Energy & Fuels 2007, 21, 2147-2155
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Advances in Modeling of New Phase Growth Seyed Jalaladdin Hashemi and Jalal Abedi* Department of Chemical & Petroleum Engineering, UniVersity of Calgary, 2500 UniVersity Dr. NW, Calgary, Alberta, T2N 1N4 Canada ReceiVed December 17, 2006. ReVised Manuscript ReceiVed May 2, 2007
The growth of the stationary new phase formed from supersaturated liquid has a significant role in many areas, such as the growth of bubbles in heavy oil and the growth of crystalline solids in hydrate engineering. The simplified governing equations for new phase growth in a solution have been formulated and solved for a single nucleate, assuming that mass transfer is the rate-limiting step. This problem is of interest to the heavyoil solution gas drive process and gas transportation. Applications to the growth of the hydrate as a new phase in a hydrate-water slurry and bubble as a new phase in heavy oil were shown. It can also be applied to the analysis of the growth of a stationary gas bubble as a result of the mass transfer of one component from the liquid to the gas forming the bubble. The growth behavior of a single nucleate is needed in order to study population growth.
1. Introduction The growth of the new phases formed in solutions has a significant role in many areas, such as heavy oil reservoirs (bubble growth) and hydrate engineering (hydrate particles). The formation and growth of new phases in a solution happens under favorable thermodynamic conditions. In the case of bubbles and solid particles in fluids, favorable thermodynamic conditions are specific pressure and temperature. The growth is a competition between kinetic, heat, and diffusion limitations. Thus, growth is limited by diffusion in the newly developed phase rather than the solution phase. The two main areas where the new phase growth has applicability are bubble growth due to solution gas diffusion and hydrate particle growth. The solution gas drive process in heavy oil reservoirs consists of nucleation and growth. It is the growth which is of interest in this work. Scriven presented a formulation of bubble growth due to heat and or mass transfer with an emphasis on heat transfer as the controlling process.1 Szekely and Martins2 presented the growth of a spherical gas bubble in a supersaturated liquid. Szekely and Fang3 showed the role of surface tension, surface kinetics, liquid inertia, and viscous forces on the bubble growth. Kumar and PooladiDarvish4 showed that the hydrodynamic forces have little or no effect on bubble growth in heavy oils at late times. An icelike substance called methane hydrate forms under suitable pressures and temperatures. The methane hydrate form of the energy is more abundant than any other forms of the fossil fuel. The abundance of methane reserves has made researchers think of methane as a long-term source of clean* Corresponding author. Fax: (403) 284-4852. E-mail:
[email protected]. (1) Scriven, L. E. On the Dynamics of Phase Growth. Chem. Eng. Sci. 1959, 10, 1-13. (2) Szekely, J.; Martins, G. P. Non-Equilibrium Effects in the Growth of Spherical Gas Bubbles due to Solute Diffusion. Chem. Eng. Sci. 1971, 26, 147-159. (3) Szekely, J.; Fang, S. D. Non-Equilibrium Effects in the Growth of Spherical Gas Bubbles due to Solute Diffusion-II The Combined Effects of Viscosity, Liquid Inertia, Surface Tension and Surface Kinetics. Chem. Eng. Sci. 1973, 28, 2127-2140. (4) Kumar, R.; Pooladi-Darvish, M. Effect of Viscosity and Diffusion Coefficient on the Kinetics of Bubble Growth in Solution-Gas Drive in Heavy Oil. J. Can. Pet. Technol. 2001, 40 (3), 1-8.
