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Langmuir 1997, 13, 1286-1296
Hard Coal Surface Heterogeneity in the Sorption Process† Janina Milewska-Duda*,‡ and Jan Duda§ Faculty of Fuels and Energy and Institute of Automatics, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krako´ w, Poland Received November 17, 1995. In Final Form: November 12, 1996X This paper discusses sources of hard coal energetic heterogeneity and its effects on sorption properties. Natural coal is viewed as a porous elastic copolymer in which absorption and adsorption phenomena can occur. The dominant role of submicropores which are about the size of a water molecule is stressed, as the effects of larger pores on sorption capacity are negligible. Bearing in mind the size of the submicropores, one must take account of swelling of the sorbent and the resultant changes in its surface area. A unified theoretical description of sorption equilibria with respect to swelling phenomena (multiple sorption model) is developed. Alternative models describing effects of geometrical and chemical factors on the sorption energy are discussed. The resultant theoretical sorption isotherms are compared with empirical data for sorption of water and benzene in two coals. The model explains differences in sorption properties of the coals in terms of the pecularities of their surfaces. Expansion was found to be of crucial importance for the sorption of benzene. It is shown that a simplified description of sorbent-sorbate energetic interactions is adequate to express the basic relations between the energies attributed to submicropores of different size.
Introduction Sorption of small molecules in natural coal has been often interpreted as a multilayer adsorption process in coal pores, yielding isotherms of type II or III (BET classification). However, it does not explain the noticeable expansion of coal (1-2%) observed during sorption.1 Present knowledge of coal structure points to a considerable fraction of elastic macromolecular and molecular compounds,2 together with a large amount of submicropores. This suggests that sorption in coal is mainly by simultaneous absorption and volumetric adsorption in pores of molecular and submolecular size. Absorption and adsorption subprocesses in such a material certainly interact and contribute to the total sorption capacity to different extents depending on the sorbent surface geometry and its chemical properties. The question is how to use sorption data to get information on coal surface structure and on sorption mechanisms. In this case a simple fitting of empirical isotherm data by an equation like the Langmuir, BET, or other adsorption formulas may be misleading, as the effect of absorption on the isotherm shape is usually significant and very involved. Hence, the coefficients determined in such a way are in general of poor physical meaning. The only way seems to be development of a more rigorous mathematical model based on analysis of the physical phenomena occurring in coal-sorbate systems. This requires specification of the basic factors which may affect sorption properties of submicroporous materials, followed by a quantitative and qualitative analysis of their contribution to the process. In ref 3 Milewska-Duda proposed a dual sorption model based on thermodynamic relations (free energy changes due to sorption), which includes absorption and adsorption as two interacting subprocesses. This model was found †
The paper has been presented at the Second International Symposium on Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland-Slovakia, September 4-10, 1995. ‡ Faculty of Fuels and Energy. E-mail,
[email protected]. § Institute of Automatics. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Milewska-Duda, J.; Ceglarska-Stefan˜ska, G.; Duda, J. Fuel 1994, 73, 975. (2) Larsen, J. W.; Kovac, J. In Organic Chemistry of Coal; Symp. Series 71; American Chemical Society: Washington, DC, 1978; p 36. (3) Milewska-Duda, J. Fuel 1993, 72, 419.
