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Virginia Tech | As is commonly understood, the two-parameter Langmuir equation is based on the assumption of
monolayer adsorption. This means if the adsorption equilibrium pressure increases and multilayer
adsorption occurs, the two-parameter Langmuir equation is no longer as accurate as in the low-
pressure scenarios. It is critical to find a new model to describe the relationship between the true
adsorption content and pressure.
It should be pointed out that under low-pressure conditions (< 10-15 MPa), the volume of the
adsorbed layer is very small and the application of equation (3) is acceptable and has been widely
used for estimating GIP for shallow coal seams and shale formations. Because of limited data for
methane adsorption in shale under high-pressure conditions (>15 MPa), the commonplace
approach uses methane adsorption measurements under intermediate-pressure conditions (10-15
MPa), on the basis of Equations (9) and (10), to predict the methane adsorption behavior in the
higher-pressure region (>15 MPa).
In practice, we can directly calculate GIP using laboratory-measured data without assuming V to
a
be negligible (Tian et al, 2016; Bruns et al, 2016):
GIP V n (5)
gas tot e
where n is the observed adsorption quantity under reservoir temperature and pressure, also called
e
Gibbs excess adsorption quantity. Equation (5) indicates that if all the observed adsorption
isotherm tests are available, we can accurately estimate the shale GIP resource in the subsurface.
Equation (5) is superior since it does not use any assumptions similar to equation (3), and can be
used under any pressure. Equation (5) also indicates that if we only want to obtain the shale GIP
content in shale formations and do not want to differentiate the exact ratio between adsorbed gas
and free gas, the observed adsorption isotherms measured under reservoir conditions is sufficient.
2.3.2.2 Shale gas transport model in shale formations
Existing studies have demonstrated that adsorbed gas accounts for 20-80% of the total shale GIP
content (Curtis et al, 2012). Therefore, shale gas production has to consider the adsorbed gas
content in both shale gas resource estimation stage and shale gas recovery stage. This also means
shale gas transport models for predicting shale gas production should take the adsorbed methane
content into consideration.
59 |
Virginia Tech | Shale gas production is a complex, coupled process from nanoscale to reservoir scale. In order to
accurately investigate gas transport in shale, the ad/desorption processes of gas in shale must be
considered (Yu et al, 2014; Akkutlu et al, 2012; Civan et al, 2011; Singh et al, 2016; Wu et al,
2015; Naraghi et al, 2015; Wu et al, 2016). Since shale is rich in nanopores (Chen et al., 2013),
the large surface area strengthens the adsorbed surface diffusion process (Ross et al, 2009;
Chalmers et al, 2012). In addition, the volume of the adsorbed layers in nanopores (<10 nm) cannot
be neglected. When gas desorbs to free gas, it increases the space for gas transport in these pores
(Ambrose et al, 2012; Singh et al, 2016). Several researchers have proposed different gas transport
models in shales by considering the gas adsorption effect (Akkutlu et al, 2012; Civan et al, 2011;
Yu et al, 2014; Singh et al, 2016; Wu et al, 2015; Naraghi et al, 2015; Wang et al, 2015). All these
models have a common characteristic that they employ the two-parameter Langmuir equation
(equation 4) to describe the adsorbed methane phase based on observed adsorption isotherms even
though it has been found the two-parameter Langmuir equation failed to describe the observed
methane adsorption isotherms in shales (Rexer et al, 2013; Tian et al, 2016; Bruns et al, 2016;
Gasparik, et al, 2014). This indicates that conclusions from current available models may not be
reliable under high pressure conditions since they always use the incorrect ratio between free gas
and adsorbed gas in their models.
In order to differentiate the exact ratio between adsorbed gas and free gas under reservoir
conditions especially when the Gibbs excess adsorption phenomena becomes obvious, some
researchers attempt to use available adsorption models (three or more parameters adsorption model)
to predict absolute adsorption isotherms (Rexer et al, 2013; Tian et al, 2016; Bruns et al, 2016;
Gasparik, et al, 2014). The typical method is to use available adsorption models to fit observed
adsorption isotherms independently and then obtain the empirical relationship between the fitting
parameters and temperatures. Based on the obtained empirical relationship, the adsorption
isotherm beyond the test data is extrapolated. The absolute adsorption isotherms is also obtained
based on either the constant density of the adsorbed layer or the constant volume of the adsorbed
layer (Rexer et al, 2013; Tian et al, 2016; Bruns et al, 2016; Gasparik, et al, 2014). However, this
commonplace method lacks theoretical support and is problematic because the physical meaning
of the fitting parameters has already changed when they fit the observed data independently. One
of the obvious and critical defects is that the obtained adsorbed density of methane is higher than
the liquid density, which should not occur (Do et al, 2003; Zhou et al, 2000 & 2001). Furthermore,
60 |
Virginia Tech | they have not realized the large difference between observed isotherms and absolute adsorption
isotherms for methane in shale under high pressure because of their limited test pressure ranges.
It is helpful to point out a historical misunderstood concept in the shale gas industry for the past
twenty years: the definition of free gas and adsorbed gas for shale gas in shale formations. In the
shale gas industry, equation (9) is the most widely used, where we call the term ( V ) as the
gas tot
“true free gas” and (n ) as the “true adsorbed gas”. This is incorrect because the true free gas
a
should be term (n V ) and true adsorbed gas should be (n V ). Since V is always
free gas free a a a tot
higher than V , this results in overestimation of the true free gas content as shown in Figure 2.3.1.
free
That means if we use equation (3) to obtain the ratio between free gas and adsorbed gas, we will
miscalculate the ratio between free gas and adsorbed gas, which has been commonly used in
published reports of shale GIP resources over the past twenty years making them unreliable.
Figure 2.3.2 Conventional shale gas research methodology
From the above discussion, it can be concluded current shale gas development theories are based
on the two parameter Langmuir equations as shown in Figure 2.3.2. The reason why two parameter
Langmuir equation is widely used is only because of its mathematical simplicity. Under low
pressure conditions where the volume of the adsorbed layer can be neglected, the usage of two
parameter Langmuir equation produces valid results. However, under high pressure conditions
where the Gibbs excess adsorption phenomenon becomes obvious, the usage of the two parameter
Langmuir equation to describe observed adsorption isotherms results in many problems, especially
in estimating shale gas resource and modelling shale gas transport behavior in shale formations.
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Virginia Tech | 2.3.3 Concept of deep shale gas reservoir and its implication
2.3.3.1 Dual-site Langmuir model for describing high pressure methane adsorption in shale
In a previous study, we measured methane adsorption isotherms in Longmaxi shale sample under
303.15, 318.15, 333.15 and 355.15K and up to 27 MPa as shown in Figure 2.3.3 (Tang et al, 2016).
It was found that the dual-site Langmuir model can not only describe observed methane adsorption
behavior using equation (6) and interpret all observed adsorption phenomenon but also can predict
absolute methane adsorption isotherms using equation (7) and extrapolate adsorption isotherms
beyond test temperatures (without using any empirical relationship). Detailed discussion of the
dual site Langmuir equation refers to Tang et al, 2016. The successful application of the dual site
Langmuir model lays the foundation to predict shale gas resource and model shale gas transport
behavior in high pressure deep shale formations.
K (T)P K (T)P
n (n V ) (1)( 1 )( 2 ) (6)
e max max g 1K (T)P 1K (T)P
1 2
K (T)P K (T)P
n n (1) 1 2 ) (7)
a max 1K (T)P 1K (T)P
1 2
where n is the observed adsorption uptake, also called Gibbs excess adsorption uptake, n is the
e a
absolute adsorption quantity under equilibrium temperature (T) and pressure (P), n is the
max
maximum adsorption capacity, V is the volume of the adsorbed phase at maximum adsorption
max
capacity, is the bulk gas density, K (T) and K (T) ( K (T)A exp( E 1 ) and
g 1 2 1 1 RT
E
K (T) A exp( 2 ) ), are equilibrium constants weighted by a coefficient α (0<α<1), E 1 and E 2
2 2 RT
are the energy of adsorption, and A and A are the pre-exponential coefficient (where both E and
1 2 0
A are independent of temperature), P is equilibrium pressure.
0
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Virginia Tech | Figure 2.3.3 High pressure methane adsorption isotherms under different temperatures;
solid squires are measured data, solid color lines are fitting curves using equation (6), dotted
color lines are fitting curves using equation (7), black solid and dotted lines are extrapolated
adsorption isotherms beyond test data
2.3.3.2 Concept of deep shale gas reservoir
Shale formations usually have a depth of 500- 3000 meters. A depth deeper than 1000 meters can
lead to a high pressure reservoir. Generally speaking, the deeper the shale formation, the higher
the reservoir pressure. High pressure is one of the major characteristics of deep shale formations.
As discussed previously, the high pressure condition results in the pronounced difference between
observed adsorption isotherms (Gibbs adsorption isotherms) and true adsorption isotherms for
methane in shale. The two-parameter Langmuir model is no longer valid to describe either the
observed adsorption isotherms or true adsorption isotherms. We can no longer use the current
shallow shale gas and coalbed methane recovery theory to guide the development of deep, high-
pressure shale gas recovery technologies. We have to develop a new theory that is suitable to high-
pressure shale gas reservoirs on the basis of the dual site Langmuir equation.
In order to differentiate deep high-pressure shale gas reservoirs and shallow low-pressure shale
and coalbed methane reservoirs, we introduce a new concept, the deep shale gas reservoir. Deep
shale gas reservoirs specifically refer to deep shale gas formations, where the in-situ reservoir
pressure and temperature cause methane adsorption to increase and reach the maximum value and
then decrease before the adsorption equilibrium pressures reach the in-situ reservoir pressure
(shown in Figure 2.3.4). The inflection pressure in Figure 2.3.4 refers to the corresponding pressure
at the maximum observed adsorption content. If the inflection pressure is higher than the reservoir
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Virginia Tech | pressure, the shale gas reservoir is a conventional shallow shale gas reservoir and all current
theories are valid. If the inflection pressure is lower than the reservoir pressure, the shale gas
reservoir becomes a deep shale gas reservoir. For deep shale gas reservoirs, the two-parameter
Langmuir model becomes invalid. The dual-site Langmuir model is available to use for assessing
the shale GIP resource and modeling shale gas transport behavior in shale formations.
Previous studies have shown that the occurrence of the pronounced Gibbs excess adsorption
behavior also depends on many other physical properties, such as moisture content, kerogens
maturity, mineral composition, pore characterization and surface area, et al. Therefore, in order to
confirm whether the shale gas reservoir is a deep shale gas reservoir or shallow shale gas reservoir,
the very first step is to conduct methane adsorption isotherm measuremens under reservoir
conditions. The pronounced Gibbs excess adsorption behavior for methane adsorption isotherms
at the reservoir pressure and temperatures (the critical pressure is lower than the reservoir pressure)
is the only necessary condition.
Figure 2.3.4 Fundamentals for shale gas development
2.3.4 Implications for shale gas development
2.3.4.1 Deep shale GIP estimation
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Virginia Tech | As is commonly known, shale gas resources typically exist in deep formations and the reservoir
pressure and temperature can be as high as 27MPa and 76 ℃, respectively (Curtis et al, 2002). The
high-pressure, high-temperature in-situ condition does affect methane adsorption behavior in shale.
If equation (5) is used to estimate the shale GIP resource, there is no need differentiate whether
the shale gas reservoir is shallow shale gas reservoir or deep shale gas reservoir. However, there
is a significant cost because one must conduct many high pressure methane adsorption tests under
different temperatures because of the increasing geothermal effect in the deep subsurface. As
revealed in the literature, few labs have the ability to conduct methane adsorption tests in shale up
to 27MPa (Zhang et al, 2012; Ji et al, 2012; Tang et al, 2016; Gasparik, et al, 2014). Therefore, in
order to assess the shale GIP resource in deep shale formations accurately and decrease the
experimental cost, the first step is to determine the type of shale gas reservoir, the shallow shale
gas reservoir or the deep shale gas reservoir. The methane adsorption isotherms under reservoir
conditions need to be measured to do these assessments. If the shale gas reservoir belongs to
shallow shale gas reservoir, the conventional two parameter Langmuir equation methodology
approximates the real shale GIP resource. If the shale gas reservoir belongs to a deep shale gas
reservoir, the dual-site Langmuir model should be used to describe the methane adsorption
behavior and predict shale GIP resource. Another feature of the dual site Langmuir equation is that
it can be used to extrapolate adsorption isotherms beyond test temperatures without using an
empirical relationship.
It is worth to note that using equation (5) can only show the total shale GIP resource but not the
ratio between the bulk gas and free gas. This means one does not know which part contributes
more for the shale gas production, the free shale gas or the adsorbed shale gas. As pointed out
earlier, it is a historical misunderstanding that we treat the ( V ) term as the free gas in the
gas tot
subsurface, which results in the overestimation of the free shale gas resource in the subsurface.
2.3.4.2 Thermodynamic analysis for methane in shales
Thermodynamics analysis can reflect the interaction between gas adsorbate and adsorbent for an
equilibrium gas sorption system. For example, the thermodynamic index such as the isosteric heat
of adsorption reflects how the enthalpy changes when the unit amount of adsorbate is adsorbed on
a certain amount of adsorbent molecular (Pan et al, 1998; Sircar et al, 1999; Shen et al, 2000).
Previous studies of methane in shales have also reported the isosteric heat of adsorption during the
65 |
Virginia Tech | adsorption process (Zhang et al, 2012; Gasparik et al, 2014; Ji et al, 2012). However, there are
several problems associated with these studies in the calculation of the isosteric heat of adsorption
(Dejardin et al, 1982; Pan et al, 1998; Stadie et al, 2013 & 2015). First, how to obtain the absolute
adsorption isotherms from observed adsorption isotherms (Pan et al, 1998; Stadie et al, 2015).
Thermodynamic analysis must use the absolute adsorption uptake instead of the observed
adsorption uptake. Generally, the liquid density of methane is used to obtain absolute methane
isotherms from observed adsorption isotherms, which is still arguable (Pini, 2010; Bae et al, 2006;
Sakurovs et al, 2007; Stadie et al, 2013 & 2015). Second, the classic Clausius–Clapeyron
approximation is on the basis of ideal gas assumption and that the contribution of the adsorbed gas
phase is ignored (Pan et al, 1998; Stadie et al, 2012; Krishna et al, 2015; Askalany et al, 2015).
When the Gibbs excess adsorption phenomenon becomes pronounced, both assumptions are
incorrect for gas adsorption isotherms. Previous studies in thermodynamics analysis do not address
these questions and there conclusions needs to be treated with cautions (Zhang et al, 2012;
Gasparik et al, 2014; Ji et al, 2012).
Fortunately, the dual site Langmuir equation provides an option to solve these problems. As shown
in Tang et al 2016, the dual site Langmuir model can reasonably address all observed adsorption
phenomenon during the adsorption tests such as interpretation of the crossover of the adsorption
isotherms, predicting absolute adsorption isotherms, and extrapolating isotherms beyond test
temperatures without using any empirical relationships. Furthermore, it can also be used to
calculate the isosteric heat of adsorption for methane in synthetic material, which may be used for
thermodynamic analysis of high pressure methane in shales (Stadie et al, 2013 & 2015).
2.3.4.3 Shale gas transport model for deep shale gas reservoir
For shallow shale gas formations, the two parameter Langmuir works well because the observed
adsorption isotherms approximate the true adsorption isotherms. Current shale gas transport
models can still describe methane transport behavior in shale and predict shale gas well production
behavior (Akkutlu et al, 2012; Civan et al, 2011; Yu et al, 2014; Singh et al, 2016; Wu et al, 2015;
Naraghi et al, 2015). However, all these models cannot be extended to the deep shale gas reservoirs
since the two-parameter Langmuir model does not represent the true adsorbed gas content. In deep
shale gas reservoirs, the observed adsorption isotherms no longer approximate the true adsorption
isotherms. In fact the observed adsorption content is much lower than the true adsorption content
66 |
Virginia Tech | as illustrated in Figure 2.3.3. Therefore, the dual site Langmuir equation for describing the absolute
adsorption isotherms (equation (7)) should be used in these shale gas transport models. By
applying the absolute adsorption isotherms, the true ratio between free gas and adsorbed gas can
be differentiated. Then, the true contribution of either free gas or adsorbed gas for the total shale
gas production can be reasonably investigated.
2.3.5 Conclusions
Based on the Gibbs excess adsorption phenomenon for high pressure methane adsorption in shale,
this work introduces a new concept, the deep shale gas reservoir. This concept offers a new theory
frame work for shale gas development and calls for more in-depth studies in shale GIP estimation,
thermodynamics analysis in high pressure gas adsorption, and shale gas transport models for deep
shale gas reservoirs. On the basis that the dual-site Langmuir model can not only describe the
methane adsorption behavior under high pressure conditions but also differentiate the true
adsorbed methane content and gaseous methane content in deep shale gas reservoirs, the dual site
Langmuir model lays the foundation for developing new techniques in deep shale gas development.
Acknowledgements
This research was supported in part by the U.S. Department of Energy through the National Energy
Technology Laboratory’s Program (No. DE-FE0006827).
References
Akkutlu, I. Y., & Fathi, E. (2012). Multiscale gas transport in shales with local kerogen
heterogeneities. SPE Journal, 17(04), 1-002.
Askalany, A. A., & Saha, B. B. (2015). Derivation of isosteric heat of adsorption for non-ideal
gases. International Journal of Heat and Mass Transfer, 89, 186-192.
Ambrose, R. J., Hartman, R. C., Diaz-Campos, M., Akkutlu, I. Y., & Sondergeld, C. H. (2012).
Shale gas-in-place calculations part I: new pore-scale considerations. SPE Journal, 17(01), 219-
229.
Andrews, I. J. (2013). The Carboniferous Bowland Shale gas study: geology and resource
estimation.
67 |
Virginia Tech | Chapter 3 Thermodynamic analysis for gas adsorption in shale and coal
3.1 Adsorption affinity of different types of coal: mean isosteric aeat of adsorption
Xu Tang*a, Zhaofeng Wangb,c, Nino Ripepia, Bo Kangb,c, Gaowei Yueb,c
(a Department of Mining and Minerals Engineering, Virginia Polytechnic Institute and State
University, Blacksburg, Virginia, 24060, U.S.b College of Safety Science and Engineering, Henan
Polytechnic University, Jiaozuo, Henan, 454000,China;c the State Key Laboratory Cultivation
Base for Gas Geology and Gas Control, Henan Polytechnic University, Jiaozuo, Henan,
454000,China)
Abstract: Understanding the sorption behavior of gas in organic-rich sedimentary rocks, and more
specifically recognizing the adsorption properties of methane in coal, is a crucial step for
evaluating the coalbed methane (CBM) gas-in-place content, gas quality and CBM recovery
potential. However, the adsorption affinity of coal on methane has not been previously considered.
This paper introduces the isosteric heat of adsorption in Henry’s region, renamed the mean
isosteric heat of adsorption, as means to evaluate the adsorption affinity of coal on methane. 18
group isothermal adsorption tests for methane in three different coals were conducted from
243.15K to 303.15K. The mean isosteric heat of adsorption for anthracite, lean coal, and gas-fat
coal is -23.31KJ/mol, -20.47 KJ/mol, and -11.14 KJ/mol, respectively. The minus signs indicate
the adsorption is an exothermal process. The mean isosteric heat of adsorption is independent of
temperature from 243.15K to 303.15K, and shows the overall heterogenous property of different
coal. Therefore, the mean isosteric adsorption of heat can serve as a quantified index to evaluate
the coal adsorption affinity on methane.
Key words: Coal; Isothermal adsorption; Affinity; Methane; Outburst
Published in Energy Fuels, 29 (6), 2015, pp 3609–3615.
73 |
Virginia Tech | 3.1.1 Introduction and background
Methane sorption properties in coal is crucial for coalbed methane (CBM) gas-in-place estimation
[1, 2], coal seam degasification in underground coal mines [3, 4], and carbon dioxide sequestration
with enhanced CBM recovery [1, 5-7]. Generally, the methane content in the coal seams consists
mainly of adsorbed gas and free gas, with the adsorbed gas accounting for 80-90% of the coal
seam content. Since the adsorbed gas plays a significant role in determining the coal seam’s content,
the adsorption properties of methane in coal is an important topic for researchers [1-13]. Even
though there are lots of models used to describe the sorption properties of coal [12, 14-17], the
affinity of methane on different types of coal has not received as much attention.
For a gas and solid sorption system, when the pressure is low, the gas adsorption is proportional
to the equilibrium pressure; this is called Henry’s law. This has been validated by classical
statistical thermodynamics. Henry’s law describes the affinity between the adsorbate molecule and
the adsorbent. In Henry’s region, each gas molecule can explore the whole adsorbent surface
independently, as the interactions among gas molecules are negligible because of low densities
[18]. Therefore, the isosteric heat of adsorption in Henry’s region obtained via Henry’s coefficient
become a unique index for evaluating the affinity between an adsorbate molecule and the adsorbent.
This relationship has already been considerably studied for gas and solid interaction [18-24] and
chromatographic measurements of retention volumes [25-26]. Surprisingly, the isosteric heat of
adsorption has not been previously considered for organic materials and gas sorption system such
as coal and methane.