burning fuel. In the case of hydrate, its first discovery goes back to the early 1800s, and research started after Davy and Faraday’s experiment with chlorine-water mixtures. The preliminary research on hydrate formation started in Russia, and the second phase of the research on hydrates began after the plugging of the natural gas pipelines in the 1930s, and the “solid natural gas” in the Western Siberia basin, in the late 1960s, dramatically changed the global view of hydrate science. Hydrate formation is viewed as a crystallization process that includes nucleation and growth processes.5 The focus was to present the various perspectives on the kinetic processes at a conceptual level. Many others have modeled nucleation and growth up to the critical size of hydrate nuclei and described the kinetics of hydrate formation as follows: (a) the formation of critically sized nuclei, (b) the growth of these critically sized nuclei to a crystal, (c) diffusion of the components to the growing hydrate surface, and (d) dissipation of the heat of crystal formation. All of the above steps are interrelated, and a satisfactory quantitative model has not yet been found. Extensive work has been done on the kinetics of hydrates.5-15 Svandal et al.16 considered simulations of CO2 hydrate growth using phase theory and a model based (5) Bishnoi, P. R.; Natarajan, V. Formation and Decomposition of Gas Hydrates. Fluid Phase Equilib. 1996, 117 (1-2), 168-177. (6) Bollavaram, P. S.; Devarakonda, M.; Selim, S.; Sloan, E. D., Jr.Growth Kinetics of Single Crystal sII Hydrates: Elimination of Mass and Heat Transfer Effects. Ann. NY Acad. Sci. 2000, 912 (1), 533-543. (7) Christianson, R. L.; Bansal, V.; Sloan, E. D., Jr. Avoiding Hydrates in the Petroleum Industry: Kinetics of Formation. University of Tulsa Colonial Petroleum Engineering Symposium, Tulsa, OK, 1994; SPE, Inc: Houston, TX, 1994. (8) Freer, E. M.; Sami Selim, M.; Dendy Sloan, E., Jr. Methane Hydrate Film Growth Kinetics. Fluid Phase Equilib. 2001, 185 (1-2), 65-75. (9) Kim, H. C.; Bishnoi, P. R.; Heidemann, R. A.; Rizvi, S. S. H. Kinetics of Methane Hydrate Decomposition. Chem. Eng. Sci. 1987, 42, 16451653. (10) Kvamme, B. A Unified Nucleation Theory for the Kinetics of Hydrate Formation. Ann. NY Acad. Sci. 2000, 912 (1), 496-501. (11) Lederhos, J. P.; Sloan, E. D. Transferability of Kinetic Inhibitors between Laboratory and Pilot Plant. Proceedings Annual Technical Conference and Exhibition, Denver, CO, Oct. 23-25, 1996; SPE: Houston, TX, 1996. (12) Leporcher, E. M. Multiphase Transportation: A Kinetic Inhibitor Replaces Methanol to Prevent Hydrates in a 12-inc. Pipeline. European Petroleum Conference, The Hague, Netherlands, 1998; SPE: Houston, TX, 1998.
10.1021/ef060643l CCC: $37.00 © 2007 American Chemical Society Published on Web 06/21/2007
2148 Energy & Fuels, Vol. 21, No. 4, 2007
on cellular automata. Mesoscopic modeling of hydrate growth has been done using phase field theory.17,18 Svandal et al.16 applied phase field theory simulations to model the growth of CH4 and CO2 hydrates from respective aqueous solutions of these hydrate formers. A number of laboratories in the world are actively seeking a solution to the problem of natural gas storage and transportation in a solid hydrate state. The use of gas hydrates in the storage and transport of natural and associated gas offers new opportunities to the oil and gas industries. Gas conversion into hydrates in the form of crude oil or water slurries can be a solution to several industrial situations. Thus, the flow leaving the separator is gas-free hydrate