Proceedings of the 7th International Conference on Gas Hydrates (ICGH 2011 ),
Edinburgh, Scotland, United Kingdom, July 17-21, 2011.
MODELING OF METHANE AND PROPANE HYDRATE FORMATION
KINETICS BASED ON CHEMICAL AFFINITY
Farshad Varaminian 
Department of Chemical Eng., Engineering Faculty, Semnan University, Semnan, IRAN
Amir Abbas Izadpanah
Department of Chemical Engineering, Engineering Faculty, Persian Gulf University,
Bushehr, IRAN
ABSTRACT
In this study, experimental data on the kinetics of methane and propane hydrate formation at
constant volume were collected. The experiments were carried out in a batch reactor at different
temperatures and pressures. The chemical affinity was used for the modeling of hydrate formation
rate in a constant volume process. In this method, the system was considered as classical
thermodynamic or macroscopic view. The results show that this method can predict constant
volume experimental data well for both crystals I and II hydrate former.
Keywords: Chemical affinity, Methane, Propane, Formation kinetics, Gas hydrates
NOMENCLATURE
a: chemical activity
A: chemical affinity
Ar : constant of proportionality
 : stoichiometric coefficient of reaction
A T ,V : affinity decay rate in constant temperature
Subscripts:
A: initial condition for hydrate formation
B: final condition for hydrate formation
i: arbitrary component
j: arbitrary data point
and volume
K: equilibrium constant
n: number of mole of gas that occupied the cavities
n w : number of water molecules
NH: number of hydrate forming gas
Ncav : number of cavities in hydrate structure
P: pressure
Q: amount of equilibrium constant in nonequilibrium conditions
R: universal gas constant
t: time
t k : time required to get equilibrium conditions.
T: temperature
V: volume
Z: compressibility factor
Greek letters:
 : chemical potential

 Q : extent of reaction based on mole
 t i : extent of reaction based on time
INTRODUCTION
Gas hydrates are crystalline solids. They are
composed of polyhedra of hydrogen bonded water
molecules. The crystalline structure forms cages
that contain at most one small guest molecule. At
low temperature close to freezing point of water
and high pressure, these guest molecules, which
are a large variety of gases or volatile liquids, can
form gas hydrates when they came into contact
with water under certain conditions
Many different thermodynamic models have
been suggested for predicting hydrate equilibrium
conditions [1, 2],but more studies must be done
Corresponding author: Phone: +98-231-3354100(3370) Fax: +98-231-3354092 E-mail: [email protected]
about their formation and decomposition rate.
There are many different views about the hydrate
formation process. A semi-empirical model was
proposed for the gas consumption rate [3,4]. Later
an intrinsic kinetics model for hydrate growth,
with only one adjustable parameter, was
formulated [5]. Hydrate formation is also proposed
by nucleation and growth processes. It is very
difficult to distinguish between nucleation and
growth rate since both processes occur
simultaneously. Many researchers work on
different models which heat and mass transfer are
control mechanisms [6,7] but because of the
complexity and stochastic nature of the process, a
model is needed that uses initial and final
conditions of the process to predict the formation
rate.
In this work, the chemical affinity was used for
modeling hydrate formation rate in a constant
volume process. More over an experimental setup
wa s set up to collect hydrate formation data.
MATERIALS AND PROCEDURES
The experiments were done in a system which
consist of a reactor, a shell for heat transfer and a
data acquisition system. A schematic of apparatus
is shown in Fig. 1 and detail description of
apparatus and procedure are given in [8].
equilibrium A  0 , and in any state in which the
reaction proceeds spontaneously A  0 .
Classically, the chemical potential  i of
component i in any arbitrary state can be related
to its chemical potential  i in some standard state
by an equation of the form
0
 i  i   i  i0  RT ln ai  i