S0743-7463(95)01049-3 CCC: $14.00
to be satisfactory for sorption of small molecules like water and methanol.4,5 It enabled us to determine energetic and structural parameters of different rank coals and to examine other properties of coal-sorbate systems. The examination showed that absorption capacity is significantly lower than might be expected, based on chemical and structural properties of coal.1,4,5 In order to give a formal explanation of such effects and obtain a model adequate for larger sorbate molecules, we elaborated a mathematical description of sorption equilibria in submicroporous elastic materials (like hard coal) including absorption and a spectrum of adsorption-like subprocesses. This is referred to as the multiple sorption model.6 It expresses the energetic heterogeneity of microporous sorbents in terms of geometrical parameters (size and shape) of pores, with the effects of active groups (chemical heterogeneity) taken into account. The basic properties of such a model (related to those of dual sorption model) are presented in ref 6. The aim of this paper is to show how the model may be used to investigate effects of coal surface properties on sorption capacity. The formal representation of coal energetic heterogeneity is discussed in more detail. Since the sorption energy in particular places of the system is immeasurable, alternative formulas were proposed and examined. They involve parameters of clear physical meaning. This paper attempts to explain basic mechanisms of sorption of vapor molecules in coal by numerical analysis of consequences of different assumptions concerning coal surface properties. Energetic Interactions in Hard Coal-Sorbate Systems From the viewpoint of the sorption process, coal may be considered as a submicroporous elastic material.3 The elastic phase itself is heterogeneous.2,4 It consists of a macromolecular structure containing relatively large arene domains linked with typical copolymer chains and filled with mobile, short chain polymer compounds. In (4) Milewska-Duda, J. Mathematical Modelling of Equilibrium States of Process of Small Molecules Substances Sorption in Coal, Sci. Bull. Acad. Min. Metall. Chem. Bull. 11 1988, No. 1236 (in Polish). (5) Milewska-Duda, J.; Duda, J. Fuel 1994, 73, 971. (6) Milewska-Duda, J.; Duda, J. Langmuir 1993, 9, 3558.
© 1997 American Chemical Society
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turn, the characteristic feature of coal porosity is a large number of micropores and submicropores which may be viewed as irregular holes of molecular and submolecular size, randomly distributed within the elastic phase. It was stated,1,4,5 that these play a very important role in sorption on coal. Larger pores (meso- and macropores) affect sorption properties to a much smaller extent7 as their surface area is practically negligible compared to that of submicro- and micropores. In such a material the sorption process may be considered as penetration of sorbate molecules among elastic chains (pure absorption) and placing of molecules in holes (pores) properly enlarged to the molecule size. In the latter case we may distinguish larger holes (micropores), which need no changes in size to hold a sorbate molecule. The corresponding process will be referred to as pure adsorption, while that in smaller holes (submicropores) is of an adsorptive-absorptive nature. Let us consider a sorption system compared to its standard state, the latter consisting of pure liquid sorbate and dry sorbent. In order to get a formal description of energy changes due to sorption, one may consider a hypothetical three-stage process: (a) separation of the sorbate molecule from its liquid state (evaporation); (b) creation of a hole large enough to have room for the sorbate molecule (expansion); (c) placing of the molecule in the hole (condensation). Let ea, eb, and ec denote the molar energy attributed to the above stages (a, b, and c), respectively, provided that the hole mentioned in the stages b and c is isolated from others. At the initial stage of sorption when the concentration of sorbate in the system is negligible, the expected molar energy of sorbate molecule in the considered site (hole) may be expressed by the energetic parameter Qa
Qa ) (1 - φh°)(eb - ec) + ea
(1)
where φh° denotes the surface fraction of holes in dry coal. In eq 1, the energies both of expansion and of condensation are related to the surface of sorbent-sorbate contact; hence the factor (1 - φh°) is introduced. It represents an effect of excluded surface due to mutual contacts of holes. Notice that in the case of a completely random distribution of holes, the excluded surface area is on average proportional to the surface fraction of holes in total surface of all particles present in the system. As yet, the above factor is the only modification of the classical formulas for mixing energy (e.g., see ref 8 ). The first stage energy ea is in fact the molar evaporation energy of the sorbate. However, in eq 1 and in further analysis, the same value is used for the energy of sorbatevacuum interactions in submicropores, which may be a bit lower. For this reason, in ref 6 we assumed ea ) Ua, where Ua was treated as a semiempirical parameter bounded by the evaporation energy and that of the surface tension. Nevertheless, it seems to be more appropriate to use the evaporation energy as the value for Ua but to modify the sorbate-vacuum energy. Stage b results in creation of a new surface, so one may take eb to be proportional to the surface expansion ratio ωa. On the other hand, it can be viewed as the creation of new free volume, with the energy eb proportional to the volume expansion ratio wa. Assuming that both the hole and sorbate molecule are spherical, we have (7) Lason˜, M., Ed. Sorption of gases and vapours, and properties of polish hard coals as dispersive systems. Sci. Bull. Acad. Min. Metall., Chem. Bull. 8, 1988, No. 1212 (in Polish). (8) Flory, J. P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