The theoretical calculation for Henry’s coefficient is based on the assumption that 1) Henry’s
coefficient is a function of temperature and the interaction energy of one adsorbate molecule with
the surrounding adsorbent, and 2) the interaction among adsorbate molecules can be neglected [23,
27]. Generally, the accuracy of Henry’s coefficient determines the accuracy of the mean isosteric
heat of adsorption. For manmade materials such as carbon nanotube, the Steele’s equation can be
used to calculate Henry’s coefficient based on the energetically homogeneous adsorbent
assumption [27, 28] (see Section 4.2). However, this theoretical calculation is not applicable for
coal because it is difficult to identify the complex, quantitative-pore system of coal and the
heterogeneous properties of coal. Also, the isosteric heat of adsorption in Henry’s region cannot
indicate the pore features (pore width, pore shape, etc.) of coal as it can with manmade carbon
74 |
Virginia Tech | nanotube [18, 24]. Because of this, the mean isosteric heat of adsorption is introduced to rename
the isosteric heat of adsorption in Henry’s region for coal in order to distinguish the isosteric heat
of adsorption for manmade materials. The mean isosteric heat of adsorption should show the
affinity of coal on methane, which results from the integrated effects of the pore size, shape,
intersection and the surface area in coal, or the overall heterogeneous property of different coal.
This index may serve as a fundamental parameter to evaluate the adsorption affinity of coal
theoretically, which requires support from the test data.
In order to explore the coal affinity on methane, 18 isothermal adsorption tests from 243.15K to
303.15K were conducted on three different types of coal (anthracite, lean coal, and gas-fat coal)
using in-house low temperature isothermal adsorption equipment. Since the low temperature
isothermal adsorption tests for coal and methane (under 273.15K) have not been reported before,
the tests are introduced in detail (see Section 2). Then, two approaches for calculating the mean
isosteric heat of adsorption are introduced (see Section 3). Finally, the test results are analyzed and
discussed (see Section 4).
3.1.2. Isothermal adsorption tests: from 243.15K to 303.15K
3.1.2.1 Sample preparation
The different types of coal used in this study were obtained from the Jiulishan coal mine, the
Xinyuan coal mine, and the Panbei coal mine in China. The physical parameters of the coal were
evaluated using Chinese national standards (Table 3.1.1). The coal specimen was then ground and
sieved using 0.17mm-0.25mm metal sifters and placed in a drying oven at 104 to 110℃ for 1 hour
to dehydrate. After dehydration the prepared sample was stored in a dehydrator for later use.
Table 3.1.1 Physical parameters of coal sample
75 |
Virginia Tech | 3.1.2.2 Isothermal test procedure
Figure 3.1.1 Schematic setup for low temperature isothermal sorption-diffusion
comprehensive device; 1-Gas Chromatograph (GC), 2-Data recording module, 3-Vaccum
pump, 4- Vaccum gage, 5- Water injection pump, 6-Measuring cylinder, 7-Sample cell; the
low temperature control system can control the temperature between 225.15K and 373.15K
with fluctuation of ±0.5 K.
The isothermal test was conducted using the in-house low temperature isothermal instrument based
on the volumetric method (Fig. 1). The general test procedures for isothermal adsorption testing
are shown below [28],
1) Calibrate the sample cell volume and double-check the tightness of the whole test
system.
2) Vacuum the sample cell and then charge the sample cell gas via the reference cell. The
adsorption gas content is calculated by the following equation,
madsorbedminjected
munadsorbed(PVM
)
(PV voidM
)
gas gas gas ZRT pump ZRT samplecell
where m is the mass of gas, P is pressure, T is temperature, M is the molar mass of the
gas species, Z is the compressibility coefficient of methane calculated using the
Redlich-Kwong equation (when pressure is less than 9 MPa), R is the universe gas
coefficient, △V is the volume change of the pump, and V is the volume of the free
void
gas in the sample cell.
3) Monitor the pressure change of the sample cell to determine the point of the sorption
equilibrium state or suspend the time for the sorpiton process. Once the equilibrium
point is reached, this phase ends and the sorption content and pressure can be obtained.
76 |
Virginia Tech | 4) Repeat step 2) and step 3) until the next defined equilibrium pressure point is reached.
5) Once all the equilibrium points are obtained, the test is suspended and the isothermal
adsorption curve can be established.
3.1.2.3 Sorption equilibrium state determination
Sorption equilibrium state determination is very important for the accuracy of test results. The
pressure and sorption time monitoring approach are the two most popular approaches for
determining the equilibrium state. The pressure monitor approach measures the change of the
pressure cell; if the pressure change of the sample cell is within a certain value, the sorption system
is treated as having reached a sorption equilibrium state. The sorption time monitor method is an
empirical based method, and different research groups use different sorption times for isothermal
adsorption testing [29-33]. However, there is no international standard can be referred for sorption
equilibrium state determination.
This is the first tentative isothermal adsorption test for coal and methane from 263.15K to 243.15K
for coal and methane. Thus, determining the equilibrium state is the key step in obtaining credible
and accurate test results. The procedure to determine the equilibrium state is introduced in detail
here. First, the sample cell with a sorption equilibrium coal-methane under 293.15 K is reached.
Second, the sorption equilibrium sample cell is put into the 253.15 K low temperature control
system, and the pressure variation with time inside the cell is recorded (shown in Fig. 3.1.2). Figure
3.1.2 shows the pressure of the sample cell decreasing over time, and a sharp decrease occurs
within the first two hours. The pressures of the sample cells at times of 1.84h, 9.49h, and 23.30h
are 6.1709, 6.1685, and 6.1660 MPa, and the pressure differences are only 0.08% and 0.04%
compared with the pressure at 23.30h. Since there is only a tiny change of the pressure in the
sample cell after it stays inside the low temperature system for 2 hours, the authors consider that
the sample cell almost approaches the equilibrium status.
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Virginia Tech | Figure 3.1.2. Sorption pressure decreases with time in low temperature control system
Based on the sorption equilibrium determination test data, the authors take two steps to ensure the
equilibrium state of the coal and methane sorption system under low temperatures: 1) the
equilibrium coal-methane sorption sample cell under 293.15 K is acquired for 12 hour sorption,
and 2) the sample cell obtained in step (1) is then put into the low temperature control system
under different temperatures (243.15 K, 253.15 K, 263.15K, 273.15K) for another 12 hours. Once
both steps are completed, the authors assume the sample cell has reached an equilibrium state
under defined low temperatures.
3.1.2.4 Test results
Figure 3.1.3 shows the isothermal adsorption of methane in anthracite, lean coal, and gas-fat coal
under different temperatures ranging from 243.15K to 303.15K. It was found that the adsorbed
methane content increases with decreasing temperature, and that coal at the temperatures lower
than 273.15K adsorbed more methane than that of above 273.15K. The maximum adsorption
content of anthracite, lean coal and gas-fat coal increases at 0.19cm3/g, 0.15 cm3/g, and 0.13 cm3/g
respectively, when temperature decreases at 1K.
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Virginia Tech | If the mean isosteric heat of adsorption is constant and independent of temperature, the integration
of equation (3) is:
H 1
ln(K') 0 ln(K') (3)
R T 0
From equation (3), if the linear relationship between lnK’ and 1/T is obtained from equation (3),
the mean isosteric heat of adsorption can be calculated using the slope of the linear line.
However, under certain circumstances, the mean isosteric heat of adsorption is influenced by
temperature and equation (3) cannot be used. In 2011, T. Galanon & V. David proposed a binomial
expression to describe the relationship between Henry’s coefficient and temperature (equation (4))
[34], where equation (5) is used to calculate the temperature influenced mean isosteric heat of
adsorption [34,35]:
b c
ln(K')a
T T2
(4)
c
H R(a2 ) (5)
0 T
where a, b and c are fitting parameters. The temperature dependent mean isosteric heat of
adsorption can be obtained by the fitting parameters.
In order to determine the mean isosteric heat of adsorption from equation (3) and (4), the Henry
coefficient (K’) under different temperatures is first calculated. To calculate Henry's law constants,
adsorption in the low-pressure region is modeled by a Virial-type equation [36-37]:
n
ln( ) A An A n2 (6)
p 0 1 2
where n is the content of adsorbed gas at pressure p, and the first virial coefficient A is related to
0
the Henry’s law constant, K’, and K’=exp(A ).
0
When n is small, the high-order term can be neglected, and equation (6) can be written in the
following form:
p
ln( )A An (7)
n 0 1
80 |
Virginia Tech | Table 3.1.3 Determination of the mean isosteric heat of adsorption in coal
Once the Henry’s coefficients are calculated under different temperatures, the relationship between
lnK’ and 1/T can be described. Figure 3.1.5 shows the linear relationship between ln (K’) and the
reciprocal of temperature for three different types of coal in the temperature range of 243.15K to
303.15K, which satisfies Equation (3). This also means the mean isosteric heat of adsorption can
be treated as a constant and is independent of temperature between 243.15K and 303.15K. The
mean isosteric heat of adsorption is shown in Table 3.1.3, and the minus sign means the adsorption
is an exothermal process.
It should be pointed out that the volumetric approach for isothermal adsorption testing is preferred
for measuring Henry’s coefficients. This is because the amount of gas adsorbed is determined by
the large difference in between the amount of gas dosed to the system and the amount of gas left
in the system after adsorption, instead of by the small weight difference under low pressure [38].
When the pressure is low, the small difference between the weight before and after adsorption
increases the experimental error via the gravimetric approach.
3.1.4.2 Discussion
When the temperature of isothermal tests ranges from 243.15K to 303.15K, the value of the mean
isosteric heat of adsorption decreases in the following order: anthracite, lean coal and gas-fat coal.
This agrees with the general theory that higher rank coal usually has higher adsorption capacity
under same sorption conditions [8, 12, 39].
Theoretically, the Henry’s coefficient can be obtained based on the energetically homogeneous
assumption of adsorbent [40]:
S Zmax (z)
H BET [exp( )1]dz
RT 0 kT (8)
where H is Henry’s coefficient, S is the BET surface area, Z is the distance perpendicular to the
BET
surface, Z depends on the structure of the solid, T is temperature, and φ(z) is the interaction
max
83 |
Virginia Tech | energy. Several approaches have been proposed for calculating the interaction energy [27, 28], and
most of these approaches are suitable for analyzing the uniform pore of man-made material [18-
24]. However, for natural material such as coal, it is hard to acquire the accurate interaction energy
φ(z) only through simplified assumptions. Surface chemistry plays an important role for the
adsorption characteristics; the heterogeneous properties of coal with complex structure, pore size
and shape distribution results in the characterized adsorption sites with different energies. The
mean isosteric heat of adsorption found using the Henry’s coefficient includes the overall effects
of heterogeneous coal properties, making it a better option for evaluating the adsorption affinity
of coal. The isothermal adsorption approach is more applicable for obtaining the Henry’s constant
because of the shortcomings of the theoretical approach.
As mentioned earlier, the experimental approach to obtain Henry’s coefficient is based on two
assumptions that 1) Henry’s coefficient is a function of temperature and the interaction energy of
one adsorbate molecule with the surrounding adsorbent, and 2) the interaction among adsorbate
molecules can be neglected. When temperature ranges from 243.15K- 303.15K, Henry’s law is
applicable for three different types of coal (Figure 3.1.4). This supports that when the sorption
content is low, Henry’s coefficient is only dependent on the interaction between the adsorbent
surface and the adsorbed gas molecules. According to the kinetic theory of gas, higher temperature
means the average kinetic energy of methane molecular is higher, and therefore the interaction
among methane molecular in higher temperature system cannot be neglected. For a low
temperature system (243.15K- 303.15K), the interaction energy between methane molecular and
coal surfaces dominates the process instead of the interaction of methane molecular within Henry’s
region. This process can be treated as a monolayer adsorption process. Under these conditions the
mean isosteric heat of adsorpion remains constant, which is also supported by the constant mean
isosteric heat of adsorption acquired from the test results.
84 |
Virginia Tech | Figure 3.1.6 The isosteric heat of adsorpion acquired via the Clausius-Clapeyron equation
(after [41] Yue, G. et al, 2014)
In the previous paper [41], the isosteric heat of the whole adsorption process is found using the
Clausius-Clapeyron equation. Figure 3.1.6 shows that 1) the isosteric heat of adsorption is
influenced by both temperature and adsorption content, and 2) when the adsorption content is the
same, the isosteric heat of adsorption under 265.15K, 253.25K and 243.15K can be treated as a
constant value.
The mean isosteric heat of adsorption value is within the isosteric heat of adsorption ranges under
different temperatures. Comparing the isosteric heat of adsorption and the mean isosteric heat of
adsorption, the mean isosteric heat of adsorption is more useful because it is independent of
temperature. This is reasonable because the mean isosteric heat of adsorption reflects the overall
heterogenous effect of coal, which should be an independent physical property of different types
of coal. The constant mean isosteric heat of adsorption confirms this point. Therefore, the mean
isosteric heat of adsorption can be used as an index to for evaluating the affinity of coal on methane.
3.1.5 Conclusions
18 group isothermal adsorption tests for methane and three different coals were conducted from
243.15K to 303.15K. The test results supports the following conclusions:
1) The maximum adsorption content of anthracite, lean coal and gas-fat coal increase at
0.19cm3/g, 0.15 cm3/g, and 0.13 cm3/g when temperature decreases at 1K.
2) The mean isosteric heat of adsorption for anthracite, lean coal, and gas-fate coal is -
23.31KJ/mol, -20.47 KJ/mol, -11.14 KJ/mol, respectively, and the minus sign indicates the
adsorption is an exothermal process.
85 |
Virginia Tech | 3.2 Thermodynamic analysis of high pressure methane adsorption in Longmaxi shale
Xu Tang*, Nino Ripepi*,†, Nicholas P. Stadie‡, Lingjie, Yu§,¶
(*Department of Mining and Minerals Engineering & †Virginia Center for Coal and Energy
Research, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24060, U.S;
‡ETH Zürich, Laboratory of Inorganic Chemistry, Vladimir-Prelog-Weg 1, 8093 Zürich,
Switzerland; § Wuxi Research Institute of Petroleum Geology of Sinopec Exploration &
Production Research Institute & ¶ Sinopec Key Laboratory of Petroleum Accumulation
Mechanisms, Wuxi, Jiangsu, 214151, China)
Abstract: Thermodynamic analyses of high pressure methane adsorption in shale are rarely
reported because of the lack of a reliable approach for obtaining the true adsorption uptake from
observed adsorption isotherms and the routinely used, oversimplified Clausius–Clapeyron (C-C)
approximation. This work extends our previously proposed dual-site Langmuir adsorption model
to calculate the isosteric heat of adsorption analytically from the observed adsorption isotherms
for high pressure methane adsorption isotherms on Longmaxi shale from Sichuan, China (up to 27
MPa and 355.15 K). The calculated isosteric heat of adsorption considers both the real gas behavior
of bulk methane and the adsorbed phase volume, which are neglected in the C–C approximation.
By this method, the temperature dependence as well as the uptake dependence of the isosteric heat
can be readily investigated, where the former cannot be revealed using the C–C approximation.
The influence of the adsorbed phase and the gas behavior (real gas or ideal gas) on the isosteric
heat of adsorption are also investigated, which shows that neglecting either the real gas behavior
or the adsorbed phase volume always results in an overestimation of the isosteric heat of adsorption.
In the Henry’s law regime of low pressure and low adsorption uptake (and up to a surface
occupancy of < 0.5), the isosteric heat of adsorption of methane on Longmaxi shale is
approximately constant at 15-17 kJ/mol, but then decreases significantly at higher pressures. This
work therefore justifies the method to obtain the true isosteric heat of adsorption for high pressure
methane in shale, which lays the foundation for future investigations of the thermodynamics and
heat transfer characteristics of the interaction between high pressure methane and shale.
Key words: methane, natural gas, shale gas, adsorption, Langmuir isotherm, isosteric heat of
adsorption
91 |
Virginia Tech | 3.2.1 Introduction
Shale gas has long been recognized as a promising alternative source of natural gas, and increasing
demands for energy have led to a widespread international effort to estimate the extent of its
resources and develop its production [1, 2]. The gas found in shale formations is fundamentally
different from conventional natural gas in that the formation itself is both the source and the
reservoir. Within the porous formation, the total shale gas content consists of bulk gas (in larger
pore spaces), dissolved gas (in the liquid brine), and adsorbed gas on the solid surface. This
adsorbed component varies widely from resource to resource, accounting for 20% to 80% of the
total shale gas content in five formations investigated in the United States [3, 4]. Nevertheless, it
is clear that the adsorbed quantity is a significant component that must be taken into account in
accurate estimations of the total gas-in-place resource and the working life of a producing well.
Methane adsorption in carbonaceous shale has been extensively studied over an intermediate range
of temperature and pressure but high pressure (>15 MPa) studies have remained less common [5-
10]. Furthermore, the thermodynamic characteristics of methane adsorbed on shale have rarely
been considered, especially at high pressure. Shale formations at depths of 1500 m to 2500 m
below the surface exist under conditions between 330-360 K and up to 38 MPa (given a pressure
coefficient of 15 MPa/km and geothermal gradient of 27.3 ℃/km as found in Longmaxi formations)
[11], where common approximations as to the ideal nature of the bulk gas are no longer applicable
and where the accurate prediction of the true adsorbed amount is not trivial. Understanding the
thermodynamic properties of the adsorbed phase is important for evaluating the value of a deep
shale resource, as in other adsorption systems such as gas separation and purification applications,
adsorption chillers, and adsorptive energy storage [12-17].
Physical adsorption (or physisorption) at the gas-solid interface is the process of gas adsorbate
accumulation on the surface of the solid adsorbent as a consequence of the weak van der Waals
forces that exist between any two species [18]. The change in heat associated with physical
adsorption is negative and significantly lower in magnitude than for chemical adsorption. Methane
and shale can only interact via London dispersion forces (neither has a permanent dipole), the
weakest type of van der Waals interactions, and the change in enthalpy is typically only 10-22
kJ/mol [5]. In this system, the quantity of adsorbed methane changes as a function of temperature
and pressure so that the adsorbed phase and the bulk fluid phase are at the same chemical potential.
The specific quantity is not only influenced by material properties of the shale (e.g., organic
92 |
Virginia Tech | components, minerals, and surface structure) but also by the composition of the gas adsorbate (e.g.,
content of moisture) [5-10]. The isosteric heat of adsorption can also vary as a function of the
amount of adsorbate and the system conditions [19-21]. It therefore serves as an important
descriptor of the physisorption system, and is directly related to the strength of the interaction
between gas adsorbate and solid adsorbent [22, 23]. The isosteric heat of adsorption typically
decreases as adsorption uptake increases because of binding site (and therefore binding energy)
heterogeneity [20, 24, 25].
Thermodynamic analysis of the properties of the adsorbed phase is possible by measuring the
properties of the bulk gas that is in equilibrium with it, and the adsorbed content is measured as a
function of bulk gas pressure at various constant temperatures. There are several issues associated
with the calculation of the isosteric heat of adsorption directly from observed adsorption isotherms
(where the observed adsorption quantity is also called the Gibbs excess adsorption uptake) [19,
26-28]. First, the Gibbs excess adsorption quantity is an underestimation of the absolute amount
adsorbed [19, 28]. At low pressure, the experimental adsorption isotherm well approximates the
absolute isotherm; however, at high pressure, the observed adsorbed content first reaches a
maximum and then decreases with increasing pressure which is not consistent with the physical
nature of adsorption [28]. Thus, it is necessary to calculate the isosteric heat of adsorption along
absolute isosteres and an effective method for their determination is needed. Second, a direct,
uniform approach for obtaining the absolute quantity of adsorption from measured adsorption
isotherms has not been developed, and the correct modeling of the physical parameters of the
adsorbed phase such as its density are complex issues that remain actively discussed [27-33].
Lastly, a consideration of the deviation of real gas behavior from the ideal gas law is necessary
when calculating the isosteric heat of adsorption, especially under high pressure and low
temperature conditions. Methane deviates significantly (>10%) from the ideal gas law at pressures
above 6 MPa at room temperature, which has a dramatic effect on the calculation of
thermodynamic parameters in this regime [27]. For methane adsorption in shale, the C-C
approximation is routinely used to calculate the isosteric heat of adsorption. However, since the
C-C approximation ignores the adsorbed phase effect and uses the ideal gas law, the calculated
results may not reveal the true thermodynamics behavior for methane in shales. In order to
reasonably analyze the thermodynamic characteristics of a gas-solid system such as methane in
shale, the above issues must be addressed [19, 27, 34, 35].
93 |
Virginia Tech | In previous work, we applied a dual-site Langmuir model to obtain absolute adsorption isotherms
from observed Gibbs excess adsorption equilibria utilizing the assumption that the density of the
adsorbed phase is an unknown constant [36]. The proposed model gives a reasonable explanation
for all observed phenomena in high pressure methane adsorption in shales, which have not been
reasonably addressed using the conventional Langmuir model, the potential theory based model,
or their revised forms [36]. Considering the justification of its use, the dual-site Langmuir model
is therefore extended in this work to calculate the isosteric heat of adsorption analytically for high
pressure methane on shale. This method considers both the real gas behavior of the bulk methane
and the volume of the adsorbed phase, both of which are neglected in the classic C-C approach.
The influence of the adsorbed phase volume and the nature of the gas behavior (real or ideal) on
the isosteric heat of adsorption are also investigated. Calculations in the Henry’s law region
(corresponding to the limit of low pressure) were also performed based on the absolute methane
adsorption isotherms to validate the above methodology. Rouquerol’s approach [37] is applied in
this case to avoid any potential for subjective judgements in determining the properties of the
adsorbed phase in the Henry’s law pressure range arising from the use of high pressure gas
adsorption isotherm data.
3.2.2 Adsorption model and thermodynamic calculations
3.2.2.1 Dual-site Langmuir model
For a pure gas and solid adsorption system, the observed adsorption content, also called the Gibbs
excess adsorption uptake (n ), is given by the Gibbs equation,
e
n n V (P,T) (1)
e a a
where n refers to the difference between the absolute quantity adsorbed (n ) and the amount that
e a
would be present in the same volume (V ) of the adsorbed phase at the density of the bulk phase
a
(P,T)
( ). In the limit of low ρ, the excess adsorbed amount well approximates the absolute
quantity. At higher pressures where the density of the bulk phase approaches that of the adsorbed
phase, the amount of gas that would be present in the volume of the adsorbed layer even in the
absence of adsorption cannot be neglected. In order to obtain the absolute quantity of adsorption,
the average density (or the total volume) of the adsorbed layer must be obtained.