slurry where the continuous phase can be either water or oil, depending on the experimental conditions. As hydrates are known to block pipelines and process equipment, great care in producing the slurries was exercised, resulting in systems in which the hydrate particles flow along with the carrying fluid and no deposition takes place. When producing hydrate-in-water slurries, a certain amount of gas will be injected into a water-filled system at hydrate subcooling temperatures of 2.5-6.5 °C. The pressure is kept constant at either 60 or 90 bar. The gas phase used to produce the slurries is pure methane, and the water phase was deareated tap water. The injection of gas results in a slurry with a certain concentration of hydrates. It was shown19,20 that hydrates can form from dissolved gas in water as the methane molar fraction C0 is greater than the molar fraction Ce corresponding to the hydrate equilibrium conditions at temperature T. In fact, experiments have proved that hydrates could form in the porous media under realistic conditions even if the gas is not present under its gas phase.21 Considering the growth process, the model requires taking into account the solubility of methane in water. 2. Governing Equations To be able to study the growth of the new phases formed in solutions analytically, a stationary spherical gas, liquid, or solid phase with an initial radius of δ must be consdiered, which is growing in a solution phase of infinite extent due to the transfer of a solute from the solution phase to the gas, liquid, or solid phase. In addition, the problem is greatly simplified by the following assumptions: (i) density changes neither due to mass transfer in the new phase nor due to the solution surrounding it; (ii) mass (13) Monfort, J. P.; Jussaumei, L.; El Hafaiai, T.; Canselieri, J. P. Kinetics of Gas Hydrates Formation and Tests of Efficiency of Kinetic Inhibitors: Experimental and Theoretical Approaches. Ann. NY Acad. Sci. 2000, 912 (1), 753-765. (14) Notz, P. K.; Bumgartner, S. B.; Schaneman, B. D.; Toddet, J. L. The Application of Kinetic Inhibitors to Gas Hydrate Problems. 27th Annual Offshore Technology Conference, Houston, Texas, May 1-4, 1995; SPE: GHouston, TX, 1995; pp 719-730. (15) Sakaguchi, H.; Ohmura, R.; Mori, Y. H. Effects of Kinetic Inhibitors on the Formation and Growth of Hydrate Crystals at a Liquid-Liquid Interface. J. Cryst. Growth 2003, 247 (3-4), 631-641. (16) Svandal, A.; Kuznetsova, T.; Kvamme, B. Thermodynamic Properties and Phase Transtions in the H2O/CO2/CH4 System. Phys. Chem. Chem. Phys. 2006, 8, 1707-1713. (17) Kvamme, B.; Graue, A.; Aspenes, E.; Kuznetsova, T.; Granasy, L.; Toth, G.; Pusztai, T.; Tegze, T. Kinetics of Solid Hydrate Formation by Carbon Dioxide: Phase Field Theory of Hydrate Nucleation and Magnetic Resonance Imaging. Phys. Chem. Chem. Phys. 2003, 6 (9). (18) Tegze, G.; Pusztai, T.; To´th, G.; Gra´na´sy, L.; Svandal, A.; Buanes, T.; Kuznetsova, T.; Kvamme, B. Multi-Scale Approach to CO2-Hydrate Formation in Aqueous Solution: Phase Field Theory and Molecular Dynamics. Nucleation and Growth. J. Chem. Phys. 2006, 124. (19) Rempel, A. W.; Buffet, B. A. Formation and Accumulation of Gas Hydrate in Porous Media. J. Geophys. Res. 1997, 105 (B5), 10151-10164. (20) Buffet, B. A.; Zatsepina, O. Y. Formation of Gas Hydrate from Dissolved Gas in Natural Porous Media. Mar. Geosci. 2000, 164, 69-77. (21) Pooladi-Darvish, M.; Firoozabadi, A. Solution Gas Drive in Heavy Oil Reservoirs. J. Can. Pet. Technol. 1999, 38 (4), 54-61.
Abedi and Hashemi
Figure 1. Initial condition for new phase growth.