eb ) ωa 4π Ra2 N Γha ) ωa‚Va Usha
(2)
eb ) wa 4π/3 Ra3 N Uh ) wa Va Uh
(3)
or
where def
ωa ) 1 - (Rh/Ra)2 def
wa ) 1 - (Rh/Ra)3
(ωa g 0)
(4)
(wa g 0)
and Ra denotes the effective radius of the sorbate molecule, Va is the molar volume of sorbate, N is Avogadro’s number, Γha and Uh are the surface tension and cohesion energy s of the sorbent, respectively, and Uha is an energetic parameter. In the ref 6 the application of both eqs 2 and 3 was considered. We showed that for coal-sorbate systems, s ) Uh yields a theoretical isotherm with a eq 2 with Uha shape closer to empirical data than that obtained from eq 3. Thus, the former will be used in this paper. Any point within the elastic phase of the sorbent matter may be viewed as a potential hole of zero size (Rh ) 0) where a sorbate molecule can be inserted after spontaneous removing of sorbent particles (segments of polymer chains) with ωa ) 1. For such a process, referred to as pure absorption, relatively small differences in the energy of a particular sites may be expected. Following the theory of solutions,9 for the pure absorption we assume that the cohesion energy is equal to the square of solubility parameter δc, which may be determined experimentally and theoretically.10,11 Any hole of nonzero size existing in the dry sorbent (a submicropore) can become a sorption site provided that it has been properly enlarged by spontaneous movement of polymer segments. However in this case the energy required for insertion of the sorbate molecule is certainly much more diversified than that for absorption, since it depends on the hole surroundings. One may expect that expansion of smaller holes requires more energy, because they can exist only in surroundings rigid enough to prevent the material from releasing the high energy of their surface. Hence, the expansion is certainly not isotropic and results in creation of larger surface (space) than that expressed by ωa. The larger the hole, the more elastic the walls that may be encountered, and so the expansion energy eb approaches that for absorption, i.e., the cohesion energy. It can be taken into account by using a correction factor f(Rh) with the form
Uh ) f(Rh) δc2
(5)
The following function f(Rh) is proposed as a reasonable representation of this effect (see Figure 1)
f(Rh) ) Cf(1 - Rh/Rf) + 1 1
{
for Rh e Rf (6) for Rh ) 0 and for Rh > Rf
where Cf is a constant (Cf > 1) and Rf is a characteristic hole size such that for larger holes the cohesion energy may be taken as the same as the average value for the elastic phase (Uh ) δc2). (9) Prigogine, I.; Bellemans, A.; Mathot, V. The Molecular Theory of Solutions; North Holland: Amsterdam, 1957. (10) Hombach, H. P. Fuel 1980, 59, 465. (11) van Krevelen, D. W. Fuel 1965, 44, 229.
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Figure 1. Correcting factor for cohesion energy vs hole size.
Let us recall the energy ec (eq 1) evolved by placing a sorbate molecule in the created hole. For mixtures of simple nonpolar liquids, the Berthelot rule (Rowlinson12) is applicable
ec ) 2(eaeb)1/2
(7)
The above rule is proposed to be applied to coal-nonpolar sorbate systems as well, but certain losses in the contact surface area of sorbate with irregular walls of the hole should be taken into account. For this, one may use a correction factor ζ(Rh,Ra) e 1, which expresses the fraction of effective contact surface area. Thus the following formula is proposed
e ) ζ(Rh,Ra)2δcxVaUa c
(8)
One may suppose that for holes of very small original size (Rh = 0) the factor ζ is relatively small (ζ ) ζ0) due to the reasons presented above in eq 5. The larger the holes, the more elastic are the walls that are encountered; hence ζ increases. At a characteristic hole size RB, it reaches a maximal value ζB. For larger Rh, the shape irregularity of the original holes is reduced by expansion, so a decrease in ζ may be expected. Finally, for pure adsorption a characteristic value ζA is reached. Notice that such a factor transforms the real interactions between sorbate molecule and coal surface (perhaps of different local intensity) into two types: full interaction (as in absorption) with reduced contact surface area, and sorbate-vacuum interaction, keeping the energy ea of the remaining surface fraction. Thus, the above correction compensates also for possible differences between the value taken for ea and the real energy of a sorbed molecule surface not in contact with any other particle (sorbate or coal). In our earlier works4,5 we stated that for polar sorbates a higher energy ec in the largest holes than that predicted by eq 7 should be assumed. This is due to specific interactions with polar groups present on the coal surface. One may suppose that such interactions become less significant in smaller holes. This effect may be taken into account by taking a larger ζ for holes whose size Rh is larger than RB. The following formula is proposed to (12) Rowlinson, J. S. Liquids and Liquid Mixtures; Butterworth: London, 1969.
