density are known (i.e. the gas composition is
not known), the following simple equation can be used
η = 0.01036 + 0.000033t + 0.000021D + 0.00000017D2
(14)
where t is given in oC, and D in kg/m3 (=g/dm3), and η is mPa.s
The estimated uncertainty of this method is about 5 % (95 % confidence interval)
(Bias=0.08 %, RMS=2.57 %; actually 95.0% of errors are within 4.7%).
Note : eq.14 was fitted to the experimental data.
Note : eq.14 correctly monotonically increases with temperature and with density.
The figures below show the distribution of the errors for the 721 data points
Figure 3 Simple equation errors in viscosity as a function of temperature pressure
Figure 4 Histogram of simple equation errors in viscosity
5
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
3
OTHER PROPERTIES
Other properties can be accurately calculated using the GERG-2008 equation of
state (as detailed in ISO-20765:2 [2] and AGA8 [3]), and implemented in the DNVGL program GasVLe.
There are no existing widely-used simple methods for these properties (unlike the
case above for viscosity), so new equations were derived.
To determine the optimal equations, a range of simulated natural gases was
generated based on the following rules:
Table 2 – Natural Gas rules
mol %
Lower limit Upper limit
N2
0.05
10
CO2
0.01
4
CH4
80
98
C2H6
0.25
9
C3H8
0.01
3.5
nC4
0.001
1
nC5
0.001
0.2
nC6
0.001
0.1
iC4/nC4
0.45
0.83
iC5/nC5
0.83
1.33
neoC5/nC5
0.01
0.015
Cn/Cn-1
0.2
0.4
CV (MJ/sm3)
35
45
The procedure was to generate N2, CO2, and C2H6 composition values uniformly
within this range; C3H8 values were generated from ratio limits (Cn/Cn-1) using the
C2H6 value, and similarly for nC4, nC5, and nC6 (using ratio limits with C3H8, nC4, or
nC5, respectively); generate iC4, iC5, and neoC5 from ratio limits. The CH4
composition is the remainder. The CH4 value and CV ranges were checked (as well
as C3H8, nC4, nC5, and nC6 values to be within their ranges). The composition was
accepted if all the above limits were satisfied.
The GERG-2008 equation of state was then used to calculate the properties for all
the mixtures in a grid of temperatures and pressures over the range of interest.
From these calculations it was observed that the compositional variation was not
significant, compared to the temperature and pressure variation, and was within
the target uncertainty outlined in the introduction. Thus, equations as a function
of T and P only were sought. (The compositional variation is accounted for in the
final overall uncertainty.)
The GERG-2008 equation of state has uncertainty with respect to reliable,
consistent, experimental data. However, the uncertainties in  (Joule-Thomson
coefficient) and  (isentropic exponent) are not directly given.
AGA8 part 2 table 7 gives the expanded uncertainty (U) (k=2) for density (D) as
<0.1 % (250 to 450 K, less than 350 bar); for speed of sound (W) as <0.1% (250
to 450 K, less than 120 bar); and for gas phase heat capacities (Cp and Cv) as <1
to 2 %. Whilst ISO 20765-2 table 7 gives the expanded uncertainty in density as
0.03 to 0.05 % and speed of sound as 0.03 to 0.05 %, both for the gas phase (less
than 300 bar).
For the conditions considered here (-20 to 40 oC, less than 100 bar) it is realistic
to take U(D) = 0.05%, U(W) = 0.05% and U(Cp) = 1%.
6
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
 can be expressed as = (R.T2/P.Cp).(Z/T)P ; so using the standard uncertainty
(quadrature) equation {U()/} = √[ {U(Cp)/Cp}2 + {U(Z/T)/(Z/T)}2 ]
(assuming that the uncertainties are uncorrelated). Assuming U(Z/T) is
essentially the same as U(D); then the uncertainty in  is about 1.0%
(=√[12+0.052]) (U(Z/T) would need to be 0.5% for the combined uncertainty to
increase to 1.1%)
 can be expressed as = W2.D/P (re-arranged equation 18 below) so
{U()/} = √[ 4.{U(W)/W}2 + {U(D)/D}2 ] , (assuming that the uncertainties are
uncorrelated). Thus, the uncertainty in  is about 0.1% (=√[4×0.052+0.052])
RMS corresponds to the standard uncertainty (k=1), so the RMS% given in sections
3.1 and 3.2 below should be increased be adding, in quadrature, half the
uncertainties above (to allow for the GERG-2008 equation of state uncertainty).
For  the RMS% below is about 7.0%, this should be √[7.02+(1.0/2)2] = 7.0%
For  the RMS% below is about 1.0%, this should be √[1.02+(0.1/2)2] = 1.0%
Thus, the GERG-2008 uncertainty does not add anything significant to the RMS%
tables in section 3.1 and 3.2
To determine the optimal equation, a bank of terms with powers of T and P
(including fractional powers, and positive and negative values) was used with the
SuperFit routine of the DNV GL Excel Add-In GasTools. In the end, the equations
chosen as achieving the requirements were very straightforward.
The recommended equations are given below, with a table of values (from the
equation), a table of bias errors, and a table of RMS errors (as absolute values and
as percent). The RMS can be interpreted as a standard uncertainty (coverage
factor k=1).
The bias and RMS (root-mean-squared) errors with respect to the calculation of
ISO-20765:2 (GERG-2008 equation of state) are (for property X):
1
gerg
eqn
bias = ∑𝑁
− 𝑋𝑖 )
𝑖=1(𝑋𝑖
𝑁
1
gerg
eqn
RMS = √ ∑𝑁
− 𝑋𝑖 )
𝑖=1(𝑋𝑖
2
(15)
𝑁
The value N is the number of test points (=5000).
The major contribution to the RMS comes from the compositional variation in the
property, rather than from the inadequacy of the simple equation.
3.1
Joule-Thomson coefficient ()
𝜕𝑇
Definition:
𝜇=( )
Equation:
 = ( 0.594 - 0.0042t ) + (-0.177 + 0.0021t )(P/100)2
𝜕𝑃 𝐻
(16)
where t is given in oC and P bar(abs)
The data were actually fitted in the range 0 to 30 oC, 10 to 100 bar(abs), but as
shown in the table below, the extrapolation outside of this range is acceptable.
Table 3 – Joule-Thomson coefficient equation value and bias
P/bar
100
80
60
40
20
t/oC
0.459
0.538
0.599
0.643
0.669
-20
Joule-Thomson coefficient value
0.438 0.417 0.396 0.375
0.509 0.481 0.452 0.424
0.565 0.530 0.496 0.461
0.604 0.566 0.527 0.488
0.628 0.587 0.546 0.505
-10
0
10
20
(K/bar)
0.354
0.395
0.427
0.450
0.463
30
7
0.333
0.366
0.393
0.411
0.422
40
0.043
0.006
-.018
-.020
-.012
-20
0.019
0.004
-.005
-.004
0.001
-10
Bias (K/bar)
0.008 0.004 0.003
0.005 0.007 0.007
0.003 0.008 0.009
0.006 0.010 0.010
0.009 0.011 0.009
0
10
20
0.003
0.006
0.007
0.006
0.003
30
0.002
0.003
0.002
-.001
-.005
40
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Table 4 – Joule-Thomson coefficient uncertainty (RMS and RMS %)
P/bar
100
80
60
40
20
t/oC
RMS (K/bar)
0.018 0.019 0.019
0.028 0.028 0.027
0.037 0.034 0.031
0.040 0.037 0.034
0.040 0.037 0.033
0
10
20
0.044
0.024
0.047
0.054
0.051
-20
0.021
0.028
0.040
0.045
0.044
-10
3.2
Isentropic exponent ()
0.019
0.025
0.028
0.030
0.029
30
0.019
0.023
0.025
0.026
0.027
40
10.6
4.7
7.2
7.8
7.2
-20
5.2
5.6
7.0
7.3
7.0
-10
RMS%
5.0
6.5
7.2
7.4
7.2
10
4.5
6.1
7.1
7.3
7.1
0
5.4
6.7
7.2
7.3
6.9
20
5.6
6.6
7.0
6.9
6.4
30
5.8
6.4
6.5
6.4
6.1
40
𝑉 𝜕𝑃
Definition:
𝜅 = − 𝑃 (𝜕𝑉)
Equation:

