NUCLETION OF GAS HYDRATES

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1 NUCLETION OF GAS HYDRATES Dimo Kashchiev Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia Detailed presentation of the results in: D.Kashchiev, A.Firoozabadi, J. Crystal Growth 241 (2002) 220 J. Crystal Growth 243 (2002) 476

2 1. What are gas hydrates 2 G + n w H 2 O G. n w H 2 O Building unit: 1 gas molecule and n w water molecules v h = v cell /n g Gas Hydrate structure n w n g v h (nm 3 ) Ar sii 17/ Kr sii 17/ N 2 sii 17/ O 2 sii 17/ CH 4 si 23/ Xe si 23/ H 2 S si 23/ CO 2 si 23/ C 2 H 6 si 23/ c-c 3 H 4 si 23/ C 3 H 8 sii i-c 4 H 10 sii Gas molecule encaged by water molecules in gas hydrate Application: - natural gas production - natural gas transportation - CO 2 sequestration - energy source n w stoichiometric hydration number n g stoichiometric number of gas molecules per unit cell of hydrate crystal lattice v h volume of hydrate building unit v cell volume of unit cell of hydrate crystal lattice

3 2. Supersaturation Dm 3 Definition: µ µ old phase - µ new phase GAS SOLUTION µ gg µ hs = µ gs + n w µ w Old phase = dissolved gas hydrate µ old phase = µ hs = µ gs + n w µ w New phase = gas-hydrate crystal µ new phase = µ h Hence: µ = µ gs + n w µ w - µ h At solution/gas equilibrium µ gs = µ gg. Then: µ = µ gg + n w µ w - µ h HYDRATE µ h Three-phase system of gas, aqueous solution of the gas and gas hydrate µ hs chem. potential of hydrate building unit in solution µ gs chem. potential of gas molecule in solution µ w chem. potential of water molecule in solution µ h chem. potential of hydrate building unit in crystal µ gg chem. potential of gas molecule in gas

4 4 hydrate formation µ>0 PRESSURE P e (T) S T=constant A P=constant B no hydrate formation µ<0 Isothermal regime TEMPERATURE T e (P) Phase diagram of gas-solution-hydrate system µ = kt ln[ϕ(p,t)p/ϕ(p e,t)p e ] + v e (P P e ) with v e = n w v w (P e ) v h (P e ) Approximation for P»P e or j»1: µ = kt ln[p/p e ] + v e (P P e ) P pressure P e equilibrium pressure T temperature T e equilibrium temperature ϕ gas fugacity coefficient k Boltzmann constant

5 ϕ = 1 Dm / kt P e P (MPa) Pressure dependence of the supersaturation for crystallization of methane hydrate at T=273.2 K

6 6 Dm / kt P e ϕ = 1 gas-to-liquid transition P (MPa) Pressure dependence of the supersaturation for crystallization of ethane hydrate at T=273.2 K

7 7 Isobaric regime µ = s e T ( c p,e /2T e ) T 2 with s e = s gg (T e ) + n w s w (T e ) s h (T e ) and c p,e = c p,gg (T e ) + n w c p,w (T e ) c p,h (T e ) T = T e T (undercooling) Approximation for T»T e : µ = s e T (Volmer) s e dissolution entropy (per hydrate building unit) s gg entropy per gas molecule in gas s w entropy per water molecule in solution s h entropy per hydrate building unit in crystal c p,gg constant-pressure heat capacity per gas molecule in gas c p,w constant-pressure heat capacity per water molecule in solution c p,h constant-pressure heat capacity per hydrate building unit in crystal

8 8 Dm / kt e ethane methane T T e e T (K) Temperature dependence of the supersaturation for crystallization of methane and ethane hydrate at P=19.4 and 1.74 MPa, respectively

9 9 3. Nucleus size n* and nucleation work W* solution Dm n* (a) s gas n* substrate Homogeneous nucleation of spherical (a) and 3D heterogeneous nucleation of cap-shaped (b) and lens-shaped (c) hydrate crystals (b) s ef n* (c) s ef n* = 32πv h 2 σ ef3 /3 µ 3 (Gibbs-Thomson equation) W* = 16πv h 2 σ ef3 /3 µ 2 σ ef = σ homogeneous nucleation σ ef < σ heterogeneous nucleation σ specific surface energy of hydrate/solution interface σ ef effective specific surface energy

10 10 n* W* / kt (a) (b) P e 60 o 90 o o o homogeneous P (MPa) homogeneous Pressure dependence of (a) the nucleus size and (b) the nucleation work for homogeneous and 3D heterogeneous nucleation of methane hydrate at T=273.2 K (in heterogeneous nucleation the substrate is with wetting angle 60 0 and 90 0 )