(2)
where R is the gas constant and a i is the activity
of component i .
Equation (2) can be expressed in terms of affinity
by substituting from equation (1) to yield
A  A 0  RT
 ln a 
i
i
i
(3)
where A 0 is the affinity of the reacting system if
the components are in their standard state and A is
only a function of temperature. If an activity ratio
Q is defined as
Q   a i 
i
(4)
i
then equation (4) can be rewritten as
A  A0  RT ln Q
(5)
at reaction initiation Q  0 , so that A   ; while at
equilibrium A  0 , so that A 0  R T ln  K  , where
K is the thermodynamic equilibrium constant.
Substituting for A 0 in equation (5) yields
A  RT ln  Q 
(6)
where  Q  Q K  . By definition, the value of  Q


is limited to the range 0   Q  1 ; and its standard
Figure 1 Experimental setup, hydrate formation
apparatus
MODELING
Chemical affinity
Chemical affinity A , is defined as a generalized
driving force for chemical reactions as [9]:
A   i  i
i
(1)
where  i is the chemical potential of component
i.
According
to
this
definition,
at
state value  Q0 is equal to 1 K . Hence,  Q is a
dimensionless measure of the extent of reaction
from A   to A  0 .
Accordingly, for any chemical reaction proceeding
spontaneously in a closed system of fixed volume
V and at constant temperatureT , the affinity
decays towards zero, so that
A T ,V  0
(7)
 A

where A T ,V  
is the affinity decay rate.
  t  T,V

The calculated values of A
T ,V were correlated
with various functions of elapsed time t i by a
regression analysis to determine the best data fit.
However as soon as the reciprocal-time
relationship was examined it was apparent that
A T ,V wa s inversely proportional to the elapsed
time [10]
A T ,V  1 t  I
(8)
 = 0 at the equilibrium (or end
Because of A
T ,V
of reaction), the intercept ( I ) must depend on the
time of achieving the equilibrium ( t K ) and it can
rearrange the Eq. (8) to:
A T ,V  Ar  1  1 
 t
t K 
(9)
where Ar is a constant of proportionality, and
denotes the affinity rate constant.
In order to directly correlate the calculated values
of the chemical affinity with the measured values
of the elapsed time Eq. (9) must be integrated,
which yields
Ai  Ar ln  ti . exp 1   ti
(10)



t
where  t i   i t  is similar to  Q , and the extent
 K
of reaction  t i is limited to the range 0   ti  1 .
However the value of  t i must be known to
correlate empirical data by Eq. (11) so as to
determine the value of t K , but  t i itself depends
on t K . This obstacle was overcome by generating
values of t K by an iterative subroutine [11].
Hydrate formation modeling
As shown in Fig.2 the experimental conditions for
hydrate formation must be far away from the 3phase equilibrium curve (like point A). In a
constant volume –
constant temperature
experiments after the formation of hydrate
crystals; pressure decreases gradually because of
gas consumption and the final pressure must be
equal to Peq (point B). At this point, hydrate
formation stops and the system reaches
equilibrium.
In this research, the main assumption is that there
is similarity between hydrate formation and
chemical reaction in constant T and V conditions.
Both processes progress until they reach
equilibrium conditions. Then we supposed that
hydrate formation is a chemical reaction similar to
the following [12]:
NH
 NH


 ni 
 gi  nwH 2 O  

 gi .nw H2 O 





N
cav 
 i1  cav 

 Nn
i 1

i

(11)
where
ni
N cav
, nw are stoichiometric coefficients
of reaction for hydrate formation gas and water
respectively.
For calculating of affinity in different conditions
we must measure the extent of reaction with time
by using the pressure of gas in each elapsed time.
As shown in Fig. 2, the amount of total gas
consumed during hydrate formation is equal to
n A  nB  and the extent of reaction can be
obtain from
(12)
To calculate the number of moles of gas by using
measured pressure at system condition, we can use
an equation of state to calculate the
compressibility of gas phase and then calculate the
number of gas moles as
n
PV
Z RT
(13)
Now, the Eq. (12) can be rewritten as
Q 
PA Z A   Pi Z 
nA  n

n A  n B  PA Z A   PB Z B 
(14)
and we obtain affinity for each time by the formula
(15)
A   RT ln 
i
 
Qi


By plotting Ai versus ln  ti . exp 1   ti
 , we can
obtain Ar ,t K .
Figure 2 Hydrate formation condition in constant
P
A
C
Pini
Peq
B
Hydrate 3 phase
Equilibrium Curve
Texp
temperature
RESULTS AND DICUSSION
T
The results at different temperatures and at initial
experimental pressure are shown in Figs. 3-7 for
methane and Figs. 8-10 for propane (because of
simplicity the letter q was shown for  t i in these
figures). Compressibility of gas phase wa s
calculated by PR equation of state.
As we have seen there is a linear relation between
data. The experiments were done for methane at
temperatures 274, 276, 278, 280, 282 K and for
propane at temperatures 274, 275, 276 K with
different initial pressures. The calculated Ar , t K
are given in Table 1 for methane and Table 2 for
propane.