𝑆
=
+
+
( 1.3028 - 0.0005794t )
(-0.08437 + 0.002658t )  (P/100)
( 0.3267 - 0.005517t )  (P/100)2
(17)
where t is given in oC and P bar(abs)
The data were actually fitted in the range 0 to 20 oC, 5 to 75 bar(abs), but as shown
in the table below, the extrapolation outside of this range is acceptable.
Table 5 – Isentropic exponent equation values and bias
P/bar
100
80
60
40
20
t/oC
1.614
1.484
1.389
1.329
1.304
-20
Isentropic exponent value
1.580 1.545 1.511 1.476 1.442
1.464 1.444 1.425 1.405 1.385
1.379 1.370 1.360 1.350 1.341
1.325 1.321 1.317 1.313 1.309
1.302 1.299 1.296 1.294 1.291
-10
0
10
20
30
1.408
1.365
1.331
1.305
1.288
40
-.224
-.058
-.003
0.002
-.001
-20
-.107
-.024
0.001
0.001
-.002
-10
-.055
-.010
0.002
0.000
-.002
0
Bias
-.033
-.005
0.002
-.001
-.002
10
-.027
-.007
0.000
-.001
-.002
20
-.032
-.011
-.003
-.001
-.001
30
2.2
1.1
1.1
1.1
1.0
30
2.9
1.6
1.1
1.1
1.0
40
Table 6 – Isentropic exponent uncertainty (RMS and RMS %)
P/bar
100
80
60
40
20
t/oC
RMS
0.034
0.009
0.014
0.015
0.013
10
0.250
0.062
0.010
0.015
0.013
-20
0.119
0.025
0.012
0.015
0.013
-10
3.3
Speed of Sound (W)
Definition:
0.060
0.010
0.013
0.015
0.013
0
1
0.027
0.012
0.014
0.014
0.013
20
0.032
0.016
0.014
0.014
0.013
30
0.043
0.022
0.015
0.014
0.013
40
12.9
4.0
0.7
1.1
1.0
-20
6.9
1.7
0.9
1.1
1.0
-10
3.7
0.7
0.9
1.1
1.0
0
RMS %
2.2
0.6
1.0
1.1
1.0
10
1.8
0.8
1.0
1.1
1.0
20
𝜕𝑃
𝑊 = √𝑀 (𝜕𝜌)
𝑆
Equation:
W = [P105/D]
m/s,
P bar(abs),
D kg/m3
(18)
 from equation (15) above, D from accurate equation of state
Because W varies with D, it is only useful to provide a table of RMS % uncertainty.
Table 7 – Speed of sound uncertainty (RMS %)
P/bar
100
80
60
40
20
t/oC
6.75
2.02
0.34
0.57
0.50
-20
3.51
0.84
0.43
0.56
0.50
-10
1.86
0.36
0.48
0.56
0.50
0
8
RMS %
1.11
0.32
0.51
0.55
0.50
10
0.91
0.42
0.52
0.55
0.49
20
1.09
0.57
0.54
0.54
0.49
30
1.48
0.79
0.57
0.54
0.49
40
-.042
-.018
-.005
0.000
0.000
40
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
USAGE
4
In order to calculate flow (according to equation (1)), the required properties (, 
and ) are calculated at the upstream temperature and pressure. However, to
calculate the upstream temperature from the measured downstream temperature
using  (references [4] and [5]) or  (references [6] and [7]), there are a number
of feasible conditions to use. There is also the option of using an averaged value
(rather than a value from the equation at a specified T & P). Using the property
equation at the measured downstream T and upstream P has the merit of not
requiring recalculation of the property. In general, the best property to use is
calculated at the average upstream & downstream temperature & pressure
(compared with an exact isentropic or isenthalpic calculation). The difference
between the various options is general insignificant. The variation in upstream
temperature is usually less than 0.01 K (whilst the difference with a proper
thermodynamic calculation is more like 0.05 K).
CONCLUSION
5
Equations have been given that are simple to implement, but that are accurate
enough to be useful (especially for high pressure orifice plate metering) for natural
gas in the temperature range -20 to 40 oC and pressure range up to 100 bar.
Some of the equations have already been implemented by some manufacturers in
their flow computer. The methods and equations are proposed to be ISO 20765
part 5 (“Natural gas – Calculation of thermodynamic properties – Part 5: Calculation
of viscosity, Joule-Thomson coefficient, and isentropic exponent”).
NOTATION
6
Symbol
Meaning
Units
C
discharge coefficient
[-]
d
orifice diameter
[m]
D
pipe (inside) diameter
[m]
D
mass density
[kg/m ]
H
enthalpy
[kJ/kg]
M
molar mass
[g/mol]
P
pressure (absolute)
[bar]
S
entropy
[kJ/kg.K]
T
temperature
[K]
t
temperature
[oC]
q
mass flowrate
[kg/s]
Re
Reynolds number,
(4.q)/(..D)
[-]
V
molar specific volume,
=1/
[dm3/mol]
W
speed of sound
[m/s]
Xi
mole fraction of component i
[mol/mol]

diameter ratio,
[-]
p
differential pressure
[bar]

permanent pressure loss
[bar]

expansibility factor
[-]

fluid dynamic viscosity
[mPa.s]

isentropic exponent
[-]
(used in Re and )
3
= d/D
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)

Joule-Thomson coefficient

= 3.141592654…

molar density
[K/bar]
[mol/dm3]
7
REFERENCES
[1]
O.Kunz and W.Wagner, “The GERG-2008 wide-range equation of state for
natural gases and other mixtures: An expansion of GERG-2004”, J. Chem.
Eng. Data, 57, 3032-3091, (2012)
ISO 20765-2, “Natural gas – Calculation of thermodynamic properties – Part
2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges
of application”, (2015)
American Gas Association Report No. 8, Part 2, “Thermodynamic Properties
of Natural Gas and Related Gases, GERG-2008 Equation of State”, (2017)
ISO 5167-1:1991, “Measurement of fluid flow in closed conduits – Part 1.
Pressure differential devices – Section 1.1: Specification for square-edged
orifice plates, nozzles and Venturi tubes inserted in circular cross-section
conduits running full”, (BS 1042:Section 1.1:1992)
BS EN ISO 5167-1:1997, “Measurement of fluid flow by means of pressure
differential devices – Part 1: Orifice plates, nozzles and Venturi tubes
inserted in circular cross-section conduits running full”, (BS 1042-1.1:1992
renumbered, incorporating Amendment No.1 (renumbering the BS as BS EN
ISO 5167-1:1997), and Amendment No.1 to BS EN ISO 5167-1:1997)
BS EN ISO 5167-1:2003, “Measurement of fluid flow by means of pressure
differential devices inserted in circular cross-section conduits running full –
Part 1: General principles and requirements”
BS EN ISO 5167-2:2003, “Measurement of fluid flow by means of pressure
differential devices inserted in circular cross-section conduits running full –
Part 2: Orifice plates”
American Gas Association Report No. 3, “Orifice Metering of Natural Gas and
Other Related Hydrocarbon Fluids”, Part 1, “General Equations and
Uncertainty Guidelines”, (1990)
American Gas Association Report No. 3, “Orifice Metering of Natural Gas and
Other Related Hydrocarbon Fluids”, Part 3, “Natural Gas Applications”,
(1992)
J.Lohrenz, B.G.Bray and C.R.Clark, “Calculating Viscosities of Reservoir
Fluids From Their Compositions”, J.Petrol.Technol., pp.1171-1176, October
(1964)
N.L.Carr, “Viscosities of natural gas components and mixtures”, Institute of
Gas Technology Research Bulletin 23, (1953)
I.F.Golubev, “Viscosity of Gases and Gas Mixtures: A Handbook”, p.214
(1959) (translation 1970)
M.Gonzalez, B.E.Eakin and A.L.Lee, Monograph on API Research Project 65,
American Petroleum Institute (1970)
H.Nabizadeh and F.Mayinger, High Temperatures-High Pressures, 31,
pp.601-612 (1999)
M.J.Assael, N.K.Dalaouti and V.Vesovic, Int. J. Thermophysics, 22(1),
pp.61-71 (2001)
P.Schley, M.Jaeschke, C.Kuchenmeister and E.Vogel, Int. J. Thermophysics,
25(6), pp.1623-1651 (2004)
L.I.Langelandsvik, S.Solvang, M.Rousselet, I.N.Metaxa and M.J.Assael, Int.
J. Thermophys. 28, pp.1120-1130 (2007)
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 1 : Methane Viscosity
Viscosity for pure methane is tabulated in the following standard reference works
:1) B.A.Younglove & J.F.Ely,
J.Phys.Chem.Ref.Data,vol.16,no..4,pp.577-798,(1987)
2) Encyclopedie des Gaz, L'Air Liquide (1976), p.296
3) N.B.Vargaftik, Y.K.Vinogradov & V.S.Yargin, p.436,
“Handbook of Physical Properties of Liquids and Gases”, 3rd ed., (1996)
4) D.G.Friend, J.F.Ely & H.Ingham,
J.Phys.Chem.Ref.Data,vol.18,no.2,pp.583-638,(1989)
5) P.Schley, M.Jaeschke, C.Kuchenmeister & E.Vogel,
Int.J.Thermophys.,vol.25,no.6,pp.1623-1652,(2004)
The figure below shows the first four references compared to the data from the last
(actually using the provided equations from ref.5 as a function of density for the
five isotherms). The figure shows that even for methane the agreement of viscosity
measurements is no better than 1 to 2 %.
Methane Viscosity Comparisons
3
% difference
2
1
Younglove
E.d.Gaz
0
Vargaftik
Friend
-1
-2
-3
250 260 270 280 290 300 310 320 330 340 350
T (K)
Thus, any method that reliably calculates viscosity with an uncertainty of about 2%
can be considered satisfactory.
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Viscosity of pure methane using the reference formulas as implemented in NIST
RefProp 10.0 (2018). This uses the equation of state of U.Setzmann and
W.Wagner, J.Phys.Chem.Ref.Data, 20:1061-1151 (1991), and the viscosity
method of S.E.Quinones-Cisneros, M.L.Huber and U.K.Deiters, unpublished work
(2011).
P/bar
100
80
60
40
20
t/°C
0.014007
0.012412
0.011247
0.010443
0.009908
-20
0.013729
0.012435
0.011450
0.010736
0.010238
−10
Reference viscosity
0.013623 0.013619
0.012532 0.012675
0.011676 0.011916
0.011031 0.011327
0.010565 0.010888
0
10
P/bar
100
80
60
40
20
t/°C
0.59
-0.65
-0.99
-0.65
0.01
-20
-0.07
-0.93
-1.05
-0.63
0.06
−10
-0.47
-1.05
-1.03
-0.56
0.14
0
P/bar
100
80
60
40
20
t/°C
-0.22
0.84
1.55
1.86
1.80
-20
-0.04
0.75
1.29
1.57
1.55
−10
(mPa.s)
0.013683
0.012850
0.012165
0.011622
0.011206
20
0.013792
0.013045
0.012422
0.011917
0.011522
30
0.013933
0.013256
0.012682
0.012210
0.011834
40
-0.92
-1.16
-0.97
-0.47
0.22
20
-1.25
-1.39
-1.18
-0.68
-0.01
30
-1.53
-1.60
-1.37
-0.88
-0.23
40
Simple equation error %
0.05
0.12
0.18
0.67
0.62
0.59
1.11
0.98
0.92
1.35
1.20
1.12
1.36
1.24
1.18
0
10
20
0.26
0.61
0.89
1.08
1.16
30
0.35
0.66
0.91
1.09
1.18
40
LBC error %
-0.68
-1.05
-0.94
-0.44
0.25
10
The reference viscosity estimated uncertainty is less than 0.3 %.
LBC equation
: Bias = -.67% , RMS = 0.87%
Simple equation
: Bias = 0.91% , RMS = 1.04%
(Neither method used this data as part of the equation development.)
The Schley et al. (2004) data for pure methane (at 260, 280, 300, 320, 340 &
360 K, 2 to 220 bar; 340 points) has been fitted to the following equation :-
η = (-0.000884 + 0.014228×τ - 0.002262×τ2) + (0.001187 + 0.00062×τ)×δ
+ (0.002908 - 0.000366×τ)×δ2 – 0.000565×δ3 + 0.000186×δ4
(mPa.s), where τ = T/298.15 (T in K), and δ = D/100 (D in kg/m3) (D calculated
using the GERG-2008 equation of state.)
Bias = 0.001%, RMS = 0.034%, Maximum error = 0.103%
This equation agrees with the Quinones-Cisneros et al. equation (250 to 340 K, 1
to 100 bar) with errors : Bias = -0.034%, RMS = 0.080%, Max = 0.273%.
Joule-Thomson coefficient and Isentropic Exponent
For pure methane, -20 to 40 oC and 0 to 100 bar (compared to Setzmann &
Wagner or GERG-2008 – the differences are less than 0.001 for  , 0.002 for ) :
Eq.16 has bias=0.056 , RMS=0.057 , max=0.076 (C/bar) . The improved
equation below has bias=0.003, RMS=0.006, max=0.013 :CH4 = ( 0.525 - 0.0036t ) + (-0.145 + 0.0016t )(P/100)2
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Eq.17 has bias=-0.023, RMS=0.024, max=0.082 . The improved equation below
has bias=-.004, RMS=0.013, max=0.025 :CH4 = (1.3288 - 0.000101t ) + (-0.1496 + 0.00314t )(P/100)
+ (0.4129 - 0.00607t )(P/100)2
Compression Factor
The compression factor (compressibility factor, Z, = (P.V)/(R.T) ) for pure
methane is accurately given as an explicit function of temperature (T) and
pressure (P) by the following equation :
Z = 1 + p1×Pr + p2×Pr2 + p3×Pr3 + p5×Pr5
where
p1 = 0.0020223 + 0.1263/Tr - 0.28396/Tr2 - 0.17754/Tr3
p2 = 0.058793/Tr - 0.32923/Tr2 + 0.70391/Tr3 - 0.50675/Tr4
p3 = 0.006748/Tr4 - 0.100907/Tr14
p5 = -0.0021805/Tr4 + 0.0067056/Tr6
and
Tr = T / 190.564
Pr = P / 45.992
(T in K, P in bar(abs))
In the range -20 to 40 oC, up to 80 bar: the difference from the GERG-2008
equation of state is : Bias = -.001%, RMS = 0.003%, Max = 0.019%
The above equation was developed using GERG-2008 for methane.
p1 and p2 were fitted to B and C virial coefficients from Setzmann & Wagner
(1991) over the range -30 to 50 oC (p1=B×Pc/(R×T) , p2=(C-B2)×Pc2/(R×T)2)
(Compared with the equation of U.Setzmann & W.Wagner, J.Phys.Chem.Ref.Data,
vol.20,pp.1061-1151,(1991) : Bias=-.001%, RMS=0.004%, Max=0.033%)
The following table gives the breakdown of the differences from GERG-2008 as a
function of temperature and pressure :P/bar
Errors in Z : 100×(Zeqn-ZGERG)/ZGERG
100
Bias%
RMS%
Max%
80
Bias%
RMS%
Max%
60
Bias%
RMS%
Max%
40
Bias%
RMS%
Max%
20
Bias%
RMS%
Max%
0
T/oC
-20
-.006
0.033
0.097
0.042
0.054
0.119
0.040
0.051
0.117
0.025
0.034
0.089
0.009
0.016
0.052
-.003
0.006
0.020
0.000
0.005
0.019
0.000
0.001
0.004
-.002
0.003
0.004
-.004
0.004
0.006
-.005
0.005
0.006
-.004
0.004
0.007
0.007
0.007
0.010
0.003
0.003
0.005
0.001
0.001
0.002
0.000
0.001
0.003
-.001
0.001
0.002
0.000
0.001
0.002
0.002
0.003
0.006
0.001
0.002
0.004
0.000
0.001
0.002
-.001
0.001
0.002
-.001
0.001
0.003
-.001
0.002
0.003
-.001
0.001
0.002
-.002
0.002
0.002
-.002
0.002
0.003
-.002
0.002
0.003
-.002
0.002
0.003
-.002
0.002
0.003
-10
0
10
20
30
(The statistics in each box comes from the comparison of 440 calculations.)
13
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 2 : Methods for the Calculation of the
Viscosity of Natural Gas
1
Introduction
The viscosity of natural gas is required for the calculation of flowrate through orifice
plate meters using the equations in ISO 5167. Viscosity is used to calculate the
Reynolds number, which is used in the equation for the discharge coefficient.
The relevant conditions are :
Temperature 260 to 340 K
(-13 to 67 oC) (8 to 152 oF)

