Chapter 3 – Methods and instrumentation
3.6 Clathrate formation - Powder 4 model
“Powder 4” model presented below has been created in a group led by Prof. A. N.
Salamatin1
1 Department of Applied Mathematics, Kazan State University, Kazan 420008, Russia.
at the Kazan State University, Russia in cooperation with our department. It is still under development but can be already applied for mono dispersed spherical powders with a radius of a few μ (frost). Ice spheres with a larger diameter can be also treated only in the initial part maximally up to 30-40 wt% where the reaction rate starts to be strongly dependent on a sample’s geometry. The treatment of the later part of the diffusion controlled transformation as well as polydispersivity is not yet implemented. Mathematical
200 240 280 320 360 400
0,5 0,6 0,7 0,8 0,9 1,0
160 200 240 280
0,96 0,97 0,98 0,99 1,00
10kPa 15kPa 20kPa 25kPa 30kPa 35kPa 40kPa 45kPa 50kPa 55kPa 60kPa 65kPa 70kPa 75kPa 80kPa 85kPa 90kPa 95kPa 100kPa 150kPa 200kPa 250kPa 300kPa 350kPa 450kPa 900kPa 950kPa 1Mpa
f/p
Temperature [K]
Fig. 81) Fugacity/ pressure ratio as a function of temperature (Angus et al., 1976). In the blow up chosen data are extended to lower temperatures through the fitting. Thick blue, orange and green curves (see arrows) are calculated for 0.6kPa, 6kPa and 170kPa, respectively.
0.6kPa
6kPa
170kPa
107
description of the transformation process is a synergy of already published phenomenological models (Salamatin and Kuhs, 2002), (Staykova et al., 2003), (Genov et al., 2004) and Johnson-Mehl-Avrami-Kolmogorv-Ginstling-Brounshtein concept (Genov, 2005). For details the reader may look into the previous PhD thesis (Genov, 2005) where both approaches were critically discussed.
3.6.1 Theoretical background
Powder 4 utilizes information on the nucleation and growth acquires during the latest interrupted runs (4.1.2). Previously considered case of the crack filling is still preserved if one feels a need to apply it. New data suggest that the initial surface coverage is in no longer limited by a nucleation rate but depends on a lateral growth of the clathrate film.
Mathematically it is expressed with Johnson-Mehl-Avrami-Kolmogorv (JMAK) model that is modified for the spherical geometry. To account for observed heterogeneity of the nucleation (4.1.2) it is necessary to introduce a time dependent expression 𝑁𝑁0𝑡𝑡𝜎𝜎−1 defining nucleation rate constant 𝑁𝑁0 at a moment 𝜏𝜏 and a phenomenological exponent 𝜎𝜎 that ranges from 0 to 1 for instantaneous and uniform nucleation respectively. Consequently the nucleation rate 𝑁𝑁̇ per unit area of free ice surface is:
𝑁𝑁̇= 𝑁𝑁0𝜏𝜏𝜎𝜎−1
For an infinite 2D nucleation domain a single nucleus will increase its radius 𝑙𝑙 from the moment of formation 𝜏𝜏 to a moment 𝑡𝑡 as follows:
𝑙𝑙 = 2𝐺𝐺(𝑡𝑡 − 𝜏𝜏)𝑚𝑚2
A new term 𝐺𝐺 stands here for a nuclei growth rate constant. Parameter 𝑚𝑚 is an empirical exponent that is related to a dimensionality of the growth varying between 1 and 3. Since in reality the growth seldom follows only one type (e.g. m=2 for 2D), it is allowed to use real numbers (e.g. m=2.3 for a mixed 2D/3D growth), (4.2.4). Modification introduced in Powder 4 limits above domain to the surface of a spherical particle:
𝑆𝑆(𝑡𝑡 − 𝜏𝜏) = 4𝜋𝜋 min{𝑟𝑟𝑖𝑖02,𝐺𝐺2(𝑡𝑡 − 𝜏𝜏)𝑚𝑚}
The limitation of the nucleation domain causes the nucleation rate 𝑁𝑁̇ after a period of 𝑑𝑑𝜏𝜏 to be actually smaller since the available area of free ice decreases with time. In order to account for that, authors introduce a “phantom nuclei” 𝛼𝛼𝑆𝑆𝑁𝑁̇𝑑𝑑𝜏𝜏 which would have been created if the surface was free. This in turn leads to a fraction 𝐴𝐴𝑒𝑒𝑒𝑒 of the ice surface 𝑆𝑆 that would have been coated by fictitious nuclei at a time 𝑡𝑡:
108 coating respectively. The rate of surface coating at a moment 𝜏𝜏 will be then:
𝐴𝐴𝑒𝑒𝑒𝑒 = ln 1
1− 𝛼𝛼𝑆𝑆 =� Ω𝑆𝑆 𝑡𝑡 0
(𝜏𝜏)𝑑𝑑𝜏𝜏
The last part of the expression can be implemented into the exponent of the JMAK general equation as follows:
𝛼𝛼𝑆𝑆 = 1− 𝑒𝑒�− ∫ Ω0𝑡𝑡 S(τ)dτ� Refined JMAK approach gives the following parameterization:
� Ω𝑆𝑆(𝜏𝜏)𝑑𝑑𝜏𝜏=
The rate of coating Ω𝑆𝑆 is here presented using a nucleation limited rate of hydrate formation 𝜛𝜛𝑠𝑠 = 4𝜋𝜋𝑟𝑟𝑖𝑖02𝑁𝑁0 for two different regimes defined by a typical time of the sphere coverage 𝑡𝑡0 = �𝑟𝑟𝐺𝐺𝑖𝑖0�
2
𝑚𝑚. The first case, 𝑡𝑡< 𝑡𝑡0, discusses a variant of a predominant nucleation in cracks, which presently seems to be not a valid scenario. The other one considers mentioned earlier case of a reaction where the initial surface coverage is in no longer limited by the nucleation rate but depends on the lateral growth of the clathrate film.