Figure 6 Affinity versus ln  ti .exp 1   ti
methane at 280 K and different initial pressures.


Figure 7 Affinity versus ln  ti .exp 1   ti


Figure 3 Affinity versus ln  ti .exp 1   ti
 for
 for
 for
methane at 282 K and different initial pressures.
methane at 274 K and different initial pressures.


Figure 8 Affinity versus ln  ti .exp 1   ti
 for
propane at 274 K and different initial pressures.


Figure 4 Affinity versus ln  ti .exp 1   ti
 for
methane at 276 K and different initial pressures.


Figure 9 Affinity versus ln  ti .exp 1   ti
 for
propane at 275 K and different initial pressures.


Figure 5 Affinity versus ln  ti .exp 1   ti
 for
methane at 278 K and different initial pressures.
274
274
274
275
275
275
276


Figure 10 Affinity versus ln  ti .exp 1   ti
 for
propane at 276 K and initial pressure.
Pinitial
 Bar 
Ar  J mol 
t K Sec
274
45.13
274
274
276
276
276
276
276
278
278
278
280
280
280
282
282
54.89
69.16
45.20
55.11
69.89
79.25
87.40
56.35
71.63
78.71
71.63
81.36
88.84
82.68
90.68
-3982
-3894
-3900
-5023
-4051
-6331
-5753
-4700
-4049
-4556
-3074
-3951
-3963
-4163
-4157
-4118
2353
2392
2254
2341
2231
1274
1363
1827
2949
2818
3197
4068
3585
3414
4763
3711
 
Texp K
Table 1. Calculated parameters of model for
methane at different conditions
 