94 |
Virginia Tech | The Langmuir equation is the simplest model for adsorption at the gas-solid interface that is
applicable over the entire range of surface occupancy, making a number of simplifying
approximations such as perfect adsorbent binding site homogeneity. For heterogeneous adsorbents,
the multi-site Langmuir model is more suitable than the single-site Langmuir model for describing
the gas adsorption behavior [28, 38, 39]. The binding energy of the adsorption sites will vary,
where the lowest energy sites will be filled first, followed by the higher energy sites. For the
purposes of many real-world adsorbents, a dual-site Langmuir model incorporating only two
different binding sites is sufficient for fitting experimental data when measured over a wide range
of pressures and supercritical temperatures [27, 38, 39]. Each site can be modelled by a separate
equilibrium constant, K 1(T)and K 2(T) (K (T) A exp( E 1 ) and K (T) A exp(E 2 ) ) [38]. The
1 1 RT 2 2 RT
dual-site Langmuir equation can then be expressed in the following form, where αis the relative
fraction of the second site (0<α<1),
K (T)P K (T)P
n (P,T)n (1)( 1 )( 2 ) (2)
a max 1K (T) P 1K (T)P
1 1 2
The same type of equation can be used to describe the volume change of the adsorbed layer as a
function of pressure, as [27, 28, 39],
K (T)P K (T)P
V V (1)( 1 )( 2 ) (3)
a max 1K (T)P 1K (T)P
1 2
By combining equations (1), (2), and (3), the excess adsorption amount and the surface coverage,
θ, in the dual-site model can be obtained, giving:
K (T)P K (T)P
n (P,T)(n V (P,T)) (1)( 1 )( 2 ) (4)
e max max 1K (T)P 1K (T)P
1 2
n (P,T) K (T)P K (T)P
a (1)( 1 )( 2 ) (5)
n 1K (T)P 1K (T)P
max 1 2
The dual-site Langmuir model described herein (equations 2, 4, and 5) is based on the assumption
that the volume of the adsorbed layer monotonically increases with increased pressure, which is
consistent with the physical nature of adsorption and does not necessitate any complex empirical
equation for the density of the adsorbed phase. This monotonic increase is further approximated
95 |
Virginia Tech | as linear with respect to the number of adsorbed species, which is a reasonable, simple assumption.
The absolute adsorption quantity as a function of temperature and pressure can be obtained via
global curve fitting of the entire set of experimental excess uptake isotherms.
3.2.2.2 Isosteric heat of adsorption
The change in enthalpy of the system due to adsorption at a specific state of surface occupancy is
referred to as the isosteric heat of adsorption (H ). It can be determined via the Clapeyron
ads
relationship which is relevant to the equilibrium between two phases (in this case the adsorbed
phase, a, and the gas phase, g) in a closed system:
dP dP
H ( ) T v ( ) T (v v ) (6)
ads dT na dT na a g
The derivative of pressure with temperature along an isostere (constant value of adsorption uptake),
dP
( ) , can be expanded in various ways for further investigation. Since the pressure in a closed
dT na
system is a function of temperature and quantity adsorbed, a general expansion may be made such
that [40],
dP P dn (lnP)
( ) ( ) a P( ) (7)
dT n a n T dT T n a
a
If the bulk fluid is approximated as an ideal gas,PvRT , it follows that,
(lnP) RT2 P dn P dn (lnP)
H RT2[( ) ] ( ) a [( ) a P( ) ]T v (8)
ads(n a) T n a P n T dT n T dT T n a a
a a
RT2 P dn
In right hand side (RHS) of equation (8), the second term, ( ) a , includes the
P n T dT
a
P dn (lnP)
behavior of the adsorbed phase mass, and the third term, [( ) a P( ) ]T v ,
n T dT T n a a
a
considers the volume effect of the adsorbed phase. If the volume of the adsorbed layer is taken to
be negligible and the influence of the adsorbed mass is ignored, the conventional Clausius-
Clapeyron (C-C) relationship is obtained:
(lnP)
H H RT2[( ) ] (9)
ads ads,cc T na
96 |
Virginia Tech | Optionally, if the relationship between the amount adsorbed and the process conditions (pressure
and temperature) is known, the derivative can be directly determined. If that relationship is taken
to have the form of a Langmuir equation, then the derivative can be expanded into three simpler
terms:
dP P K
( ) ( ) ( ) ( ) (10)
dT na na K na T na
By combining equation (6) with equation (10), the isosteric heat of adsorption is then directly
obtained. There are numerous ways to handle the difference between the molar volume of the gas
and adsorbate as required to solve equation (6). For example, by applying the ideal gas law
(Pv RT ), one form of the isosteric heat of adsorption can be obtained simply as,
g
P K RT
H ( ) ( ) ( ) T( v ) (11)
ads(na),IGL na K na T na P a
Ignoring the volume of the adsorbed phase (in other words, assuming v << v ), another form of
a g
the isosteric heat of adsorption can be obtained,
P K RT
H ( ) ( ) ( ) T( ) (12)
ads(na),IGL0V na K na T na P
If equation (6) and equation (10) are combined and the true gas density is applied (the true gas
density can be obtained using NIST REFPROP database) instead of the ideal gas law density, the
isosteric heat of adsorption is obtained as:
P K
H ( ) ( ) ( ) T(1v ) (13)
ads(na),RGL na K na T na g a
Ignoring the volume of the adsorbed phase in equation (13), an additional form of the isosteric
heat of adsorption can be obtained,
P K
H ( ) ( ) ( ) T(1) (14)
ads(na)RGL0V na K na T na g
Equations (11)-(14) are much easier to solve using an analytical approach than equation (8). The
P K
analytical form of ( ) ( ) ( ) in the case of the dual-site Langmuir equation is:
na K na T na
97 |
Virginia Tech | (1)P EK (T) P E K (T)
1 1 2 2
P K (1K (T)P)2 RT2 (1K (T)P)2 RT2
( ) ( ) ( ) 1 2 (15)
n a K n a T n a (1)K (T) K (T)
1 2
(1K (T)P)2 (1K (T)P)2
1 2
The merits of the second approach (equation (10) over equation (7)) are twofold. Firstly, both the
true gas behavior and the ideal gas law can easily be implemented as shown in equations (11) and
(13), to determine the effect of assuming gas phase ideality in the result. Secondly, the volume of
the adsorbed layer can also be taken into consideration, an especially important feature to account
for outside of the low-pressure (Henry’s law) limit. On the contrary, the conventional C-C equation
inherently adopts the ideal gas law and does not consider the density of the adsorbed layer to be
significant compared to the bulk gas.
3.2.2.3 The Henry’s law limit
An approximate approach is provided here to extrapolate the isosteric heat of adsorption to low
pressures (the Henry’s law region) from the as-collected high pressure adsorption isotherms. In
this way, an unbiased isosteric heat of adsorption can be calculated without the dependence on any
model or specific methodology, for comparison to the method described above.
In the limit of low pressure, gas adsorption behavior follows Henry’s law,
n K P (16)
a H
where P is the pressure of the bulk gas, n is the absolute adsorption content, and K is the Henry’s
a H
law constant. Together with the van’t Hoff equation (which relies on the ideal gas law, also
applicable in the Henry’s law regime), the relationship between K and the thermodynamic
H
quantities of adsorption is expressed as,
H S
lnK H H (17)
H RT R
The isosteric heat of adsorption in the Henry’s law region can be obtained from the linear
relationship between ln(K ) and the reciprocal of T. The key step at this point is to obtain Henry’s
H
constant using a reasonable pressure range wherein the linear relationship between absolute
adsorption content and pressure is valid.
98 |
Virginia Tech | In order to calculate Henry’s law constants, the relationship between adsorption uptake and
pressure in the low-pressure region can be expressed by a virial expansion,
ln(n /P) A An An2 (18)
a 0 1 2
where n is the absolute content of adsorption at bulk gas pressure P, and the first virial coefficient
a
A is related to the Henry’s law constant, K , as K = exp(A ). When n is small, the higher-order
0 H H 0 a
terms can be neglected, and equation (18) can be written as,
ln(P/n )A An (19)
a 0 1
From equation (19), A can then be obtained by fitting the linear region of ln(P/n ) as a function
0 a
of n , where n is approximated by measured n isotherms. Rouquerol’s recommended approach
a a e
[37] is applied herein to avoid any subjective judgements in determining the Henry’s pressure
range:
a. the application of equation (19) should be limited to the pressure range where the term
n (1-P/P ) continuously increases with P/P (P is the maximum pressure
a max max max
investigated).
b. an apparent linear relationship must be obtained, i.e., the correlation coefficient (R2) should
be above 0.95.
Once Henry’s law constant values are obtained, the isosteric heat of adsorption in Henry’s region
can be obtained from the linear relationship between ln(K ) and the reciprocal of temperature as
H
shown in equation (17).
3.2.3 Experimental data and analysis
Four high-pressure adsorption isotherms of methane on Longmaxi shale (China) were measured
using the gravimetric method: at 303.15 K, 318.15 K, 333.15 K, and 355.15 K (Figure 3.2.1) [36].
All four isotherms were then fitted simultaneously to the dual-site Langmuir model (equation 4)
by a least-squares residual minimization algorithm. The seven independent fitting parameters were
varied to achieve the global minimum of the residual-squares value within the following limits: 0
< n < 100 mmol/g, 0 < V < 10 mL/g, 0 < α < 1, 0 < E < 100 kJ/mol, 0 < E < 100 kJ/mol,
max max 1 2
A > 0, and A > 0. Once the best-fit parameters were determined, absolute and excess adsorption
1 2
uptake could be expressed at arbitrary temperatures and pressures by use of equations (2) and (4).
99 |
Virginia Tech | As shown in Figure 3.2.1, the dual-site Langmuir adsorption model (equation 4) gives a good
global fit to the observed data with the residual sum of squares, 0.000263, and the corresponding
best-fit parameters are: n = 0.1715 mmol/g, V = 0.0097 mL/g, α = 0.2640, E = 16.706 kJ/mol,
max max 1
A = 0.0002 1/MPa, E = 15.592 kJ/mol, and A = 0.0032 1/MPa. Detailed experimental methods
1 2 2
and material properties of the shale are described in our previous work [36].
Figure 3.2.1. Equilibrium adsorption uptake of methane on Longmaxi shale between 303-
355 K and 0.5-25 MPa: solid symbols are measured Gibbs excess uptake, solid lines are
modeled Gibbs excess uptake (equation (4)), and open symbols and dashed lines are modeled
absolute uptake (equation (2)). The data are reproduced from a previous study [36].
The well-known phenomenon that the observed Gibbs excess adsorption uptake increases with
increasing pressure up to a maximum value and then decreases, as well as the corresponding
crossover of isotherms, can be seen in Figure 3.2.1. This is attributed to the increasing volume of
the adsorbed phase with increasing pressure, leading to a maximum in the Gibbs excess adsorption
at each temperature [36]. This crossover in high pressure methane adsorption isotherms on shale
has not been reasonably addressed using other commonly used adsorption models in literature.
Furthermore, the temperature dependence of the adsorption uptake is built into the model in this
work [36], where previously only empirical relationships or no relationship at all was addressed,
making this globally fitted model more descriptive of temperature-related phenomena. As is
characteristic of the physical nature of adsorption, the absolute adsorption quantity increases
monotonically up to 27 MPa at all temperatures. The temperature dependence of the absolute
adsorption uptake is also clear: the higher the temperature the lower the absolute adsorption uptake.
100 |
Virginia Tech | These features ensure that an accurate thermodynamic analysis can be achieved using the absolute
adsorption equilibria calculated in this work.
3.2.4 Thermodynamic analysis and discussion
In the thermodynamic analysis of methane adsorption on shale performed in this work, the
robustness of the isosteric heat of adsorption is first confirmed. Then, the various quantities
describing the isosteric heat of adsorption (equations (11)-(14)) are compared to understand how
the real gas behavior and the adsorbed phase volume influence the isosteric heat of adsorption.
The temperature influence on the isosteric heat of adsorption is also compared. Finally, the
isosteric heat of adsorption within Henry’s pressure region is calculated to validate the above
methodology using the high pressure methane absolute adsorption isotherms.
3.2.4.1 Robustness of the isosteric heat of adsorption calculation
The isosteric heat of adsorption is best calculated by including all measured adsorption data in the
fitting routine, obtaining the best-fit parameters, and then directly solving equation (13) to obtain
-ΔH , which not only considers the real gas behavior but also takes the volume of the adsorbed
ads
phase into consideration. The isosteric heat of adsorption of methane on shale, as a function of
absolute quantity of methane adsorbed, is shown as solid isothermal lines in Figure 3.2.2. The
isosteric heat varies from 16.5 kJ/mol at low pressure and high temperature (355 K), down to <5
kJ/mol at high pressures, indicating a heterogeneous distribution of adsorption sites in the porous
shale structure.
To demonstrate the robustness of the analytical calculation of the isosteric heat of adsorption by
our method, the results obtained using different processing approaches are compared: (1) using all
measured data, (2) using only the data between 0-15 MPa, and (3) using only the data between
303.15-333.15 K as fitting data to obtain a best fit. Method (1) represents the best approach as
described previously, and method (2) and (3) demonstrate the effects of collecting less
experimental data (e.g., at <15 MPa as in a majority of previous investigations). As shown in
Figure 3.2.3, the resulting isosteric heats are approximately the same except in the low pressure
region. While this may seem counterintuitive, it is important to note that subtle changes in the best-
fit parameters lead to large changes in the dP/dT term in equation (10) and (15), and the best-fit
parameters can only be achieved by using a large range of measured data (typically requiring
numerous isotherms and data extending well beyond the Gibbs excess maximum).
101 |
Virginia Tech | Figure 3.2.2. Isosteric heat of adsorption of methane on shale between 303-355 K (blue to red)
as a function of absolute adsorption uptake up to 30 MPa (solid lines). For comparison, the
isosteric heat calculated by including experimental data from restricted ranges of pressure
and temperature is also shown (as small and large dashes, respectively).
3.2.4.2 Effect of real gas behavior and adsorbed phase volume
The second step is to investigate the effects of real gas behavior and the volume of the adsorbed
phase on the isosteric heat of adsorption beyond the Henry’s law region. A summary of the
assumptions included within each isosteric heat of adsorption investigated herein is shown in Table
3.2.1 and a detailed discussion of the comparison with Henry’s law analysis is given in Section
4.4.
Table 3.2.1. Definition of various isosteric heats of adsorption
A comparison of the isosteric heats of adsorption calculated according to Table 3.2.1 is shown in
Figure 3.2.3 The different isosteric heats of adsorption follow a similar behavior irrespective of
temperature and the isosteric heat of adsorption at 303.15 K is taken as an example to interpret the
102 |
Virginia Tech | influence of real gas behavior and adsorbed phase volume. The C-C approximation employs the
ideal gas law and does not consider the adsorbed phase volume. Both assumptions become less
valid with increasing adsorption content and cause a significant overestimation of the isosteric heat
of adsorption. Regardless of the gas law employed, the adsorbed phase volume significantly affects
the behavior of the calculated isosteric heat of adsorption, especially under high pressure
conditions: without considering v , the isosteric heat of adsorption is always overestimated. For
a
cases considering the finite volume of the adsorbed phase, the difference between the real gas and
ideal gas density also affects the isosteric heat of adsorption significantly: the ideal gas law always
corresponds to a higher isosteric heat. Comparing these effects at different temperatures, it can be
found that as temperature increases, the influence of both the equation of state of the gas and the
volume of the adsorbed phase decreases.
Figure 3.2.3. Isosteric heat of adsorption of methane on shale as calculated using four
different methods: ∆H (equation (12)) as solid lines, ∆H (equation (14))
ads(na), IGL-OV ads(na), RGL-0V
103 |
Virginia Tech | as dashed lines, ∆H (equation (11)) as single dotted lines, and ∆H (equation
ads(na), IGL ads(na), RGL
(13)) as double dotted lines. The isosteric heat of adsorption calculated in the C-C
approximation (equation (9)) is also shown as filled black symbols.
3.2.4.3 Effect of temperature
Isosteric heats of adsorption calculated in different ways (according to equations (9) and (11)-(14))
are shown at all temperatures investigated in Figure 3.2.4. In all cases, temperature generally has
a negative effect on the isosteric heat of adsorption for both real gas and ideal gas cases: the higher
the temperature, the lower the isosteric heat of adsorption. Using the real gas equation of state, if
the adsorbed phase volume is ignored, isosteric heats of adsorption first decrease and then increase
and the temperature-dependence reverses at high pressure. In the ideal gas assumption, if the
adsorbed phase volume is ignored, the isosteric heat of adsorption becomes the C-C approximation
and the temperature-dependence disappears altogether. Figure 3.2.4 also shows that in the low
pressure region (i.e., low adsorption uptake), the isosteric heats of adsorption merge at a constant
value. However, because limited data were measured in this region, extrapolation to the limiting
value is difficult. This problem can be solved using the approach discussed in Section 4.4.
Figure 3.2.4. Comparison of isosteric heat of adsorption of methane on shale: ∆H
ads(na), IGL-OV
(equation (12)) as solid lines, ∆H (equation (11)) as single dotted lines, ∆H
ads(na), IGL ads(na), RGL-
(equation (14)) as dashed lines and ∆H (equation (13)) as double dotted lines. The
0V ads(na), RGL
isosteric heat of adsorption calculated in the C-C approximation (equation (9)) is also shown
as filled black symbols.
3.2.4.4 Determination of Henry’s law limit
104 |
Virginia Tech | The linear range of equation (19) determined using Rouquerol’s approach shown in Figure 3.2.5.
The mean isosteric heat of adsorption over the entire temperature range, calculated according to
equation (17) is 16.5 kJ/mole (see Figure 3.2.6), which is consistent with the value determined by
both the analytical and conventional approaches as described above (Figures 3-4).
Figure 3.2.5. Equilibrium adsorption uptake of methane on shale (n ) between 303-355 K
a
and 0.5-25 MPa, as measured (solid symbols) and as fitted by a virial-type equation (solid
lines, equation (19)). (left) Adsorption uptake is shown as a product of n and 1-P/P , as a
a max
function of P/P . (right) Adsorption uptake is shown in the linear region for ln(P/n ) as a
max a
function of n .
a
Figure 3.2.6. Mean isosteric heat of adsorption calculated by equation (17)
3.2.5 Discussion
Understanding the isosteric heat of adsorption is useful in accurate estimations of the temperature
evolution process during essentially isothermal adsorption processes. It has been previously
reported that during methane adsorption on coal, the temperature change induced by adsorption
105 |
Virginia Tech | first shows a sharp increase to a peak, and then a gradual decrease to the environmental (bath)
temperature at equilibrium [41]. Moreover, this change in temperature was not the same at different
temperatures; at higher temperatures, a smaller temperature change occurred upon adsorption of
methane on coal. Considering the fact that the physical adsorption of methane on either coal or
shale should exhibit similar characteristics (both are bulk, naturally occurring carbonaceous
organic-rich materials), these previous results are readily comparable to those obtained in this work.
Herein it is found that the isosteric heat of adsorption decreases with increasing adsorption content
for methane in shale, which is also affected by the temperature of the isotherms. Specifically, the
isosteric heat of adsorption decreases as a function of temperature, which is qualitatively consistent
with the observed temperature change upon methane adsorption on coal. The precise dependence
of the isosteric heat of adsorption on both adsorption uptake and temperature must be taken into
consideration for accurate modeling of the heat transfer process during methane extraction from
shale.
The isosteric heat of adsorption of methane on shale in the Henry’s law region is consistent with
similar reports on coal samples [42]. In the Henry’s law pressure range, each adsorbed molecule
can explore the entire adsorbent surface independently because of the extremely low adsorbate
concentration at low pressure. The adsorption sites of highest energy will be occupied first (in this
work, “highest energy” refers to lowest (negative) absolute energy), and the interactions between
adsorbed methane molecules and the gas molecules themselves can both be neglected. When the
interaction among adsorbed methane molecules and/or gas molecules becomes significant with
increasing pressure, the most accurate calculation of the isosteric heat of adsorption must consider
both the real gas behavior and the finite adsorbed phase volume as in equation (13). Then,
considering the interaction between the adsorbate molecules and the solid porous adsorbent as the
only remaining significant interaction in the system, the evolution of the isosteric heat of
adsorption as a function of pressure or adsorption uptake can reflect the overall heterogeneity of
the adsorbent which is a very relevant property for comparison between materials [27, 42]. In
addition, since the isosteric heat of adsorption in the Henry’s law region is independent of
temperature, as shown in equation (13), it can be used as a unique index to evaluate the adsorption
affinity of the highest energy sites in porous adsorbents such as coal and shale.
106 |
Virginia Tech | 3.2.6 Conclusions
In this work, the isosteric heat of adsorption of methane on Longmaxi shale at geologically relevant
pressures is obtained by considering both the real gas behavior of bulk methane and the finite
adsorbed phase volume. The effects of real gas behavior, adsorbed phase volume, and temperature
on the isosteric heat of adsorption are investigated, facilitated by the use of a two-site Langmuir
adsorption model. Three conclusions can be drawn.
First, for high pressure methane adsorption on shale, the isosteric heat of adsorption decreases with
increasing adsorption uptake (or equilibrium pressure) and the dependence on temperature is
negative.