An expansion of the initial film into an ice particle generally follows a shrinking core concept introduced in previous phenomenological models e.g. (Salamatin and Kuhs, 2002).
Knowing that clathrates have somewhat lower density (ignoring gas molecules in the structure) then a thickness of the initial hydrate film 𝑑𝑑0 can be written in the following form:
𝑑𝑑0 =𝛿𝛿0(1 +𝐸𝐸)
𝛿𝛿0 describes a thickness of the ice layer converted during the initial mantling process. E is
109
a hydrate phase expansion coefficient defined as a ratio between the mole density of ice 𝜌𝜌𝑖𝑖
and empty clathrate structure 𝜌𝜌𝑘𝑘𝑘𝑘 corrected for observed sub-μ porosity 𝜀𝜀ℎ (see: 4.2.2):
𝐸𝐸 = 𝜌𝜌𝑖𝑖 𝜌𝜌𝑘𝑘𝑘𝑘(1− 𝜀𝜀ℎ)
The ice core radius after the initial coating Δ(𝑡𝑡) will undergo gradual consumption during the formation process. Sharp clathrate/ice interface will move toward the cores’ center:
𝑑𝑑Δ
𝑑𝑑𝑡𝑡 =− 𝜔𝜔𝑉𝑉
𝜌𝜌𝑖𝑖𝑟𝑟𝑖𝑖0(1 +𝑣𝑣02) ,Δ ≥0
A new parameter 𝜔𝜔𝑉𝑉describes a number of ice moles transformed to clathrate in a unit of time and ice surface under the initial coating. Consequently ice radius is:
Δ|𝑡𝑡=0 = 1− 𝛿𝛿0
𝑟𝑟𝑖𝑖0(1 +𝑣𝑣02)
The decrease of the ice core’ radius in the period of time (t− τ) leads to a fraction of clathrates in this increment:
𝑑𝑑𝛼𝛼 = [1− Δ3(t− τ)]𝑑𝑑𝛼𝛼𝑆𝑆
𝑑𝑑𝛼𝛼𝑆𝑆defines here the fraction of a surface covered in the discussed time period. After integration over time one obtains the reaction degree 𝛼𝛼 in a time interval 𝑑𝑑𝜏𝜏:
𝛼𝛼 =�[1− Δ3(𝑡𝑡 − 𝜏𝜏)]𝑑𝑑𝛼𝛼𝑆𝑆(𝜏𝜏)
𝑡𝑡
0
The typical time for coating 𝑡𝑡0 and the rates of the surface coating 𝜔𝜔𝑉𝑉 as well as of the volume transformation 𝜔𝜔𝑉𝑉are related to a driving force defined as a supersaturation of the system in gas ln𝑓𝑓𝑓𝑓 kD permeation constant is related to the gas/water transport through the hydrate shell.
