Ind. Eng. Chem. Res. 1994,33,3247-3255
3247
A Scheme for Reducing Experimental Heat Capacity Data of Gas Hydrates Dimitrios Avlonitis' Division of Chemistry, Aero-engines Factory, Elefsis Airport, Elefsis, Greece
Experimental heat capacity data of simple gas hydrates on xenon, methane, ethane, and propane are reduced by application of classical thermodynamics and the ideal solid solution theory. It is shown that calculated heat capacities of the empty hydrate lattices of the structure I and I1 hydrates can be higher or lower than the heat capacity of ice. Similarly, the calculated partial molar heat capacity of the enclathrated gases are higher or lower than the corresponding experimental ideal gas heat capacity. These differences depend on the size of the guest relative to the cavity, the hydrate number, and the temperature. For estimation of the thermodynamic properties of the empty hydrate lattice, further experimental work is recommended. Within the present limitations, a consistent methodology is applied for the prediction of the heat capacity of a natural gas hydrate.
Introduction
Previous Implementations of the Classical Statistical Model
Accurate knowledge of the thermal properties of natural gas hydrates is required for the development of novel technology for the recovery of natural gas from hydrates in the earth and for storing of natural gas in hydrate form (Sloan, 1990; Berecz and Balla-Achs, 1983). Though the enthalpies of dissociation can be obtained reliably by the use of the Clapeyron equation (Fleyfel and Sloan, 1991; Rueff et al., 1988), no general scheme has been documented for the prediction of heat capacities of natural gas hydrates. Handa (1986a,b)has measured the heat capacities of simple gas hydrates of xenon, krypton, methane, ethane, and propane with a calorimetric technique from 85 t o 230-270 K with accuracy better than 1%.The reduction of experimental heat capacity data could reveal the nature of molecular interactions in the crystal. Handa and Tse (1986) have analyzed the data for the first three gases, and they have concluded that over the temperature range of 100270 K the heat capacity of the empty hydrate lattice of both structures is essentially equal t o that of ice, within experimental uncertainties. These authors did not consider the data for ethane and propane. In a subsequent study, Handa (1986~)presented further heat capacity data for the hydrate of xenon and calculated the molar heat capacity of the enclathrated Xe guest to be about 2.5R and 1.7R in Xe5.90Hz0 and Xe.6.29HzO hydrates, respectively. The latter is clearly an unexpectedly too low a value. In the present work, a thermodynamic method, based on the ideal solid solution theory of van der Waals and Platteeuw (1959), is used for a new reduction of the experimental data of Handa and the prediction of heat capacities of natural gas hydrates. No additional assumptions are made, but the present work has been enabled by the previous publication of unique values of potential parameters (Avlonitis, 1994). The formulae are particularly simple for ethane and propane hydrates, where the gas occupies the large cavity only. For multicomponent mixtures, the system of equations is solved numerically.
The statistical mechanics theory of van der Waals (1956) provides a formula for the calculation of the isothermal energy of formation, A@Lgg-H, of a clathrate per mole of gas. Parsonage and Staveley (1958) were the first to apply the statistical mechanics theory of van der Waals (1956) to calculate the heat capacity of the argon quinol clathrates. Briefly, by differentiation of AU,,, they arrive at an expression for the constant volume heat capacity difference, ACF+H,between the heat capacities of the occupied cage and the empty cage plus that of the guest molecule per mole of gas. The heat capacity contribution of the enclathrated guest was calculated as the difference between the heat capacity of the hydrate minus the heat capacity of the empty lattice:
* Present address: Miriofitou 16,Amfiali, Piraeus 187 57, Greece. 0888-5885/94l2633-3247$04.50/0
CH = cy - c: "g
Equation 1is independent of composition, and this is an implicit crucial assumption in the above model, suggesting that guest-guest interactions are negligible, the lattice is not distorted by the guest, and there is no coupling between the guest and lattice modes (Handa and Tse, 1986). Assuming ideal gas heat capacity for the gas phase gives the constant volume heat capacity contribution of the guest molecule in the clathrate lattice as:
Parsonage and Staveley (1958) have reduced their experimental data by observing that the heat capacity of the clathrate is a linear function of composition a t constant temperature. Then, a plot of the heat capacity of the clathrate versus composition would give the heat capacity of the empty lattice at the intercept x = 0 and the heat capacity of the encaged molecule from the slope of the curve. The agreement between theory and experiment was considered satisfactory. This model has been used t o predict successfully the heat capacities of the P-quinol clathrates of Ar, Kr, N2,02, CO, and CHI, all of which were linear functions of composition (Parsonage and Staveley, 1984), indicating validity of eq 1. The same model has also been used by Handa and Tse
0 1994 American Chemical Society
3248 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 Table 1. Parameters for Calculation of the Quantity c;, (J mol-' K-') Xe CH4 CzHs C3He C4H10 COz
Nz
19.71681 0.00309 0.82386 32.24370 -1.11157 1.97359 3.87388 27.27646 15.46896 19.90132 -0.13930 11.04932 32.35889 -1.00001 8.01671 -0.73171 18.77030 0.07422 29.17562 -0.27748
-1.01172 -14.97040 -0.15156 -0.26675 3.11012 -0.01603
0.07 0.54 0.10 0.37 0.40 0.17 0.03
(1986) as part of their analysis of the heat capacity data of Handa (1986a) for xenon krypton, and methane gas hydrates.
