This is Part 8 of a series.
In Part 7, I predicted the properties of mixtures of methane with helium and found that predictions worsened when using the 'classical' critical constants of Messerly et al. [1] in place of the true critical constants for helium. In this part, I'll attempt an independent confirmation of those results.
My desire for independent confirmation of the results stems mainly from two concerns:
The first step in getting TREND to recognise the modified critical
parameters for helium is to write a new EOS block in
the helium.fld file. This is not necessarily straightforward
as the formatting of the .fld files can at best be described
as semi-documented. However, having worked with the files for a few years,
I think the model developed in
Part 6 should take the following form:
#EOS !equation of state specification FEK Kunz and Wagner (2004) with modified Tc, rhoc (Rowland) ?LITERATURE REFERENCE \ ?Kunz, O., Klimeck, R., Wagner, W., Jaeschke, M. ? "The GERG-2004 Wide-Range Equation of State for Natural Gases ? and Other Mixtures," GERG Technical Monograph 15, ? Fortschritt-Berichte VDI, VDI-Verlag, Düsseldorf, 2007. ? ? Tc and rhoc from Messerly et al., J. Chem. Eng. Data, 2020, 65, 1028 ?\ !end of info section 2.1768 !lower temperature limit [K] 1500.0 !upper temperature limit [K] 100000.0 !upper pressure limit [kPa] 88.73 !maximum density [mol/L] PHK !pointer to Cp0 model 4.002602 !molecular weight [g/mol] 2.1768 !triple point temperature [K] 4.8565 !pressure at triple point [kPa] 36.537 !density at triple point [mol/L] 4.22 !normal boiling point temperature [K] 0.0 !acentric factor 13.0 930.0 28.40 !Tc [K], pc [kPa], rhoc [mol/L] 13.0 28.40 !reducing parameters [K, mol/L] 8.314472 !gas constant [J/mol-K] 12 5 0 0 0 0 !# terms, # coeff/term for: "normal" terms, critical, spare -0.743976252538267 0.0 1. 0 0. 1.8217197841740673 0.125 1. 0 0. -1.2666075253340454 0.75 1. 0 0. 0.05806414920264701 1.0 4. 0 0. -0.2828551289729836 0.75 1. 1 1.6322777171101788 -0.008166594510321824 2.625 3. 1 1.6322777171101788 0.16765943748299286 0.125 5. 1 1.6322777171101788 -0.21159689697444667 1.25 5. 1 1.6322777171101788 0.03601650584651393 2.0 5. 1 1.6322777171101788 -0.03545220670850833 1.0 2. 2 2.664330545774417 -2.859365459511357d-4 4.5 1. 3 4.348927380883582 -6.022212928636659d-4 5.0 2. 3 4.348927380883582To test whether the above block is correct, I'll compare the reduced residual Helmholtz energy (and its derivatives with respect to reduced density and inverse reduced temperature) for pure helium using the original GERG-2008 EOS [4,5] and the new model. The following values were calculated at ρ = 1000 mol·m−3 and T = 300 K (note, d*ard stands for δαδr etc.):
Property GERG-2008 EOS this work
ar 1.20005500359150E-02 1.20005500359150E-02
d*ard 1.20500282599112E-02 1.20500282599112E-02
t*art 1.56736012943912E-03 1.56736012943912E-03
dt*ardt 1.60398523785441E-03 1.60398523785441E-03
d2*ardd 9.61312094957221E-05 9.61312094957221E-05
t2*artt -3.77729419317927E-03 -3.77729419317927E-03
td2*artdd 7.11032421838233E-05 9.54864223031825E-05
t2d*arttd -3.78692216821771E-03 -3.78692216821770E-03
d3*arddd -8.12325937389497E-06 -8.12325937389496E-06
t3*arttt 7.68347391216978E-03 7.68347391216977E-03
The results from the original GERG-2008 EOS are virtually identical
to the results from the model developed in Part 6 using the
Messerly et al. [1] critical parameters. This is encouraging
and means I can move on to mixture comparisons.
