Moment free energy analysis ofhydrocarbons phase equilibria
A. Speranza
I2T3, Dipartimento di Matematica “U.Dini”
Universita degli studi di Firenze
F. Di Patti
I2T3, Dipartimento di Matematica “U.Dini”
Universita degli studi di Firenze
A. Terenzi
Snamprogetti S.p.A., Fano
VIII Congresso Simai, Baia Samuele May.’06
Moment free energy analysis of hydrocarbons phase equilibria – p. 1/28
MultiPhase
MultiPhase project
Ente strumentale per il trasferimento tra Università e impresa
dell’Università degli Studi di Firenze
Moment free energy analysis of hydrocarbons phase equilibria – p. 2/28
OverviewHydrocarbons equations of states: SRK and PR
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results
Applications and developments
Moment free energy analysis of hydrocarbons phase equilibria – p. 3/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results
Applications and developments
Moment free energy analysis of hydrocarbons phase equilibria – p. 4/28
The SRK equation of stateSoave-Redlick-Kwong equation of states
Cubic EOS, i.e.:
P =NκBT
V −B− N2A(T )
g(V,B)
whereB = bκB
Tc
Pc
A(T ) = aκ2
Bα2(T )T 2
c
Pc
g(V,B) = V (V + B)
α(T ) = 1 + σ(1−
√TTc
)
σ = C1 + C2ω − C3ω2
ω = acentric factor, a, b, C1, C2, C3 empirical const.V ∼ excluded vol., A(T ) ∼ Van Der Waals attraction
Moment free energy analysis of hydrocarbons phase equilibria – p. 5/28
Polydisperse SRK EOSAssume L species, with Nk particles
Now get:
βP =
∑k Nk
V − B−
β∑
j,k A(j, k)NjNk
V (V + B)
where
β = 1/κBT B =∑
k
B(k)Nk A(j, k) =√
A(j, T )A(k, T )
with quadratic mixing rules
Moment free energy analysis of hydrocarbons phase equilibria – p. 6/28
The PR equation of statesPeng-Robinson equation of states, similar to SRK:
B = b′κBTc
Pc
A(T ) = a′κ2
Bα2(T )T 2
c
Pc
g(V,B) = V (V + B)
α(T ) = 1 + σ(1−
√TTc
)+B(V − B)
σ = C ′
1+ C ′
2ω − C ′
3ω2
with a′, b′, C ′
1, C ′
2, C ′
3different empirical constants
Now get, for a polydisperse system
βP =
∑k Nk
V − B−
β∑
j,k A(j, k)NjNk
V (V + B) + B(V −B)
as before, B =∑
k B(k)Nk, A(j, k) =√
A(j, T )A(k, T )
Moment free energy analysis of hydrocarbons phase equilibria – p. 7/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results
Applications and developments
Moment free energy analysis of hydrocarbons phase equilibria – p. 8/28
Phase equilibria 1/2From the EOS P = P (N , V, T ) of a polydisperse systemwith N = (N1, N2, . . . , NL),→ the Helmoltz free energyF (N , V, T ) from
P = −∂F
∂V
Legendre transform F → get Gibbs free energyG(N , P, T ) = F + PV
Phase equilibria: equality of chemical potentials
µ(k) =∂G
∂Nk
in all the phases i.e.
µa(k) = µb(k) for k = 1 . . . L a, b = 1 . . . P
Moment free energy analysis of hydrocarbons phase equilibria – p. 9/28
Phase equlibria 2/2Can always divide free energy into ideal and excesspart F (N , V, T ) = Fid(N , V, T ) + F (N , V, T ) whereβFid(N , V, T ) =
∑k Nk(ln Nk − ln V Λ−3 − 1)
Define density distribution ρ(k) = Nk/V and intensivefree energy density f(ρ, T ) = βF (N , V, T )/V , get
f(ρ, T ) =∑
k
ρ(k) (ln ρ(k)− 1) + f(ρ, T )
Similarly, if n(k) = V ρ(k), g(n, P, T ) = βG(N , P, T )/N :
g(n, P, T ) =∑
k
n(k) ln n(k) + ln βP + g(n, P, T )
Phase eq.: eq. of chemical potentials µ(k) = ∂g/∂n(k)
Moment free energy analysis of hydrocarbons phase equilibria – p. 10/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results
Applications and developments
Moment free energy analysis of hydrocarbons phase equilibria – p. 11/28
Truncatable systems
System of L species is truncatable if f is functionf(ρ1, . . . , ρM , T ) of M < L moments of densitydistribution ρi =
∑k wi(k)ρ(k)
Gibbs free energy g inherits moments structure from f ,but normalized moments, g = g(m1, . . . ,mK , T ), withmi =
∑k wi(k)n(k) = ρi/ρ0
However, fid and gid, still functions of ρ(k) and n(k)
Phase equilibria might be very hard to solvenumerically, as for P phases and L spcies, haveL× (P − 1) strongly coupled equations given by thechemical potential equalities
IDEA: express fid and gid as functions of the M momentsonly, rather than the whole distribution ρ(k) or n(k)(?)