The Langmuir type constant, Cmi, of component i in the cavity m is a function of temperature only, and it is evaluated from the ideal solid solution theory. Considering that the mole fraction of water in the hydrate phase is x: = 1 - DF,the following relation is obtained for the fractional occupancies of the two types of cavities:
Proposed ThermodynamicModel
Cy,@,,= ni,with ni
The heat capacity, C,, of any system at constant pressure P is defined by:
m
&i
U and i 2 w
(6c)
X;;
Fugacities need not be calculated for the scope of the present work, and they are eliminated simply by dividing Oli by Ozi: where H, S, and G are the enthalpy, entropy, and free energy of the system and Tis the absolute temperature. By partial differentiation of the last relation we obtain
(4) where xj are the mole fractions of componentsj and pi and Cpi are the chemical potential and the partial molar heat capacity of component i, respectively. The heat capacity of the mixture is given by the weighed sum: C , = XixiC,,. From eq 4 and the defining equation of fugacity, fi, of a component in a mixture, we obtain
where R is the gas constant and cii is a unique function of temperature only, characteristic of the pure substance i. It is the same for all states of the substance, and it may be determined by forcing agreement of the model eq 5 to pure component experimental heat capacity data. We have fitted ideal gas heat capacity experimental data (MI Technical Data Book, 1977) t o the equation cii = ai biT ciTL d i p . The parameters of this equation, for the temperature range 120-600 K, are listed in Table 1 together with the attained quality of the fit. Calculation of E: of a Gas Component in a Hydrate Phase. The composition in mole fraction of a homogeneous hydrate phase comprising NH gas hydrate-forming components may be calculated from the following relations:
+
+
+
m
1
+ ccvmen2j
a l n Omi d In Cmi
-
aT
dT
d l n Cmi
alnfi
+--aT alnfi
+-z v m e m i [7 aT
d lnCmi
-)]a h 4
+ aT
cvmemi
x; =
Obviously, for a guest, k , not entering the small cavity, Olk is equal to zero and 0 2 =~ nk/vp. If the composition of the hydrate phase is known from previous phase equilibrium calculations or it has been experimentally measured, then the system of W eqs 6c and 7 can be solved for the unknown quantities @mi. An appropriate numerical method is detailed in the Appendix. From eqs 6b,c, by taking logarithms and differentiating while keeping all xi constant, we obtain
, i , j = 1, ...,NH and m = 1 , 2
where vm is the number of cavities of type m per water molecule in the unit cell and Omi is the fractional occupancy of cavity m by component i, given by the following relationship
= 0 (10)
The last system of equations (10) may be solved for the derivatives of the logarithms of the fugacities of gas components as functions of T and the composition of the hydrate phase. It is noted that these equations are independent of pressure, and the constant pressure requirement for differentiation is dropped. For simple gas hydrates, the system of equations is reduced to the following analytical expression:
Ind. Eng. Chem. Res., Vol. 33,No. 12,1994 3249 d In C,,
CYmOm,(l-
alnfi -
m
om,>d T
(13) (loa)
aT
CYmOm1(l- om,)
The ideal solid solution theory gives
m
= -RTXvm ln(1
If the gas component enters the large cavities only, the equation is particularly simple:
a In f , -
d In C,, dT
-- --
aT
(lob)
Equations 10 can be further differentiated t o obtain the second derivative of the logarithm of the fugacities of gas components. The corresponding final expressions are
+ XCmfJ)
m
(14)
j
It is noted that if the composition of the hydrate is fxed, eq 14 is independent of pressure. The final general expression for the calculation of ZFWis
RlaCY,CQ, m~
a2 lnfi
d2 In C,
d l n C, dT
+-+(-+ ala
dla
a lnfi
+
-)]aT
2
(15)
For simple gas hydrates, eq 15 is simplified
a In Omi ala d In Cmi
[?
aT dlnC,
-
a In fl 2
-)]aT
[
d2 In C,, = 0 (11)
RlaCvmemi m
d p
d p
+-
EFW=
ala
d2 In C,, d p
The Langmuir type constants are functions of temperature only and so are the derivatives of their logarithms. For multicomponent systems, eqs 10 and 11 can be solved easily for the unknown values of the first and second derivatives of the logarithms of fugacities by application of the method of successive substitutions. In the present implementation, the procedure is intialized by assigning zero values t o all unknowns. Solution is considered to be achieved when the Euclidian norm of the unknowns is less than a tolerance set equal to 10-8. Calculation of of Water in a Hydrate Phase. The partial molar heat capacity of water in the hydrate phase is calculated from the expression: (12)
4w
+-)a2
aT
lnfl
ala
+ (1- om,>
Considering eqs 10b and l l b , the heat capacity of water for simple gas hydrates where the gas enters the large cavity only is equal to that of the empty lattice:
d In Cml
+
a2 In f , -
alnfi
d l n C,, E~ = c { ~- ~ R T C V ~ O , , Pw dT m
alnfi
+
d2 In Cml
X
where is the heat capacity of water in the empty hydrate lattice and AcFW-’ is the difference between heat capacities of water in the hydrate phase and water in the empty lattice at the same temperature. AC;~-’ can be calculated by application of the thermodynamic eq 4:
4w
(15b)
Equations lob, l l b , and 15b imply that the heat capacity of hydrates with only one type of cavity occupied is a linear function of the hydrate number. This is an immediate consequence of the assumptions of the ideal solid solution model.