In Part 7 I calculated both
The set up in my code looks like:
methane = AdvancedThermo.get_fluid_component("methane")
pc = 0.93e6
Tc = 13.0
ρc = 28.4e3
helium = AdvancedThermo.get_fluid_component("helium")
helium = AdvancedThermo.new_critical!(helium,pc,Tc,ρc)
mixture = AdvancedThermo.create_mixture([methane,helium])
mixture.γv[1,2] = mixture.γv[2,1] = 1.0
mixture.γT[1,2] = mixture.γT[2,1] = 1.0
ρ = AdvancedThermo.ρ(mixture,1.0e6,300.0,[0.5,0.5])
T [K] 300.0 p [Pa] 1.e6 x(methane) 0.5 x(helium) 0.5 ρ [mol/m3] (this work) 398.7822170165558 ρ [mol/m3] (TREND 4.0) 398.7822170026990so it is evident that the results are the same to roughly 10 significant figures, which is more than close enough to convince me that the results for mixtures in Part 7 were correct.
Re-examining the assumptions I made in the mixture model when comparing to experimental data, there is one choice I made which stands out as potentially suspect. In the model with modified critical parameters, I have been setting γv and γT equal to unity; this was also the choice made by Kunz et al. [4,5] for binary systems with helium where no or few data were available. However, for some other binary systems (e.g. hydrocarbon binaries), Kunz and co-workers used a different predictive scheme based on arithmetic means (see Eq. (39) of Kunz and Wagner [5]). The code implementing those rules is as follows:
function arithmetic_γ(fluid1::FluidComponent, fluid2::FluidComponent)
v1 = inv(fluid1.ρc)
v2 = inv(fluid2.ρc)
T1 = fluid1.Tc
T2 = fluid2.Tc
# Kunz & Wagner (2012) Eq. (39)
γv = 4.0*(v1+v2)/(v1^(1/3)+v2^(1/3))^3
γT = 0.5*(T1+T2)/sqrt(T1*T2)
return (γv,γT)
end
Using the above function to calculate γv and γT yields
julia> AdvancedThermo.arithmetic_γ(methane, helium) (1.0866943852641635, 2.044932014885648)The code to test the model against the density data from Hernández-Gómez et al. [6] is essentially the same as that in Part 7 so just the statistics are posted here:
Results for γv = γT = 1
-----------------------
Statistic 0.95 CH4/0.05 He 0.9 CH4/0.1 He 0.5 CH4/0.5 He
AAD 4.297307697812139 6.526167267213767 3.014670748518208
Bias 4.297307697812139 6.526167267213767 3.014670748518208
RMS 4.845306516449933 7.366166894559518 3.594373906836211
Max 7.544565086586879 11.297293518608473 7.405915926226708
Results for γv = 1.086694, γT = 2.044932
----------------------------------------
Statistic 0.95 CH4/0.05 He 0.9 CH4/0.1 He 0.5 CH4/0.5 He
AAD 1.731476222987046 2.642525341930066 1.2820511747756336
Bias 1.731476222987046 2.642525341930066 1.2810221778628397
RMS 1.9784506861516693 3.023755441124555 1.5600218615654666
Max 3.056653371342909 4.710285585769213 3.411976327692629
All of the statistics have improved compared to the γ = 1 case, and
the predictions for the 50:50 mixture are actually better than those from the
GERG-2008 EOS [5]. So there's some reason to be optimistic. I think
next time it
will be interesting to re-visit the phase equilibria predictions with this new
model. The code so far can be obtained by
downloading the zip file.
[1] Messerly et al., J. Chem. Eng. Data, 2020, 65, 1028 (link to publisher)
[2] Rowland et al., J. Chem. Eng. Data, 2017, 62, 2799 (link to publisher)
[3] Span et al., TREND. Thermodynamic Reference and Engineering Data 4.0. Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, 2019
[4] Kunz et al., GERG Technical Monograph 15: The GERG-2004 Wide-Range Equation of State for Natural Gases and Other Mixtures, Groupe Européen de Recherches Gazières, 2007
[5] Kunz and Wagner, J. Chem. Eng. Data, 2012, 57, 3032 (link to publisher)
[6] Hernández-Gómez et al., J. Chem. Thermodyn., 2016, 101, 168 (link to publisher)
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