Pressure up to 120 bar
(up to 1740 psia)
This is also essentially the primary range given for the calculation of Z in ISO 20765,
and AGA8.
2
Viscosity methods
Viscosity can be calculated from the composition, temperature and density.
(Pressure is an alternative to density, but, theoretically and empirically, density is
the better input to use). In this report P and Z were used for convenience as the
input (density = P/(ZRT) ), where Z was calculated using the AGA8 (detail
characterization) method.
The books by Reid et al. (1977), Reid et al. (1987) and Poling et al. (2001) discuss
viscosity and list a number of methods that are considered here.
The units for viscosity,  , used here, are [mPa.s] (=cP , centipoise).
There are three principle approaches to calculating viscosity :1. General corresponding states methods, which can be applied to gases,
liquids and dense fluids, and are quite general. Such methods are NBS, PFT
and CLS as implemented in GasVLe.
2. Semi-theoretical gas phase methods. Such methods are DIL and LBC as
implemented in GasVLe.
3. Empirical methods. Often given in engineering books, which are simple to
use but are limited in use – both in T & P range, and composition.
NBS : a modified form of the NBS program TRAPP.
J.F.Ely & H.J.M.Hanley, Ind.Eng.Chem.Fundam., vol.20,pp.323-332,(1981)
J.F.Ely & H.J.M.Hanley, Ind.Eng.Chem.Fundam., vol.22,pp.90-97,(1983)
M.L.Huber & H.J.M.Hanley, chapter 13, "Transport Properties of Fluids",
Eds. J.Millat, J.H.Dymond & C.A.Nieto de Castro, (1996)
PFT : method of Pedersen, Fredenslund, Christensen & Thomassen.
K.S.Pedersen, A.Fredenslund, P.L.Christensen & P.Thomassen,
Chem.Eng.Sci.,vol.39,pp.1011-1016,(1984)
K.S.Pedersen & A.Fredenslund, Chem.Eng.Sci.,vol.42,pp.182-186,(1987)
J.K.Ali, J.Pet.Sci.& Eng.,vol.5,pp.351-369,(1991)
CLS : method of Chung, Lee & Starling,
T.H.Chung, L.L.Lee & K.E.Starling,
Ind.Eng.Chem.Fundam., vol.23,p.8,(1984)
T.H.Chung, M.Ajlan, L.L.Lee & K.E.Starling,
Ind.Eng.Chem.Res., vol.27,p.671,(1988)
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
DIL : method based on dilute gas methods; generally cubic polynomial (in T) for
dilute gas viscosity, Wilke mixture rule, and Stiel & Thodos density
correction.
LBC : method of J.Lohrenz, B.G.Bray & C.R.Clark,
J.Pet.Technol.,pp.1171-1176,Oct.1964
The first methods are rather complicated and involved. They are not considered
suitable for use in an online flow computer, and as shown in the comparisons, are
not any better than the other methods for natural gas at the conditions of interest.
The second methods are considered in detail in this report. These methods are
made up of three stages :a) Calculation of the individual pure component dilute gas viscosities (i) at the
temperature T
b) Calculation of the mixture dilute gas viscosity (mix) using mixing rules for
i
c) Calculation of the density correction to get the final gas mixture viscosity
() at the temperature and density required (or equivalently at the required
T & P)
(The dilute gas is the low or moderate pressure range where viscosity is
independent of pressure (and density). E.g. for methane at 290 K, viscosity
changes by less than 1% up to 8 bar (for gas, viscosity increases with P, and
increases with T). There is a low P limit when the mean free path of the gas
molecules is comparable to the size of the gas container – but this isn’t usually
important in practice.)
This is the approach recommended in API (2005) :a) Procedure 11B1.3 : “Viscosity of Pure Gases at Low Pressure” uses Stiel &
Thodos (alternative procedure 11B1.1 uses DIPPR equation),
b) Procedure 11B2.1 : “Viscosity of Gaseous Mixtures at Low Pressure” uses
Wilke,
c) Procedure 11B4.1 : “Viscosity of Pure Hydrocarbon Gases and their Gaseous
Mixtures at High Pressure” uses Dean & Stiel.
3
Equations
3.1
Critical Data
Required data for each component are :
Molecular weight
MWi
[g/mol]

Critical temperature
Tci
[K]

Critical pressure
Pci
[bar]

Critical compressibility factor
Zci
[-]
3.1.1 VIPAN
Critical parameters from British Gas program VIPAN (from HPMIS source code).
3.1.2 DIPPR
Design Institute for Physical Property Data, A.I.Ch.E., “Physical
Thermodynamic Properties of Pure Chemicals”, databooks, 1994 revision.
15
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
3.1.3 GasVLe
Critical parameters currently used in the DNV GL program GasVLe ; typically from
the CRC Handbook (as recommended by D.Ambrose).
3.2
Critical Mixing Rules
N
MWmix   X i MWi
In all cases
Xi is component mole fraction
i 1
Tcmix and Pcmix (atm units) are required for the viscosity reducing parameter; Vcmix
is required for calculating the reduced density (units of K/bar, i.e. R cancels).
3.2.1 Kay
Kay, Ind.Eng.Chem.,(1936)
N
Vc mix   X i
i 1
N
Zc i Tc i
Pci
N
Tc mix   X i Tc i
Pcmix   X i
i 1
i 1
Pci
1.01325
3.2.2 Lorentz & Berthelot
Assael et al.,(1996) p.85 (based on the Lorentz-Berthelot combining rules for the
length and energy parameters in the intermolecular potential, Lorentz (1881) and
Berthelot (1898)).
 