transfer, not chemical reaction or heat transfer, controls the new phase growth (thus, growth is limited by diffusion, and therefore diffusion is the rate-limiting step); (iii) the solution is restricted to two-component (solute and solvent) systems at constant temperature and pressure, with no chemical reaction; (iv) there is no mass transfer other than ordinary diffusion with constant diffusivity coefficients; (v) equilibrium conditions exist at the interface (the interface between the newly developed phase and the solution phase); (vi) there is no surface tension or body forces, and (vii) there is a constant mass fraction of the solute in the new phase. Consider a stationary spherical new phase growing in a quiescent solution phase of infinite extent, due to the transfer of a dilute component from the solution phase to the new phase. Let us assume further that the viscosity of the solution is constant, the system is isothermal, and there exists thermodynamic equilibrium at the newly developed phase-solution phase interface. Finally, let us consider that the solute within the newly developed phase is homogeneous and is at a uniform pressure. These assumptions, which are thought to provide a reasonable representation of the newly developed phase, allow us to present the governing equations in a relatively simple form: by using a spherical coordinate system, sketched in Figure 1, the equations of continuity and motion may be combined in the following form: (a) For solvent surrounding the new phase 1 ∂ 2 (r u) ) 0 r2 ∂r
r > R(t)
(1)
(b) For the solute surrounding the new phase
(
)
∂2C 2 ∂C ∂C ∂C )D 2 + -u ∂t r ∂r ∂r ∂r
r > R(t)
(2)
(c) For the solute in the new phase
(
)
{
}
∂C(R,t) d 4 3 πR mFn ) 4πR2 C(R,t) [R˙ - u(R)] + D dt 3 ∂r After simplification ∂C(R,t) ∂r
mFnR˙ ) C(R,t) [R˙ - u(R)] + D
(3)
(d) For the solvent in the new phase d 4 3 πR Fh ) 4πR2FL[R˙ - u(R)] dt 3
[
]
or u(R) ) R˙
(4)
where )1-
Fn ,ω)1- F
(5)
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Energy & Fuels, Vol. 21, No. 4, 2007 2149
Table 1. Oil-Gas Data at Constant Temperature (T ) 50 °C) P0
Pe
24.7 atm
Ce)Pe/H
H
24.19 atm
79.698
(psim3)/kg
4.404 kg
C0)P0/H
/m3
4.555 kg
Figure 1 shows the notation and the initial condition of the problem. The initial (IC) and boundary (BC) conditions can be written as follows: IC: R(0) ) δ
(6)
C(r,0) ) C0
(7)
C(R,t) ) Ce
(8)
C(∞,t) ) C0
(9)
BC:
D 5×
/m3
(Rr)
2
C0 - C ) C0 - C e
∫
∞
r/ [2xD(t + R)]
∫
∞ -2 υ β
φ(,β) )
υ-2 exp(-υ2 - 2β3υ-1) dυ 3 -1
2
∫
β
∞ -2
υ
1
Fn 152.66
kg/m3
FL 800 kg/m3
152.66 ) 0.809 800
C0 - Ce
) mFn - (1 - )Ce 4.555 - 4.404 ) 9.95 × 10-4 152.66 - (1 - 0.809) × 4.0404
β ) 0.023 can be found from the literature;1 then, from eqs 14 and 13
(11)
R)
exp(-υ - 2β υ ) dυ 2
(2 × 10-8)2 δ2 ) ) 3.78 × 10-4 2 -10 2 4Dβ 4 × 5 × 10 × 0.023
and
where β can be obtained from φ(,β) ) 2β3 eβ (1+2∈)
)1-
(10)
Replacing for u and u(R) in eqs 2 and 3 and using the method of combined variables to solve the obtained differential equation with the relevant boundary and initial conditions for the solute concentration at any time t and distance r (details in the appendix), we get
m2/s
to the following situations: (1) bubble growth in heavy oil solution gas drive processes and (2) carbon dioxide and methane hydrate particle growth in gas hydrate engineering, where is approximately zero and the accuracy of the approximate solution can be evaluated. Case 1. Bubble Growth. Bubble growth is of particular interest for the solution gas drive process.4 Consider saturated oil-gas in equilibrium with a gas bubble, at a given temperature and pressure (T, P). Let the pressure be dropped to Pe, leading to the growth of a bubble. The required data are taken from the literature21 and are tabulated in Table 1. Calculate and φ from eqs 5 and 12 as follows:
The solution has been obtained following Scriven’s work1 by integrating eq 1 and using eq 4 to find u ) R˙
m
10-10
R ) 2βxD(t + R) )
exp(-υ2 - 2β3υ-1) dυ ) γ ) C0 - Ce mFn - (1 - )Ce
(12)
0.1029 × 10-5xt + 3.78 × 10-4 (18)
(17)
Figure 2 shows bubble growth in the oil solution gas drive process for two different cases. Bubble growth is a moving boundary problem. We have solved the set of equations in a coordinate system that move with the liquid bubble interface. Case 2. Hydrate Particle Growth. (a) CO2. Global warming due to the increase in CO2 concentration in the atmosphere necessitates its removal from the flue gases in power plants. Liquid-CO2 deep-sea ocean disposal is considered to be one of the most promising mitigation strategies. The CO2 clathratehydrate, CO2(H2O)5.75, is formed under conditions of temperature less than 10 °C and pressure greater than 44 atm.22 Figure 3 shows bubble growth for the carbon dioxide gas hydrate. (b) CH4. The CH4 clathrate-hydrate, CH4(H2O)5.75 is formed under specific conditions of temperature and pressure. Consider a methane and water solution at 60 atm and 275 K. At 275 K, the equilibrium pressure is 28.4 atm. The required data for both cases are taken from the literature and tabulated in Table 2. Assuming that nucleation has been started for both cases and that the formed hydrate particles have gorwn to the critical radius of 2 × 10-8, the growth of the hydrate particle has been obtained, and this is summarized in Table 3. Figure 4 shows bubble growth for the methane gas hydrate. Comparisons of
3. Results In order to see the behavior of the new phase growth, the solution of the obtained governing equations has been applied
(22) Kato, M.; Iida, T.; Mori, Y. H. Drop Formation Behaviour of a Hydrate-Forming Liquid in a Water Stream. J. Fluid Mech. 2000, 414, 367378. (23) Sloan, E. D.; Clathrate Hydrates of Natural Gases, 2nd ed.; Marcel Dekker: New York, 1998.
The numerical solution to the above equation is given in the literature;1 therefore, the growth radius of the new phase at any time t can be calculated from R ) 2βxD(t + R)
(13)
where R)
δ2 4Dβ2
(14)
For the special cases where ∈ may be assumed to be approximately, eqs 12 and 13 can be simplified, and the solutions for the solute concentration and growth radius become
( )
r-R R C(r,t) ) C0 + (Ce - C0) erfc r x4Dt
(15)
R ) Bt1/2 + δ
(16)
where
B)
D C0 - C e + π mFn - Ce
x
x
D(C0 - Ce)2
π(mFn - Ce)
2
+
2D(C0 - Ce) (mFn - Ce)
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Abedi and Hashemi
Figure 2. Growth of new phase particles for heavy oil solution gas drive process (experimental data23).
Figure 3. Growth of new phase particles for carbon dioxide gas hydrate (experimental data23). Table 2. Data Taken from the Literature for Methane and CO2 Hydrate23
CH4 CO2
P0 Mpa
Pe Mpa
H Mpa
6 7.38
2.84 4.4
2.3 × 103 2.93 × 103
Ce)Pe/H kg /m3
C0, kg/m3
D m2/s
m
Fn kg/m3
FL kg/m3
1.235 1.502
1.337 1.799
8.5 × 10-10 3.4 × 10-10
0.134 0.298
919.7 1100
1002 1002
Table 3. Estimated Parameters for Oil Solution Gas Drive Process, Methane, and Carbon Dioxide Hydrates
oil solution gas drive CH4 CO2
(5)
ω (5)
γ (12)
β [1]
R (14)
B (17)
0.809 0.083 -0.098
0.191 0.917 1.098
9.95 × 10-4 0.00191 0.0009
0.023 0.033 0.022
3.78 × 10-4 1.080 × 10-4 1.368 × 10-5
0.1022 × 10-5 0.1850 × 10-5 0.7963 × 10-6
bubble growth in the heavy oil solution gas drive process, carbon dioxide, and methane gas hydrate are shown in Figure 5. Modeling of bubble growth showed that the power law model
may adequately describe bubble growth behavior. We can deduce that bubble growth may be modeled by the relationship R(t) ) ta.
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Figure 4. Growth of new phase particles for methane gas hydrate.
Figure 5. Growth of new phase particles for heavy oil solution gas drive process, carbon dioxide, and methane gas hydrate.