Heat Capacity of the Empty Hydrate Lattice Effect on Calculated Dissociation Pressures and Hydrate Compositions. Several authors have assumed in the past that the heat capacity difference between the empty hydrate lattice and ice is equal to zero. As it will be demonstrated below, this assumption is not strictly true. Nevertheless, it can be shown that whatever the value of this difference is it leaves predictions of dissociation pressures and hydrate compositions practically unaffected, a t least at temperatures close to the ice point. It has been reported previously (Avlonitis, 1994)that the present accuracy of experimental hydrate dissociation pressure and compositional determinations can not lead to accuracy of the Kihara potential parameters E/K and u* of xenon better than f0.3 K and f0.6 pm, respectively. On the basis of this observation, we vary at fixed temperatures the value of ACs-a and we calculate the corresponding deviation in E&? which is required to exactly match the experimental dissociation point of xenon hydrate, while u* is kept constant. By rearranging these data, we plot in Figure 1 values of
3250 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
-0
Caltd from data of Barrer and Edge, 1967.
Extrapolated
20 180
190 200 2 1 0 220 230 240 250 260 270 Temperature, K
Figure 2. Comparison of calculated heat capacities of the empty hydrate lattice of several structure I hydrates to the experimental heat capacity of ice.
A d - * versus temperature which cause a constant de&ection of -0.3 K in the value of EIK.It is noted that the corresponding differences in the values of the calculated hydrate number and the cavity occupancy ratio are less than their experimental uncertainties. From Figure 1 it is seen that above about 230 K, AC;-" should take unrealistic values before it has any appTeciable effect on calculated dissociation pressures. Identical results are obtained when o* is varied while keeping c/K constant. It is concluded therefore that, for the determination of Kihara potential parameters from dissociation pressure data, AC;;" can be safely neglected in the vicinity of the ice point. Structure I Hydrates. Handa (1986a) argues that the composition of the hydrate sample during his experiments should be considered constant for the whole range of the scan temperatures, although hydrate, ice, and gas coexisted at pressures higher than the hydrate dissociation pressure. We consider, therefore, the hydrate as a closed homogeneous phase having the reported experimental composition at any fixed temperature, although theoretical calculations show that for xenon, krypton, and methane the composition of the hydrate as well as the distribution of the guest between the two types of cavities should change appreciably. The heat capacity of the hydrate is calculated as the weighted sum of the partial molar heat capacities of gas and water. Unique values of the Kihara potential parameters of these gases were retrieved from Avlonitis (1994). Subsequently, the value of ctWis adjusted until agreement is obtained between experimental and predicted heat capacities of the simple gas hydrates. The results of this procedure are presented in Figure 2 together with the experimental heat capacity data for ice reported by Handa et al. (1984). It is seen that of the structure I hydrate calculated from ethane hydrate heat capacity data is always higher than that from methane data, which in turn is very close to ice and the one calculated from Xe.5.90H20. On the contrary, the data of Xe6.29Hz0 lead to values of the empty hydrate heat capacity which are appreciably lower than those of ice. Propane Gas-Water Potential Parameters and Heat Capacity of the Empty Hydrate Structure I1
Lattice. The heat capacity of the empty structure I1 hydrate has been obtained from the propane hydrate heat capacity data. Krypton hydrate is excluded from the present study, mainly because there are inadequate data for dissociation pressures and compositions to allow undisputed determination of the Kihara potential parameters (Avlonitis, 1994). For the case of propane hydrate, heat capacity calculations are straightforward by application of eqs 5, lob, l l b , and 15b. It is seen that only the derivatives of the Langmuir type constants are required from the ideal solid solution model, which at any fixed temperature depend exclusively on the constant values of the potential parameters. Since unique values of the potential parameters of propane cannot be obtained from any dissociation pressure and compositional data (Avlonitis, 19941, to fix these parameters we adopt the following procedure. The HIV dissociation point data of Holder and Godbole (1982) and Deaton and Frost (1946) were used to fit the equation p k P a = exp(-8.900 0.051533TX) by the method of least squares. Next, this equation was used t o obtain a best function, EIK = f(o*),of propane, in the domain of o* [340,3801pm, from the corresponding values at three different temperatures as depicted in Figure 3. Subsequently, the heat capacity data of Handa (1986b) for propane hydrate in the region 180-260 K were used to regress simultaneously the best values of the parameters o* and AC;,". The so obtained values of the Kihara potential parameters for propane-water interactions are o* = 360.8 pm and d K = 199.8 K. These parameters are employed t o predict the three-phase H L V dissociation pressures of propane hydrate above the ice point and up t o the upper quadruple point. Fugacities in the water-rich liquid and the vapor phases are calculated with the correlation proposed by Avlonitis et al. (1994). The results appear in Figure 4. For all points appearing in the figure, the average absolute deviation of predictions was 4.6%. In Figure 5 we plot the heat capacity of the empty hydrate structure I1 lattice, as determined from the heat capacity data of propane. It is seen that it is always higher than that of ice and on average it is similar in magnitude to that of the structure I hydrate derived from the ethane hydrate heat capacity data, but it exhibits a different temperature dependence.