1  Zc Tc
Vc ij    i i
 2  Pci
 

N



N
Vc mix   X i X j Vc ij
1
3
 Zc j Tc j 


 Pc 
j


Tc mix 
i 1 j1
1
1
Vc mix
3





N
3
N
Pcmix   X i
i 1
Pci
1.01325
N
 X X Vc
i 1 j1
i
j
ij
Tc i Tc j
3.2.3 Prausnitz & Gunn
Prausnitz & Gunn, AIChE J.,(1958)
N
Vc mix   X i
i 1
3.3
Zc i Tc i
Pci
N
Tc mix   X i Tc i
Pc mix 
i 1
Tc mix
Vc mix
N
X
i 1
i
Zc i
1.01325
Viscosity at low pressure
3.3.1 API (1981)
Used in VIPAN, Gasunie book p.82, API (1981) procedure. Based on the ChapmanEnskog equation, using intermolecular potential parameters from general
equations.
1.8617  Zc i  0.08314510 Tc i
σi 
 
Pci
Zc1.2

i



1
3


10  T

α  Ln 
3.6 
65.3

Tc

Zc
i
i


i = 1/( 0.91426362  - 1.068936 2 + 0.68077797 3 -0.21208677 4
+ 0.034487186 5 - 0.0028188225 6 + 0.000091590342 7)
η i  0.002669
MWi T
σ i2 Ω i
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American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
3.3.2 Assael et al.
Gasunie book p.83, Assael et al. (1996), p.283, using tabulated values for i and
(/k)i (determined from viscosity data).
 = T / (/k)i
i = 1.16145 / 0.14874
+ 0.52487 Exp(-0.7732 ) + 2.16178 Exp(-2.43787 )
- 0.0006435 0.14874 Sin(18.0323/0.7683 - 7.2371)
ηi 
MWi T
0.3125
8314.51

6022.1367
π
σ i2 Ω i
3.3.3 Stiel & Thodos
Stiel & Thodos, AIChE J.,(1961) :
Tr = T / Tci
Tr  1.5 :
 = 0.00034 Tr0.94
Tr > 1.5 :
 = 0.0001778(4.58 Tr - 1.67)0.625
1
2
MWi 2  Pci  3
ηi 

 α
1
6  1.01325
Tc i
3.3.4 DIPPR
Component parameters A, B, C and D for semi-empirical equation :i = Ai TBi /( 1 + Ci / T + Di / T2 )
3.3.5 Reichenberg
Reichenberg (1971).
Experimental component viscosities at 290 K (from various J.Phys.Chem.Ref.Data
sources) used with Reichenberg temperature dependence function.
T290 = 290 / Tci
ηi  η
3.4
290
i
Tr = T / Tci
1  0.36T290 T290  116
T290
Tr
1  0.36Tr Tr  116
Viscosity of Mixture
3.4.1 Wilke
Wilke, J.Chem.Phys.,(1950); Bromley & Wilke, Ind.Eng.Chem.,(1951)
   12 MW 14 
j 
1   η i  
 



  η j   MWi  

φ ij  
12
  MWi 

81 

MW
 
j 

2
η mix
3.4.2 Herning & Zipperer
Herning & Zipperer, Gas-Wasserfach,(1936)
17




 X i ηi 
  N

i 1
  X j φ ij 
 j1

N
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
N
η mix 
X η
i 1
N
i
X
i 1
i
i
MWi
MWi
(Which is Wilke method with ij set to (MWj/MWi)1/2)
This method doesn’t require the I to be stored.
3.4.3 Reichenberg
Reichenberg (1974) and (1977).
Tri = T / Tci
Trij = T / (Tci Tcj)
1
1
MWi 4 
Tri 2
Ci 
1
1