Second, accurate calculations of the isosteric heat of adsorption are always lower than the quantity
calculated using the C–C approximation. Neglecting either the real gas behavior or the adsorbed
phase volume always results in an overestimation of the isosteric heat. These results are consistent
with the temperature evolution phenomenon that occurs during methane adsorption on other
carbonaceous adsorbents.
Finally, the isosteric heat of adsorption in the Henry’s law region, which is independent of
temperature, can be used as a unique index to evaluate the gas adsorption affinity of adsorbents
such as coal and shale in the limit of very dilute adsorption. For all thermodynamic analysis outside
of this regime, a more sophisticated method such as fitting the data to a two-site Langmuir model
must be employed.
Acknowledgements
Financial assistance for this work was provided by the U.S. Department of Energy through the
National Energy Technology Laboratory’s Program under Contract No. DE-FE0006827, the State
Key Development Program for Basic Research of China (Grant No. 2014CB239102) and
Department of Science and Technology at China Petroleum & Chemical Corporation (Grant
No.P12002, P14156). The first author also wants to thank Prof. Matthew R Hall for his valuable
discussions on this work.
107 |
Virginia Tech | 3.3 High pressure supercritical carbon dioxide adsorption in coal: adsorption model and
thermodynamic characteristics
Xu Tanga, Nino Ripepia,b
(a.Department of Mining and Minerals Engineering & b Virginia Center for Coal and Energy
Research, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24060, U.S)
Abstract: This work uses a dual-site Langmuir model to describe supercritical carbon dioxide
(scCO ) adsorption in coal up to 20MPa and 253K by considering both the absorbed (penetrated)
2
and adsorbed carbon dioxide phase. The isosteric heat of adsorption for scCO adsorption in coal
2
is calculated analytically by considering both the real gas behavior and the behavior of the
adsorbed phase, which are ignored in the classic Clausius-Clapeyron approximation. It was found
that the proposed model can not only reasonably interpret observed test phenomena but also has
the intrinsic ability to extrapolate adsorption isotherms under different temperatures beyond test
data. The crossovers of the observed adsorption isotherms under different temperatures are caused
by the changing volume of the adsorbed phase during adsorption process. Both the temperature
dependence and adsorption uptake dependence of isosteric heat of adsorption are revealed; the
higher the temperature and the adsorption uptake, the lower the isothersteric heat of adsorption.
Using ideal gas law always overestimates isosteric heat of adsorption for scCO adsorption in coal.
2
For scCO adsorption in coal, there exists an abnormal phenomenon that the higher the temperature
2
the higher the isosteric heat of adsorption when the adsorption uptake approaches the maximum,
which has not been reported in literature. The dual-site Langmuir model and the extended method
for calculating heat of adsorption lays the foundation for accurately estimating carbon dioxide
storage capacity, differentiating bulk gas phase and adsorbed phase content, and analyzing
thermodynamic (heat transfer) characteristic of scCO2 and coal interaction.
Key words: carbon dioxide, adsorption, coal, Langmuir, isosteric heat of adsorption
112 |
Virginia Tech | 3.3.1 Introduction
Because of global warming and climate change concerns, global efforts have been made to
decrease the concentration of carbon dioxide in the atmosphere (Grubb et al., 1995; Stern, 2009;
Morgan et al., 2014; Paris Protocol, 2015). Carbon dioxide capture, utilization and storage are
considered crucial ways to meet the carbon dioxide emission reduction targets. Onshore geologic
sequestration of carbon dioxide typically involves collecting and placing carbon dioxide into
suitable underground formations for storage such as depleted oil and conventional gas reservoirs,
unconventional natural gas reservoir (unminable coal seam and shale formation), and deep
formations containing salty water and basalt formations (Herzog et al., 2001; White et al., 2003;
Metz et al., 2005; Benson et al., 2008; Orr, 2009; Figueroa et al., 2008). Among these geological
formations, unminable coal seams are one of the promising sites because of their potential for
enhancing coalbed methane (ECBM) recovery while simultaneously sequestering carbon dioxide
(White et al., 2005; Mazzotti et al., 2009; Busch et al., 2011; Godec et al., 2014). The enhanced
natural gas will help to offset the cost of carbon dioxide sequestration. However, there are still
some concerns about the long time effect of the permanent sequestration of CO in the coal seam
2
and its negative effects upon the environments (White et al., 2005; Hedges et al., 2005). Carbon
dioxide injection issues may occur along during the field injection process because of the
permeability decrease induced by the coal swelling (Reucroft, et al., 1987; Pan et al., 2007 & 2010;
Cui et al., 2007; Day et al., 2008). The existing phase of the injected carbon dioxide in the
subsurface is also important to know, because it is crucial for the carbon dioxide storage capacity
estimation. Therefore, field tests of ECBM with carbon sequestration were and are being
conducted across the world, which will be helpful to understand its potential benefits and practical
issues (Steven et al., 1998; Yamasaki, 2003; Gunter et al., 2004; Sams et al., 2005; White et al.,
2005; Van Bergen, et al., 2006; Wong et al., 2007; Ripepi, 2009; Connell et al., 2013; Gilliland et
al., 2013; Wei et al., 2015).
It is well known that the affinity of carbon dioxide in coal is higher than methane and the carbon
dioxide adsorbed on coal is always higher than methane. In order to evaluate the carbon dioxide
storage potential of unminable coal seam, the first step is to evaluate the adsorption capacity of
carbon dioxide in coal via isothermal sorption tests. However, because of the heterogenous
properties of coal, scCO adsorption behavior in coal has not attracted researcher’s attentions like
2
manmade materials. There are inconsistent test phenomena reported regarding scCO adsorption
2
113 |
Virginia Tech | in coal. Some researchers show peculiar test phenomena for scCO in coal such as the bimodal
2
Gibbs excess adsorption isotherms (Krooss et al., 2002; Toribio et al., 2004; Busch et al., 2007;
Siemons et al., 2007; Busch et al., 2008). Other researchers exhibit smooth CO excess adsorption
2
isotherms in coal when the pressure goes up to 20MPa (Sudibandriyo et al., 2003; Fitzgerald et al.,
2005; Bae et al., 2006; Ottiger et al., 2006; Sakurovs et al., 2007; Day et al., 2008; Pini et al., 2010;
Weniger et al., 2010; Song et al., 2015; Luo et al., 2015). Since the latter test phenomena can be
reproduced and the former cannot, the peculiar test phenomenon is attributed to artificial test errors
(Ottiger et al., 2006; Pini et al., 2010). Even though the smooth CO excess adsorption isotherms
2
have been observed, an optimized model for modeling both the Gibbs excess and absolute
adsorption uptake is still needed. Despite extensive researches for scCO adsorption in manmade
2
materials, a number of semi-empirical models have been adopted by revising the classic D-A, D-
R, Langmuir and Toth equations to describe the sorption behavior of scCO in coal by fitting each
2
isothermal adsorption curve independently (Sudibandriyo et al., 2003; Fitzgerald et al., 2005; Bae
et al., 2006; Ottiger et al., 2006; Sakurovs et al., 2007; Day et al., 2008; Pini et al., 2010; Schell et
al., 2012; Weniger et al., 2010; Song et al., 2015; Luo et al., 2015). However, all these models are
cling to either the empirical density of adsorbed carbon dioxide (density of liquid carbon dioxide)
in coal or the assumed constant volume of adsorbed carbon dioxide in coal. Even though good
fitting results are obtained by each researcher, the physical meaning of the obtained parameters
from these revised classical equations still needs to be confirmed. Furthermore, even though coal
swelling phenomena has been observed, few models take the coal swelling into consideration to
obtain absolute isotherms from Gibbs excess isotherms (Ozdemir et al., 2003; Romanov et al.,
2006; Pini et al., 2010). Considering both the adsorption and penetration (absorption) CO in coal
2
may explain the coal swelling effect during sorption process. The penetration CO can not only
2
compensate the elastic energy change of coal associated with volume change but also can change
the macromolecular structure of coal (Jakubov et al., 2002; Larsen, 2004; Pini et al., 2010). In
addition, these models cannot be used to extrapolate isotherms beyond test temperatures. The
prediction of adsorption isotherms under high pressure and high temperature geological conditions
are critical for carbon dioxide storage capacity estimations in deep subsurface. Therefore, to
reasonably interpret the adsorption behavior of scCO in coal and predict adsorption isotherms
2
under high temperature and high pressure in-situ conditions, a more concise and robust adsorption
model is needed.
114 |
Virginia Tech | As a crucial evaluation index for thermodynamic processes in physical adsorption tests, the
isosteric heat of adsorption for scCO in coal has rarely been considered by researchers. There are
2
still several issues existing regarding the calculation of this index for scCO adsorption in coal
2
(Pan et al., 1998; Chakraborty et al., 2006; Stadie, 2012; Stadie et al., 2013 & 2015). First, the
absolute adsorption isotherms must be obtained from the observed adsorption isotherms in order
to calculate the isosteric heat of adsorption, where a robust model is needed (Herbst et al., 2002;
Bae et al., 2006). Under low pressure conditions, the observed adsorption isotherm approximates
absolute adsorption isotherm and there is no need to obtain absolute adsorption isotherms.
However, when the Gibbs excess adsorption behavior becomes obvious under higher pressure
conditions, a reliable adsorption model is necessary to obtain absolute adsorption uptake from
observed adsorption isotherms. Second, even though the absolute adsorption isotherms can be
obtained, the classic Clausius-Clapeyron approach is not appropriate because it cannot take the
real gas behavior of scCO and the adsorbed gas phase into consideration (Pan et al., 1998;
2
Chakraborty et al., 2006; Stadie et al., 2015). Figure 3.3.1 shows the deviation of CO behavior
2
compared with idea gas under different temperatures and pressures. The usage of the Clausius-
Clapeyron approach, therefore, will hide the true behavior of the isosteric heat of adsorption
because of the ideal gas assumption and the neglect of the volume of the adsorbed gas phase
(Chakraborty et al., 2006; Stadie et al., 2015). Therefore, in order to obtain the true behavior of
the isosteric heat of adsorption, the above mentioned issues must be reasonably addressed. On the
one hand, a robust adsorption model to obtain absolute adsorption isotherms from observed
adsorption isotherms is needed. On the other hand, the conventional Clausius–Clapeyron
approximation needs to be improved by taking the real gas law of carbon dioxide into consideration
to obtain the true isosteric heat of adsorption.
115 |
Virginia Tech | Figure 3.3.1 Deviation behavior of the CO under different temperatures and pressures
2
(Data is obtained from the NIST Standard Reference Database 23 (REFPROP: Version 8.0.))
To tentatively solve the above-mentioned issues, this work first uses a dual-site Langmuir model
to describe scCO adsorption behavior in coal by considering both the adsorbed phase and
2
absorbed (penetrated) phase. Then, based on the assumption that the density of the adsorbed gas
phase is an unknown constant, the authors build in the concept of the Gibbs excess adsorption by
applying the dual-site Langmuir model. Next, the published data of high pressure scCO adsorption
2
in five different coals are retrieved from literature to validate the proposed model. Last, isosteric
heats of adsorption for scCO are calculated analytically by considering the real gas behavior of
2
scCO and the contribution of the adsorbed and absorbed phase.
2
3.3.2 Absolute adsorption model
Coal swelling is a pronounced phenomenon associated with CO injection into coal seams, which
2
results in the injection issues in field tests (White et al., 2005; Van Bergen, et al., 2006).
Researchers used different approaches such as dilatometric, optical or strain gages, X-ray and
small-angle scattering techniques to study the coal swelling effects in a laboratory scale but have
not reached an agreement on whether the coal swelling is universal or not. Radlinski et al..(2009)
found that coal microstructures were unaffected by exposure to CO pressure over a period of days
2
based on the SANS and USANS tests. Most other researchers agree that the uptake of CO in coal
2
will results in the swelling monotonically with pressure which can be modeled by a Langmuir-
type curve based on laboratory evidence as shown in equation (1) (Levine, 1996; Palmer and
Mansoori, 1998; Shi and Durucan, 2004a; Cui et al., 2007; Pini et al., 2009 & 2010),
k P
s (1)
s 0 1k P
s
where is the swelling strain under specific pressure, and k are fitting coefficients.
s 0 s
Based on the assumption that coal swelling is induced by the CO penetration (absorption) in
2
macrostructure of coal, a dual-site Langmuir model is used for describing both the conventional
adsorption of CO in coal and the CO absorption in coal as shown in equation (2) (Fornstedt et
2 2
al., 1996; Graham et al., 1953; Larsen, 2004; Pini et al., 2010). Figure 3.3.2 shows how the CO
2
phase changes before and after CO sorption: the absolute adsorption is the sum of net adsorption
2
116 |
Virginia Tech | and absorption uptake. The dual-site model therefore combines both the adsorption and absorption
content of CO in coal and both of them show a Langmuir-type term. In addition, the absorption
2
term essentially has similar properties with the coal swelling behavior.
K P K P
n n 1 n 2 (2)
a ad1K P ab1K P
1 2
where n is the absolute adsorption content under specific pressure, n is the net adsorption
a ad
content, n is the absorption content, K and K are corresponding Langmuir contents
ab 1 2
E E
((K (T)A exp( 1 ) and K (T) A exp( 2 ) )), P is pressure.
1 1 RT 2 2 RT
Figure 3.3.2 The difference of CO -coal sorption system before and after CO adsorption.
2 2
If the absorbed and adsorbed phase can be weighted by a parameter, and equation (2) is rearranged,
the following form can be obtained,
K (T)P K (T)P
n n (1) 1 2 ) (3)
a max 1K (T)P 1K (T)P
1 2
It should be pointed out that the physical meaning of n in equation (3) is different from the
max
conventional Langmuir equation, which is composed of both the net adsorption content and the
absorption content. The term “adsorbed” will be used in the following sections as a simplification
which actually refers to both adsorbed and absorbed phases.
3.3.3 Gibbs excess adsorption model and isosteric heat of adsorption
3.3.3.1 Gibbs excess adsorption model
117 |
Virginia Tech | For a pure gas and solid sorption system, the excess adsorbed amount (n ) can be shown by the
e
Gibbs equation (4),
n n V n (1 g)
(4)
e a a g a
a
where n is the excess adsorption content, n is the absolute adsorption content, V is the total
e a a
volume of both adsorbed and absorbed phase, is the density of adsorbed phase and is the
a g
density of bulk phase under specific temperature and pressure,. When the V is very small, the
ad
contribution of the adsorbed gas phase can be neglected and n is the approximation of the n (Zhou
e a
et al., 2001). However, for a high pressure sorption system, the contribution of the adsorbed phase
must be taken into consideration to physically interpret the adsorption behavior such as the
decreasing observed adsorption uptake with increasing pressures after the maximum observed
adsorption uptake. Therefore, it is imperative to find a reasonable approach for obtaining the
absolute isotherms from Gibbs excess isotherms considering properties of adsorbed gas phase for
gas-solid sorption system under high pressures and temperatures.
Considering the accepted assumption that the density of adsorbed gas phase can be treat as an
unknown constant (Agarwal et al., 1988; Zhou et al., 2001; Do et al., 2003; Stadie et al., 2012;
Schell et al., 2012; Stadie et al., 2013 & 2015; Tang et al., 2016), the volume of the adsorbed gas
phase can be obtained by equation (5),
n
V a (5)
a
a
Combining equation (3), equation (5) can be rewritten as,
K (T)P K (T)P
V V (1) 1 2 ) (6)
a max 1K (T)P 1K (T)P
1 2
where V is the total volume of adsorbed gas phase under specific temperature and pressure, V
a max
is the maximum volume at maximum adsorption content, V n . Combining equation (3),
max max a
(4) and (6), both excess adsorption (n ) and surface coverage () equation can be obtained as
e
shown in equation (7) and (8)
118 |
Virginia Tech | K (T)P K (T)P
n (P,T)(n V ) (1)( 1 )( 2 ) (7)
e max max g 1K (T)P 1K (T)P
1 2
K (T)P K (T)P
(1) 1 2 ) (8)
1K (T)P 1K (T)P
1 2
If the observed adsorption isotherms (Gibbs excess adsorption isotherms) are obtained through
isothermal adsorption tests, equation (8) is able to describe the adsorption behavior where the
parameters (V ,n ,K(T)) can be easily obtained via curve fitting. The absolute adsorption
max max
content can then be obtained via equation (5).
3.3.3.2 Isosteric heat of adsorption
Based on the Clapeyron relationship, it is known,
dP dP
H ( ) T v( ) T (v v ) (9)
ad dT na dT na a g
Where H is the isosteric heat of adsorption, v((v v )) is the volume change of phase change,
ad a g
v is the molar volume of bulk gas phase, v is the molar volume of adsorbed gas phase, and T is
g a
temperature.
dP
In order to obtainH , the ( ) must be obtained first (Chakraborty et al., 2006; Stadie et al.,
ad dT na
2015). Based on the surface coverage concept (equation (3)), the following relationship can be
obtained (Stadie et al., 2014 & 2015),
dP P K
( ) ( ) ( ) ( ) (10)
dT na na K na T na
Combining with equation (10) and (11) and applying the ideal gas law (Pv RT ), we can obtain
g
one form of isosteric heat of adsorption mathematically (equation 11),
P K RT
H ( ) ( ) ( ) T ( 1) (11)
ads(na),IGL na K na T na P a
If we combine equation (10) and (11) and apply the real gas law, another analytical form of the
isosteric heat of adsorption can be obtained (equation (12),
119 |
Virginia Tech | P K
H ( ) ( ) ( ) T (11) (12)
ads(na),RGL na K na T na g a
Equation (12) and (13) are much easier to solve using the analytical approach. Equation (13) shows
dP
the analytical solution of ( ) (Stadie et al., 2013 & 2015),
dT na
(1)P EK (T) P E K (T)
1 1 2 2
P K (1K (T)P)2 RT2 (1K (T)P)2 RT2
( ) ( ) ( ) 1 2 (13)
n a K n a T n a (1)K (T) K (T)
1 2
(1K (T)P)2 (1K (T)P)2
1 2
Using dual-site Langmuir adsorption model, absolute adsorption isotherms can be obtained
without any subjective assumptions from the excess adsorption isotherms. The isosteric heat of
adsorption can be calculated considering the real gas behavior and the contribution of adsorbed
and absorbed phase, and both of them are not considered in the classic Clausius-Clapeyron
approximation.
3.3.4 Data set acquisition and processing
The study of scCO adsorption in coals are limited in literature. Two data sets are directly retrieved
2
from the literature (Song et al. (2015); Ottiger et al., 2006). All these data were measured using
gravimetric approach to obtain the scCO sorption isotherms in dry coal under different
2
temperatures and pressure (>=15MPa). Detailed information about these tests are referred the
original publications. The pertinent data from these papers are shown in Figure 3.3.3.
Figure 3.3.3 Experimental data retrieved from Song et al.(2015), Ottiger et al.(2006); the
dotted line is to connect data points for visualization.
The observed adsorption isotherms under different temperatures are fitted simultaneously using
equation (7) within the limits of the fitting parameters (0<n <100 mmol/g, 0 <V < 100 cm3/g,
max max
120 |
Virginia Tech | For both coals, a consistent observed phenomenon is that there are crossovers of the observed
adsorption isotherms under different temperatures after the observed adsorption content reached
the maximum value. After the crossover, the observed adsorption content at higher temperature is
higher than that at low temperatures under same pressure, which cannot reflect the nature of
adsorption under different temperatures. However, the nature of adsorption under different
temperatures can be shown via the absolute adsorption uptake, where the higher the temperature
the lower the adsorption uptake. The abnormal crossover phenomenon of the observed adsorption
isotherms appears because the pressure is used as the independent variable, instead of the density
of scCO . Two components of Gibbs excess adsorption uptake (equation (7)), surface coverage (ϴ)
2
and the term (n -V *ρ (P,T), as a function of pressure and density are shown in Figure 3.3.5.
max max
When the pressure is treated as a variable, the term (n -V *ρ(P,T)) show temperature as well
max max
as pressure dependency feature because of the density of scCO , ρ(P,T) . However, when the
2
pressure is treated as a density, the term (n -V *ρ (P,T) only show pressure-independency
max max
feature. Therefore, the cross-over is simply caused by the PVT behavior of scCO .
2
Figure 3.3.5 Surface coverage (solid line, left axial) and the term (n -V *ρ (P,T)) (dotted
max max
lines, right axial) with increasing pressure (density) for both Chinese and Sulcis coals
3.3.5.2 Adsorption isotherm prediction
122 |
Virginia Tech | For geological storage of carbon dioxide in the subsurface, such as unminable coal seams and shale
formations, one of the crucial questions is how to estimate the storage capacity of carbon dioxide
under real geological conditions. With increasing depth, both in-situ reservoir pressure and
geothermal effects become pronounced, and there will be a phase change of carbon dioxide from
subcritical status to supercritical status. Thus, carbon dioxide storage capacity must take high
pressure and high temperature conditions into consideration. However, it is impractical to measure
all isotherms under different temperatures. Therefore, finding an adsorption model to predict
higher temperature adsorption isotherms using lower temperature data also arises researcher’s
interests.
Predicting isotherms under different temperatures is possible using the proposed model because
the temperature dependency of adsorption isotherms are only shown through the Langmuir
constant (K (T) and K (T)). Figure 3.3.6 (left part) shows the predicted adsorption isotherms have
1 2
the same feature of the observed adsorption isotherms, where both the crossovers and the abnormal
phenomenon (the higher the temperature the higher the observed adsorption uptake) occur. When
the adsorption isotherms are plotted as a function of bulk density, the crossovers disappear, which
means the higher the temperature the lower the observed adsorption uptake. This behavior can also
be shown in the predicted adsorption isotherms (Figure 3.3.6, right part).