+
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3251
- - 260.0 K K 205 - 265.0 270.0 K . .
200
x
-
Li
3
1 a5
330
340
350
360
370
180 190 200 210 220 230 240 250 260 270
a*, pm
Temperature, K
Figure 3. Optimum pairs of Kihara potential parametrs for propane-water interactions.
..I.II...I..I.II.I.,...Il.I..
600
.
0 V e r m a , 1974 . H Deaton a n d F r o s t , 1946
400
. v
-
v.
-
Miller a n d S t r o n g , 1941
e,
loo-'
273
"
'
I
"
274
' '
I " '
275
'
I
'
276
"
'
I . '
277
" I
"
278
'
I
-
279
Temperature, K
Figure 4. Comparison with experimental data of predicted H L V dissociation points of propane hydrate. -, Predicted.
The heat capacity differences between the empty hydrate lattice and ice, in the temperature region 180 to 260-270 K, were fitted to the equation:
where TOis the ice point. The regressed parameters from each hydrate are given in Table 2. Heat capacities are never linear functions of temperature, and it is understood that these equations are approximations only and they may not be valid outside the specified temperature region.
Figure 5. Comparison of the calculated heat capacity of the empty lattice of the structure I1 propane hydrate to the experimental heat capacity of ice. Table 2. Regressed Parameters for Calculation of ACfi;" (J mol-' K-l) parameter gas hydrate Xe5.90HzO Xe6.29HzO CH&.OOHzO CzHs.7.67HzO C3H8.17.01HzO
temp range (K) 180-270 180-230 180-270 180-260 180-260
c1
cz
-0.08560 -1.04182 -0.59831 1.04474 1.15353
-0.00485 -0.00296 -0.00552 -0.00207 0.00594
used previously t o calculate reliably fugacities of enclathrated molecules, and (3) since the parameters of the model are unique, calculated derivatives of fugacities are necessarily correct. The results of application of eq 5 for the hydrates Xe5.90Hz0, Xe6.29Hz0, CH4.6.00Hz0, CzHg7.67Hz0, and C3Hg17.01HzO are presented in Figure 6. For comparison, the experimental ideal gas heat capacities (API Technical Data Book, 1977)of the respective gases are also plotted in the same figure as dotted lines. Figure 6 shows that, in the temperature range 180270 K, enclathrated xenon in Xe6.90HzO and methane in CH4.6.00HzO have the same heat capacity, within experimental error, as the corresponding ideal gases. However, enclathrated ethane gas has a heat capacity 3.6% higher than that of the corresponding ideal gas, while the propane guest appears having a heat capacity 2.4% more than that of the propane ideal gas. Though these differences are comparable to the experimental error of the ideal gas heat capacity determinations (f1.5%),they might represent real situations. However, the partial molar heat capacity of enclathrated Xe in Xe.6.29Hz0 hydrate is calculated to be 4.1% lower than the corresponding heat capacity of the ideal gas, and this deviation is definitely outside experimental error.
Partial Molar Heat Capacity of Guest Molecules in the Hydrate Phase
Results and Discussion
The heat capacity contributions of guest molecules to their hydrate heat capacity can be calculated by direct application of eq 5 with derivatives obtained from eqs 10 and 11. These calculations are considered trustworthy, because (1)eq 5 is a generally valid thermodynamic relation while the single fitting constant c* is not extrapolated, (2) the ideal solid solution model kas been
The thermodynamic model is applied to predict the heat capacities of the natural gas hydrate studied by Cherskii et al. (1983). It is not clear whether the composition of the gas used by the experimentalists to produce the hydrate is given in volume percent at some unspecified temperature and pressure conditions or in mole percent [see, also, Cherskii et al. (198211. We
3252 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 9.0
- Partial molar h e a t capacity '
,
,
,
.
,
1
1
'
1
.
,
.
,
.