11 11 n* W* / kt e (a) homogeneous 90 o 60 o (b) homogeneous 90 o 60 o T (K) T e Temperature dependence of (a) the nucleus size and (b) the nucleation work for homogeneous and 3D heterogeneous nucleation of methane hydrate at P=19.4 MPa (in heterogeneous nucleation the substrate is with wetting angle 60 0 and 90 0 )

12 4. Nucleation rate J 12 Nucleation model of Szilard: J n*-1 n* n*+1 subnuclei nucleus supernuclei Definition: J (m - 3 s - 1 or m - 2 s - 1 ) is the frequency of appearance of supernuclei per unit volume or area. J = z f* C* = z f* C 0 e -W*/kT J = A e µ/kt e -W*/kT with A = zf e *C 0 A m - 3 s - 1 for homogeneous nucleation A<<10 35 m - 3 s - 1 or m - 2 s -1 for heterogeneous nucleation z Zeldovich factor f* frequency of attachment of hydrate building unit to nucleus f e * value of f* at µ=0 C* concentration of nuclei C 0 concentration of nucleation sites A kinetic parameter

13 13 Example: homogeneous or 3D heterogeneous nucleation (spherical or cap-shaped nuclei). 1) isothermal regime Then µ=ktlns with S=[ϕ(P,T)P/ϕ(P e,t)p e ]exp[ v e (P P e )]. Hence: J = A S exp( B/ln 2 S) with B=16πv h 2 σ ef3 /3(kT) 3 2) isobaric regime Then µ= s e T with T=T e T. Hence: J = A exp( s e T/kT) exp( B /T T 2 ) with B =16πv h 2 σ ef3 /3k s e 2 A kinetic parameter (dimension m 3 s 1 or m 2 s 1 ) B thermodynamic parameter (dimensionless) B thermodynamic parameter (dimension K 3 )

14 J (m -3 s -1 ) P e 60 o 90 o HON P (MPa) Pressure dependence of the nucleation rate in crystallization of methane hydrate at T=278.3 K: line HON homogeneous nucleation, lines 60 0 and D heterogeneous nucleation on nucleation-active particles with wetting angle 60 0 and 90 0, respectively

15 J (m -3 s -1 ) HON 90 o 60 o T (K) T e Temperature dependence of the nucleation rate in crystallization of methane hydrate at P=19.4 MPa: line HON homogeneous nucleation, lines 60 0 and D heterogeneous nucleation on nucleation-active particles with wetting angle 60 0 and 90 0, respectively

16 16 ln (J / m-2 s -1 ) - Ds e DT / kt σ ef = 3.0 mj/m 2 n* = 2 to / TDT2 (K -3 ) Temperature dependence of the rate of methane hydrate nucleation in aqueous solution: points data of Y.F.Makogon, Hydrates of Hydrocarbons, Pennwell, Tulsa, 1997, line best fit to the first five points according to theory

17 17 5. Effect of additives In general: J = z f* C 0 e -W*/kT Case: Additive molecules adsorbing on the C 0 nucleation sites by Langmuir adsorption. SOLUTION Adsorption-inhibited and free nucleation sites Then C 0 decreases and becomes C 0 /(1 + k n C a ). Hence: J = z f* [C 0 /(1 + k n C a )] e -W*/kT or with J = [A/(1 + k n C a )] e µ/kt e -W*/kT A = z f e * C 0 k n adsorption constant C a concentration of additive

18 J (m -3 s -1 ) P e 60 o 90 o HON P (MPa) Pressure dependence of the nucleation rate in crystallization of methane hydrate at T=278.3 K: line HON homogeneous nucleation; lines 0 3D heterogeneous nucleation on nucleation-active particles with wetting angle 60 0 and 90 0 (as indicated) in blank solution; lines and the same kind of heterogeneous nucleation, but in the presence of an inhibiting additive with concentration and m -3

19 J (m -3 s -1 ) HON 90 o 60 o T (K) T e Temperature dependence of the nucleation rate in crystallization of methane hydrate at P=19.4 MPa: line HON homogeneous nucleation; lines 0 3D heterogeneous nucleation on nucleation-active particles with wetting angle 60 0 and 90 0 (as indicated) in blank solution; lines and the same kind of heterogeneous nucleation, but in the presence of an inhibiting additive with concentration and m -3

20 20 6. Conclusion - The general theory of nucleation is directly applicable to nucleation of gas hydrates. - There is need of experiments on nucleation of gas hydrates in order to check the theory and obtain data for the hydrate/solution surface energy σ.

Driving force for crystallization of gas hydrates

Journal of Crystal Growth 241 (22) 22 23 Driving force for crystallization of gas hydrates Dimo Kashchiev a, Abbas Firoozabadi b, * a Institute of Physical Chemistry, Bulgarian Academy of Sciences, ul.