123 |
Virginia Tech | Figure 3.3.6 Prediction of adsorption isotherms (black lines) beyond test data; left:
adsorption uptake as a function of pressure, right; adsorption uptake as a function of bulk
density.
3.3.5.3 Confidence of the predicted adsorption isotherms
In order to validate the prediction ability of the proposed model, the Chinese coal sample is taken
as an example to show the confidence of the predicted adsorption isotherms. First, only three
adsorption isotherms from four isotherms are fitted using the proposed method. Then, the predicted
adsorption isotherms are compared with the test data to see the difference. Table 3.3.2 shows the
fitting parameters use only three adsorption isotherms. Figure 3.3.7 shows all predicted absolute
isotherms are consistent with test data. The predicted adsorption isotherms beyond test data are
approximately the same. Comparing with Figure 3.3.4, it can be seen the fitting results using four
isotherms are always better than predictions using three isotherms. This also means the best-fit
parameter can only be achieved using a large set of measured data.
Table 3.3.2 Fitting parameters (equation (7)) for observed adsorption isotherms using only
three adsorption isotherms
Predicted
Samples Fitting isotherms (K)
isotherms (K)
n max(mmol/g) V max(cm3/g) α E1(kJ/mol) E2(kJ/mol) A1(MPa-1) A2(MPa-1)
293.29, 311.11, 332.79 352.55, 392.57 1.6837 0.0760 0.5345 26.593 27.244 6.08E-06 1.54E-04
Chinese coal 293.29, 332.79, 352.55 311.11, 392.57 1.6521 0.0745 0.5033 27.632 24.941 1.66E-04 1.44E-05
293.29, 311.11, 352.55 332.79, 392.57 1.6471 0.0742 0.5050 26.962 25.192 2.05E-04 1.41E-05
124 |
Virginia Tech | Figure 3.3.7 Adsorption isotherms comparison between fitting data using three low
temperature isotherms (dotted dark lines) and fitting data using four temperature
adsorption isotherms (solid color lines). Solid symbol represents test data.
3.3.5.4 Thermodynamic analysis of isotherms
Figure 3.3.8 shows isosteric heats of adsorption calcualted using equations (11) and (12) and it can
be seen that (i) the isosteric heat of adsorption decreases with increasing absolute adsorption
uptake and (ii) temperature dependence of the isosteric heat of adsorption. It is also clear that the
isosteric heats of adsorption using the ideal gas law are always higher than that using the real gas
law at the same temperature. This means the isosteric heats of adsorption using the ideal gas law
always overestimate the isosteric heat of adsorption. Both the temperature dependency and
adsorption uptake dependency of the isosteric heat can also be readily shown using the proposed
approach.
Figure 3.3.8 Comparison of different isosteric heats of adsorption for scCO in coal: isosteric
2
heat of adsorptions using ideal gas law (dotted lines) and real gas law (solid lines), and the
gas phases are supercritical and liquid in the magnified area.
125 |
Virginia Tech | After carbon dioxide becomes scCO (or liquid CO ), the heat of adsorption shows different
2 2
behaviors as shown in the magnified area in Figure 3.3.8. When the CO is in liquid phase, the
2
isosteric heat of adsorption is the lowest compared with scCO . When the CO becomes scCO ,
2 2 2
the isosteric heat of adsorption decreases with increasing absolute adsorption uptake but there are
crossovers of isosteric heat of adsorption under different temperatures, which has not been reported
for other gases in literature. This can be attributed to the enhanced interaction among adsorbed
phase over the coal surface at higher density (Schaef et al, 2013). Figure 3.3.9 shows the density
at low temperature is much higher than that at high temperature for scCO , which may strengthen
2
the enhanced interaction effect.
Figure 3.3.9 Density of liquid and scCO ; dotted line represents liquid carbon dioxide and
2
solid line represents scCO
2
3.3.6 Implications for geological carbon dioxide storage
Caron dioxide storage in deep unminable coal seams and deep shale formations are two of the
promising geological sites for onshore carbon sequestrations. Since the adsorption phase of carbon
dioxide in coal is one of the main components of the total carbon dioxide gas-in-place in the
subsurface, the accurate prediction of adsorption carbon dioxide in coal is crucial. With increasing
depth, high pressure and high temperature geological situation has to be accounted for. Such
geological conditions result in the phase change of carbon dioxide from subcritical phase to
supercritical phase, and an optimized model is needed to model such behavior of carbon dioxide.
The proposed dual-site Langmuir model can describe both the observed adsorption and the
absolute adsorption of sub- and super-critical carbon dioxide adsorption in coal. In order to
accurately assess the storage capacity of scCO in coal, the equation (14) should be used for the
2
total carbon dioxide gas-in-place (GIP) in the subsurface (Tang et al., 2016),
126 |
Virginia Tech | GIPn V (14)
e tot g
Where n is the observed adsorption uptake (Gibbs excess adsorption uptake), V is the total pore
e tot
space of coal, is the density of bulk gas. Equation (14) supports that if the observed adsorption
g
isotherms are obtained from laboratory tests, the carbon dioxide storage capacity can be accurately
assessed and there is no need to differentiate absolute adsorption content from observed adsorption
isotherms. However, this is only valid for estimating the total storage capacity of coal seams. In
order to understand the existing status of injected carbon dioxide in coal and carbon dioxide
transport behavior under reservoir conditions, the true ratio between bulk carbon dioxide and
adsorbed phase must be known. Using the observed adsorption isotherms (Gibbs excess adsorption
isotherms) will always underestimate the true content of adsorbed phase as shown in Figure 3.3.4.
The dual-site Langmuir model can solve this problem by extrapolating the true content of the
adsorbed phase (equation (3)) from observed adsorption isotherms. Furthermore, for large scale
carbon dioxide injection test in subsurface coal seams, one of the fundamental questions is to
understand how the injected carbon dioxide transports in the coal seams (Tang et al., 2015). Since
carbon dioxide existed in coal seams mainly in two phases (free gas and adsorbed gas), the
adsorbed phase does influence the transport behavior of carbon dioxide because it not only
occupies spaces in nanopores but also interacts with pore walls. It has been found that the adsorbed
gas phase significantly affects adsorptive gas (methane) transport in coal seams and shale
formations (Yu et al., 2014; Akkutlu et al., 2012; Civan et al., 2011; Singh et al., 2016; Naraghi et
al., 2015; Wu et al., 2016). This situation will occur for carbon dioxide transports in coal. Therefore,
in order to build a reasonable gas transport model for carbon dioxide transport in coal seams, the
very first step is to obtain the true ratio of free phase to adsorbed phase (Tang et al., 2016). The
proposed single-site Langmuir adsorption model will make this possible.
3.3.7 Conclusions
This work uses a dual-site Langmuir adsorption model to describe high pressure carbon dioxide
adsorption in coal by considering both adsorbed phase and absorbed phase based on the
assumption the density of the adsorbed and absorbed phase is an unknown constant. Then, the
isosteric heat of adsorption are calculated analytically by considering both the real gas behavior
and the adsorbed gas phase. Last, the published data for high pressure carbon dioxide adsorption
in coal are retrieved to verify the proposed model.
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Virginia Tech | Modeling results reveal the proposed model can not only reasonably address observed test
phenomena but also has the intrinsic ability to extrapolate adsorption isotherms under different
temperatures beyond test data. It was found that the crossovers of the observed adsorption
isotherms under different temperatures are caused by the changing volume of the adsorbed and
absorbed gas phase during adsorption process.
Both the temperature dependence and absolute adsorption uptake dependence of the isosteric heat
of adsorption are readily investigated for scCO adsorption in coal; for subcritical carbon dioxide,
2
the higher the temperature the lower the isothersteric heat of adsorption and the higher the absolute
adsorption uptake the lower the isothersteric heat of adsorption. The gas behavior significantly
affects the quantity of isosteric heat of adsorption: the isosteric heat of adsorption using ideal gas
law always overestimates isosteric heat of adsorption for scCO adsorption in coal. For scCO
2 2
adsorption in coal, there exists an abnormal phenomenon that the higher the temperature the higher
the isosteric heat of adsorption, which has not been reported in literature.
The proposed adsorption model lays the foundation for accurately estimating the storage capacity
of carbon dioxide, differentiating the true ratio between adsorbed phase and bulk phases and
developing gas transport by considering adsorbed phase effect. The thermodynamic analysis is
helpful for interpreting temperature related phenomena associated with carbon dioxide adsorption
in coal.
Acknowledgements
Financial assistance for this work was provided by the U.S. Department of Energy through the
National Energy Technology Laboratory’s Program under Contract No. DE-FE0006827.
References
Grubb, M. (1995). Seeking fair weather: ethics and the international debate on climate change.
International Affairs (Royal Institute of International Affairs 1944), 463-496.
Stern, N. (2009). The global deal: Climate change and the creation of a new era of progress and
prosperity. PublicAffairs.
Morgan, J., Dagnet, Y., & Tirpak, D. (2014). Elements and ideas for the 2015 Paris agreement.
Washington, DC: Agreement for Climate Transformation.
128 |
Virginia Tech | Chapter 4 Gas adsorption kinetics analysis and pore characterization of coal
4.1 Isothermal adsorption kinetics properties of carbon dioxide in crushed coal
Xu Tanga*, Nino Ripepia, Ellen Gillilanda,b
(a Department of Mining and Minerals Engineering, Virginia Polytechnic Institute and State
University, Blacksburg, Virginia, 24060, USA; b Virginia Center for Coal and Energy Research
(0411), Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA)
Abstract: Understanding the dynamic response of coal to carbon dioxide sorption is crucial for
optimizing carbon dioxide sequestration in unmineable coal seams and enhanced coalbed methane
recovery. In order to explore the adsorption kinetics of carbon dioxide in coal, fifteen isothermal
adsorption tests were conducted on bituminous and subbituminous coals at 50℃ for increasing
equilibrium pressures (up to 4 MPa). The pseudo-second order (PSO) model is introduced to
approximate the carbon dioxide sorption kinetics in coal, and the kinetics properties are then
investigated via the PSO model. The linear relationship between (t/q) and (t) is validated and
confirmed with a high correlation coefficient (> 99%). The kinetics parameter, k2, decreases with
both increasing equilibrium sorption pressure and increasing pressure difference. The sorption
equilibrium content, Qe, in each sorption stage depends on both the final equilibrium pressure and
the pressure difference. Based on the relationship between sorption content and time, the sorption
content for different pressure ranges is predicted using different time intervals. The analysis
indicates that the adsorption process for carbon dioxide in coal is a combination of both bulk
diffusion-controlled and surface interaction-controlled processes; the former dominates the initial
stage while the latter controls the majority of the overall process.
Key words: Coal, Carbon dioxide, Kinetics, Adsorption, Pseudo-second order model
Published in Greenhouse Gases: Science and Technology, DOI: 10.1002/ghg.1562.
136 |
Virginia Tech | 4.1.1 Background and introduction
Understanding the carbon dioxide sorption properties of coal is crucial for carbon dioxide
sequestration in unmineable coal seams and enhanced coalbed methane recovery.1 Extensive
studies have been conducted to determine what influence moisture, coal rank, temperature and
pressure have on the sorption properties of coal gases.2-8 However, most of the studies focus on
the thermodynamic properties of the sorption process (sorption content of gas in coal), which only
relates to the ultimate state of the sorption system. Few studies consider the sorption kinetics,
which describe changes in the sorption process with respect to time, or the gas transport rate.
Having an understanding of sorption kinetics is critical for understanding the sorption mechanism
of gas in coal.
Many key research questions surrounding the geological sequestration process relate to the
sorption kinetics of carbon dioxide in coal. For example, how quickly the injected CO plume will
2
migrate through a coal seam during injection, how the sorption process will affect the
transportation of carbon dioxide in the coal seam, whether continuous injection or intermittent
injection is more effective for maximizing storage, and how long it takes for the reservoir to reach
new gas-coal sorption equilibrium. All of these issues are related to the kinetics characteristics of
gas and coal interactions. Field studies of these variables are costly and time-consuming.
Controlled laboratory studies of system kinetics, including isothermal tests, provide the best way
to investigate the mechanisms behind these phenomena.
To understand the kinetic properties of the coal and gas sorption system, the gas transport
characteristics of the system must be defined in order to determine which factors are influential.
Generally, the gas transport in coal can be divided into three stages: ① gas flow in the
macropore/cleat system of the coal, ② gas diffusion within the cleat system of coal, and ③
physical interaction of the coal and gas (gas adsorption and desorption on the coal surface). The
overall rate of the sorption process may be controlled by any of these three steps or a combination
of them. At the laboratory level, the time associated with gas transport in the macropore system
can be neglected because of the millimeter-scale coal sample in the test. Thus, step ② and step
③ or their combination controls the entire process. How gas is transported through the micropore
system of coal, especially at a nanolevel, is still unknown. Even though the CO desorbs from the
2
137 |
Virginia Tech | coal surface in only 10 to 50 ms,9 whether it will influence the overall dynamic equilibrium
sorption process is still unknown.
The purpose of this study is to investigate the adsorption processes for carbon dioxide and two
types of coal (bituminous and subbituminous coal). Fifteen isothermal adsorption tests were
conducted at 50℃ for increasing equilibrium pressures, and the test data were gathered at specific
time intervals for up to 14 days. The test data were analyzed using the pseudo-second-order (PSO)
kinetics model.
4.1.2 PSO sorption kinetics model
There are only three isothermal adsorption kinetics models which have been applied for gas and
coal interactions (shown in Table 4.1.1): the unipore model,10-19 the bidisperse model,3, 20-23 and
the dynamic diffusion model.24-26 The first two models are widely used in the CBM industry. The
unipore model is used as a theoretical foundation for estimating the lost content during drilling via
the Square-Root-Time method. The unipore model is better for high rank coal and the bidisperse
model is better for low rank coal.3
Table 4.1.1 Comparison of different adsorption kinetics models for gas in coal
Many kinetics models for the solid/solution interaction system exist in chemical engineering.27-31
The kinetics models used in physical chemistry may be used for gas-solid sorption system. There
are currently three classic and widely used kinetics models used to describe the adsorption rate for
different sorption systems: the pseudo-first-order model, 28,32-33 the pseudo-second-order model,33-
138 |
Virginia Tech | 34 and the intraparticle diffusion model (unipore model). The PFO and PSO models have been
widely used for the solid/solution interaction system to explain the kinetics phenomena occurring
in a chemical reaction. This interaction is based on the assumption that the surface interaction (Step
③) dominates the kinetics process, and the rate at which molecule of the adsorbate enter the
adsorbed phase either controls the overall rate of the sorption process or is involved in it. Here, the
surface interaction includes the actual chemical bond reaction on the surface of the adsorbent and
the physical interaction such as van der Waals forces.27 Both Azizian (2004)28 and Liu et al
(2008)33 show that the PFO and PSO models are special cases for the Langmuir rate equation. They
also point out that these models can be used to describe chemical or biosorption systems in addition
to other sorption systems. 28-30, 33
(1) Pseudo-first-order model (PFO)
The PFO, also called Lagergren equation has the following differential form, 32
dQ
t k (Q Q ) (1)
dt 1 e t
where Q is the amount of gas adsorbed on the surface of adsorbent, Q is the equilibrium
t e
value of Q, t is time, and k is the PFO rate coefficient, or the time-scaling factor describing
t 1
how fast equilibrium can be reached in the system. In Integral equation (2), the PFO model
is obtained.
Q Q (1exp(kt)) (2)
t e 1
The published literature shows that the PFO model is more reasonable when the change in
adsorbed gas density is small or the change of surface area covered in sorbent is small.28-33
This condition does not apply to the gas and coal sorption system. However, as a parallel
model, the PSO model is more applicable when the change of the gas adsorbate density is
significant. The PSO model may be used for the gas-solid interaction system.
(2) Pseudo-second-order model (PSO)
The PSO model was first empirically proposed by Blanchard et al (1984)35 and was later
theoretically proven by Azizian (2004)28. The PSO model has the following form,34
dQ
t k (Q Q )2 (3)
dt 2 e t
139 |
Virginia Tech | where Q is the amount gas adsorbed on the surface of adsorbent, Q is the equilibrium
t e
value of Q, t is time, and k is the PSO rate coefficient, a complex function of the density
t 2
of adsorbed gases. Integral equation (3) produces the PSO model.
k Q2t
Q 2 e (4)
t 1k Q t
2 e
Equation (4) can be rearranged in the following form, which is most favorable,
1 1 1 1
( ) (5)
Q k Q2 t Q
t 2 e e
Plotting 1/Qt and 1/t gives a linear relationship, where 1/Q is the intercept of the obtained
e
line and 1/(k Q 2) is the slope.
2 e
Generally, the PFO and PSO models are used for describing the adsorption/desorption kinetics
(Step③) when the entire sorption process is controlled by the surface interaction, instead of by
the adsorbate mass transfer, as with the unipore model (Step②). Rudzinski et al (2007)36 proposed
a theoretical interpretation for the difference between the diffusion and surface controlled
processes via the statistical rate theory method. Miyake (2013)37 also pointed out that a relationship
exists between the PSO rate coefficient, k , and the diffusion coefficient within spherical
2
homogenous adsorbent microspheres in the unipore model. This implies that the PSO model may
be used to interpret the sorption kinetics for carbon dioxide in coal. Therefore, the PSO model is
used here to analyze the kinetics data throughout the sorption process.
4.1.3 Experimental section
4.1.3.1 Sample preparation
The blocks of coal used in this study were obtained from the Pocahontas No. 3. coal seam
(bituminous coal) and Eagle Butte coal mine (subbituminous coal). The coal specimens were
ground and sieved by 1.0 mm-1.7mm (12-18 U.S. mesh) metal sifters with natural weathering for
the isothermal testing. The proximate and ultimate analysis results for the test samples are shown
in Table 4.1.2.
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Virginia Tech | Table 4.1.2 Proximate and ultimate analysis of coal (Unit: %)
4.1.3.2 Test approach
All tests were conducted using high temperature and pressure (HTHP) isothermal test equipment
made by GoldenAPP of China. The experimental design is based on the manometric method
(similar with volumetric approach) using the Sieverts apparatus.38 The schematic of the setup and
the parameters of the sorption system are shown in Figure 4.1.1, and the test parameters are shown
in Table 4.1.3. In this approach, a defined amount of gas is transferred from a calibrated reference
cell to a test cell containing the sample. The sorption content of gas is the difference between the
mass of the transferred gas and the mass of the reference gas. The mass of transferred gas is
calculated based on the pressure change of the reference cell. The mass of the reference gas is
calculated by multiplying the density of the gas in the test cell and the void volume, which is
determined prior to the test using a helium displacement method.
The detailed test procedure is as follows:
1) The weight-measured coal sample is put into the test cell, and the desired sorption pressure
equilibrium points are defined using the preset software.
2) Prior to the sorption test, the void volume of the test cell is determined by the helium
displacement method. The volume and density of the coal sample can also be obtained here.
3) During the sorption test, predefined amounts of gas are continuously transferred from a
calibrated reference volume into the test cell containing the sample.
4) The pressure and temperature of both cells is continuously measured and recorded at certain
time intervals throughout the test. These values are used for calculating the mass of the transferred
141 |
Virginia Tech | 4.1.3.3 Data collection and processing
The first two sorption stages (Ⅰ) and (Ⅱ) are used as examples in Figure 4.1.2 to help explain the
data measurement and recording process described in Step (4) of Section 3.2. In the first sorption
stage (Figure 4.1.2), the values of both the pressure and temperature of the test cell and the
reference cell are recorded at the incremental point (t =0, q ), then (t , q ) …… and so on (t , q )
1 1 2 2 n n
(n>1) until the process reaches the first sorption equilibrium points defined in Step 1 of 3.2 Test
procedure . The second stage begins at the end point of the first and proceeds in the same way.
The time interval between the first two points (∆T=t -t ) and other points (∆t=t -t ) (n>1) is
1 0 n+1 n
different (Figure 4.1.2); ∆T is around 20 minutes and ∆t is around 12 minutes. The ∆T is spent on
transferring gas from the reference cell to the test cell, which is necessary for instrument operation.
The gas refill process causes the small increase of sorption content in the ∆T. Since the first time
interval ∆T is small in comparison to the entire test process, the sorption process analysis in this
report will begin at the point t instead of t , and end at the next t . The total sorption content (Q)
1 0 0 t
in each sorption stage consists of two parts. Values for Q, Q and ∆Q ( where Q =Q +∆Q , for
t n n t n n
n>1) are directly obtained from data recorded during this test. The test data are used to evaluate
the quality of the predicted data derived from the kinetics model.
Figure 4.1.2 Time dependent sorption data recording process
4.1.3.4 Determination of sorption equilibrium
Standards for the determination of sorption equilibrium in the coal and gas sorption system have
only recently been established. The true equilibrium state for sorption between the coal and gases
(CO2, N2, CH4) may never be reached due to kinetics restrictions of gases in coal, but “technical
143 |
Virginia Tech | equilibrium” or “quasi-equilibrium” can be defined, reached and applied during the test.41-42
Currently, there are two popular methods to determine the equilibrium state: a pressure monitoring
approach and a sorption time monitoring approach43-44. The former monitors the change of the
pressure in the sample cell; if the pressure change of the test cell is within a certain range, the
system is treated as having reached its equilibrium state. The latter approach, determined from
sorption time, is an empirical method, and different research groups use different sorption times
for the isothermal adsorption test. However, both approaches are limited. The shortcoming of the
pressure monitoring approach is that it neglects the temperature influence on sorption content
during the sorption process. Since the temperature cannot be fully controlled during the sorption
process (accuracy of temperature measurement), it may happen that the temperature variation will
affect the sorption content change. Another issue is that applying the same equilibrium criteria
under different pressures is not reasonable since the sorption pressure interval and sorption
equilibrium pressure are different. For the empirical, time determined approach, the equilibrium
time varied from laboratory to laboratory and is hard to evaluate. Generally, a longer time produces
more accurate results, but too long of a waiting time cannot provide the quick turnaround needed
to serve the industry. In this test, the sorption equilibrium status is determined by a sorption time
monitoring approach based on the author’s technical experience. The equilibrium times for
subbituminous coal and bituminous coal are 9-10h and 5-6 h, respectively.