4. Conclusions Key issues for research in this area are identified and some possible directions for future work are suggested. The general case of diffusion-controlled new phase growth has been formulated and solved. Its industrial application in areas such as the heavy-oil solution gas drive process and hydrate particle growth has been shown. The special case of particle growth has a simplified solution with reasonable accuracy. The power law model may adequately describe bubble growth behavior. The next step would be the study of population growth, which is of interest to the gas and petroleum industries. Nomenclature B ) parameter defined in eq 17 C ) solute concentration (kg/m3)
D ) solute diffusivity coefficient (m2/s) m ) mass fraction of the solute in the new phase R ) new phase radius (m) R ) spherical coordinate (m) S ) combined variable T ) time (s) Greek Letters R, β, γ, and ) defined parameters in eqs 14, 12, 12, and 5, respectively F ) density (kg/m3) δ ) initial radius (m) φ ) defined function (eq 12) Subscripts 0 ) initial value E ) equilibrium N ) new phase
2152 Energy & Fuels, Vol. 21, No. 4, 2007
Abedi and Hashemi
{
Appendix Equation 1 is integrated
r2u ) f(t)
(A-10)
Equation 4 is considered to determine f(θ)
Css β3 ) -2s - 2s-1 + 2 2 Cs s
R u(R) ) R R˙ ) f(t) 2
}
-s 1 2 C )D C C + 2 ss 2(t + R) s 4s[D(t + R)] s [2xD(t + R)] β3 Cs (A-22) 2(t + R)s2
2
(
or
()
(A-11)
(
) ()
mFnR˙ ) CeR˙ (1 - ) + D
r > R(θ)
(A-12)
(A-24)
For both t ) 0 and R ) ∞, we have s ) ∞; therefore
In eqs 2 and 3, u and u(r) from eqs A-11 and 4 are replaced
∂2C 2 ∂C R 2∂C ∂C )D 2 + - R˙ ∂t r ∂r r ∂r ∂r
)
β3 s
Cs ) As-2 exp -s2 - 2 R2 u ) R˙ r
(A-23)
∫∞sCs ds ) A∫∞s υ-2 exp(-υ2 - 2β3υ-1) dυ
(A-25)
∫∞sυ-2 exp(-υ2 - 2β3υ-1) dυ
(A-26)
or
∂C(R,t) ∂r
C0 - C ) -A
A can be obtained from the second boundary condition 8: For r ) R, we have
or
R˙ )
∂C(R,t) D mFn - Ce(1 - ) ∂r
(A-13)
The method of combined variables is used to solve eqs A-12 and A-13 as follows: Let
R ) 2βxD(t + R)
s)
2xD(t + R)
(A-14)
C0 - C ) C0 - C e (A-16)
x
(A-17)
∂C dC ∂s -s dC ) ) ∂t ds ∂t 2(t + R) ds
(A-18)
R˙ ) β
d2C 1 ∂2C ) ∂r2 [2xD(t + R)]2 ds2
(A-20)
Substituting back into eq A-12 and integrating twice
∫s∞υ-2 exp(-υ2 - 2β3υ-1) dυ ∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ
or
C0 - C ) C0 - C e
Cs ) (A-19)
d2C ) Css ds2
(A-28)
∫r/[2∞ xD(t + R)]υ-2 exp(-υ2 - 2β3υ-1) dυ ∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ
(A-29)
Substitute eqs A-17 and A-29 into eq A-13 to find β
dC ∂C dC ∂s 1 ) ) ∂r ds ∂r 2xD(t + R) ds
