Moment free energy analysis of hydrocarbons phase...

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


OverviewHydrocarbons equations of states: SRK and PR



Polydisperse version of SRK and PR EOS




Phase equilibria of polydisperse systems





Truncatable systems





Truncatable systems

Moment free energy method





Truncatable systems


Main properties of the moment free energy





Truncatable systems



Numerical results





Truncatable systems



Numerical results

Applications and developments





Truncatable systems



Numerical results



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


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


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 )





Truncatable systems



Numerical results



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


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)





Truncatable systems



Numerical results



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)(?)


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)

)


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


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)


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





Truncatable systems



Numerical results for SRK and PR



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”


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)

]


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


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

%


Oman gas phse behaviour


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)


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

%


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)


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!


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?


To be continued . . .


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

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