4.1.4 Test results
As shown in Figure 4.1.3, changes in the isothermal sorption content with time are recorded during
the sorption process in eight different pressure ranges (MPa) under 50℃. To compare the sorption
processes over different pressures, each sorption process is extracted (right side of Figure 4.1.3).
Figure 4.1.4 shows the isothermal adsorption curves of both bituminous coal and subbituminous
coal. The adsorption capacity of bituminous coal is higher than the subbituminous coal.
144 |
Virginia Tech | Figure 4.1.3 shows there are sorption content fluctuations when the sorption pressure increases
during stages (5-8) for the bituminous coal and during stages (6-7) for the subbituminous coal.
These fluctuations are inevitable and are caused by pressure measurement error and temperature
influence. The pressure transducer is of very high precision (0.05% of full scale) and is able to
monitor and record slight changes occurring in the test cell. When the test cell pressure increases,
the corresponding measurement error also increases, ultimately causing the fluctuation in sorption
content (Figure 4.1.5). It is also clear that the sorption process in stages 7-8 for the bituminous coal
and stages 6-7 for the subbituminous coal are different from other processes (discussed in Section
5.1.). Another factor contributing to the fluctuations is related to the calculation of the sorption
content, which is based on the density of gas under different pressures and temperatures. The
fluctuation of the sorption content is the integrated effect of both temperature and pressure
variation. In the later sorption stages, as the absolute sorption pressure increases, the absolute
sorption content becomes smaller. The measurement error resulting from both the pressure
measurement error and the temperature error will be magnified compared to the previous stages.
Figure 4.1.5 Measurement error of the pressure transducer
4.1.5 Discussion
4.1.5.1. PSO model application
First, the physical meaning of the kinetics parameter in equation (5) of the gas-solid system needs
to be explained. Here, Q represents the equilibrium sorption content in each sorption stage under
e
different pressure ranges, and the k represents the other kinetics parameter used to evaluate the
2
sorption process. The linear relationship between t/Q and t is used to fit the data, which correspond
t
146 |
Virginia Tech | very well to the test data (Figures 4.1.6 and 4.1.7, Table 4.1.4). The equilibrium sorption content
(Q ) and kinetics parameter (k ) in each sorption stage are also obtained (Table 4.1.4).
e 2
Figure 4.1.6 Linear relationship between t/Qt and t: stage (1) and stage (8) for bituminous
Figure 4.1.7 Linear relationship between t/Qt and t: stage (1) and stage (8) for sub-
bituminous coal
Figures 4.1.6 and 4.1.7 show that, even though the curves show a very highly fitting trend
(R2>0.98), there are deviations in the initial test stage and the final test stage (blue circles). In the
initial test stage, the diffusion-controlled process dominates the process; carbon dioxide molecules
are adsorbed on the surface of coal and/or fill pore spaces as a result of the increase of pressure in
each stage. At this time, the increasing pressure increases the density of the gas molecules inside
the pores of the coal and is the controlling factor in the sorption system compared to interactions
among gas molecules. This is supported by the unipore model, which is applicable when
Q/Q <0.545 and which has bulk diffusion as the controlling factor in the initial adsorption stage.
t ∞
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Virginia Tech | In the following stages, the interaction among the gas molecules becomes dominant (linear
relationship between sorption content and time), which means the dynamic process of gas
adsorption/desorption controls the system. This is the point where the PSO model can be applied.
As shown in Figure 4.1.6 and 4.1.7, the diffusion-controlled process is only a small part of the
total sorption process and does not significantly influence the overall sorption process.
Table 4.1.4 PSO model fitting data
Reviewing the sorption content curves in Figure 4.1.3, it can be found that the sorption content in
stages (7-8) of bituminous coal and stages (6-7) of subbituminous coal shows an increase
compared to previous stages. This could be due to capillary condensation the mesopore system of
the coal or the pore-filling phenomena which occurs in nanopores.46-48 In addition, since the coal
sample is dried naturally, the presence of residual water vapor inside the coal may also contribute
to the carbon dioxide condensation.49-50 Therefore, the conventional sorption equilibrium
determination approach cannot be applied in these stages.
4.1.5.2 Kinetics parameters analysis using PSO model
As shown in Figure 4.1.8 (A & C), as the equilibrium sorption pressure increases, the equilibrium
sorption content in each stage appears to initially increase and then begins to fluctuate. No
consistent quantitative relationship can be obtained from current data. Figure 4.1.8 (B &D) shows
that an increase in pressure difference causes a general but unstable increase in the equilibrium
sorption content. When the pressure differences are approximately the same, the stage of low
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Virginia Tech | sorption equilibrium pressure results in a higher sorption content than the stage of high sorption
equilibrium pressure (red ellipse in Figure 4.1.8 (B)). Figure 4.1.8 (A-D) also show the sorption
content in each sorption process (1-8) is controlled by both the final equilibrium pressure and the
pressure difference. For the bituminous coal, the maximum sorption content occurs at a sorption
equilibrium pressure of 1.0884MPa with a pressure difference of 0.6018MPa, and the minimum
sorption content occurs at the first sorption pressure point of 0 MPa with a pressure difference of
0.064MPa. For the subbituminous coal, the maximum sorption content occurs at a sorption
equilibrium pressure of 3.1822MPa with a pressure difference of 1.5608MPa, and the minimum
sorption content occurs at the first pressure point of 0 MPa with a pressure difference of
0.0889MPa.
Figure 4.1.8 Relationship between pressure and kinetics parameter Qe
The kinetics parameter (k ) decreases as the equilibrium sorption pressure and the pressure
2
difference increase for both bituminous and subbituminous coal (Figure 4.1.9). The highest k
2
value occurs at a pressure of 0.064 MPa for bituminous coal and at a pressure of 0.0368 MPa for
subbituminous coal. The lowest k is 0.0815 for bituminous coal at the sorption equilibrium
2
149 |
Virginia Tech | pressure of 4.0451MPa with a pressure difference of 0.9018MPa, and is 0.0458 for subbituminous
coal at a sorption equilibrium pressure of 3.1822MPa with a pressure difference of 1.5608MPa.
Figure 4.1.9 also shows that the k of bituminous coal is generally higher than the k of
2 2
subbituminous coal, which implies that the value of k may be influenced by the coal rank or the
2
affinity of different types of coal.43 Since the highest k value is obtained in the first sorption stage
2
where the sorption pressure is low (0.064 MPa for bituminous coal and 0.0368 MPa for
subbituminous coal), it may indicate the affinity of different types of coal or the retention
properties of carbon dioxide on different types of coal.
Figure 4.1.9 Relationship between pressure and kinetics parameter k
2
4.1.5.3 Sorption capacity estimation using PSO model
Since sorption equilibrium content (Q ) is easily obtained from the time and (t/q) linear relationship,
e
it may be possible to determine the (Q ) using different time ranges under each pressure step. To
e
validate this hypothesis, the initial portion of the sorption-time relationship is used to predict the
150 |
Virginia Tech | final sorption equilibrium status. The different sorption time ranges are analyzed using the PSO
model for bituminous and subbituminous coal (same procedure used in Section 5.2). Figure 4.1.10
shows predicted curves for adsorption equilibrium content based on the PSO model for different
time intervals (120, 180, 240, 360, 480, 600 min) compared to the measured test data.
The fitting coefficient of the PSO model for each sorption stage is extremely high (> 95%)
(Detailed comparison between predicted accumulated sorption content and measured accumulated
sorption content in each sorption stage is shown in the Supporting documents). Figure 4.1.10
shows the error analysis for fitting the measured data curves to the PSO model, where the error is
the ratio of the difference of the calculated value and test value to test value. For bituminous coal,
the lowest error is achieved using the 360-minute predicted sorption-time curve to match the
measured test data. The best match for subbituminous coal is the 600-min predicted sorption-time
curve. The best-fit curve for each coal type is plotted with the associated measured test data in
Figure 4.1.10 and confirms the low error of each match. The predicted results are credible, and
the accuracy of the maximum error is within 0.2 percent. The test results support that the sorption-
time relationship can be used to predict the final equilibrium sorption content in each sorption
stage.
151 |
Virginia Tech | Figure 4.1.10 Isothermal adsorption curves and error analysis; Q is the predicted value
tc
using PSO value, Q is the measured test value.
tm
Figure 4.1.11 Comparison between predicted values and measured test data
4.1.5.4 Implication of the PSO kinetics model
Generally, the pore system of coal is complicated and consists of macropores (50nm < pore width),
mesopores (2nm <pore width< 50nm), micropores (< 2 nm) (IUPAC, 2001), and submicropores
(< 0.4 nm) 51. The micropore dominates and determines the specific surface area in coal 1,52. The
different types and shapes of pores in coal also complicate gas transport in coal (Figure 4.1.12).
The two-ended open pore is easy to access while the one-ended open pore (dead end pore and ink-
bottle pore) and closed pore are hard to reach. For the ink-bottle pore, a high external force is
needed to push the gas molecule into the pore. This may also be true for narrow channels within
the pore system induced by pore wall effects.
152 |
Virginia Tech | Figure 4.1.12 Generalized pore system in coal
The pressure influenced kinetics parameters (Q and k ) can reveal and describe important physical
e 2
properties of coal, including the complex pore system, heterogeneous properties of the coal surface,
and the approximation of monolayer adsorption or pore-filling effects under low pressures. As
shown in Figures 4.1.8 and 4.1.9, there is no consistent relationship between the maximum
equilibrium sorption content Q and kinetic parameter k . This may be attributed to the different
e 2
size and amount of pores in coal. It could also indicate that the gas easily accesses the coal but
does not remain securely stored; the higher kinetic parameter does not mean the highest sorption
content. The different sorption contents associated with different pressures also imply a
heterogeneous nature of the coal surface. When the carbon dioxide is first exposed to the coal, it
can be easily adsorbed in the high potential energy sites induced by pore wall effects. When the
pressure is increased, the low sorption potential energy sites are occupied and the interaction of
carbon dioxide molecules increases. Once the low sorption potential energy sites are filled, the
higher external force is needed to force the carbon dioxide molecule to access the available
sorption site and stay stable. The highest kinetics parameter (k ) in the initial pressure stage implies
2
that monolayer adsorption or pore-filling dominates the adsorption process, which is different from
the following sorption stages. Multilayer adsorption or capillary condensation may occur because
of the increasing external force and the interaction among gas molecules as the sorption pressure
increases.
4.1.5.5 Discussion on PSO model application for the carbon dioxide-coal sorption system
The successful application of the PSO model provides a new viewpoint to understand the carbon
dioxide-coal sorption system. The PSO model is based on a surface interaction assumption, which
has only two kinetics parameters (Q and k ) and is different than other kinetics models (shown in
e 2
153 |
Virginia Tech | Table 4.1.1). The high correlation coefficient (>99%) of the fitted data supports that this model
can be used to analyze the sorption kinetics data. This model can also be used to accurately predict
equilibrium sorption content under different pressures.
However, does the successful use of the PSO model mean that this model can explain how the
carbon dioxide interacts with coal during the adsorption process? Is the simple fitting procedure
using the PSO model sufficient for describing carbon dioxide adsorption kinetics in coal? The
answer is still unclear. The experimental results suggest the adsorption process for coal and carbon
dioxide to be a combination of both bulk diffusion-controlled and surface interaction-controlled
processes. For the non-isobaric process associated with the carbon dioxide-coal sorption system,
the initial stage is controlled by the bulk diffusion process. This is supported by the deviation of
the PSO model fitting process in the initial part of each stage (Figure 4.1.7). It should be noted that
this bulk diffusion-controlled process is only a small part of the total process considering the long
equilibrium time (Figure 4.1.7). Following the bulk diffusion stage is the surface interaction -
controlled sorption process, which requires a long time for the sorption system to reach equilibrium.
The surface interaction process dominates the sorption process when the bulk diffusion process
becomes less important. As mentioned earlier, the PSO model is based on the surface interaction-
controlled sorption That is to say, the surface interaction dominates the sorption process, which is
supported by strong curve fitting derived from the PSO model.
It should be noted that, even though the PSO model has been widely used and can reasonably
explain most of the liquid-solid sorption processes, there exist some situations where the PSO
model is applicable but surface interaction does not dominate the process.53-54 Therefore, even
though the PSO model may not reveal the mechanism behind the carbon dioxide-coal sorption
system, it provides an accurate description of the main part of the adsorption process, which is the
surface diffusion-controlled process. The relationship between the kinetics parameter (k ) and
2
pressure shows both the heterogeneous properties of the coal surface and the approximation of
monolayer adsorption for low pressures. The linear relationship between sorption content and time
can be used to predict the equilibrium sorption content (Q ) during the sorption process.
e
4.1.6 Conclusions
To understand the kinetic properties of the dynamic coal-carbon dioxide sorption process, 15
isothermal adsorption tests were conducted at 50℃ for increasing equilibrium pressures (up to 4
154 |
Virginia Tech | MPa) for bituminous and subbituminous coal. The PSO model was used to approximate carbon
dioxide adsorption kinetics in crushed coal. Analysis of the measured test data and comparison to
model predictions produced the following results:
1) The PSO model can be used to predict credible equilibrium sorption content under different
pressures for the carbon dioxide-coal sorption system.
2) A high correlation coefficient (>99%) was obtained for the linear relationship between
(Time/Sorption content) and (Time) using the PSO model.
3) The kinetics parameter (k ) decreases with the increase of both equilibrium sorption
2
pressure and the pressure difference. The sorption equilibrium content (Q ) in each sorption
e
stage depends on both the final equilibrium pressure and pressure difference.
4) The adsorption diffusion process for carbon dioxide in coal is a combination of both bulk
diffusion-controlled and surface interaction-controlled processes; the former is clear in the
initial stage while the latter dominates the overall process.
Acknowledge
Financial assistance for this work was provided by the U.S. Department of Energy through the
National Energy Technology Laboratory’s Program under Contract No. DE-FE0006827.
References
1. White C. M., Smith D. H., Jones K. L., Goodman A. L., Jikich S. A., LaCount R. B., ... &
Schroeder K. T. Sequestration of carbon dioxide in coal with enhanced coalbed methane
recovery a review. Energy & Fuels, 19(3), 659-724 (2005).
2. Busch A., Gensterblum Y., Krooss B. M., & Siemons N. Investigation of high-pressure
selective adsorption/desorption behaviour of CO< sub> 2</sub> and CH< sub> 4</sub>
on coals: An experimental study. International Journal of Coal Geology, 6(1), 53-68 (2006).
3. Busch A., & Gensterblum Y. (2011). CBM and CO2-ECBM related sorption processes in
coal: a review. International Journal of Coal Geology, 87(2), 49-71 (2011).
4. Goodman A. L., Busch A., Duffy G. J., Fitzgerald J. E., Gasem K. A. M., Gensterblum
Y., ... & White C. M. An inter-laboratory comparison of CO2 isotherms measured on
Argonne premium coal samples. Energy & fuels, 18(4), 1175-1182 (2004).
5. Goodman A. L., Busch A., Bustin, R. M. Chikatamarla, L Day, S. Duffy, G. J., ... & White
C. M. Inter-laboratory comparison II: CO< sub> 2</sub> isotherms measured on moisture-
155 |
Virginia Tech | 4.2 How different coal particle sizes generate unreliable pore characterization from gas
adsorption test
Xu Tanga, Nino Ripepia, Matthew R Hallb,c, Lee A Stevensb, David Meeb
(a Department of Mining and Minerals Engineering, Virginia Polytechnic Institute and State
University, Blacksburg, Virginia, 24060, U.S.; b Nottingham Centre for Geomechanics, Faculty
of Engineering, University of Nottingham, Nottingham, NG7 2RD UK; c British Geological
Survey, Environmental Science Centre, Keyworth, Nottingham, NG12 5GG UK)
Abstract: In gas adsorption analysis, the effect of particle size distribution in coal samples is
known to result in unreliable pore characterization. This experimental artefact has led researchers
to a misunderstanding of the pore network features in coals and the hysteresis mechanism that
occurs during gas sorption. In this study, CO and N adsorption tests on five sizes of crushed
2 2
Pocahontas No. 7 coal were conducted at 0 °C and -196°C under sub-atmosphere pressures. Our
results show that the adsorbed uptake of CO and N increases non-linearly with decreasing
2 2
particle diameter under the same test conditions. This is caused by the increase in volume of
accessible pores and fresh surface area created by comminution during sample preparation.
Comparative results from five different coal particles show that it is difficult to characterize the
pore features of coal using only one coal particle size. The difference in integral area between
desorption and adsorption isotherms (hysteresis loop) decreased as a function of particle diameter.
The force closed desorption phenomenon was observed for all N sorption tests, which may be
2
attributed to the instability of the meniscus condensation inside pores. Unclosed hysteresis loops
were observed in both CO and N sorption tests which may be attributed to the plasticization of
2 2
un-rigid coal during gas sorption.
Key words: Coal, Carbon dioxide, Nitrogen, Pore, Particle size, Hysteresis
161 |
Virginia Tech | 4.2.1 Introduction
Pore characterization of coal and organic-rich shale is of research interest because of the recent
exploitation of unconventional natural gas resources and the corresponding large storage potential
for carbon dioxide [1-9]. Since the pore features of coal and shale influence the gas transport in
the coal seam and shale formation, it is crucial to understand how the pore features may affect (i)
the production of gas and liquids, (ii) the development of enhanced gas recovery techniques, and
(iii) effective deployment of carbon dioxide storage tests [5, 10-13]. Several state-of-the-art
techniques have been applied to characterize the pore system of coal and shale [14-23]. Among
these techniques, gas adsorption under sub-atmosphere conditions is the most common one for its
applicability to the pore size range, simplicity, effectiveness and low cost [24-27]. For manmade
porous materials, there are fewer issues for pore characterization (than for heterogenous natural
materials) using gas sorption since the pore system can be designed in advance and the test
procedure has been standardized [ISO 15901-3; ISO9277 1-1, 2, 3]. Several issues persist when
the gas adsorption technique is applied for natural materials (coal and organic-rich shale) such as
sorption equilibrium status evaluation and the proper particle size of sample used for testing. In
particular, there is no standard for geo-materials such as gas sorption tests in coal using the
volumetric approach under sub-atmosphere and high pressure [28-31], making it difficult to
independently evaluate and compare published pore characterization data. The pore size
distribution (PSD) data from gas adsorption tests should ideally be validated by parallel techniques
such as small angle X-ray diffraction (SAXRD) or scanning transmission electron microscopy
(STEM) brightfield image analysis. In addition, it is unclear how to determine whether the sorption
system reaches equilibrium. An important issue is the lack of consistency in the selection of coal
particle diameter for gas adsorption analysis, along with sufficient understanding of how this can
affect the reliability of porosity characterization (Table 4.2.1). Therefore, it is imperative to clarify
how both the different coal particle size and pseudo-equilibrium state of the sorption system will
influence the pore characterization of coal.
162 |
Virginia Tech | Table 4.2.1 Different coal particle sizes used in low temperature gas adsorption analyses
In this paper, the authors have designed a series of gas adsorption tests to parametrically assess
how particle size influences the results of porosity analysis. Five different sizes of crushed coal (<
106 µm, 106-150 µm, 150-180 µm, 180-300 µm, 300-600 µm) were used to conduct CO full
2
adsorption/desorption isotherms at 0 °C, and N adsorption/desorption isotherms at -196°C at sub-
2
atmosphere pressures (4 to 750mmHg).
4.2.2 Experimental methodology
4.2.1 Sample preparation
The coal samples used in this study were all extracted from a core extracted from the Pocahontas
No.7 coal seam (Buchanan County, VA, United States). The composition of the coal was evaluated
(Table 4.2.2) in accordance with ASTM D7582 – 15 and ASTM D3176 - 15. The coal specimens
were then ground using an agate mortar and pestle and sieved through 106, 150, 180, 300, 600 µm
aperture metal sifters. Five different particle size distributions were obtained by separation as
follows: (i) <106 µm, (ii) 106-150 µm, (iii) 150-180 µm, (iv) 180-300 µm, (v) 300-600 µm. The
163 |
Virginia Tech | crushed samples were placed in a drying oven (at 104℃ to 110℃) for 1 hour to dehydrate. After
dehydration, the samples were stored in a vacuum desiccator for later use in accordance with
ASTM D3173 - 11.
Table 4.2.2 Composition of coal samples
4.2.2.2 Test approach and data processing
Gas sorption was conducted using a Micromeritics 3Flex volumetric analyzer (Norcross GA, USA)
using ultrapure N and CO adsorbates (BOC gases, Nottingham, UK). All samples were degassed
2 2
for 15 hours at 110 ℃ using a VacPrep Degasser (Micrometritics, Norcross GA, USA).
Approximately 1-2.3 gram coal samples were used for each N sorption isotherms including using
2
a filler rod. Sample tube bulbs were immersed in liquid N at approximately -196 °C. For CO
2 2
isotherms, approximately 0.2 gram of coal sample was weighed in to a sample tube. The tubes
were immersed in 50% ethylene glycol solution inside an isothermal controller (in place of the
liquid nitrogen dewar) during the tests which was set to 0℃. The equilibrium state for each test
was set based on the pressure measurement approach, i.e. when the partial pressure fluctuation
inside the test tube is within 0.3%.