dC ) Cs ds
∫β∞ υ-2 exp(-υ2 - 2β3υ-1) dυ
Substitute back into eq A-26
Then
D (t + R)
C0 - C e
(A-15)
R(0) ) δ ) 2βxD(0 + R) δ2 4Dβ2
(A-27)
∫∞βυ-2 exp(-υ2 - 2β3υ-1) dυ
Then, from initial condition 6
R)
)β
C0 - Ce ) -A A)
r
2xD(t + R)
Therefore, eq 8 changes to C(β) ) Ce
and
s)
R
(A-21)
C e - C0
(
∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ
x
)
β3 S
S-2 exp -S2 - 2
dC | ∂C | D 1 | ) | )β ) ∂r |r)R 2xD(t + R) ds |S)β (t + R) D 1 C (β) mFn - (1 - )Ce 2xD(t + R) s 1 -2 1 β) Aβ exp(-β2 - 2β2) (A-30) mFn - (1 - )Ce 2 A ) 2β3[mFn - (1 - )Ce] exp[β2(1 + 2)]
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Energy & Fuels, Vol. 21, No. 4, 2007 2153
Replacing for A from A-28 gives
∂2CA
C 0 - Ce
∂r2
∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ
)
2β3[mFn - (1 - )Ce] exp[β2(1 + 2)]
2
φ(,β) ) 2β3 eβ (1+2∈)
∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ ) γ
2
(A-40)
∂CA 1 ∂ν ) ∂t r ∂t
(A-41)
∂ν ∂2ν )D 2 ∂t ∂r
∫s∞υ-2 exp(-υ2 - 2s3υ-1) dυ
(A-42)
The initial and boundary conditions become
where
γ)
C0 - Ce
(A-32)
mFn - (1 - )Ce
Equation A-31 can be solved for β. The function φ(,β) is given in Table 1 of ref 1 for different values of ∈ and β. Therefore, for any value of γ and , β can be obtained from Table 1 of ref 1; then, the new phase radius R can be evaluated at any time t from eq A-14
R ) 2βxD(t - R)
(A-29)
()
where β can be obtained from
(A-44)
ν(∞,t) ) 0
(A-45)
(A-46)
T(z,0) ) 0
(A-47)
T(0,t) ) 1
(A-48)
T(∞,t) ) 0
(A-49)
z x4Dt
(A-50)
Introducing
φ(,β) ) γ
η)
The special case of ) 0, or Fn/F ) (1 - m)-1 simplifies eqs A-12 and A-13, which may be solved more simply as follows: Equations A-12 and A-13 reduce to
(
ν(R,t) ) RCA*
with the following initial and boundary conditions:
β 3 (1 + 2)(β2-s2) e φ(,s) C0 - C ) s
)
2
(A-43)
∂T ∂2 T )D 2 ∂t ∂z
(A-14)
∫r/[2∞ xD(t + R)]υ-2 exp(-υ2 - 2β3υ-1) dυ ∫β∞υ-2 exp(-υ2 - 2β3υ-1) dυ
ν(r,0) ) 0
When new variables z ) r - R and T ) (ν/RCA*) are introduced, then eqs A-24-A-27 reduce to
and C at any time t and distance r from eq A-29
C0 - C ) C0 - C e
2 2ν 2 ∂ν 1 ∂ CA + + r3 r2 ∂r r ∂r2
Substituting in the main PDE (eq A-16) gives
(A-31)
φ(,s) ) 2s3 es (1+2∈)
)
∂ C 2 ∂C ∂C )D 2 + ∂t r ∂r ∂r
r > R(t)
∂C(R,t) D R˙ ) mFn - Ce ∂r
(A-34)
)
∂2CA 2 ∂CA ∂CA )D + ∂t r ∂r ∂r2 CA(r,0) ) 0
(A-51)
∂T dT ∂η dT 1 ) ) ∂z dη ∂z dη x4Dt
(A-52)
∂2T 1 d 2T 1 d2T ∂η ) ) 2 2 ∂z 4Dt dη ∂z x4Dt dη
(A-53)
(A-33)
When the variations of R(t) are neglected, to be able to assume fixed boundary conditions for eq 33, it can be solved as follows: Let CA ) C - C0; then, eq A-33 and its initial and boundary conditions 7-9 reduce to