,
.
of enclathrated gas
0 Exptl h y d r a t e d a t a of Cherskii e t a1.,1983
Exptl ideal g a s heat capacity
-Predicted
0 Exptl d a t a of ice ( H a n d a e t al.,1984)
c 4 5
-
1.5 235
CH,.G.OOH,O
3.5
-
3.0
-
---------
Xe.5.90H20 Xe.6.29Hz0
'
1
"
'
1
'
"
'
'
"
"
'
'
-
Temperature, K
Figure 6. Partial molar heat capacities of enclathrated gases in comparison to heat' capacities of respective ideal gases. Table 3. Composition" of the Natural Gas of Cherskii et al. (1983)
a
CzHs 3.94
C3Hs 1.22
C4Ko 0.59
CsHiz
Nz
0.20
2.00
Oz 0.40
See text.
Table 4. Calculated Composition of the Unit Cell (136 HzO Molecules) of the Gas Hydrate of Cherskii et al. (1983) at the H"IV Hydrate Point at 268 K" cavity small large
"
1
"
"
240
"
"
"
'
"
'
245
~
'
'
'
'
250
255
260
K
Figure 7. Experimental and calculated heat capacities of a synthetic gas hydrate prepared from the Mastakh field natural gas by Cherskii et al. (1983).
25----
CH4 91.51
"
Temperature,
4.0
20
i
CHI
Nzb
CzHfi
CaHs
nC4Hi0
10.45 0.31
0.0548 0.0014
2.09
5.08
0.49
a Calculated dissociation pressure is 0.544 MPa. Includes oxygen.
assume therefore the composition of the gas given in Table 3 in mole percent. Moreover, the composition of the synthetic gas hydrate has not been reported, and the details of its preparation are rather diffuse. We assume that the hydrate has been produced at 258 K and 5 MPa (conditions for the preparation of their silica sandhydrate mixture) in equilibrium with a large excess of the natural gas. Considering that Handa in his sophisticated experiments was unable to fully convert ice into hydrate, we assume that some ice was also present in the preparation of Cherskii et al. (1983). Accordingly, the composition of the hydrate is calculated at the three-phase VIH hydrate point at 258 K, and it is reported in Table 4. We further assume that this composition of the hydrate did not change during the heat capacity measurements. The experimental data were reported in the form of a fitted equation with unspecified range of validity. We assume that this equation is valid in the temperature range 235-260 K. The preceding analysis of experimental simple gas hydrate heat capacity data shows that we may not
assign a unique value to the heat capacity of the empty hydrate lattice, and this should be particularly true for multicomponent systems. Considering that cavities of the structure I1 hydrate are larger or equal to the corresponding cavities of the structure I hydrate and that the heat capacity of the empty hydrate lattice of simple gas hydrates of small occupants is generally very close to that of ice, we may attribute a zero value to the heat capacity difference AC{;", due to the contribution of methane and nitrogen in the structure I1 hydrate. Similarly, a zero contribution should be assumed for ethane, which occupies large cavities in the structure I1 hydrate. For n-butane, where there are no quantitative data, we assume a contribution equal t o that of propane, though the larger size of this guest implies a higher effect. Since energetic effects and any deformation of the hydrate lattice caused by the guest molecules should depend on their concentration in it, the heat capacity difference AC{-" should be a function of their concentration. While the theoretical (maximum permissible) occupation is 8 guest moleculeshnit cell, the calculated combined concentration of propane and butane in the gas hydrate of Cherskii et al. (1983) is 5.57, and since the hypothesized function of AC{ia versus guest concentration cannot be established on the basis of the current experimental data, we assume the same value as that of propane hydrate. Figure 7 presents the experimental data of the hydrate heat capacity and the associated error bars as reported by Cherskii et al. (1983). For comparison, the heat capacities of ice (Handa et al., 1984) and the predictions of our model are also plotted in Figure 7. The predictions of the model are about 10%lower than the experimental line, and they are higher but very close to the experimental heat capacity data of ice. Such a deviation may be considered satisfactory considering the uncertainties associated with the experimental data. It is noted that Rueff et al. (1988) are also considering the data of Cherskii et al. (1983) to be too high and have questioned their reliability. An experimental study of the heat capacities of naturally occurring hydrates has been carried out by Handa (1988). The gas in the first sample was almost pure methane with a composition slightly different from the synthetic sample studied before by the same author
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3253 (1986b). The second sample was a structure I1 hydrate with high propane content. Unfortunately, precise heat capacity measurements could not be reported for either sample, due to the amount of impurities in the samples. No other experimental heat capacity data of multicomponent hydrate systems have been reported in the open literature. Empirical or semiempirical models, based on adjustable parameters t o fit experimental data, may provide erroneous results when they are extrapolated or used for calculation of properties which depend on the first or higher order derivatives of the fitted property. The determination of unique and theoretically meaningful values of the Kihara potential parameters by the present author (Avlonitis, 1994) renders the ideal solid solution theory, on which the present model is based, independent of fitting constants. The present implementation, especially, is also independent of the values of the reference thermodynamic properties of the empty hydrate lattice. There is adequate evidence that the physical theory of van der Waals and Platteeuw (1959) is sound and that the values of pertinent parameters are correct, as follows: First, good results are generally obtained when this model is tested against any type of hydrate phase equilibrium data. Second, dissociation enthalpies, which depend on the derivative of the dissociation pressure versus temperature, have also been accurately predicted. Third, for hydrates of small molecules such as methane and xenon, the present predictions of the heat capacity of the empty hydrate lattice and the heat