Moment free energy analysis of hydrocarbons phase equilibria – p. 12/28
The moment method 1/2For truncatable systems, the moment method allows toexpress, the ideal part of free energy as a function ofthe M moments only.
Principle: Minimize fid with respect to ρ(k), imposing thedefinition of the M moments ρi in f , as constraints
If λi are the lagrange multipliers, the minimum is
fmom(ρ) =∑
i
λiρi − ρ0 + f(ρ)
The minimum is reached for
ρ(k) = r(k) exp
(∑
i
λiwi(k)
)
Moment free energy analysis of hydrocarbons phase equilibria – p. 13/28
The moment method 2/2Function r(k) is obtained by imposing the “lever rule”,
ρ0(k) =∑
a
vaρa(k)
where ρ0(k) is the “parent” distribution (i.e., the wholesystem “before” phase split), and va = V a/V
Same for gmom, with ρi, ρ(k), va, replaced by themi = ρi/ρ0, n(k) = ρ(k)/ρ0 and φa = Na/N
With fmom, the chemical potentials becomeµ(k) = ∂fmom/∂ρ(k) =
∑i (∂fmom/∂ρi) (∂ρi/∂ρ(k))
Thus µ(k) =∑
i
µiwi(k) with µi = ∂fmom/∂ρi
Can prove that ρa(k) solves µa(k) = µb(k)⇔ µai = µb
i
Moment free energy analysis of hydrocarbons phase equilibria – p. 14/28
Main properties of fmom 1/2Get right value for P , from Gibbs-Duhem relation:
P =∑
k
µ(k)ρ(k)− f =∑
i,k
µiwi(k)ρ(k)− f =∑
i
µiρi − f
Two phases are in equilibrium⇔ in equilibrium for fmom
Thus, if retain lever rule, get exact solution
However, lever rule involves the whole distribution ρ(k),i.e., L equations coupled with others; still hard to solve
Idea: Retain lever rule, only for the moments, i.e., impose
ρ0i =
∑
a
vaρai for i = 1, . . . ,M
Replace r(k) with ρ0(k)→ Get exact cloud point andexact spinodal, as ρ0(k) is always a solution.
Not exact inside coexistence region where ρa(k) 6= ρ0(k)
Moment free energy analysis of hydrocarbons phase equilibria – p. 15/28
Main properties of fmom 2/2The approximation made can be controlled by retainingextra moments so to “enlarge the family”
ρ(k) = ρ0(k) exp
(∑
i
λiwi(k)
)× exp
∑
j
λextj wext
j (k)
in order to include (possibly) the exact solution
Use “adaptive method”, to retain only 2 extra moments
Start with two independent functions
Adaptive extra weight function is obtained iteratively aswext
1 (k) = λext1 wext
1 (k) + λext2 wext
2 (k)
wext2 (k) = ln
ρ0(k)∑a vaρa(k)
Throw away old functions and keep new at every step
Moment free energy analysis of hydrocarbons phase equilibria – p. 16/28
OverviewHydrocarbons equations of states: SRK and PR
Polydisperse version of SRK and PR EOS
Phase equilibria of polydisperse systems
Truncatable systems
Moment free energy method
Main properties of the moment free energy
Numerical results for SRK and PR
Applications and developments
Moment free energy analysis of hydrocarbons phase equilibria – p. 17/28
SRK and PR EOS are truncatableFor SRK get, after a bit of rearranging
f =N
Vln
1
1− B/V− D
B/Vln (1 + B/V )
where B/V =∑
k B(k)ρ(k) andD =
∑j,k
√A(j)A(k)ρ(j)ρ(k)
Thusf = −ρ0 (1− ρ1)−
ρ22
ρ1
ln (1 + ρ1)
g = lnρ0
βP− 1 +
βP
ρ0
− ln (1− ρ0m1)−m2
2
m1
ln (1 + ρ0m1)
ρ0 overall density, ρ1 average “excluded volume”, ρ2
average “Vand Der Waals attraction”
Moment free energy analysis of hydrocarbons phase equilibria – p. 18/28
SRK and PR EOS are truncatableFor PR get, after a bit of rearranging
f =N
Vln
1
1− B/V−√
2
4
D
B/Vln
[1 + (1−
√2B/V )
1 + (1 +√
2B/V )
]
where B/V =∑
k B(k)ρ(k) andD =
∑j,k
√A(j)A(k)ρ(j)ρ(k)
Thusf = −ρ0 (1− ρ1)−
√2
4
ρ22
ρ1
ln
[1 + (1−
√2ρ1)
1 + (1 +√
2ρ1)
]
g = lnρ0
βP− 1 +
βP
ρ0
− ln (1− ρ0m1) +
−√
2
4
m22
m1
ln
[1 + (1−
√2ρ0m1)
1 + (1 +√
2ρ0m1)
]
Moment free energy analysis of hydrocarbons phase equilibria – p. 18/28
Numerical resultsResults compared with PVTsim (Snamprogetti s.p.a.)