η i 2  1  0.36Tri Tri  1 6


MWi MW j
H ij  

3
 32MWi  MW j  
Ki 
1
2




1
2
1  0.36T T
rij
rij
1
 C
 1
1
6
Cj 
2
i
Trij2
X i ηi
N
2MWj 


X i  η i  X j H ij  3 
MWi 
j1 i

N
i 1
N
N


η mix   K i 1  2 H ij K j    H ij H ik K j K k 
i 1
j1
j1 i k 1 i


3.5
Viscosity at high density
2
1
ξ
3
MWmix2 Pc mix
1
where MWmix [g/mol], Pcmix [atm], Tcmix [K]
6
Tc mix
r = VcmixP /( ZT )
P [bar], T [K], gas Z calculated by AGA8
( is a group of parameters with, in principle, the same dimensions as viscosity
[Pa.s] and is commonly used as a reducing parameter that brings viscosities to an
equivalent, dimensionless value. Actually  is (MW/NA)1/2(Pc)2/3/(RTc/NA)1/6, where
NA = 6.02213671023 mol-1, R = 8.314510 J/mol.K and MW [kg/mol], Pc [Pa], Tc
[K]. Although, as above, it is defined with MW, Pc & Tc having engineering units,
which means it isn’t dimensionless and thus care must be taken with the
equations which have units conversion factors ‘hidden’ in the coefficients.)
3.5.1 Dean & Stiel
Dean & Stiel, AIChE J.,(1965), API (2005) procedure
 = 0.000108 [ Exp(1.439 r) - Exp(-1.11 r1.858) ]
 = mix +  
3.5.2 Jossi, Stiel & Thodos
Jossi, Stiel & Thodos, AIChE J.,(1962) : = 0.1023 + 0.023364 r + 0.058533 r2 - 0.040758 r3 + 0.0093324 r4
 = mix +  (4 – 0.0001 )
18
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
3.5.3 Reichenberg
Reichenberg (1975).
Tr = T / Tcmix
Pr = P /(1.01325Pcmix)
A = (0.001982358 / Tr)Exp(5.26827 / Tr0.576687)
B = A(1.655163Tr - 1.276002)
C = (0.131889 / Tr) Exp(3.703471 / Tr79.867774)
D = (2.949592 / Tr) Exp(2.918976 / Tr16.616883)
E = BPr + 1 / (1 + CPrD)
 = mix ( 1 + APr1.5 / E)
This method doesn’t require gas Z to be calculated.
3.6
Empirical Methods
Quoted in engineering books, such as McCain (1990) (p.514), Danesh (1998)
(p.83)
The methods below require only the MW of the gas.
3.6.1 Lee, Gonzalez & Eakin
A.Lee, M.H.Gonzalez & B.E.Eakin, J.Petrol.Trans.,pp.997-1000,August (1966)
T [K] & P [bar] input, and Z is the calculated value for the gas.
A = (9.379 + 0.01607MWmix)(1.8T)1.5 /(209.2 + 19.26MWmix + 1.8T)
B = 3.448 + 986.4 / (1.8T) + 0.01009MWmix
C = 2.447 - 0.2224B
 = MWmixP0.1 / (Z8.31451T)
 = A0.0001Exp(BC)
There are enhancements which have additional terms for CO 2 , e.g. Elsharkawy
(2004).
3.6.2 Starling & Ellington
K.E.Starling & R.T.Ellington, AIChE J.,(1964).
A = (7.77 + 0.0063MWmix)  (1.8T)1.5 / (122.4 + 12.9MWmix + 1.8T)
B = 2.57 + 1914.5 / (1.8T) + 0.0095MWmix
C = 1.11 + 0.04B
 = MWmixP0.1 / (Z8.31451T)
 = A0.0001Exp(BC)
4
Comparisons
627 experimental values selected to compare the various combinations outlined
above. Data sources :1) Golubev
(1959)
92 mol% CH4
17 points
2) Nabizadeh et al.
(1999)
95 mol% CH4
32 points
3) Assael et al.
(2001)
85 mol% CH4
22 points
4) Gonzalez et al.
(1970)
72 to 98% CH4
35 points
5) Schley et al.
(2004)
19
100 mol% CH4
201 points
90 mol% CH4
160 points
84 mol% CH4
160 points
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Error% = 100  (calc – expt )/ expt
Bias% = (  error% )/ 627
RMS% = [ { (error%)2 }/ 627 ]
There are 33533 = 405 combinations of methods. The table below lists some.
The number in the first column refers to the method given in the subsection of
section 3.1, e.g. section 3.1.3 for the first row. Similarly for the other columns.
crit.data
3.1.?
3
3
3
3
3
3
3
3
3
1
2
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
2
3
1
2
3
1
2
3
1
2
2
crit.mix
3.2.?
1
1
1
1
1
1
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
3
3
3
1
1
1
vis.lowP
3.3.?
3
3
3
3
3
3
3
3
3
3
3
3
1
2
3
4
5
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
1
3
5
vis.mix
3.4.?
2
2
2
1
2
3
2
2
2
2
2
2
2
2
2
2
2
1
1
1
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
vis.den
3.5.?
1
2
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
3
1
2
3
1
2
3
3
3
3
3
3
3
3
3
3
1
1
1
2
2
5
3
2
3
1
3
3
3
3
1
2
3
3
5
5
4
5
3.6.1
3.6.2
Bias%
RMS%
-.53
-.18
-.95
-.47
-.18
-.53
-.18
-.23
-.18
-.05
-.18
-.18
-.97
1.36
-.18
1.84
0.44
1.16
1.50
0.91
1.48
1.83
1.27
1.12
1.46
0.87
-.72
-.71
-.71
-.60
-.59
-.59
-.73
-.72
-.72
-1.22
-.82
-.26
1.26
1.37
1.69
1.32
1.37
1.33
1.37
1.37
1.37
1.36
1.37
1.37
1.73
1.93
1.37
2.29
1.47
1.49
1.97
1.53
1.81
2.28
1.86
1.45
1.93
1.48
1.40
1.39
1.39
1.38
1.38
1.38
1.39
1.39
1.39
1.66
1.30
1.03
1
-.35
1.02
3
3
-2.29
2.56
1
2
1
1
1
1
1
1
-.25
0.05
1.18
-.24
0.17
-2.38
1.03
1.16
1.51
1.04
3.56
3.99
20
change
of
vis.den.
change
of
vis.mix
change
of
crit.mix
change
of
crit.data
change
of
vis.lowP
change
of
vis.mix
and
vis.den.
change
of
crit.data
and
crit.mix
smallest
RMS
largest
RMS
Best
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
NBS
PFT
CLS
DIL
LBC
0.91
1.22
-1.14
0.61
-.18
1.49
1.51
2.06
1.10
1.37
GasVLe
(For ease of comparison some combinations are listed more than once.)
In general, there is not a big difference between any of the methods and
combinations. The type 1 methods (NBS, PFT & CLS) are no better than the other
methods. Only the type 3, empirical methods (3.6.1 & 3.6.2) are not so good, but
they are not a lot worse.
LBC, (Lohrenz et al., 1964)) is the same as method (3,1,3,2,2).
DIL is similar to (3,3,4,1,1) or (3,3,5,1,1). The difference is due to a different
component low pressure viscosity equation being used. (DIL uses polynomial
equations from ESDU data items vol. 6e & 6f.)
API (2005) corresponds to (2,1,3,1,1) or (2,1,4,1,1)
API(1981), VIPAN corresponds to (1,1,1,1,1) (although test results from the
VIPANII.exe DOS program (Oct.1990) do not agree exactly – for example, VIPAN
uses API (Lee-Kesler) procedure to calculate Z rather than AGA8)
From all the 405 combinations, biases go from -2.29% to +1.92%. The minimum
RMS is 1.02%. The (3,1,5,1,1) method is arguably the best since the bias is slightly
less and it is computationally moderate.
Overall the recommended method is LBC (3,1,3,2,2). Figure 1 in the main text
show the errors as a function of temperature and pressure.
5
Recommendations
The empirical methods are simplest, but not the best. The recommendation is one
of the type 2 method combinations, chosen by the following reasons :1. The critical data used is not important. Any reasonable set could be used.
2. No critical mixing rule is significantly better. Thus, the simplest method,
3.2.1 (Kay) is recommended.
3. Using experimental pure component viscosity does not appear to be
significantly better than general equations. Hence, for simplicity and
minimal data storage, method 3.3.3 is recommended
4. All mixture viscosity methods are similar. The simplest method 3.4.2 is thus
recommended, also i does not needed to be stored.
5. Either method 3.4.1 or 3.4.2 looks to be satisfactory.
Thus, the recommendation is (3,1,3,2,2) - Lohrenz-Bray-Clark (LBC). This is also
a standard method used in reservoir engineering, e.g. Pedersen et al. (1989),
Danesh (1998), Pedersen et al. (2007), and is generally accepted as being an
industry standard method and widely available. The only disadvantage is that the
calculated viscosity is sensitive to the value of the density (or Z) used. However,
this is not relevant for metering since using the AGA8 method will provide an
accurate Z.
6
References
1) API Technical Data Book, 7th ed., The American Petroleum Institute and
EPCON International, (2005)
21
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
2) M.J.Assael, J.P.M.Trusler & T.F.Tsolakis, “Thermophysical Properties of
Fluids”, Imperial College Press, (1996)
3) M.J.Assael, N.K.Dalaouti & V.Vesovic, “Viscosity of Natural-Gas Mixtures:
Measurements and Prediction”, Int.J.Thermophysics, vol.22,no.1,pp.6171,(2001)
4) D.Berthelot, Compt.Rend.Acad.Sci., vol.126,pp.1703-6,1857-8,(1898)
5) L.A.Bromley & C.R.Wilke, “Viscosity behaviour of gases”,
Ind.Eng.Chem.,vol.43,no.7,pp.1641-1648,(1951)
6) A.Danesh, “PVT and phase behaviour of petroleum fluids”, Elsevier, (1998)
7) D.E.Dean & L.I.Stiel, “The viscosity of nonpolar gas mixtures at moderate
and high pressures”, AIChE J.,vol.11,no.3,pp.526-532,(1965)
8) DIPPR, Design Institute for Physical Property Data, A.I.Ch.E., “Physical and
Thermodynamic Properties of Pure Chemicals”, databooks, 1994 revision.
9) A.M.Elsharkawy, “Efficient methods for calculation of compressibility,
density and viscosity of natural gases”, Fluid Phase Equil.,vol.218,pp.113,(2004)
10) Encyclopedie des Gaz, L'Air Liquide (1976), p.296
11) D.G.Friend, J.F.Ely & H.Ingham, “Thermophysical properties of methane”,
J.Phys.Chem.Ref.Data, vol.18,no.2,pp.583-638,(1989)
12) Gasunie, “Physical Properties of Natural Gas”, N.V.Nederlandse Gasunie,
(June 1988), section 2.5.1.3
13) I.F.Golubev, “Viscosity of Gases and Gas Mixtures : A Handbook”, p.214,
(1959), Israel Program for Scientific Translations, Jerusalem, (1970)