capacity of the enclathrated gas may be reasonably accepted as correct. Our analysis of the heat capacity data of Handa (1986a,b) for methane and xenon leads essentially to the same conclusion as that reached by Handa and Tse (1986): For hydrates of small molecules, the heat capacities of the guest and water are-within experimental error-equal to those of the ideal gas and ice phase, respectively. According to our calculations, the heat capacity of the enclathrated Xe in Xe629HzO hydrate is about 2.4R. Handa (1986~) calculated the latter value of the heat capacity of the Xe to be about 1.7R in the temperature range 150-230 K. This result is clearly unrealistic, and it is a direct consequence of the assumption that the heat capacity of the empty hydrate lattice is nearly the same as that of ice. It was not possible to reduce the experimental data of krypton hydrate for the reasons outlined elsewhere (Avlonitis, 1994). The scheme used by Handa and Tse (1986) to reduce the heat capacity data for xenon, krypton, and methane is not applicable to ethane and propane. No analysis of the data for the latter components has come to our attention. The application of the methodology proposed in this paper leads to the conclusion that the ethane guest causes a small but definite increase of the heat capacity of water in the hydrate lattice as well as an increase of the heat capacity of the guest itself. For large molecules, such as ethane in the hydrate crystal of structure I and propane in structure 11, it has been stated (Avlonitis, 1994) that the values of their Kihara potential parameters for gas-water interactions incorporate a number of uncertainties, including the deformation of the crystal and the nonsphericity of the guest, and it was implied that these nonidealites should rather be incorporated into the values of the thermodynamic properties of the empty gas hydrate lattice. However, the extent to which these effects affected the present values of the potential parameters is not known. For
example, unique values of the potential parameters of ethane were determined from mixed data of dissociation pressures of pure ethane gas (structure I) and ethanepropane mixtures (structure 11), and it may be noted that ethane should not cause a deformation of the structure I1 lattice so that the potential parameters of ethane are not too much biased by nonidealities. Thus, heat capacity predictions for the gas hydrate of ethane should not be far from the correct values. For propane, potential parameters were adjusted in the present work to fit concurrently dissociation pressure as well as hydrate heat capacity data. Since the contribution of the encaged guest to the heat capacity of the clathrate is minimal, predictions are not very sensitive to the values of the potential parameters. Considering eq 15b, this means that in this case the calculated heat capacity of the empty hydrate lattice should be nearly correct. Davidson et al. (1977) have recorded nuclear magnetic resonance spectra of the clathrate hydrates andfor deuteriorates of methane, ethane, propane, and isobutane and conducted dielectric relaxation measurements in hydrogen sulfide, propane, isobutane, and n-butane hydrogen sulfide hydrates. While they estimate that there are no reorientation barriers for the hydrogen sulfide- and methane-encaged molecules, they report average barriers to reorientation (“activation energies”) of 1.2 kcaVmol for ethane (structure I) and 0.6 kcaVmol for propane in structure I1 hydrates. These experimental findings corroborate the results of our calculations for methane, ethane, and propane. It has not been possible to reach a unique function for the heat capacity of the (hypothetical) empty hydrate lattice, but different approximations were obtained from hydrates of different guest molecules and different compositions. Although this finding is contrary to the conventional approach of assigning unique values to the thermodynamic properties of the empty hydrate lattice (irrespective of the guest), it does support the suggestion of Holder et al. (1988) for different values of the reference chemical potential for different guests. Added evidence is taken again from Davidson et al. (1977) who were able t o detect wide distributions of reorientation rates in all hydrates studied by them. They attribute this phenomenon to variations of distortion from cage to cage. Thus, the varying values of the heat capacity of the empty hydrate lattice calculated here are in accord with this experiment. The chemical potential difference between the empty hydrate lattice and ice could be regressed as a function of guest concentration from variable composition experimental hydrate dissociation pressure and compositional data in the neighborhood of the ice point. For structure I, the choice of the system xenon-ethane is recommended, since any ratio of these components in the hydrate phase is obtainable. For structure 11, the mixture of ethane-propane is the preferred system, though in this case the range of achievable compositions is limited by the structural transition at higher ethane concentrations. In either of these systems, the first component is relatively small and any disturbance of the host lattice should be attributed solely to the larger guest. Measurement of the heat capacity of the above systems will permit regression of the heat capacity of the empty hydrate lattice as a function of the larger guest concentration, and it could provide insight into guest-host as well as any possible guest-guest interactions. Heat capacity measurements are also needed for the simple gas hydrates of carbon dioxide and nitrogen,
3254 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
which have not been studied so far. The first gas is expected to interact with the lattice a t lower temperatures where it is believed that it enters both types of the structure I cavities. Data for nitrogen are useful for comparison to those of propane and krypton. As it has already been noted, dissociation pressure and compositional data are also required for krypton hydrate. Finally, heat capacity data of multicomponent systems of both structures are required for testing the predicting capability of this model.