Since this process is tightly related to the geometry and packing of ice it must be also included in the model. The description of starting spheres follows published consideration (Staykova et al,. 2003) on a random dense packing of monodispersed spheres. Generally each ice particle is represented as a truncated sphere of a radius 𝑟𝑟ℎ initially equal to 𝑟𝑟𝑖𝑖0 that is surrounded by other particles. A number of contact, the coordination number 𝑍𝑍 can be expressed as a linear function of a relative hydrate shell radius 𝑅𝑅ℎ =𝑟𝑟𝑟𝑟ℎ
𝑖𝑖0:
110
𝑍𝑍=𝑍𝑍0+𝐶𝐶(𝑅𝑅ℎ −1)
𝑍𝑍0and 𝐶𝐶 are empirical values of the initial coordination number and the slope of the random density function respectively. For ice powders 𝑍𝑍0~ 7 and 𝐶𝐶 ~ 15.5. The specific surface of the starting material can be obtained experimentally (3.4) or through following equation:
𝑆𝑆𝑖𝑖0 = 3 𝑟𝑟𝑖𝑖0𝜌𝜌𝑖𝑖(1 +𝑣𝑣02)
During the formation ice core radius 𝑟𝑟𝑖𝑖0 decreases but the overall 𝑟𝑟ℎ in fact increases due to lesser density of clathrates. The normalized volume of the transforming particle is related to the reaction degree 𝛼𝛼 through a relative radius of the ice-hydrate interface 𝑅𝑅𝑖𝑖:
𝑅𝑅ℎ3−𝑍𝑍0
4 (𝑅𝑅ℎ −1)2(2𝑅𝑅ℎ + 1)− 𝐶𝐶
16(𝑅𝑅ℎ −1)3(3𝑅𝑅ℎ + 1) = 1 +𝐸𝐸(1− 𝑅𝑅𝑖𝑖3) The fraction of the hydrate surface 𝑠𝑠 in contact with gas is:
𝑠𝑠= 1−𝑍𝑍0
Growing spheres improve contacts with surrounding particles and form new ones decreasing the porosity 𝜀𝜀𝑚𝑚 and total surface area of empty voids 𝑆𝑆𝑚𝑚:
𝜀𝜀𝑚𝑚 =𝜀𝜀𝑚𝑚0− 𝛼𝛼(1− 𝜀𝜀𝑚𝑚0)𝐸𝐸 , 𝑆𝑆𝑚𝑚 = (1− 𝛼𝛼𝑆𝑆+𝛼𝛼𝑆𝑆𝑅𝑅ℎ2𝑠𝑠)𝑆𝑆𝑖𝑖0
The normalized distance 𝑅𝑅 from the center of an ice particle to an averaged contact plane is:
𝑅𝑅 =𝑅𝑅ℎ�1−2(1− 𝑠𝑠)
𝑍𝑍 �
Finally the permeation constant kD is calculated as follows:
𝑘𝑘𝐷𝐷 =𝜌𝜌𝑖𝑖𝐷𝐷
𝑟𝑟𝑖𝑖0 √𝑠𝑠𝑅𝑅ℎ𝑅𝑅
Δ�√𝑠𝑠𝑅𝑅ℎ(𝑅𝑅 − Δ) +Δ(𝑅𝑅ℎ − 𝑅𝑅)�
𝐷𝐷 is the apparent gas/water mass transfer coefficient that changes with temperature. All four constants (𝑘𝑘𝐺𝐺,𝑘𝑘𝑁𝑁, kR,𝑘𝑘𝐷𝐷) are assumed to be Arrhenius-type functions of temperature: energy of the J-type step of the hydrate formation. The gas constant is denoted here as 𝑅𝑅𝑔𝑔.
111 3.6.2 Formation from frost - “Frost” module
Frost particles create a special case of the general model described above, since a major volume of a few μm size ice is transformed already at the initial coating stage.
Limited reaction volume sets the maximum size of a single nucleus to the 4
3𝜋𝜋𝑟𝑟𝑖𝑖03 and its evolution from the moment 𝜏𝜏 to 𝑡𝑡 can be written as:
𝑉𝑉(𝑡𝑡 − 𝜏𝜏) =4
3𝜋𝜋 min{𝑟𝑟𝑖𝑖03,𝐺𝐺3(𝑡𝑡 − 𝜏𝜏)𝑚𝑚}
Defining the nucleation domain as a volume changes also the expression of the nucleation rate 𝑁𝑁̇:
𝑁𝑁̇ = 3𝑁𝑁0𝑡𝑡𝜎𝜎−1 𝑟𝑟𝑖𝑖0
Consequently the general JMAK equation is also redefined:
𝛼𝛼= 1− 𝑒𝑒�− ∫ Ω0𝑡𝑡 V(τ)dτ�
Similarly to the previous definition of Ω𝑆𝑆 a new term Ω𝑉𝑉 describes the rate of clathrate formation as a volume fraction of non reacted ice changed to hydrate in a time unit.
Refined JMAK approach gives:
The driving force and activation energy is calculated as previously.
112
113