14) M.Gonzalez, B.E.Eakin & A.L.Lee, “Viscosity of Natural Gases”, Monograph
on API Research Project 65, American Petroleum Institute,(1970)
15) F.Herning & L.Zipperer, “Calculation of the viscosity of technical gas
mixtures from the viscosity of the individual gases”, Gas u.
Wasserfach,vol.79,pp.69-73,(1936)
16) J.A.Jossi, L.I.Stiel & G.Thodos, “The viscosity of pure substances in the
dense gaseous and liquid phases”, AIChE J.,vol.8,no.1,pp.59-63,(1962)
17) W.B.Kay, “Gases and Vapors at high temperature and pressure – density
of hydrocarbons”, Ind.Eng.Chem.,vol.28,no.9,pp.1014-1019,(1936)
18) A.L.Lee, M.H.Gonzalez & B.E.Eakin, “The Viscosity of Natural Gases”,
J.Petrol.Technol.,pp.997-1000,August (1966)
19) J.Lohrenz, B.G.Bray & C.R.Clark, “Calculating Viscosities of Reservoir
Fluids From Their Compositions”, J.Petrol.Technol.,pp.1171-1176, October
(1964)
20) H.A.Lorentz, Ann.Physik, vol.12,pp.127-136,(1881)
21) W.D.McCain, “The properties of petroleum fluids”, 2 nd ed., PennWell books,
(1990)
22) H.Nabizadeh & F.Mayinger, “Viscosity of binary mixtures of hydrogen and
natural gas (hythane) in the gaseous phase”, High Temperatures-High
Pressures,vol.31,p..601-612,(1999)
23) K.S.Pedersen, A.Fredenslund & P.Thomassen, “Properties of oils and
natural gases”, Gulf publishing, (1989)
22
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
24) K.S.Pedersen & P.L.Christensen, “Phase behavior of petroleum reservoir
fluids”, Taylor & Francis, (2007)
25) B.E.Poling, J.M.Prausnitz & J.P.O’Connell, “The Properties of Gases and
Liquids”, 5th ed., McGraw-Hill, (2001)
26) J.M.Prausnitz & R.D.Gunn, “Volumetric properties of nonpolar gaseous
mixtures”, AIChE J.,vol.4,no.4,pp.430-435,(1958)
27) D.Reichenberg, “The viscosities of organic vapours at low pressures”, NPL
DCS Report 11, August (1971)
28) D.Reichenberg, “The viscosities of gas mixtures at moderate pressures”,
NPL Report Chem 29, May (1974)
29) D.Reichenberg, “The viscosities of pure gases at high pressures”, NPL
Report Chem 38, August (1975)
30) D.Reichenberg, “New simplified methods for the estimation of the
viscosities of gas mixtures at moderate pressures”, NPL Report Chem 53,
May (1977)
31) R.C.Reid, J.M.Prausnitz & T.K.Sherwood, “The Properties of Gases and
Liquids”, 3rd ed., McGraw-Hill, (1977)
32) R.C.Reid, J.M.Prausnitz & B.E.Poling, “The Properties of Gases & Liquids”,
4th ed., McGraw-Hill, (1987)
33) P.Schley, M.Jaeschke, C.Kuchenmeister & E.Vogel, “Viscosity
Measurements and Predictions for Natural Gas”,
Int.J.Thermophysics,vol.25,no.6,pp.1623-1651,(2004)
34) K.E.Starling & R.T.Ellington, “Viscosity correlations for nonpolar fluids”,
AIChE J.,vol.10,no.1,pp.11-15,(1964)
35) L.I.Stiel & G.Thodos, “The viscosity of nonpolar gases at normal
pressures”, AIChE J.,vol.7,no.4,pp.611-615,(1961)
36) N.B.Vargaftik, Y.K.Vinogradov & V.S.Yargin, p.436,
“Handbook of Physical Properties of Liquids and Gases”, 3rd ed., (1996)
37) C.R.Wilke, “A viscosity equation for gas mixtures”,
J.Chem.Phys.,vol.18,no.4,pp.517-519,(1950)
38) B.A.Younglove & J.F.Ely, “Thermophysical Properties of Fluids. II.
Methane, ethane, propane, isobutane and normal butane”,
J.Phys.Chem.Ref.Data, vol.16,no..4,pp.577-798,(1987)
23
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 3 : High Pressure Viscosity of Natural Gas
The focus of the methods in this paper have been for natural gas metering in
transmission and distribution pipelines, which are generally below 100 bar.
However, metering sometimes occurs at higher pressure, often offshore at
reservoir conditions. This appendix examines the errors in the LBC (Lohrenz-BrayClark) viscosity method (section 2.2 above) for high pressure natural gas (using
the GERG-2008 equation of state for density).
References :
[C]
[GO]
[G]
[NM]
[AD]
[S]
[L]
[A]
N.L.Carr, “Viscosities of natural gas components and mixtures”, Institute of
Gas Technology Research Bulletin 23, (1953)
Mixtures : [C1], [C2], [C3], [C4] & [C5]
I.F.Golubev, “Viscosity of Gases and Gas Mixtures : A Handbook”, p.214,
(1959) (translation 1970)
M.Gonzalez, B.E.Eakin & A.L.Lee, “Viscosity of Natural Gas”, Monograph on
API Research Project 65, American Petroleum Institute, (1970)
Mixtures : [G1], [G2], [G3], [G4], [G5], [G6], [G7] & [G8]
H.Nabizadeh & F.Mayinger, High Temperatures-High Pressures, vol.31,
pp.601-612, (1999) (table with Tn=263.15 K should be Tn=363.15 K)
M.J.Assael, N.K.Dalaouti & V.Vesovic, Int. J. Thermophys., vol.22, pp.6171, (2001)
P.Schley, M.Jaeschke, C.Küchenmeister & E.Vogel, Int. J. Thermophys.,
vol.25, pp.1623-1652, (2004)
Mixtures : [S1], [S2] & [S3]
L.I.Langelandsvik, S.Solvang, M.Rousselet, I.N.Metaxa & M.J.Assael, Int. J.
Thermophys., vol.28, pp.1120-1130, (2007)
Mixtures : [L1], [L2] & [L3]
M.Atilhan, S.Aparicio, R.Alcalde, G.A.Iglesias-Silva, M.El-Halwagi &
K.R.Hall, J. Chem. Eng. Data, vol.55, pp.2498-2504, (2010)
Mixtures : [A1], [A2] & [A6] (this is pure methane)
([A2] 450 K data looks wrong, and the 12 points were ignored)
M.Atilhan, S.Aparicio, G.A.Iglesias-Silva, M.El-Halwagi & K.R.Hall, J. Chem.
Eng. Data, vol.55, pp.5117-5123, (2010)
Mixtures : [A3], [A4] & [A5]
Mixture compositions :
Mol%
N2
CO2
CH4
C2H6
C3H8
iC4
nC4
nC5
He
Mol%
N2
CO2
CH4
C2H6
C3H8
iC4
[C1]
15.8
0
73.1
6.1
3.4
0.2
0.6
0
0.8
[G1]
0.21
0.23
97.80
0.95
0.42
0
[C2]
0.3
0
95.6
3.6
0.5
0
0
0
0
[G2]
5.20
0.19
92.90
0.94
0.48
0.01
[C3]
0.6
0
73.5
25.7
0.2
0
0
0
0
[G3]
0.55
1.70
91.50
3.10
1.40
0.67
[C4]
0.4
0
99.0
0.5
0
0
0
0
0
[C5]
0
0
99.8
0.1
0.1
0
0
0
0
[GO]
5.0
0
91.5
1.8
0.8
0
0.6
0.3
0
[NM]
1.83
0
94.67
3.50
0
0
0
0
0
[AD]
5.60
0.66
84.84
8.40
0.50
0
0
0
0
[G4]
0.04
2.04
88.22
5.08
2.48
0.87
[G5]
0
3.20
86.3
6.80
2.40
0.43
[G6]
0.67
0.64
80.90
9.90
4.60
0.76
[G7]
4.80
0.90
80.70
8.70
2.90
0
[G8]
1.40
1.40
71.70
14.00
8.30
0.77
24
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
nC4
nC5
nC6
nC7
He
0.23
0.09
0.06
0.03
0
0.18
0.06
0.06
0
0
Mol%
helium
carbon_dioxide
nitrogen
oxygen+argon
hydrogen
methane
ethane
propane
n-butane
isobutane
n-pentane
isopentane
neopentane
hexanes
heptanes
octanes
nonanes
decanes+
benzene
toluene
xylenes
Mol%
Hydrogen
Helium
Water
Argon+oxygen
Nitrogen
Carbon_dioxide
Methane
Ethane
Propane
n-Butane
i-Butane
n-Pentane
i-Pentane
neo-Pentane
Hexanes
Heptanes
Octanes
Nonanes
Decanes+
Benzene
Toluene
Xylene
mix1
0
0.0068
0.0006
0.0514
0.7587
1.7947
90.1584
6.3077
0.8010
0.0643
0.0446
0.0044
0.0054
0.0003
0.0014
0.0005
0.0001
0
0
0.0002
0
0
0.50
0.28
0.26
0.08
0
H gas
0.0137
0.7740
1.5324
0.0419
0.0007
89.5669
6.1464
1.2532
0.1924
0.2857
0.0324
0.0565
0.0032
0.0572
0.0340
0.0038
0.0010
0.0009
0.0021
0.0009
0.0006
mix2
0.0010
0.0084
0
0
0.6601
2.1902
80.0079
9.3063
4.9630
1.2791
0.7188
0.2499
0.2556
0.0055
0.1793
0.1010
0.0197
0.0086
0.0063
0.0173
0.0157
0.0061
0.58
0.41
0.15
0.13
0
0.48
0.22
0.10
0.04
0
L gas
0.0520
1.4523
9.7520
0.0100
0.0005
84.3322
3.4085
0.6023
0.1282
0.1033
0.0350
0.0357
0.0056
0.0388
0.0174
0.0041
0.0021
0.0013
0.0250
0.0031
0.0010
Used
N2
CO2
CH4
C2H6
C3H8
iC4
nC4
iC5
nC5
nC6
nC7
nC8
nC9
nC10
He
H2
O2
mix3
0.0005
0.0168
0
0
1.3916
1.0030
92.2045
4.3373
0.5396
0.0771
0.2562
0.0198
0.0468
0.0033
0.0606
0.0364
0.0038
0.0009
0.0002
0.0007
0.0006
0.0003
25
Used
N2
CO2
CH4
C2H6
C3H8
iC4
nC4
iC5
nC5
nC6
nC7
nC8
nC9
nC10
He
H2
O2
H2O
1.35
0.60
0.39
0.11
0.05
1.70
0.13
0.06
0.03
0.03
[S1]
1.5324
0.7740
89.5669
6.1464
1.2532
0.2857
0.1924
0.0597
0.0324
0.0593
0.0349
0.0044
0.0010
0.0009
0.0137
0.0007
0.0419
[L1]
0.7587
1.7947
90.1584
6.3077
0.8010
0.0446
0.0643
0.0057
0.0044
0.0016
0.0005
0.0001
0
0
0.0068
0
0.0514
0.0006
1.90
0.39
0.09
0.01
0.03
[S2]
9.7520
1.4523
84.3322
3.4085
0.6023
0.1033
0.1282
0.0413
0.0350
0.0638
0.0205
0.0051
0.0021
0.0013
0.0520
0.0005
0.0100
[L2]
0.6601
2.1902
80.0079
9.3063
4.9630
0.7188
1.2791
0.2611
0.2499
0.1966
0.1167
0.0258
0.0086
0.0063
0.0084
0.0010
0
0
[S3]
0
0
100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
[L3]
1.3916
1.0030
92.2045
4.3373
0.5396
0.2562
0.0771
0.0501
0.0198
0.0613
0.0370
0.0041
0.0009
0.0002
0.0168
0.0005
0
0
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Mol%
methane
ethane
propane
isobutane
n-butane
isopentane
n-pentane
n-octane
toluene
methylcyclopentane
nitrogen
carbon_dioxide
mix1
84.990
5.529
2.008
0.401
0.585
0.169
0.147
0.152
0.090
0.102
3.496
2.331
mix2
90.260
5.828
2.106
0.412
0.641
0.214
0.162
0.161
0.110
0.111
0
0
mix3
80.340
5.189
1.878
0.384
0.573
0.188
0.140
0.145
0.092
0.092
6.596
4.380
mix4
84.700
5.584
1.962
0.416
0.553
0.214
0.155
0.150
0.098
0
3.711
2.457
mix5
85.094
5.529
2.009
0.401
0.612
0.171
0.141
0.152
0
0.099
3.496
2.296
Used
N2
CO2
CH4
C2H6
C3H8
iC4
nC4
iC5
nC5
nC6
nC7
nC8
[A1]
3.496
2.331