Figure 2. Correcting factor ζ for adhesion energy vs hole size Rh and fraction ξ of effective contact surface area vs hole size.
{
express the above effects, both geometrical and chemical
1 ζ0 + (ζB - ζ0) Rh/RB ζ(Rh,Ra) ) ζB + (ζACP - ζB)× (Rh - RB)/(Ra - RB) ζACP
for Rh ) 0 for 0 < Rh e RB for RB < Rh < Ra for Rh g Ra
(9)
where the coefficients RB, ζ0, ζB, and ζA parameterize the geometric heterogeneity (ζ0 < ζB e ζA e 1; RB e Ra) and the parameter Cp expresses the effect of specific interactions (Cp ) 1 for nonpolar sorbates and Cp > 1 if a polar sorbate is considered). See Figure 2. As seen in eq 9, the pores whose size exceeds that of the sorbate molecule are treated with the averaged parameter ζ ) ζACp representing generally their surface properties and with no expansion being assumed (ωa ) 0). This means that the model does not account for the more complicated adsorption phenomena typical for rigid sorbents, such as multilayer adsorption, energetic heterogeneity of large pore surface, etc. There are no formal obstacles to take these into account by proper classification of larger pores according to their size and taking suitable ζ values for such classes. However, in the case under consideration the above simplification seems to be acceptable. Formulas 2 and 3 point to differences in original size of submicropores as the first source of the energetic heterogeneity of the submicroporous-elastic sorbent. This seems to be evident if we accept the hypothesis that volumetric adsorption in submicropores makes a significant contribution to the sorption process. In turn, the correction factors defined by formulas 6 and 9 for the expansion and condensation energy are proposed to take into account the heterogeneity of the elastic phase itself. It should be noticed that the corresponding effects cannot be measured directly, so the proper formulas are only intuitive. For this reason the correction factors are expressed in the form of very simple functions of the hole size relative to that of sorbate molecule. They only handle the main effects but seem to be adequate for further analysis.
{
Hard Coal Surface Heterogeneity
Langmuir, Vol. 13, No. 5, 1997 1289
1 ZB Rh/RB Z(Rh,Ra) ) ZB + (ZA - ZB) × (Rh - RB)/(Ra - RB) ZA
for Rh ) 0 for 0 < Rh e RB for RB < Rh < Ra for Rh g Ra
(12)
The question whether such a model is sufficient for practical use will be discussed in the sequel on the basis of computational results. Multiple Sorption Model Let us consider the sorption system as consisting of a number of subsystems a, each containing mpa moles of sorbate molecules with the same molar energy. Let T and p be temperature and pressure, respectively, and po ) the pressure of saturated vapor of sorbate. In the equilibrium state of such a system the following general formula is valid3 Figure 3. Simplified correcting factor for adhesion energy vs hole size.