Conclusions Classical thermodynamics and the ideal solid solution theory were applied to reduce the heat capacity experimental data for the simple gas hydrates of xenon, methane, ethane, and propane obtained by Handa (1986a,b,c), and a consistent methodology was detailed for the prediction of the heat capacity of simple and multicomponent gas hydrates. The heat capacities of the empty hydrate lattices of structure I and I1 simple hydrates of the present gases were regressed as functions of temperature, and it was shown that the heat capacities of the empty lattice of propane and ethane hydrates are always larger than those of xenon and methane, which in turn are very close to that of ice. The calculated partial molar heat capacities of the enclathrated gases of methane and xenon (in Xe5.90H20) were always ideal-gas-like, while ethane demonstrated a definite increase larger than that of propane in its hydrate. A small but definite decrease has been calculated for the heat capacity of the empty hydrate lattice as well as the heat capacity of enclathrated Xe in Xe6.29H20 hydrate. These calculations indicate that variable reference thermodynamic properties, according to the type and concentration of the larger guests, should be used for application of the van der Waals and Platteeuw theory for dissociation pressure predictions of natural gas mixtures. To obtain such correlations of the thermodynamic properties of the empty hydrate lattice, certain experimental studies of dissociation pressures, compositions, and heat capacities of simple and mixed gas hydrates are suggested.
H = enthalpy of a system I Z ~= mole of guest component i per mole of water in the hydrate phase IVH = number of guest components in the hydrate lattice R = gas constant S = entropy of a system T = absolute temperature xj = mole fraction of component j
Greek Symbols AC{+kH = constant volume heat capacity difference A@’gH m = energy of formation of a clathrate per mole of gas E = Kihara energy parameter K = Boltzmann’s constant pi = chemical potential of component i v, = number of cavities of type m per water molecule in the unit cell 0,i = fractional occupancy of type m cavities by component i CJ = Kihara distance parameter Superscripts /3 = index of the empty hydrate lattice
g = index of a gas phase H = index of a hydrate phase Subscripts i ,j = gas component index m = cavity type index w = index of water
Appendix Calculation of Hydrate Cage Occupancies. The solution of an equivalent problem, i.e., the calculation of guest gas fugacities from the composition of a homogeneous hydrate phase, has been addressed previously by Cole and Goodwin (1990) and Michelsen (1991). Either of these methods might be employed, but the first is not advantageous for the present scope since guest gas fugacities are initially calculated. A simpler method is proposed below, which is similar in concept to the original of Michelsen (1991). Equation 7 may be written in the following form:
Acknowledgment The author wishes to thank the Department of Petroleum Engineering of Heriot-Watt University, Edinburgh, UK, for supporting this study. Thanks are also due to Prof. A Danesh for commenting on the manuscript.
where
Nomenclature cii = constant, a function of temperature only character-
istic of the pure substance i CF = constant volume heat capacity of a clathrate CF = constant volume heat capacity of enclathrated gas Cf = constant volume heat capacity of the empty hydrate lattice = partial molar heat capacity of component i C1 = fitting constant, eq 16 Cz = fitting constant, eq 16 C,i = Langmuir type constant for a gas molecule, i , in a hydrate cavity, m C, = constant pressure heat capacity fi = fugacity of component i G = Gibbs free energy of a system
1- C O l j j
e= 1-
c@2j
(A31
.i
From eqs A1 and 6c we obtain
Substitution of eqs A4 into eq A3 leads to the following
Ind. Eng. Chem. Res., Vol. 33,No. 12, 1994 3255 relation:
)
N H n j ( l - rj)
1-
-
+ VP
j Vlrje
1 = 0 , with e
=- 0
(A5)
which enables the determination of e. For the simple hydrate of a gas distributing in all cavities, eq A5 reduces t o a quadratic with only one positive root. If the single guest occupies exclusively large cavities, then e =V ~ / ( VZ n). In the general case of a multicomponent gas, e is computed by application of the one-dimensional Newton-Raphson method. The first derivative @e)/ de may be calculated from the following analytical expression:
Me)
NH nj(l
- rj)
-=1-x
dQ
j VlrjQ
Nn njrj(l - rj)
+vlep -
+ ~2
j
+
1
(vlrje v212 (A61
An initial estimate of e may be obtained by assuming that guests not entering small cavities are the sole occupants of large cavities, i.e.,
e= I--
1
NH - '
nj
'%=NsH+l
where NsH is the number of hydrate forming gases entering both types of cavities. For the gas of Cherskii et al. (1983),the proposed algorithm converges to the root of eq 5 , within a tolerance of lo+, in four iterations.