The BET specific surface area using N was calculated by the linear BET relationship under the
2
relative pressure ranges from Rouquerol’s approach, which includes (1) both the resulting
parameter C is positive and the intercept on the ordinate of the BET-plot is positive, and (2) the
BET
term V (p -p) should continuously increase with p/p [52; ISO 9277: 2010]. Detailed calculation
ad o o
procedure is referred to in ISO 9277: 2010.
The pore size distribution was obtained using the non-local density functional theory (NLDFT)
model applied to the adsorption branch [53-55]. The calculation approach of the pore size
distribution using NLDFT and DFT method is based on the integral adsorption equation (Eq. 1).
164 |
Virginia Tech | Dmax
N(p/ p ) N(p/ p ,D)f(D)dD Eq. 1
0 0
Dmin
where, D is the pore size (diameter or width), N(p/p ,D) is the kernel of the theoretical isotherms
0
of different pore widths; f(D) is the PSD. Once the isothermal adsorption data is obtained, the PSD
can be derived numerically by solving the integral adsorption equation. The NLDFT model is
available for describing the PSD to the whole range of micro- and meso-pore sizes. Since several
DFT kernels have been applied for both mesoporous activated carbons and other organic materials,
it is logical to have confidence that this approach will work for coal [54-55]. In this paper, the
carbon slit pore model of NLDFT kernel was applied for meso- and macro-pore using N
2
adsorption data, and the CO -DFT model was applied for micropore using CO adsorption data.
2 2
4.2.3 Test results and discussion
4.2.3.1 Adsorption behavior of carbon dioxide for different coal particle size ranges
Figure 4.2.1 Adsorption behavior of carbon dioxide and nitrogen
From Figure 4.2.1, it can be seen that the adsorbed volume of both CO and N in coal is greater
2 2
for smaller particle diameters under the same test conditions. This can be attributed to the increased
surface area as well as access to fresh pores resulting from grinding. Since coal typically has a
high proportion of micropores, it is important to understand how the adsorbate enters the micropore
network. For microfluidic flow, the Knudsen number, Kn is given by the ratio of mean free path
length to pore diameter [56] is used to classify different flow patterns. Four distinct Kn regimes
have been measured as shown in Table 4.2.3. From Figure 3.3.2, it can be seen that in the
micropore (0-2nm) range the free molecular flow dominates the process because the Kn is
extremely high (above 10). Since surface diffusion is insignificant and configuration diffusion can
be neglected after monolayer completion, Knudsen diffusion is assumed to dominate the free
165 |
Virginia Tech | molecular flow process. In the mesopores (2-50nm), both the free molecular and continuum flow
with molecular diffusion occur, where the concentration-driven bulk diffusion is also included. In
the macropores (50 nm<pore size), Fickian bulk diffusion may occur. Since diffusion is a time-
consuming process, a longer time is needed for the sorption system of large coal particle sizes to
reach equilibrium state. This is because the gas molecular moves a longer distance to access
adsorption sites in larger coal particles compared with smaller coal particles. In addition, the
reduction of coal particles via grinding exposes more access pores and surface area of coal particles,
which increases the portion of available adsorption sites or pores during the sorption test [57].
Table 4.2.3 Microfluidic regime classified by Knudsen number (revised from [56])
Figure 4.2.2 Knudsen number of CO and N in different size of pore under different test
2 2
conditions
4.2.3.2 BET surface area
Even though the CO adsorption test is reliable for measuring the pore size distribution of
2
micropores in coal (based on the micropore-filling assumption), the specific surface area calculated
from CO adsorption isotherms is less so due to (i) uncertainties in monolayer capacity, (ii) the
2
cross-sectional area of an adsorbed molecule in a molecular-scale pore, and (iii) the coal swelling
166 |
Virginia Tech | phenomenon [58]. Thus, the BET specific surface area obtained from CO adsorption data can be
2
unreliable for coals and so the N isotherms were used instead for this study in order to minimize
2
the error compared with the absolute value [59,60]. Following Rouquerol’s approach, the
maximum relative pressure for linear BET fitting in each tests ranged from 0.0088 to 0.2 p/p .
0
As shown in Figure 4.2.3, it can be seen that the BET surface area increases exponentially with
decreasing coal particle size in the test results; the BET area is four times higher for coal particles
less than 106 µm than for coal particles in the range of 300-600 µm. This can be attributed that the
comminution of the coal particle provides more fresh surface area of the tested particles during the
sample preparation stage. Figure 4.2.3 also shows the surface area of 30-50 µm and 50-80 µm coal
particle are very close, and the surface area increases rapidly especially when the coal particle is
less than 106 µm.
Figure 4.2.3 N BET surface area comparison of different sizes of coal particles
2
4.2.3.3 Pore size distribution (PSD)
Figures 4.2.4-4.2.5 show the comparison of the PSD of different coal particles from CO and N
2 2
adsorption data. From Figure 4.2.4, it can be seen that the micro pore size distribution has a similar
trend with different pore width for all tested samples but the incremental pore volume of each pore
size is different. The measured pore width ranges from 0.4nm to 0.9nm in all tested samples, and
the smaller particle size samples have higher cumulative pore volume. The cumulative pore
volume increases with the reduction of particle size as shown in the test data (Figure 4.2.4). There
is a large gap for accumulative pore volume; the accumulative pore volume of coal is less than 106
µm, which is around two times that of coal particles in the range of 300-600 µm.
167 |
Virginia Tech | Figure 4.2.4 Comparison of micropore size distribution of different coal particles from CO
2
adsorption test
Figure 4.2.5 Comparison of pore size distribution of different coal particle size from N2
adsorption test
From Figure 4.2.5, the PSD obtained from N adsorption data has a similar trend in the range of
2
10 nm to 80 nm, but there are marked differences for PSD in pore ranges from 2 nm to 8 nm. The
measured PSD decreases with increasing particle size, and there are no pores ranging from 2 to
8nm detected for particles in 180-300 µm or 300-600 µm range. The reason why no pores less than
1 nm were detected during N nitrogen adsorption is that the nitrogen molecule is prevented from
2
entering micropores because of the diffusion restriction issue at a low temperature of -195.95℃
[ISO 15901-3]. The accumulated pore volume in micro-, meso- and macro-pores exponentially
decreases and is inversely related to particle size, as shown in Figure 4.2.5. The cumulative pore
volume of coal particles >105 µm diameter is almost twice that of particles in the range 300-600
µm. This can be attributed to the fact that the apparent pore volumes in smaller coal particles
increased following comminution.
Comparing both Figure 4.2.4 and Figure 4.2.5, the micropores detected using carbon dioxide
sorption in coal is one order of magnitude higher than that of the mesopores and macropores
detected using nitrogen. It can reasonably be expected that the micropore volume measured by
nitrogen does highly underestimate the micropore volume in coals [61].
168 |
Virginia Tech | 4.2.3.4 Effect of particle size distribution on hysteresis
Physisorption hysteresis is mostly associated with capillary condensation in mesopores, and its
apparent form is dependent on the pore feature of adsorbents and the sorption environment such
as pressure and temperature [24-26, 62]. In addition, it is possible that the CO is partly
2
chemisorbed to the coal adsorbent [15,17; 63, 64].
From Figure 4.2.6, it can be seen that the extent of hysteresis decreases with particle diameter, as
observed by a separate recent study [57]. The desorption isotherms do not close with their
corresponding adsorption branches even under very low pressures (Figure 4.2.6). For the unclosed
hysteresis loop in Figure 4.2.6, it is difficult to attribute a reason, though partial chemisorption
during the test is possible. Whether the plasticization of un-rigid coal occurs or not is arguable
since the unaffected coal structure has been directly observed during adsorption/desorption tests
in previous studies using SAXS and USANS [15,17, 65,66]. Another reason is that the adsorbate
affinity to active sites along the pore walls of coal is likely to be variable due to the heterogeneous
nature of the material composition [30, 67]. CO is more easily trapped in micropores compared
2
with the meso- and macro-pores because of the accumulated adsorption potential energy.
Figure 4.2.6 Hysteresis behavior of carbon dioxide in coal
169 |
Virginia Tech | Figure 4.2.7 Hysteresis behavior of nitrogen in coal
Figure 4.2.7 shows the nitrogen isotherms of all tested samples and the hysteresis loops does not
close under very low relative pressures (around 0.01 p/p in this work), and the hysteresis loop may
0
close under extremely low pressures up to 0.001 p/p . It is also hard to classify them to the standard
0
H3 or H4 type hysteresis loop. For H3 hysteresis loop, the sorption isotherms do not exhibit any
limiting adsorption at high relative pressure, which is usually observed with aggregates of plate-
like particles with slit-shaped pores [24]. H4 hysteresis loop is similar to H3 hysteresis loop but
the adsorption branch is a composite of Type 1 and Type 2. The force closed phenomenon that
occurs at a relative pressure ranging from 0.4 to 0.6 p/p was observed for all isotherms, which is
0
caused by (i) the tensile strength effect, (ii) the interconnected pore features of coal, and (iii) the
potential existence of the “ink bottle” pore [68]. The large particle size has a relative sharp drop
compared with that in a small particle size. This can be attributed to the different stability state of
multilayer nitrogen adsorption for different particle sizes [25].
From Figure 4.2.6 and Figure 4.2.7, it can be seen that both CO and N ad/desorption curves show
2 2
specific features for different coal particle sizes. Specifically, in Figure 4.2.6, the hysteresis loops
of CO in the sample of 180-300 and 300-600 µm are larger than that in the sample of 150-180,
2
106-150, and <106 µm. In Figure 4.2.7, N sorption isotherms in the sample of 300-600 and 180-
2
300 µm are different from that in the sample of 150-180, 106-150, and less than 106 µm, i.e. the
170 |
Virginia Tech | general slope of the former is lower than the latter when the relative pressure is between 0.2 and
0.3. Therefore, it may be possible that for large coal particle sizes (up to 600 µm), the additional
‘fresh’ pore volume and associated surface area, due to comminution, are insignificant. However,
for smaller coal particle sizes (below 180 µm), the ‘fresh’ pore volume and surface area due to
comminution are more pronounced, as shown by the increasing adsorption content with pressure.
4.2.4 Conclusions
Carbon dioxide and nitrogen adsorption tests on five sizes of crushed Pocahontas No. 7 coal were
conducted at 0 °C and -196°C in sub-atmosphere pressures to clarify how different coal particle
sizes influence the pore characterization of coal. Several conclusions can be made based on the
test results:
(1) The adsorption content of carbon dioxide and nitrogen in coal increases with decreasing particle
size, which is mainly caused by the increasing accessible pores and fresh surface area created via
comminution. The hysteresis loop decreases with the reduction in coal particle size.
(2) The measured micro-, meso- and macro-pore volume and nitrogen BET surface area all
increase non-linearly as particle size is reduced. This makes it difficult to accurately characterize
the real pore features of coal using only one particle size during gas adsorption analysis. Therefore,
we cannot recommend the optimum coal particle size for pore characterization of coal. However,
we do find that the fresh pores and surface area created via comminution significantly influence
our understanding about the real pore features of natural coals. Our preliminary recommendation
is that larger particle sizes are preferred, and that mean particle diameters less than 180 µm should
be removed by sifting in order to minimize the error in apparent micropore volume.
(3) The ‘force closed desorption phenomenon’ was observed for all N sorption tests, which may
2
be attributed to the instability of the meniscus condensation inside pores. An unclosed hysteresis
loop was observed in both CO and N sorption tests, which is perhaps due to the plasticization of
2 2
un-rigid coal during gas sorption process.
Acknowledgements
This research was supported in part by the U.S. Department of Energy through the National Energy
Technology Laboratory’s Program (No. DE-FE0006827). The first author wants to acknowledge
the Pratt Grad Study Abroad Scholarship from Virginia Tech for the travel support.
171 |
Virginia Tech | Chapter 5 Conclusions and future work
5.1. Conclusions
In this dissertation, high pressure gas (methane and carbon dioxide) adsorption in different shale
and coal samples under different temperatures were measured, modelled and analyzed to
understand gas adsorption behavior, thermodynamic characteristics, and gas adsorption kinetics.
Some tentative conclusions can be obtained.
The dual-site Langmuir adsorption model can simulate methane adsorption behavior in
shale under high pressure (up to 27MPa) and high temperature (up to 355.15K) conditions
as well as supercritical carbon dioxide adsorption in coals under high pressure (up to
20MPa) and high temperature (up to 352.57K).
The dual-site Langmuir adsorption model can not only interpret all observed adsorption
phenomena, such as how the observed adsorption uptake first increases, reaches the
maximum and then decreases with increasing pressure and the crossover of adsorption
isotherms under different temperatures, but also it can extrapolate adsorption isotherms
beyond test data.
The dual-site Langmuir adsorption model can be used to differentiate the true ratio between
adsorbed phase and bulk gas phase for shale gas under reservoir conditions. This can be
used to obtain an accurate shale GIP resource estimation as a function of reservoir pressure
and geothermal gradients.
Based on the dual-site Langmuir adsorption model, it was found the maximum gas
adsorption capacity of shale and coal is independent of temperature, and the temperature
dependence of observed and absolute adsorption uptake are confirmed.
The concept of the deep shale gas reservoir is proposed to provide a new perspective on
shale gas development on the basis of the successful application of the dual-site Langmuir
adsorption model.
Neglecting either the real gas behavior or the adsorbed phase volume, such as the Clausius–
Clapeyron approximation, results in an overestimation of the isosteric heat of adsorption.
179 |
Virginia Tech | Based on the dual-site Langmuir model, the isosteric heat of adsorption for high pressure
gas adsorption in shale and coal can be calculated analytically by considering both the real
gas behavior and the volume effect of the adsorbed phase.
The true isosteric heat of adsorption exhibits adsorption uptake as well as temperature
dependence for high pressure gas adsorption in shale and coal, which can be readily
investigated using the dual-site Langmuir adsorption model.
The isosteric heat of adsorption in Henry’s region for methane in anthracite, lean coal, and
gas-fat coal is -23.31KJ/mol, -20.47 KJ/mol, -11.14 KJ/mol, respectively, are independent
of temperature and can display the overall heterogenous property of different types of coal.
Carbon dioxide adsorption kinetics in coal can be modeled by the pseudo-second order
model. Modelling results indicates that the adsorption process for carbon dioxide in coal is
a combination of both bulk diffusion-controlled and surface interaction-controlled
processes; the former dominates the initial stage while the latter controls the majority of
the overall process.
Particle size of coal samples can significantly influence the sorption behavior of gas in coal,
which affects the pore characterization of coal. It is difficult to characterize the pore
features of coal using only one coal particle size.
The differences in integral area between desorption and adsorption isotherms (hysteresis
loop) for gas in coal decreases as a function of particle diameter.
Unclosed hysteresis loops were observed in both low pressure carbon dioxide and nitrogen
sorption tests which may be attributed to the plasticization of un-rigid coal during gas
sorption.
These scientific conclusions provide an option for accurate estimation of the shale gas-in-place
resource (total gas, adsorbed gas and free gas) in deep subsurface, accurate estimation of carbon
dioxide storage capacity in coal seams, heat transfer analysis during shale gas production process,
pore characterization of geo-materials such as coal and shale, and gas adsorption kinetics
properties in geo-materials.
180 |
Virginia Tech | 5.2. Future work
While the results and conclusions from this study provide valuable information for our
understanding of high pressure gas (methane and carbon dioxide) adsorption behavior,
thermodynamics and kinetics in shale and coal, these topics are far from being exhaustive. The
published data for high pressure (> 15MPa) gas adsorption in shale and coal is still very limited.
More experimental work should be continued to include more shale and coal samples to verify and
validate the applied method in this work. Followings are some recommendations for future
research work.
Water influence on gas adsorption capacity and transport in shale and coal
Water treatments, such as water based drilling, hydraulic fracturing and water production with
coalbed methane, accompany shale gas and coalbed methane development for subsurface
reservoirs. However, water influence on these reservoirs is complex. On the one hand, water can
displace the adsorbed phase of methane because of the high affinity of water on shale and coal. On
the other hand, water can damage the reservoir because of the retention of water in shale and coal
caused by the spontaneous imbibition effect, which can significantly impair the formation
permeability and reduce the productivity. Therefore, it is critical to study the interaction among
water, gas and shale/coal by simulating the real field scenarios to further enhance natural gas
production in shale gas and coalbed methane reservoirs.
Mixture gas competitive adsorption in shale and coal
This dissertation mainly focuses on pure gas adsorption in coal and shale. However, for enhanced
coalbed methane and shale gas recovery by carbon dioxide injection and supercritical gas
fracturing technique, the competitive adsorption between methane and carbon dioxide are
important to recognize. Furthermore, since natural gas is a mixture gas composed of other heavier
hydrocarbons like ethane, butane et al., how the natural gas quality (natural gas composition)
changes with time for a shale gas well is still unclear. Since field tests for competitive adsorption
behavior of mixture gases are expansive and sometimes impractical, controlled laboratory studies
can provide an effective way to investigate the mechanisms behind these phenomena.
Validation of thermodynamic characterization of gas adsorption in shale and coal
181 |
Virginia Tech | The thermodynamic modeling results of this work reveal that the isosteric heat of adsorption
depends on both adsorption uptake and temperature. Even though the results are supported by
robust theoretical derivation, direct measurement of heat release are needed to verify the modeling
results. Considering the unique feature of differential scanning calorimetry in measuring heat
release, heat measurement for gas adsorption in shale and coal are feasible. This future study will
help further in understanding the interaction between gas and shale/coal during the adsorption
process.
Adsorption mechanism investigation of gas in shale and coal using molecular
simulation
Shale and coal are pore-rich natural compounds containing pores from nanoscale to macroscale
fractures. Therefore, it is impossible to understand the true gas adsorption behavior in different
scale of pores through laboratory tests. Furthermore, there is no equation of states available for
obtaining the physical properties of gas such as density in a confined nanoscale space. Considering
the controllability and robust theoretical background of the molecular simulation approach,
investigation of gas adsorption behavior in different scales of pore under extreme pressure and
temperatures are possible. This future study will be helpful to understand and interpret observed
gas adsorption behavior in shale and coal from laboratory tests.
182 |
Virginia Tech | Introduction
This paper introduces a dual site Langmuir model to describe and predict methane adsorption
behavior under 303.15K, 318.15K, 333.15K, and 355.15K and up to 27 MPa. The dual site
Langmuir is used for accurate prediction of adsorbed methane in deep shale gas reservoirs under
high pressure and temperature conditions. The shale gas-in-place (GIP) content is estimated by
considering the volume of the adsorbed layer at in-situ conditions. Our findings show that for shale
formations deeper than 1000 m (> 15 MPa) below the subsurface, the GIP has historically been
significantly overestimated. Also, the ratio of the adsorbed phase compared to the free gas has
been significantly underestimated.
In order to support our findings in the paper, we provide the following information:
(1) Physical properties of Longmaxi shale (China)
(2) High pressure methane adsorption test data
(3) Data processing approach
(4) Geological gas-in-place estimation using conventional approach
1. Physical properties of Longmaxi shale (China)
Shale samples from the Lower Silurian Longmaxi Formation (collected at a depth of 2400.8 m)
were obtained from the Fuling #1 well in the Fuling region, Sichuan Province, China. The vitrinite
equivalent reflectance (Ro) of the sample is 2.2% - 2.5%. The physical properties of the shale are
shown in Table A-1 (Note: TOC – total organic carbon (%), S1 – hydrocarbons evolved at 300°C
(mg/g), S2 – hydrocarbons evolved between 300 and 600°C (mg/g) upon heating at 25°C/min, S3
– organic carbon dioxide evolved at 300°C and up to 390°C (mg/g), T is the maximum
max
temperature for obtaining S2). The rock pyrolysis measurement was conducted using a Rock-Eval
6 analyzer (Vinci Technologies, France).
184 |
Virginia Tech | Table A-1 Properties of shale
The shale specimen was ground and sieved using 0.38-0.83 mm metal sifters and placed in a drying
oven at 105 °C for 24 hour to dehydrate. After dehydration, the prepared sample was stored in a
desiccator prior to adsorption measurements.
2. High pressure methane adsorption tests
Methane adsorption measurements were conducted using a Rubotherm Gravimetric Sorption
Analyzer IsoSORP. The methane density was obtained via the NIST package using the Setzmann
& Wagner equation (1). The instrument can achieve pressures of up to 35 MPa and temperatures
up to 150°C ±0.2℃. Ultrapure methane gas (99.99%) was used as the adsorbate. Equilibrium was
defined as when the adsorption time was longer than 2 hours or when the weight change of the
sample was within 30 μg over a span of 10 min. The detailed characteristics of the instrument has
been extensively described anywhere else (2).
The test procedure used was as follows:
(1) Mass of the adsorption cell
The blank test (without shale samples) was first conducted in order to obtain the mass and volume
of the adsorption cell (shown in Figure A-1). The whole system is pumped down to vacuum
conditions, and then the measurement is conducted by dosing pure nitrogen into the adsorption
cell up to 5MPa. The apparent weight of the adsorption cell can be recorded from magnetic
suspension balance (MSB), which is the interaction between the weight of the adsorption cell and
the buoyancy induced by the dosing N . Through the linear relationship between the apparent
2
weight of the adsorption cell and the density of the nitrogen, the mass and the volume of the
adsorption cell can be obtained,
mN2 m V (S-1)
sc s N2 s
185 |
Virginia Tech | where mN2is the apparent of the adsorption cell, m is the mass of the adsorption cell, is the
sc s N2
density of nitrogen obtained from NIST package, and V is the adsorption cell volume.
s
Figure A-1 Blank test results
(2) Mass of test shale sample
After the shale sample is put in the adsorption cell, the whole system is pumped down to vacuum
conditions. Then, the non-adsorbed pure Helium is dosed into the system up to 5MPa (shown in
Figure A-2). The apparent weight of the adsorption cell with shale can be recorded from MSB. It
should be noted even though Helium adsorption in shale is very small, it still has some influence
on the test results, which cannot be avoided. The Helium intrusion test is also the routine method
for measuring the skeletal density of porous material. Through the similar relationship in equation
(S1), the total mass of the adsorption cell and the shale sample can be obtained,
mHe m V (S-2)
sc sc He sc
where mHeis the apparent of the adsorption cell, m is the mass of the adsorption cell, is the
sc sc He
density of nitrogen obtained from the NIST package, and V is the adsorption cell volume.
sc
Figure A-2 Helium test results
186 |
Virginia Tech | Then, the mass and volume of the shale sample can be obtained using equation (S-3) and (S-4),
m m m (S-3)
shale sc s
V V V (S-4)
shale sc s
(3) Methane adsorption test
Once the mass of the shale sample is obtained, the system is then pumped to vacuum conditions.