Oman gas: 3.002% mole N2, 1.001% mole CO2,84.045% mole C1, 7.154% mole C2, 2.862% mole C3,0.66% mole i-C4, 0.726% mole n-C4, 0.22% mole i-C5,0.2% mole n-C5, 0.13% mole n-C6
Moment free energy analysis of hydrocarbons phase equilibria – p. 19/28
Oman gas phase behaviour
C1
Com
posi
tion
in m
ole
%
0
10
20
30
40
50
60
70
80
90
100
180 190 200 210 220 230 240 250 260 270 280T (k)
GasLiq
Exp gasExp liq
1e−05
1e−04
0.001
0.01
0.1
1
10
180 190 200 210 220 230 240 250 260 270 280T (k)
GasLiq
Exp gasExp Gas
C6
Com
posi
tion
in m
ole
%
Moment free energy analysis of hydrocarbons phase equilibria – p. 20/28
Oman gas phse behaviour
Moment free energy analysis of hydrocarbons phase equilibria – p. 21/28
Kashagan gas phase behaviourKashagan gas: 1.134% mole N2, 5.046% mole CO2, 16.987% H2S, 60.513% mole
C1, 8.802% mole C2, 4.285% mole C3, 0.650% mole i-C4, 1.299% mole n-C4, 0.360%
mole i-C5, 0.358% mole n-C5, 0.289% mole n-C6, 0.006% mole benzene, 0.146 n-C7,
0.008% mole toluene, 0.071% mole n-C8, 0.001% mole et-benzene, 0.006% mole
p-xylene, 0.019% mole n-C9, 0.010% mole n-C10, 0.004% mole n-C11, 0.002% mole
n-C12, 0.001% mole n-C13, 0.0015% mole n-C14, 0.0015% mole n-C15
0
20
40
60
80
100
120
100 150 200 250 300 350T (k)
envelopeexp
mphase
P (
bar)
Moment free energy analysis of hydrocarbons phase equilibria – p. 22/28
Kashagans gas and PR EOS
5
10
15
20
25
30
35
40
45
200 220 240 260 280 300 320 340
H2S
Com
posi
tion
in m
ole
%
T (k)
PR gasPR liq
Exp gasExp gas
1e−07
1e−06
1e−05
1e−04
0.001
0.01
0.1
1
200 220 240 260 280 300 320 340T (k)
PR gasPR liqexp liq
Tol
uene
Com
posi
tion
in m
ole
%
Moment free energy analysis of hydrocarbons phase equilibria – p. 23/28
Kashagan gas
0.01
0.1
1
10
200 220 240 260 280 300 320 340
Mol
ar v
olum
e (m
3/km
ole)
T (k)
PR gasPR liq
exp gasexp liq
Gas
Liquid
0
0.2
0.4
0.6
0.8
1
1.2
200 220 240 260 280 300 320 340
Mol
e fr
actio
n
T (k)
Moment free energy analysis of hydrocarbons phase equilibria – p. 24/28
Summing upThe moment method turns out to be an excellentapproximation
Limited number of variables (M + 2) allows to solvephase equilibria of polydisperse systems with manyL > M components
Approximation is good even with heavy components aswell as well as light ones
Monte Carlo method allows minimization of the freeenergy in the moments-space, thus allowing stabilityanalysis of the phase equilibrium solution found
Approximation made is efficiently reduced by “adaptiveextra weight functions”
Surprising the most: the moment method works!
Moment free energy analysis of hydrocarbons phase equilibria – p. 25/28
Applications and developementsAllow continuous dependence of species on a“polydispersity parameter” σ (molar weight, molecularradius, . . . )
Analyse phase behaviour for different parentdistributions (Schultz, Log-normal, bimodal etc.)
What happens, varying the width of the distribution?
Can SRK and PR give multiple gas or liquid phases?
What about experiments?
What about other equations of states?
Can we apply the moment method to other phasetransitions like liquid/solid ones?E.g., can we use it in, say, metallurgy?
Moment free energy analysis of hydrocarbons phase equilibria – p. 26/28
To be continued . . .
Moment free energy analysis of hydrocarbons phase equilibria – p. 27/28
Proof µa(k) = µb(k)⇒ µai (k) = µb
i(k)
The free energy of polydisperse system is
f [ρ(k)] =∑
k
ρ(k) (ln ρ(k)− 1) + f [ρ(k)]
dependence on T is omitted
Thus for a truncatable system, the chemical potentials
µ(k) =∂f
∂ρ(k)= ρ(k) + µ(k) = ρ(k) +
∑
i
wi(k)µi(k)
Thus, equality of chem. potential µa(k) = µb(k) = r(k):
ρa(k) = r(k) exp
[−∑
i
µiwi(k)
]
which belongs to “the family” r(k) exp [∑
i λiwi(k)], withλa
i = µai + θ(k), solution of µa
i = µbi for fmom
← BackMoment free energy analysis of hydrocarbons phase equilibria – p. 28/28