84.990
5.529
2.008
0.401
0.585
0.169
0.147
0.102
0.090
0.152
[A2]
0
0
90.260
5.828
2.106
0.412
0.641
0.214
0.162
0.111
0.110
0.161
[A3]
6.596
4.380
80.340
5.189
1.878
0.384
0.573
0.188
0.140
0.092
0.092
0.145
Calculated results (for all data) :
[C1]
[C2]
[C3]
[C4]
[C5]
[GO]
[G1]
[G2]
[G3]
[G4]
[G5]
[G6]
[G7]
[G8]
[NM]
[AD]
[S1]
[S2]
[S3]
[L1]
[L2]
[L3]
[A1]
[A2]
[A3]
[A4]
[A5]
[A6]
T range
(K)
299-394
303-398
298-339
294-340
295-367
273-523
311-444
311-444
311-444
311-444
311-444
311-444
344-444
311-444
299-399
241-455
260-320
260-320
260-360
263-303
263-304
263-304
250-450
250-440
250-450
250-450
250-450
250-450
P range
(bar)
1-661
1-584
1-692
1-554
1-277
1-456
14-552
28-207
28-552
1-276
14-276
14-483
14-172
48-552
1-67
2-140
1-201
1-202
1-292
45-251
133-253
50-252
100-650
100-650
100-650
100-650
100-650
100-700
Overall
241-523
1-700
points
Bias%
RMS%
Max%
35
33
53
36
41
112
38
34
27
31
30
26
24
34
59
40
224
224
345
45
34
45
248
236
248
248
246
77
1.6
1.0
1.1
-0.1
2.4
1.5
-3.6
-6.0
-4.3
-2.5
-3.8
-5.5
-4.5
-0.4
-1.1
-0.5
1.1
1.6
0.0
-2.0
-2.6
-1.5
-1.5
-0.8
1.2
-1.8
-1.6
2.4
3.2
3.0
3.3
2.4
4.5
3.7
5.1
6.9
6.4
6.1
4.2
5.9
4.9
3.7
1.3
0.9
2.3
2.5
1.7
2.8
3.5
2.2
2.9
2.8
2.8
3.1
3.0
4.3
7.8
6.7
11.9
5.3
7.1
11.1
12.1
15.7
18.0
14.4
6.1
10.8
7.9
6.4
2.5
2.3
5.8
7.0
5.8
5.4
8.3
4.2
5.5
6.3
7.0
6.2
5.8
7.6
2873
-0.4
3.1
18.0
The figure below shows the distribution of the errors for the 2,873 data points :
26
[A4]
3.711
2.457
84.700
5.584
1.962
0.416
0.553
0.214
0.155
0
0.098
0.150
[A5]
3.496
2.296
85.094
5.529
2.009
0.401
0.612
0.171
0.141
0.099
0
0.152
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Close-up of the below 200 bar results :
(Gonzalez data looks to be incorrect – either in measurement of viscosity etc., or
reporting of the composition)
27
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Calculated results for below 105 bar, and above 105 bar :
[C1]
[C2]
[C3]
[C4]
[C5]
[GO]
[G1]
[G2]
[G3]
[G4]
[G5]
[G6]
[G7]
[G8]
[NM]
[AD]
[S1]
[S2]
[S3]
[L1]
[L2]
[L3]
[A1]
[A2]
[A3]
[A4]
[A5]
[A6]
overall
Pressure below 105 bar
points Bias% RMS% Max%
17
-0.1
1.5
3.1
17
-0.6
1.6
3.6
28
0.4
2.0
4.3
25
-0.9
2.0
3.9
11
-0.6
2.2
4.3
42
0.7
3.7
11.1
17
-6.0
6.6
12.1
22
-5.6
6.2
9.9
8
-8.5
8.8
12.1
25
-4.9
5.1
6.5
18
-4.1
4.4
5.9
12
-4.8
5.1
10.0
21
-4.2
4.6
7.4
8
-4.8
4.9
6.1
59
-1.1
1.3
2.5
34
-0.6
0.9
2.3
147
0.0
0.8
2.0
147
0.7
1.0
2.0
222
-0.7
1.1
2.4
14
-4.1
4.3
5.4
0
0.0
0.0
0.0
15
-3.4
3.4
4.2
17
0.0
1.4
2.7
16
-3.4
3.9
5.9
17
3.0
3.0
3.1
17
0.1
1.4
2.8
15
-0.3
1.3
1.9
11
-2.8
3.3
4.7
1002
-1.0
2.6
12.1
Pressure above 105 bar
points Bias% RMS% Max%
18
3.2
4.3
7.8
16
2.8
4.0
6.7
25
1.8
4.3
11.9
16
1.2
3.0
5.3
25
3.8
5.2
7.1
70
2.0
3.7
8.8
21
-1.8
3.3
6.7
12
-6.8
8.0
15.7
19
-2.5
5.1
18.0
6
7.7
9.1
14.4
12
-3.4
3.9
6.1
14
-6.1
6.5
10.8
3
-6.4
6.5
7.9
26
1.0
3.3
6.4
0
6
0.1
0.6
1.2
77
3.1
3.7
5.8
77
3.3
4.0
7.0
123
1.1
2.5
5.8
31
-1.1
1.8
4.8
34
-2.6
3.5
8.3
30
-0.6
1.1
2.8
231
-1.6
3.0
5.5
220
-0.6
2.7
6.3
231
1.1
2.8
7.0
231
-1.9
3.2
6.2
231
-1.7
3.0
5.8
66
3.3
4.4
7.6
1871
-0.1
3.3
18.0
For the over 105 bar data, 95% of the errors are less than 5.9%. (5% of errors
are less than -4.9%, and 5% of errors are greater than 5.5%; hence, 90% of errors
are actually between -4.9% and 5.5%).
Thus, below 105 bar the uncertainty (given in section 2.2 above) is about 4% (95%
confidence limit); whilst above 105 bar it is about 6% (although there are some
large errors). It is still within the target uncertainty (given in section 1 above) of
17%.
For the simple equation (eq.14) the results are :
Eq.14
All data
Below 105 bar
Above 105 bar
points
2873
1002
1871
Bias%
-3.4
0.6
-5.6
RMS%
7.5
2.9
9.1
Max%
33.7
10.9
33.7
Thus, eq.14 is reasonable for all conditions – although there some large errors.
28
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 4 : Viscosity of Hydrogen and Natural Gas
Currently it is being considered to replace natural gas with hydrogen, or hydrogen enriched
natural gas (up to 20 mol% H2). This Appendix looks at the accuracy of the LBC viscosity
method (using GERG-2008 density) for hydrogen and hydrogen in natural gas.
Experimental data has been reported by H.Nabizadeh & F.Mayinger, “Viscosity of binary
mixtures of hydrogen and natural gas (hythane) in the gaseous phase”, High TemperaturesHigh Pressures, vol.31, pp.601-612, (1999)
This has experimental viscosity data for mixtures of Natural gas + 5, 15, 30 & 75 mol%
Hydrogen, and pure H2; temperature 298 to 400 K, pressure 1 to 70 bar (total 333 points,
including Natural gas – see Appendix 3, [NM])
At 300 K, the reduced temperature (Tr) for hydrogen is about 9; thus eq.7 is not appropriate,
hence eq.8 was developed for hydrogen (and helium) (modified LBC).
Eq.8 was fitted to pure H2 and He viscosity (at 1 bar) as calculated by NIST RefProp (version
10.0, database 23, 2018) from 150 to 1000 K, (H2 RMS%=1.0, He RMS%=1.5).
Calculated results :
Nat.Gas
+5% H2
+15% H2
+30% H2
+75% H2
Pure H2
Original LBC
Bias% RMS% Max%
-1.1
1.3
2.5
-1.8
1.9
3.2
-3.4
3.5
4.6
-5.7
5.7
6.9
-12.8
12.8
14.0
-13.7
13.7
15.7
Modified LBC
Bias% RMS% Max%
-1.1
1.3
2.5
-1.6
1.8
3.0
-2.8
2.9
4.0
-4.3
4.4
5.5
-7.0
7.0
8.0
0.0
0.8
1.5
Thus, showing the improvement in the modified LBC, and also showing the suitability of LBC
for hydrogen containing natural gas. Although for less than 50 mol% hydrogen, there is
actually little effect of hydrogen on the natural gas mixture viscosity.
For hydrogen mixtures the Wilke equation for the (low pressure) mixture viscosity (Appendix
2, section 3.4.1) is better than the LBC eq.10 (Appendix 2, section 3.4.2). This can be written
as follows, with the calculated results :
𝜑ij =
2
1/4 MW1/4
MW
j
i
[ 1/2 + 1/2 ]
η
η
i
j
1⁄2
8
8
[
+
]
MW𝑖 MW𝑗
X MWi
i
ηmix = ∑N
i=1 (∑N
j=1 Xj φij
)
29
Nat.Gas
+5% H2
+15% H2
+30% H2
+75% H2
Pure H2
Modified LBC with Wilke
Bias% RMS% Max%
-1.3
1.5
2.6
-1.1
1.3
2.5
-1.0
1.2
2.2
-0.7
0.9
1.8
1.3
1.4
1.9
0.0
0.8
1.5
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 5 : Python routine for LBC Viscosity
def VisLBC(Tin, Din, Xin):
# Lohrenz-Bray-Clark method for viscosity,
# Tin degC , Din mol/dm3 : Vis mPa.s = cP
# MW g/mol , Tc K , Pc bar , Dc mol/dm3
# 1:CH4 2:N2 3:CO2 4:C2H6 5:C3H8 6:nC4 7:iC4 8:nC5 9:iC5 10:nC6 11:nC7
# 12:nC8 13:nC9 14:nC10 15:H2 16:O2 17:CO 18:H2O 1 9:H2S 20:He 21:Ar 22:neoC5
M=(0.0, 16.04246, 28.0134, 44.0095, 30.06904, 44.09562, 58.1222, 58.1222,
72.14878, 72.14878, 86.17536, 100.20194, 114.22852, 128.2551, 142.28168,
2.01588, 31.9988, 28.0101, 18.01528, 34.08088, 4.002602, 39.948, 72.14878)
Tc=(0.0, 190.564, 126.192, 304.1282,305.322, 369.825, 425.125, 407.817, 469.7,
460.35, 507.82, 540.13, 569.32, 594.55, 617.7, 33.19, 154.595, 132.86,
647.096, 373.1, 5.1953,150.687, 433.75)
Pc=(0.0, 45.992, 33.958, 73.773, 48.718, 42.4661, 37.9053, 36.3729, 33.7098,
33.7823, 30.4293, 27.3107, 24.9781, 22.8198, 21.0137, 13.15, 50.3895,
34.9821, 220.64, 89.9873, 2.2746, 48.5963, 31.94)
Dc=(0.0,10.139342719,11.1839,10.624978698, 6.870854540, 5.000043088, 3.920016792,
3.860142940, 3.215577588, 3.271, 2.705877875, 2.315324434, 2.056404127, 1.81,
1.64,14.94,13.63,10.85,17.873716090,10.19,17.399, 13.407429659, 3.24397)
NC = 0
X = [0.0]*23
sumX = 0.0
for i in range(1,23):
X[i] = Xin[i]; sumX = sumX + X[i]
if (X[i]>0.0): NC = i+1
if (sumX<0.01): sumX = 1.0; X[1] = sumX; NC = 2
for i in range(1,NC): X[i] = X[i]/sumX
T = Tin + 273.15 #degC to K
Mmix = 0.0; Vcmix = 0.0; Tcmix = 0.0; Pcmix = 0.0; Vis1 = 0.0; Vis2 = 0.0
for i in range(1,NC):
Mmix = Mmix + X[i] * M[i]
#eq.3
Vcmix = Vcmix + X[i] / Dc[i]
#eq.4
Tcmix = Tcmix + X[i] * Tc[i]
#eq.4
Pcmix = Pcmix + X[i] * Pc[i]/1.01325
#eq.4
Tr = T / Tc[i]
#eq.5
if (Tc[i]<40.0):
alpha = 0.0001 * (7.08 * Tr + 2.26)**0.72 #eq.8
else:
if (Tr <= 1.5):
alpha = 0.00034 * Tr**0.94
#eq.6
else:
alpha = 0.0001778 * (4.58 * Tr - 1.67)**0.625 #eq.7
SM = math.sqrt(M[i])
Vis = SM * Tc[i]**(-1.0/6.0) * (Pc[i]/1.01325)**(2.0/3.0) * alpha #eq.9
Vis1 = Vis1 + X[i] * SM
#eq.10
Vis2 = Vis2 + X[i] * SM * Vis
#eq.10
Vismix = Vis2 / Vis1
#eq.10
xi = math.sqrt(Mmix) * Tcmix**(-1.0/6.0) * Pcmix**(2.0/3.0) #eq.11
Dr = Vcmix * Din
#eq.12
delta = 0.1023 +Dr*(0.023364 +Dr*(0.058533 +Dr*(-0.040758 +Dr*0.0093324))) #eq.12
Vis = Vismix + xi *( delta**4 - 0.0001)
#eq.13
if (Vis < 0.0): Vis= 0.0
return (Vis)
30
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 6 : Equations for Orifice Plate Flow Metering
6.1 Introduction
The International, European and British standard ISO 5167 has undergone revisions. The
standard has been issued in 1991(ref.1), 1997 (ref.2) and 2003 (refs.3 & 4). This appendix
outlines the methods for orifice plate flow measurement.
6.2 Symbols
C
d
D
P
T
q
Re