Then, the adsorption cell is dosed with methane and the apparent weight of the adsorption cell can
be recorded at each equilibrium point:
mCH4 m m (V V ) (S-5)
t sc a CH4 sc a
Where mCH4is the apparent weight of the adsorption cell, V is the volume of the adsorbed
t a
methane layer, m is the absolute adsorption uptake, is the density of methane obtained
a CH4
from NIST package. If we rewrite equation (S-5)
m m V mCH4 m V (S-6)
Gibbs a CH4 a t sc CH4 sc
Where the term (m V ) is called Gibbs excess mass (m ), which can be easily obtained
a CH4 a Gibbs
from equation (S-6). Then, we can obtain the molar excess adsorption uptake (n ) and M is
ex CH4
the molar mass of methane (16.04 g/mol).
mCH4
n Gibbs (S-7)
ex M m
CH4 shale
This process is repeated at predefined measuring equilibrium pressure points.
187 |
Virginia Tech | from the NIST REFPROP database (1). The seven independent fitting parameters were varied to
achieve the global minimum of the residual-squares value within the following limits: 0<nmax
<100 mmol/g, 0< Vmax<10 cm3/g, 0<α<1, 0< E1 <100 kJ/mol, 0< E2 <100 kJ/mol, A1 > 0, A2 >
0). Minimization was performed in excess of 100 unique times by changing the random seed in
order to assure that a global minimum was achieved. Once the seven fitting parameters were
determined, absolute and excess adsorption uptake could be easily calculated at any temperature
and pressure by use of equations 5 and 6.
4. Geological gas-in-place estimation using conventional approach
The most widely used approach for estimating the adsorbed methane content is as follows the
Langmuir equation is used to fit the adsorption isotherms under intermediate pressures and
temperatures (6-11). Then, based on the relationship between Langmuir constants (n and K(T))
max
and temperature, the adsorbed methane content is predicted using equation (S-8) at in-situ
temperatures and pressures (6-11). The test isotherm data before the observed the maximum value
is used to obtain the relationship between Langmuir constants and temperature (test data is shown
in Table A-2), and the fitting parameters are shown in Table A-3.
K(T)P
n n (S-8)
a max 1K(T)P
Table A-3 Fitting parameter using two parameter Langmuir equation
Langmuir Fitting
Temperature (K) a (mmol/g) K(T)(1/MPa)
303.15 0.104 0.782
318.15 0.095 0.663
333.15 0.093 0.499
355.15 0.088 0.3999
Langmuir constant-a (T) a(T)=0.189-0.000287*T
Langmuir constant - K(T) K(T)=0.00723*EXP(1423.77/T)
It is worth pointing out that in Figure 2.2.7 and 2.2.8 in the paper, there are crossover between
the Absolute Adsorption curve and Conventional Absolute prediction, which should not occur
theoretically. This deviation can be attributed to the empirical equation for Langmuir constants
with temperatures as shown in Table A-3. The obtained empirical equation induces new
uncertainties for the adsorbed methane content when it is used for prediction.
189 |
Virginia Tech | The Ferrous Regeneration Process
for Use in Alternate Anode Reaction Technology
in Copper Hydrometallurgy
Emily Allyn Sarver
Abstract
The Fe(II) regeneration process is an important aspect of Alternate Anode Reaction
Technology (AART) using Fe(II)/Fe(III)-SO reactions for copper hydrometallurgy; however
2
little has been done to study it specifically. The process regenerates Fe(II) via Fe(III)
reduction by SO , catalyzed by activated carbon particles. To better understand and
2(aq)
improve the process, two studies have been conducted with respect to variable factors and
their affects on the regeneration.
A study of fundamental kinetics confirms that the regeneration reaction is mass
transfer-controlled, requiring adsorption of reactants onto the catalyst surface for reaction.
The reaction rate is limited by the diffusivity of Fe(III). Initial Fe(III) concentration and
carbon particle size are determined to be the most influential factors on the rate under the
condition studied. Furthermore, it is observed that flow rate may inhibit the reaction by
reducing ion diffusivity. The experimentally validated rate expression for the regeneration is
below, where the Fe(III) diffusivity is 1.1x10-7 cm2/s.
⎡ 1 2 ⎤
dFe2+ 6M 2D V 2D 3
= ⎢ f +0.6 t f ⎥[C ]
dt ρdV ⎢ d 1 1 ⎥ Fe3+
d 2v 6
⎣ ⎦
An optimization problem is also developed and solved for the process, constrained by
the requirement that negligible SO could be present in the process effluent. Before
2
optimization, a relationship is developed between regeneration rate and variable factors.
Again, carbon size and initial Fe(III) are the most influential factors on the regeneration rate,
related to it linearly; temperature is significant with a squared relationship to the rate; initial
SO is insignificant. Optimal conditions are found with minimum carbon particle size,
2
maximum initial Fe(III) concentration, and moderate temperature. |
Virginia Tech | Chapter 1: Introduction
The Copper Industry
Copper has been used for millenniums due to its unique properties and, in fact, is
currently the third most used metal, after iron and aluminum. In addition to being malleable
and ductile, copper forms favorable alloys because it is corrosion resistant, biostatic and
easily cast, and is an exceptional electrical and thermal conductor. Today, copper is nearly
exclusively exploited for its conductivity and resistance to corrosion, and the rapid
progression of construction and technology maintains a high demand for the metal. In 2003,
about 17 million metric tons of copper were produced to supply the significantly growing
global demand (ICGS, 2003).
Like in all commodity markets, as demand has grown so have copper prices due to
depleted stocks and the fact that increased production inherently lags demand. In April 2005,
the copper spot price reached a high of $1.54 per pound, up from $0.65 just two years earlier
(LME, 2005). High prices are good for producers – so good that many have re-opened
existing capacities or brought new capacities online to take advantage of the supply deficit –
but how long will they last?
Over the next 10 to 15 years, worldwide copper demand is expected to increase with
growing economies (Demler, 2005). This is especially of interest when considering the
major copper producing and consuming nations and their respective economies. For
example, Chile accounts for nearly 35% of global production and only about 1% of
consumption, the U.S. nearly balances its production and consumption at 18% and 16%
respectively, and China accounts for only 5% of production and over 17% of consumption
(ICGS, 2003). Of the three, the Chinese economy (and demand for copper) stands out with
enormous recent growth, a trend which is expected to continue for quite some time.
While copper supplies are also expected to increase, causing the market to cycle to
balance supply and demand, increasing production costs will likely shift the average cycle
1 |
Virginia Tech | price upwards. If prices tend to remain elevated with production costs, a real opportunity is
presented for low-cost producers; increased profits are available for corporate expansion and
for investments, and investments in technological improvements and development will
perpetuate growing profit margins.
Hydrometallurgy: Past, Present and Future
Progression in Copper Processing
Hydrometallurgy was integrated into commercial copper production in the late 1970’s
as a means of processing additional ore types at lower costs than pyrometallurgy. The
processing method can be applied to both oxide and oxidized sulfide ores or wastes, unlike
pyrometallurgy, which only be applied to sulfide ores.
The processing costs of hydrometallurgy are also more attractive than those of its
counterpart because of the difference in overall power consumption. Hydrometallurgy has an
associated power requirement of about 10 to 35MJ/kg of copper produced; the wide variation
depends largely on whether the source is previous waste or newly mined ore. Comparatively,
pyrometallurgy requires about 65MJ/kg, utilizing much of the power for the energy intensive
comminution needed to reduce particles to floatable sizes (Dresher, 2001).
Additionally, hydrometallurgy is considered relatively neutral with respect to
environmental effects, generating air emissions only from power consumption and wastes
that can be safely disposed in excavated areas. Due to its many benefits, hydrometallurgical
processing now accounts for over 21% of global copper production, mostly in Chile and the
U.S., the two largest producing countries, and continues to grow rapidly.
The General Hydrometallurgical Process
Copper hydrometallurgy is typically considered to encompass three major phases:
leaching, solvent extraction (SX) and electrowinning (EW). Leaching refers to the
dissolution and extraction of copper-bearing minerals from coarsely crushed copper ore using
acidic solutions. Following leaching, the solution containing extracted copper minerals is
collected and sent to the SX phase.
2 |
Virginia Tech | During SX, aqueous leach solution is mixed with an organic phase specifically
developed to remove the dissolved copper, leaving most of the dissolved impurities behind.
The copper-deficient leach solution is recycled back to the leaching phase; the copper-rich
organic is treated with another chemical called a stripper, which transfers the copper minerals
from the organic to an aqueous phase. The stripped organic is then recycled back to the SX
phase, and the copper-rich aqueous electrolyte is sent to the EW phase.
EW refers to the process of plating metallic copper from copper electrolyte, in which
it is present as copper (II) sulfate (CuSO ). This is done by filling large tanks that have an
4
alternating series of anodes and cathodes with the electrolyte and passing current. The
electrical potential provided drives an oxidation reaction at the anodes that is balanced by a
reduction reaction at the cathodes, onto which the copper plates. The conventional reactions,
which are discussed in further detail below, are water oxidation and copper reduction,
whereby divalent copper is reduced to the neutral metallic species.
Conventional Electrowinning vs. Alternate Anode Reaction Technology
Two main areas of concern in conventional copper EW are relatively high power
consumption and acid misting, both of which are associated with the oxidation of water as
the anode reaction. The EW power requirement, mostly due to the cell voltage requirement,
accounts for 30 to 50% of the total requirement for hydrometallurgical processing, which can
be problematic from a financial perspective and also with respect to growing concern for the
environment. Acid misting is a product of oxygen gas (O ) generation at the anode, which
2
causes a hazardous and uncomfortable environment for those working in EW tank houses, as
well as corrosive conditions for structures and equipment.
To address both of the above issues, alternate anode reaction technology (AART) has
been suggested as a possible solution. AART refers to changing the conventional anode
reaction, preferably to one that has a considerably lower equilibrium potential, such that
power consumption is reduced, and which also does not produce a gas that causes acid
misting. For copper EW, a ferrous [Fe(II)] to ferric [Fe(III)] iron oxidation reaction appears
3 |
Virginia Tech | quite feasible as an alternate anode reaction for several reasons (Sandoval and Dolinar,
1996):
• it has a significantly smaller equilibrium potential;
• iron is already present in commercial electrolytes (due to its abundance in most
copper ores); and
• oxidized Fe(II) may be regenerated by reducing Fe(III) such that electrolyte solutions
can remain recyclable – the proposed reducing agent is sulfur dioxide (SO ), possibly
2
from burners at nearby smelting operations due to its availability and low cost.
A comparison of the conventional and proposed AART reactions is given in Table 1.1,
highlighting the difference between the anode and overall equilibrium potentials.
Table 1.1 – Conventional and Proposed AART Electrode Reactions
Conventional AART
Reaction Equilibrium Potential Reaction Equilibrium Potential
Oxidation
1
half-reaction H 2O→ 2O 2+2H++2e− E ox0 =−1.229V 2×(Fe2+ → Fe3+ + e−) E ox0=−0.770V
(anode):
Reduction
half-reaction CuSO 4+2e−→SO 42−+Cu E red0 =0.345V CuSO
4
+2e− →SO 42− +Cu E red0 =0.345V
(cathode):
Balanced
reaction CuSO +H O→1 O +H SO +Cu E total0=−0.884V CuSO 4+2Fe2+ →2Fe3++SO 42−+Cu E total0=−0.425V
(total): 4 2 2 2 2 4
As highlighted in green, the water oxidation anode reaction requires a considerably higher
energy input (1.229V) than does the Fe(II)/Fe(III) reaction (0.770V), while the cathode
reactions are identical for the conventional and AART processes. Based on thermodynamics,
AART reduces the overall voltage input for electrode reactions by over 50% (from 0.884 to
0.425V). Also, because there is no gas generation in the AART anode reaction, acid misting
is completely eliminated. These results have been demonstrated and documented in several
different studies, each of which noted the importance of using dimensionally stable anodes
(DSA’s) to achieve the highest reduction in required cell voltage (Sandoval and Lei, 1993),
(Sandoval et al., 1995) and (Sandoval and Dolinar, 1996). DSA’s promote uniform current
density and, consequently, uniform cathode plating.
4 |
Virginia Tech | It should be noted that total power consumption for EW includes both that associated
with the cell voltage and pumping power. In conventional EW, O generation provides some
2
electrolyte mixing, which is important for maintaining ion concentrations around the
electrodes for reaction. Since mixing caused by O is absent in AART pumping power must
2
be increased to compensate. In a 1995 study by Sandoval and Dolinar, this aspect was
examined specifically and results showed that at optimal cell injection configuration the total
power associated with AART was roughly 60% of conventional EW. Even considering
additional operations (e.g., Fe(II) regeneration and acid recovery), presumably, the total cost
of commercial copper production using AART will be substantially lower than using
conventional EW, however published cost data is currently unavailable.
A key factor for utilizing AART is the regeneration of Fe(II), as mentioned above,
because electrolyte iron concentrations must be maintained for a balanced process. The
proposed method of regeneration is by reduction of the Fe(III) produced at the anode by SO ;
2
the Fe(II)/ Fe(III) anode reaction combined with the Fe(II) regeneration using SO is
2
abbreviated FFS. This process should occur during electrolyte recycling as shown in the
generalized schematic in Figure 1.1.
Figure 1.1 – General AART Process Using FFS
5 |
Virginia Tech | An additional benefit of AART using FFS is recoverable H SO , a byproduct of the
2 4
Fe(II) regeneration that may be used in subsequent leaching operations. The Fe(II)
regeneration reaction is written as shown in Equation 1,
SO +2Fe3+ +2H O → 2Fe2+ +SO 2− +4H+ (1)
2 2 4
whereby, when considering an overall reaction with the copper sulfate (CuSO ) in the
4
electrolyte, two moles of H SO are formed for every mole of metallic copper (Cu) plated;
2 4
this is twice the amount of acid needed to maintain the electrolyte concentration, so one mole
should be extracted to balance the system. Different schemes have been proposed for acid
recovery, including an SX extraction demonstrated by Sandoval et al. in 1990.
The major benefits of AART using FFS as compared to conventional EW can be
summarized as follows:
• substantial power reduction and, hence, cost savings and environmental benefits for
copper hydrometallurgy;
• elimination of acid misting and associated problems;
• by-production of recoverable acid, which might be used for leaching operations.
As further indication for the potential success of this technology, a pilot study has been
undertaken by Phelps Dodge with positive results, though specifics are protected by
confidentiality agreements.
While AART using FFS appears promising, several areas necessitate further research
before commercialization is likely. Such areas include optimization of acid recovery
methods, anode types, and the Fe(II) regeneration process. This process has several variables
and is complicated by the fact that the reduction of Fe(III) by SO is known to be naturally
2
slow, thus requiring a catalyst (Sandoval and Dolinar, 1996). Further description of the
process, the importance of understanding and optimizing the system, and two studies, one
from an industrial and the other from a more fundamental perspective, follow.
6 |
Virginia Tech | Ferrous Iron Regeneration Using Sulfur Dioxide
Fe(II) regeneration is critical to AART using FFS in order to achieve a balanced
system with sufficiently constant concentrations of Fe(II) and Fe(III); there must be adequate
Fe(II) supplied to EW cells for the anode reaction, and the Fe(II)/Fe(III) ratio must be
maintained to control cell voltage.
The Reaction
As stated above, the method of Fe(II) regeneration for AART using FFS uses SO to
2
reduce the Fe(III) generated by the EW anode reaction. Based on stoichiometry, one mole of
SO can reduce two moles of Fe(III) to Fe(II) via a two-electron transfer as shown in the
2
addition of the two half reactions below in Equation 2:
SO +2H O →2e− +SO 2− +4H+
2 2 4
(2)
2Fe3+ +2e− →2Fe2+
+
SO +2Fe3+ +2H O→2Fe2+ +SO 2− +4H+
2 2 4
Equilibria are very complex for sulfur oxides and constitute an entire field of study
for inorganic chemists, but the following is the general solution chemistry for dissolved SO .
2
For the AART using FFS process, SO is introduced as a gas into the (airtight) Fe(III)-rich
2
electrolyte, upon which it dissolves into the solution as aqueous SO and [theoretically]
2
forms sulfurous acid (H SO ), an anhydrous oxide. H SO is very unstable; in fact, it cannot
2 3 2 3
be detected spectroscopically in solution, and de-protonates and dissociates immediately
(Shriver and Atkins, 2003). The degree to which the first de-protonation to bisulfite (HSO -)
3
occurs is much greater than that of the second de-protonation to sulfite (SO 2-), as dictated by
3
their respective equilibrium constants. The following (Equations 3-6) represents the
behavior of SO in aqueous solution as is applicable to FFS:
2
SO ⇔ SO (3)
2(g) 2(aq)
SO + H 0 ⇔ H SO (4)
2(aq) 2 2 3
H SO ⇔ HSO − +H+............K =1.4×10−2 (5)
2 3 3
HSO − ⇔ SO 2− +H+..............K =6.3×10−8 (6)
3 3
7 |
Virginia Tech | Since H SO is known to be so unstable and undetectable in solution, it is assumed
2 3
that it dissociates completely to SO and water, such that the acid really only exists as a
2(aq)
way of explaining the formation of HSO - and SO 2- with common concepts in acid
3 3
chemistry. Upon SO dissolving into the electrolyte, the potential sulfur containing species
2(g)
in solution are then SO , HSO - and SO 2-; given the equilibrium constants, the amount of
2(aq) 3 3
SO 2- is very small as compared to the other species, and given the high acidity of electrolyte,
3
SO will be far more prevalent than HSO -, as is discussed below. The above suppositions
2(aq) 3
are supported by several studies of aqueous phase SO oxidation (Kumar et al., 1996) and
2
(Govindarao and Gopalakrishna, 1995).
The sulfur species are met by Fe(III) ions in the electrolyte solution and subsequently
react, reducing the Fe(III) and oxidizing the S(IV). Since the homogeneous reaction occurs
slowly, the reaction must be catalyzed in practice by passing the electrolyte containing the
reactants through a bed of activated carbon. Also worth considering is the possibility that
Fe(II) ions might react with dissolved O in the solution and be oxidized, opposing the
2
desired increase in Fe(II) concentration; however, this does not seem likely during Fe(II)
regeneration since the reactor should be airtight.
Important to note are the conditions under which Fe(II) regeneration must occur, most
of which are dictated by the typical nature of EW electrolyte solutions:
• high temperature,
• high molar acidity,
• specific Fe(III):Fe(II) ratios,
• and high flow rate.
Literature Review
While Fe(II) regeneration by SO has been successfully managed in both bench and
2
pilot scale testing of AART using FFS, there is little published about the regeneration
reaction itself, as testing has tended to focus on issues surrounding EW (i.e., cell voltage,
pumping power, current density). Few studies outside of the realm of AART are found that
8 |
Virginia Tech | examined the reaction specifically, although one in particular had some significant
similarities despite being conducted under differing conditions and with differing goals.
Additionally, several articles were found in the literature that examined similar reactions
(e.g., Fe(III) reduction, separately, SO oxidation, catalyzed by carbon or a comparable
2
catalyst), however, none at anywhere near the extreme conditions encountered in EW
electrolytes (e.g., high acidity and temperature).
Sandoval and Lei reported in 1993 that the Fe(III) reduction by SO could be
2
accelerated by increasing temperature and also that increasing total iron concentration
negatively affected the reaction. The reaction was later observed by Sandoval and Dolinar to
be catalyzed by passing the reactants (contained in electrolyte) through a bed of activated
carbon, although little information was reported on specific reaction mechanisms or kinetics.
Other similar reduction-oxidation reactions are also known to be catalyzed this way, notably,
one of the relevant half-reactions in Fe(II) regeneration, the reduction of Fe(III) to Fe(II).
Thomas and Ingraham studied this reaction using oxygen as a reducer, and tested both
CuSO and activated carbon as catalysts. The reaction rate was reported to have increased
4
14-fold and 2400-fold, respectively, in the presence of the listed catalysts.
The Thomas and Ingraham study perceived the general mechanism of catalysis to be
the provision of surface area onto which the reactants could adsorb, thereby allowing the
reaction to occur. As such, the effect of surface area was examined as part of the study and,
as expected, increased surface area greatly increased the rate of reaction. (Further
explanation regarding the mechanism of catalysis by materials such as activated carbon is
addressed below.) Other variables determined to be of importance to the reaction rate were
temperature, acidity, and concentrations of reactants and products, which seem likely as
influences on the Fe(II) regeneration process also. The study also concluded that the activity
of the carbon catalyst could be retained for relatively long periods of time.
With respect to the other relevant half-reaction, SO oxidation, there are also
2
published studies in which the surface of a solid was utilized as a catalyst; differing reaction
limitations were reported. Seaburn and Engel concluded in a 1973 study that the rate of
9 |
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