Texp K
274
Pinitial
 Bar 
Ar  J mol 
t K Sec
2.84
-9700
10267
3.38
4.00
4.28
3.39
4.08
4.28
4.43
-9543
-9989
-10000
-10732
-11876
-9048
-9390
6936
5061
6410
12977
10325
9890
15800
Table 2. Calculated parameters of model for
propane at different conditions
The variation of the reactor pressures with time
wa s ca lculated using averaged Ar , t K for each
temperature from results in Table 1 and Table 2
for the experiments. In these experiments, the
variation of the pressures in the hydrate formation
reactor with time wa s measured and shown in
Figures 11(a to q) for methane and 12(a to h).
Figs. 11 and 12 demonstrate that there is a good
agreement between calculated results and
experimental data at low pressure near the 3-phase
equilibrium curve in most experiments. However
for the high pressure experiments, such
agreements exist at the beginning of the
experiments. It can be observed from high
pressure experiments that the final pressure of the
reactor is not Peq (point B in Fig.2). Theoretically,
the final pressure must be equal to the pressure of
point B. We know if the conditions of hydrate
formation were far from equilibrium (high driving
force or high initial pressure) the nuclei from
crystallization would become very small and the
rate of hydrate formation would be high, then a
large amount of hydrates would form in a short
time. If the mixing of the reactor were not
sufficient (in our experiments mixing is prepared
by movement of mercury in the reactor), hydrates
wo uld aggregate in the gas-liquid inter-phase and
reduce the mass transfer area and the kinetics of
hydrate formation would be affected by mass
transfer of gas into liquid. These are the
mainsprings
of
the
differences
between
experimental and modeling results at high initial
pressures and if there was a good mixing within
the reactor, the agreement of model will be good.
We ignored the last data of experiments that don’t
get the Peq in each experiments and calculation of
Ar and tk was based on the first rate data. Also this
model can be applied for crystals I and II gases.
Methane T=274K,P=45.13bar,Peq=29.5bar
46
44
Methane T=276K,P=79.25bar,Peq=35.67bar
80
calculated
experimental
calculated
experimental
75
42
70
65
38
36
P(bar)
P(bar)
40
34
32
45
0
500
1000
1500
2000
2500
40
ti(sec)
Figure 11.a Calculated and experimental rate data
for methane (T=274 K and Pinitial=45.13 bar)
70
35
0
500
1000
1500
2000
2500
ti(sec)
Figure 11.g Calculated and experimental rate data
for methane (T=276 K and Pinitial =79.25 bar)
Methane T=274K,P=69.16bar,Peq=29.5bar
calculated
experimental
65
Methane T=278K,P=71.63bar,Peq=43.3bar
75
60
P(bar)
55
50
30
28
60
calculated
55
70
50
65
experimental
45
P (bar)
60
40
35
50
30
25
0
55
500
1000
1500
2000
45
2500
ti(sec)
Figure 11.c Calculated and experimental rate data
for methane (T=274 K and Pinitial =69.16 bar)
55
Methane T=276K,P=55.11bar,Peq=35.67bar
calculated
experimental
40
0
500
1000
1500
ti(sec)
2000
2500
3000
Figure 11.j Calculated and experimental rate data
for methane (T=278 K and Pinitial =71.63 bar)
90
Methane T=278K,P=87.96bar,Peq=43.3bar
calculated
experimental
85
50
80
70
P(bar)
P(bar)
75
45
65
60
40
55
50
35
0
45
500
1000
1500
2000
2500
ti(sec)
Figure 11.e Calculated and experimental rate data
for methane (T=276 K and Pinitial =55.11 bar)
40
0
500
1000
1500
2000
2500
ti(sec)
Figure 11.l Calculated and experimental rate data
for methane (T=278 K and Pinitial =87.96 bar)
Methane T=280K,P=81.36bar,Peq=52.7bar
85
calculated
experimental
80
Propane T=274K,P=3.38bar,Peq=1.95bar
3.6
calculated
experimental
3.4
3.2
75
P(bar)
P(bar)
3
70
65
2.8
2.6
2.4
60
2.2
55
2
50
0
500
1000
1500
2000
ti(sec)
2500
3000
3500
4000
Figure 11.n Calculated and experimental rate data
for methane (T=280 K and Pinitial =81.36 bar)
1.8
0
1000
4000
ti(sec)
5000
6000
7000
8000
Propane T=274K,P=4bar,Peq=1.95bar
4
calculated
experimental
85
3000
Figure 12.b Calculated and experimental rate data
for propane (T=274 K and Pinitial =3.38 bar)
Methane T=280K,P=88.84bar,Peq=52.7bar
90
2000
calculated
experimental
3.5
80
3
70
P(bar)
P(bar)
75
65
2.5
60
2
55
50
0
500
1000
1500
2000
ti(sec)
2500
3000
3500
4000
Figure 11.o Calculated and experimental rate data
for methane (T=280 K and Pinitial =88.84 bar)
Methane T=282K,P=90.68bar,Peq=64.6bar
95
1.5
0
2000
3000
4000
ti(sec)
5000
6000
7000
Figure 12c Calculated and experimental rate data
for propane (T=274 K and Pinitial =4.0 bar)
calculated
experimental
90
1000
Propane T=275K,P=4.08bar,Peq=2.4bar
4.5
calculated
experimental
85
4
P(bar)
P(bar)
80
75
3.5
70
3
65
60
0
500
1000
1500
2000
ti(sec)
2500
3000
3500
4000
Figure 11.q Calculated and experimental rate data
for methane (T=282 K and Pinitial =90.68 bar)
2.5
0
2000
4000
6000
ti(sec)
8000
10000
12000
Figure 12.f Calculated and experimental rate data
for propane (T=275 K and Pinitial =4.08 bar)
Propane T=276K,P=4.43bar,Peq=3bar
4.5
calculated
experimental
P(bar)
4
3.5
3
0
2000
4000
6000
8000
ti(sec)
10000
12000
14000
16000
Figure 12.h Calculated and experimental rate data
for propane (T=276 K and Pinitial =4.43 bar)
CONCLUSIONS
There are many models in literature for gas
hydrate formation that using a microscopic driving
force like mass transfer from gas phase to water or
heat transfer between solid particles and the bulk
of liquid. Such models need experimental
parameters like the mass transfer coefficients or
heat transfer coefficients or population of particles
and these coefficients differ for each experiment.
But in this research a conceptual model wa s
proposed that define a macroscopic driving force
that only need the initial condition (experimental
condition, temperature and pressure) and final
conditions (equilibrium conditions). The basic idea
is that there is a unique path for each experiment
which the crystallization process decays the
affinity. Microscopic phenomena such as mass
transfer, heat transfer, nucleation occurs in their
special rates and causes the overall formation rate
so that the time for this affinity decay is minimal.
Because of the thermodynamic relation between
affinity and Helmholtz free energy, this model can
directly and easily be used for gas mixture and
predicting the Ar for pure gases.
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