p







discharge coefficient
orifice diameter
pipe (inside) diameter
pressure (absolute) (upstream)
temperature
(upstream)
mass flowrate
Reynolds number, (4.q)/(..D)
diameter ratio, d/D
differential pressure
permanent pressure loss
expansibility factor
fluid dynamic viscosity (at T,P)
isentropic exponent
Joule-Thomson coefficient
= 3.141592654…
fluid density (at T,P)
[-]
[m]
[m]
[Pa]
[K]
[kg/s]
[-]
[-]
[Pa]
[Pa]
[-]
[Pa.s]
[-]
[K/Pa]
[kg/m3]
6.3 Basic Equations
The basic mass flowrate equation is :-
q
π
ε d 2 2.p.ρ
1 β4 4
C
(1)
C is function of  and Re, and of the type of orifice pressure tappings,
 is a function of , P, p and .
The standards differ in the functions for C and .
Note that although q is given by equation (1) iteration is required since C is a function of Re
and Re is a function of q. Similarly, given q, equation (1) does not directly give p since  is a
function of p.
6.3.1
Discharge coefficient
ISO-5167:1991 uses the Stolz equation :-
C  0.5959 0.0312β
2.1
Corner tappings
D & D/2 tappings
Flange tappings
:
:
:
 106 

 0.1840β  0.0029β 
 Re 
8
2.5
L1 = 0
L1 = 0.4333
L1 = 0.0254/D
0.75
 0.0900L1
β4
 0.0337L2β 3 (2)
4
1 β
L2 = 0
L2 = 0.47
L2 = 0.0254/D
ISO-5167:1997 and 2003 use the Reader-Harris/Gallagher equation :-
 106 β 

C  0.5961 0.0261β  0.216β  0.000521
 Re 
2

8
 0.043 0.080e10L1  0.123e7L1
0.7
 106 

 0.0188 0.0063A β 
 Re 
β4
1  0.11A 4  0.031 M 2  0.8M21.1 β1.3
1 β


31
0.3
3.5

(3)
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
when D < 0.07112 m, the following term is added :
 19000β 
A

 Re 
Where
Corner tappings
D & D/2 tappings
Flange tappings
:
:
:
0.8
and
M2 
L1 = 0
L1 = 1
L1 = 0.0254/D
D 

 0.0110.75  β  2.8 

0.0254

2L 2
1 β
L2 = 0
L2 = 0.47
L2 = 0.0254/D
6.3.2 Expansibility factor
ISO-5167:1991 and 1997 use the following equation : = 1 – (0.41 + 0.35 4) p/(P)
ISO-5167:2003 uses the following equation : = 1 – (0.351 + 0.2564 + 0.938 )(1 - (1 - p/P)1/)
(4)
(5)
6.3.3 Permanent pressure loss
ISO-5167:1991 and 1997 use the following equation :-
Δω 
1  β 4  Cβ 2
1  β 4  Cβ 2
Δp
(6)
ISO-5167:2003 uses the following equation :-
Δω 
6.4
 
1  C   Cβ
1  β 4 1  C 2  Cβ 2
1 β4
2
2
Δp
(7)
Upstream temperature
Metering installations typically measure the upstream pressure, the differential pressure
across the orifice, whilst only the downstream temperature is measured – in order not to
disturb the upstream flow. The downstream pressure is calculated from the permanent
pressure loss equation.
ISO-5167:2003 assumes that the overall process is isenthalpic (ref.3 section 5.4.4.1) (i.e. T
is approximately given by Tdownstream + .). Whilst ISO-5167:1991 and 1997 do not specify
what the overall process is – merely that the upstream T can be calculated (refs.1 & 2 section
5.4.2); the general assumption is that it is isentropic (i.e. also assuming constant Z
(=P.V/R.T) (e.g. ideal gas Z=1), T is approximately given by Tdownstream.(1 - /P)1/-1).
6.5 References
1) ISO 5167-1:1991, “Measurement of fluid flow in closed conduits – Part 1. Pressure
differential devices – Section 1.1: Specification for square-edged orifice plates, nozzles
and Venturi tubes inserted in circular cross-section conduits running full”, (BS
1042:Section 1.1:1992)
2) BS EN ISO 5167-1:1997, “Measurement of fluid flow by means of pressure differential
devices – Part 1: Orifice plates, nozzles and Venturi tubes inserted in circular crosssection conduits running full”, (BS 1042-1.1:1992 renumbered, incorporating
Amendment No.1 (renumbering the BS as BS EN ISO 5167-1:1997), and Amendment
No.1 to BS EN ISO 5167-1:1997)
3) BS EN ISO 5167-1:2003, “Measurement of fluid flow by means of pressure differential
devices inserted in circular cross-section conduits running full – Part 1: General
principles and requirements”
4) BS EN ISO 5167-2:2003, “Measurement of fluid flow by means of pressure differential
devices inserted in circular cross-section conduits running full – Part 2: Orifice plates”
32
American Gas Association Operations Conference, Chicago, May 21, 2020 (cancelled)
Appendix 7 : Alternate Joule-Thomson coefficient equation
ISO/TR 9464 (2008), “Guidelines for the Use of ISO 5167:2003”, gives an alternative
equation for Joule-Thomson coefficient :
 = ( 0.35 - 0.00142t ) + ( 0.231 - 0.00294t + 0.0000136t2) 
(0.998+0.00041P-0.0001115P2+0.0000003P3)
Range of applicability : 0 to 100 oC, 1 to 200 bar, natural gas with CH4 > 80 mol%.
The expanded uncertainty, U (k=2, 95% interval) is given as 0.066(1-t/200) (for P<70 bar),
0.066(1-t/200)(1-(290-t)/4(1/70-1/P)) (for P>70 bar), (K/bar).
For 40,001 simulated natural gas mixtures generated according to the procedure outlined in
section 3 (9,781 rejected as not satisfying all constraints), the results compared to the
calculations using GERG-2008 are :
Joule-Thomson coefficient uncertainty (RMS)
P/bar
100
80
60
40
20
t/oC
0.042
0.025
0.040
0.046
0.044
-20
0.023
0.028
0.036
0.041
0.041
-10
Eq.16
0.019
0.028
0.035
0.039
0.040
0
RMS (K/bar)
0.019 0.020
0.028 0.027
0.034 0.032
0.037 0.034
0.038 0.034
10
20
0.019
0.025
0.029
0.030
0.028
30
0.013
0.022
0.024
0.025
0.024
40
0.027
0.024
0.046
0.049
0.045
-20
ISO/TR 9464 eq. RMS (K/bar)
0.012 0.015 0.018 0.019 0.018
0.028 0.028 0.026 0.024 0.022
0.039 0.034 0.030 0.027 0.025
0.041 0.036 0.032 0.029 0.026
0.039 0.036 0.032 0.029 0.026
-10
0
10
20
30
0.017
0.020
0.023
0.024
0.024
40
The two equations are very similar, and both certainly satisfy the requirement given in section
1; eq.16 being somewhat simpler. The source of the RMS is largely from the composition
variation, rather than from the simple form of the equation.
The table below gives the average values (for the same mixtures) and the standard deviation
(SD) of the values. The SD arises from the variation in compositions. Thus, SD is the limit (for
the RMS) to which any equation that ignores the composition effect can possibly achieve.
Joule-Thomson coefficient values
P/bar
100
80
60
40
20
t/oC
0.414
0.532
0.620
0.667
0.685
-20
0.419
0.507
0.573
0.612
0.630
-10
Average value (K/bar)
0.409 0.393 0.373 0.352
0.478 0.447 0.418 0.391
0.530 0.490 0.455 0.422
0.563 0.520 0.481 0.446
0.581 0.538 0.498 0.462
0
10
20
30
0.332
0.365
0.392
0.414
0.430
40
0.011
0.024
0.047
0.055
0.054
-20
Standard deviation (K/bar)
0.010 0.016 0.020 0.021 0.021
0.029 0.030 0.030 0.028 0.026
0.044 0.040 0.036 0.033 0.030
0.049 0.044 0.039 0.035 0.032
0.048 0.043 0.039 0.035 0.032
-10
0
10
20
30
0.020
0.024
0.027
0.029
0.029
40
The above figures are based on calculations using GERG-2008. Section 3 gives the estimated
uncertainty in Joule-Thomson coefficient of 1 % (i.e. about 0.005 K/bar). Compared to the
composition variation this is small and has no significant effect on the RMS values in the tables
(which can be interpreted as the (standard, k=1) uncertainty of eq.16).
33
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