(
( ) ( )
z ∂T dT ∂η dT ) )∂t dη ∂t dη 2tx4Dt
Substituting in eq A-28 gives
dT d 2T + 2η ) 0 dη dη2
(A-35) (A-36)
(A-54)
The boundary conditions become
CA(R,t) ) Ce - C0 ) CA*
(A-37)
T(0) ) 1
(A-55)
CA(∞,t) ) 0
(A-38)
T(∞) ) 0
(A-56)
dT )p dη
(A-57)
Introducing a new variable ν ) rCA gives
2 ∂CA 2ν 2 ∂ν )- 3 + 2 r ∂r r r ∂r
Let
(A-39)
2154 Energy & Fuels, Vol. 21, No. 4, 2007
Abedi and Hashemi
Then, eq A-36 becomes
R˙ )
dp -2ηp ) dη
(A-58)
(
D(C0 - Ce) 1 1 + mFn - Ce R xπDt
)
(A-72)
To solve this nonlinear first-order ordinary differential equation, let
integrating twice to get
∫0η exp(-V2) dV + C2
T ) C1
R ) Bxt
(A-59) then
Boundary conditions A-37 and A-38 are used to find constants C1 and C2 as follows:
1 R˙ ) Bt-1/2 2
C2 ) 1
(A-60)
Replace for R and R˙ in eq A-72 and solve for the constant B
(A-61)
D(C0 - Ce) 1 B 1 ) + mFn - Ce Bxt xπDt 2xt
∫0∞ exp(-V2) dV + 1
0 ) C1
(
Noting that
∫0
∞
B2 - 2
xπ exp(-V2) dV ) 2
(A-62) B)
then
2 xπ
C1 ) -
)
(A-74)
2D(C0 - Ce) D C0 - C e B) 0 (A-75) π mFn - Ce mFn - Ce
x
D C0 - Ce + π mFn - Ce
x
x
(A-63)
D(C0 - Ce)2
π(mFn - Ce)
Therefore
(A-73)
2
+
2D(C0 - Ce) (mFn - Ce)
(A-76)
Therefore, the final solution for R will be
T(η) ) 1 2 xπ
2 xπ
∫ η0 exp(-V2) dV
R ) Bt1/2 + δ
(A-64)
∫0η exp(-V2) dV ) erf(η)
(A-65)
(A-77)
Sample of Calculations (a) ∈ ) 0 Approximate Method
T(η) ) 1 - erf(η) ) erfc(η)
B ) 0.1846 × 10-5
or
( )
r-R T(r,t) ) erfc x4Dt
R ) Bt1/2 + δ ) 0.1846 × 10-5xt + 2 × 10-8
(A-66)
Exact Method
and
ν(r,t) ) RCA* erfc
φ(∈,β} ) γ )
( ) ( ) ( ) r-R x4Dt
(A-67)
R r-R CA(r,t) ) CA* erfc r x4Dt
(A-68)
r-R R C(r,t) ) C0 + (Ce - C0) erfc r x4Dt
(A-69) R)
r-R R ∂C ) - 2(Ce - C0) erfc ∂r r x4Dt (r - R)2 R (A-70) (Ce - C0) exp 4Dt rxπDt
[
ω ) 1 - ) 1 - 0.0 ) 1 β ) 0.03
Replacing for C in eq A-15 and solving for R
( )
C 0 - Ce ) mFh - (1 - e)Ce 1.337 - 1.1007 ) 0.001 935 0.134 × 919.7 - 1.1007
]
∂C 1 1 (R,t) ) - (Ce - C0) (Ce - C0) (A-71) ∂r R xπDt
(2 × 10-8)2 δ2 ) ) 1.307 × 10-4 4Dβ2 4 × 0.85 × 10-9 × 0.032
R ) 2βxD(t + R) ) 0.178 × 10-5xt + 1.307 ×10-4 (b) ) 1 - (Fh/FL) ) 1 - 0.917 ) 0.083 Approximate Solution
B ) 0.1846 × 10-5 R ) Bt1/2 + δ ) 0.1846 ×10-5xt + 2 ×10-8
AdVances in Modeling of New Phase Growth
Exact Method
Energy & Fuels, Vol. 21, No. 4, 2007 2155
R) φ(,β} ) γ ) 0.001 919
ω ) 1 - ) 1 - 0.083 ) 0.917 β ) 0.033
(2 × 10-8)2 δ2 ) ) 1.08 × 10-4 2 -9 2 4Dβ 4 × 0.85 × 10 × 0.033
R ) 2βxD(t + R) ) 0.1924 × 10-5xt + 1.08 × 10-4 EF060643L