Literature Cited A P Z Technical Data Book, Petroleum Refining, 4th ed.; American Petroleum Institute, Division of Refining: Washington, DC, 1977;Vol. 11, Procedure 7Al.l. Avlonitis, D. The determination of Kihara potential parameters from gas hydrate data. Chem. Eng. Sci. 1994,49,1161-1173. Avlonitis, D.; Danesh, A.; Todd, A. C. Prediction of VL and VLL equilibria of mixtures containing petroleum reservoir fluids and methanol with a cubic EoS. Fluid Phase Equilib. 1994,94, 181-216. Barrer, R. M.; Edge, A. V. J. Gas hydrates containing argon, krypton and xenon: kinetics and energetics of formation and equilibria. Proc. R . Soc. London 1967,A300, 1-24. Berecz; Balla-Achs. Gas Hydrates; Elsevier: Amsterdam, 1983. Cherskii, N. V.; Groysman, A. G.; Nikitina, L. M.; Tsarev, V. P. First experimental determination of heats of decomposition of natural-gas hydrates. Dokl. Akad. Nauk SSSR 1982,265,185189. Cherskii, N. V.; Groysman, A. G.; Tsarev, V. P.; Nikitina, L. M. Thermophysical properties of natural gases. Dokl. Akad. Nauk SSSR 1983,207,949-952. Cole, W. A.; Goodwin, S. P. Flash calculations for gas hydrates: A rigorous approach. Chem. Eng. Sei. 1990,45,569-573.
Davidson, D. W.; Garg, S. K.; Gough, S. R.; Hawkins, R. E.; Ripmeester, J. A. Characterization of natural gas hydrates by nuclear resonance and dielectric relaxation. Can. J . Chem. 1977,55,3641-3650. Deaton, W. M.; Frost, E. M., Jr. Gas hydrates and their relation to the operation of natural gas pipelines. US.Bureau of Mines Monograph 1946,8. Fleyfel, F.; Sloan, E. D. Prediction of natural gas hydrate dissociation enthalpies. Proceedings, First International Offshore and Polar Engineering Conference, Edinburgh, U.K.,Aug. 11-16; ISOPE, 1991. Handa, Y. P. Calorimetric determinations of the compositions, enthalpies of dissociation, and heat capacities in the range 85 to 270K for clathrate hydrates of xenon and krypton. J . Chem. Thermodyn. 1986a,18,891-902. Handa, Y. P. Compositions, enthalpies of dissociation and heat capacities in the range 85 to 270K for clathrate hydrates of methane, ethane, and propane, and enthalpy of dissociation of isobutane hydrate, as determined by a heat-flow calorimeter. J . Chem. Thermodyn. 198613,18,915-921. Handa, Y. P. Composition dependence of thermodynamic properties of xenon hydrate. J . Phys. Chem. 1986c,90,5497-5498. Handa, Y. P. A calorimetric study of naturally occurring gas hydrates. Znd. Eng. Chem. Res. 1988,27,872-874. Handa, Y. P.; Tse, J. S. Thermodynamic properties of the empty lattices of structure I and I1 clathrate hydrates. J . Phys. Chem. 1986,90,5917-5921. Handa, Y. P.; Hawkins, R. E.; Murray, J. J. Calibration and testing of a Tian-Calvet heat-flow calorimeter. Enthalpies of fusion and heat capacities for ice and tetrahydrofuran hydrate in the range 85 to 270K. J . Chem. Thermodyn. 1984,16,623-632. Holder, G. D.; Godbole, S. P. Measurement and prediction of dissociation pressures of isobutane and propane hydrates below the ice point. AIChE J . 1982,28,930-934. Holder, G. D.; Corbin, G.; Papadopoulos, K.Thermodynamic and molecular properties of gas hydrates from mixtures containing methane, argon and krypton. Znd. Eng. Chem. Fundam. 1980, 19,282-286. Holder, G. D.; Zetts, S. P.; Pradhan, N. Phase Behavior in systems containing clathrate hydrates. A review. Rev. Chem. Eng. 1988,5,1-70. Michelsen, M. L. Calculation of hydrate fugacities. Chern. Eng. Sei. 1991,46,1192-1193. Parsonage, N. G.; Staveley, L. A. K. Thermodynamic properties of clathrates: The heat capacity and entropy of argon in the argon quinol clathrate. Mol. Phys. 1958,2,212-222. Parsonage, N. G.; Staveley, L. A. K.Thermodynamic studies of clathrates and inclusion compounds. In Inclusion Compounds; Academic Press: London, 1984;Vol. 3, Chapter 1. Rueff, R. M.; Sloan, E. D.; Yesavage, V. F. Heat capacity and heat of dissociation of methane hydrates. AIChE J . l988,9,14681476. Sloan, E. D. Clathrate hydrates of natural gases; Marcel Dekker: New York, 1990. van der Waals, J. H. The statistical mechanics of clathrate compounds. Trans. Faraday Soc. 1956,52,184-193. van der Waals, J. H. Some observations on clathrates. J . Phys. Chem. Solids 1961,18,82. van der Waals, J. H.; Platteeuw, J. C. Clathrate compounds. In Advances in Chemical Physics; Prigogine, I., Eds.; Interscience: New York, 1959;pp 1-57. Received for review October 4, 1993 Revised manuscript received March 19, 1994 Accepted August 3, 1994" Abstract published in Advance ACS Abstracts, October 1, 1994. @