The absorption energy needs to be expressed by special values in both formulas 6 and 9 because of the qualitative difference between the energetic properties of the compact elastic phase and the very small holes. The absorption needs also special treatment in consideration of entropy effects as it has an infinite number of possible sites, while any adsorptive subprocess is attributed to a finite number of holes (see next section). By virtue of eqs 2-9, eq 1 may be rewritten as 2
Qa ) (1 - φh°)[ωa Va f(Rh) δc ζ(Rh,Ra) 2δc xVaUa] + Ua (10) For nonporous sorbents where only pure absorption is present, we have φh° )0, ωa ) 1, ζ ) 1, and f ) 1 and so formula 10 defines the Flory-Huggins parameter,8 i.e. the energetic parameter for polymer solutions. Notice that the presence of submicropores leads to a significant increase in this parameter due to the factor (1 - φh°) because the term in brackets in eq 10 is negative and close to Ua. It explains the fact that absorption capacity of coal is much lower than that of typical polymers.1,4,5 However, effects of a highly dispersed surface on sorption properties are more complex because of additional phenomena such as sorbate-sorbate interactions and changes in the surface area due to swelling of the sorbent. This problem will be discussed in the next sections of the paper. To compute the value for Qa, model 10 requires seven parameters to be given or identified, i.e., Rf, Cf, RB, ζ0, ζB, ζA, and Cp. The multiple sorption model presented in ref 6 uses the following simplified formula
Qa ) (1 - φh°)[ωa Va δc2 - Z(Rh,Ra) 2δc xVaUa] + Ua (11) where the correction function Z(Rb,Ra) absorbs the effects of both f(Rb) and ζ(Rb,Ra) in such a way as to get relatively high energy in the smaller holes. The function Z(.) is computed similar to ζ with ζ0 ) 0 (see eq 10) on the basis of three parameters RB, ZB, and ZA corresponding to RB, ζB, and ζACp (see Figure 3)
RT ln
( )∑ p
p0
mpa ) ∆H - T ∆S
(13)
a
where ∆H and ∆S denote total energy and entropy change due to sorption. The value for ∆H is a sum of quantities Qa (see eq 10) for all sorbate molecules, corrected for sorbate-sorbate interactions and changes in the sorbent surface area. It may be expressed in the following form6
∆H )
∑a mpa(Qa - ∑b φpb Eab -
(φh + δφh)(1 - φh°)ωaVaδc2 - δφh(Qa - Ua)) + δAh(1 - φh - δφh)(1 - φh°)Vaδc2 (14) where the indices a and b denote subsystems with sorption sites of different energy (in this paper they mean the holes of different size), φpa and φpb denote the surface fraction of sorbate molecules placed in holes of the ath and bth subsystems, respectively, φh is the surface fraction of holes in the sorption system, δAh and δφh are the changes in surface area Ab of pores and in the fraction φh due to sorption, and Eab is the energy constant defined for each pair of molecules (a,b)
Eab ) ωaQa + (ξaξbηab - ωa)Ua
(15)
ξa and ξb is fraction of effective contact surface area between hole and sorbate in ath and bth subsystems computed by using eq 9 with Cp ) 1 and ηab is a parameter representing intensity of sorbate-sorbate energetic interactions
ηab ) 1 - Uab/Ua
(16)
where Uab denotes the interaction energy of neighboring sorbate molecules placed in holes of ath and bth type. If direct contact occurs, the mutual energy is the same as in the liquid; hence Uab ) 0 and so ηab ) 1. If molecules placed in neighboring sites are separated, one may assume that Uab ) Ua (no interaction); thus ηab ) 0. The parameter ηab is introduced to express the fact that because of the irregularity of surface shape, direct contact of sorbed molecules is less probable than might be expected for a completely random distribution of sorbate molecules in the system. It is common practice to assume ηab ) 1 for absorption and ηab ) 0 for pure adsorption. Hence, for intermediate subprocesses in expanded holes we have taken the following formula:
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Milewska-Duda and Duda
(17)
ηab ) ωa + ωb - ωaωb
In order to examine the effect of changes in the surface area δAh, we assume (as in ref 6) that a free surface of holes (which is not in contact with any sorbate molecule) is enlarged like the surface of a spherical body due to an increase of bulk volume of the sorbent (swelling). This leads to the formula:
2 δAh ) [Ah 3
∑b
(1 - ωb)mpbξb]
∑b
vpbwb
(18)
vc
where xe denotes average length of cross-linked chains relative to the volume of the sorbate molecule and other symbols are as in eqs 1-19. By solving the set of equations (20) written for all subsystems with an assigned pressure p, one obtains values for mpa(p/po). The values found for a sequence of p constitute the set of individual theoretical sorption isotherms, in particular, the absorption isotherm. Then, by summing up the sorption capacities mpa for all the subsystems, one may obtain the total sorption isotherm mp(p/po). Computation Results
where vpb and vc denote the volume fraction of sorbate in the bth subsystem and the sorbent, respectively. The value for Qa will be computed using eq 10 or eq 11. In the latter case formula 12 is applied to compute the quantities ξa and ξb (the fraction of effective contact surface area between hole and sorbatessee eq 15) but the resultant value is limited to 1, if polar sorbates are considered (ZA > 1). The entropy term in eq 13 may be approximately expressed as the configurational entropy of all particles and polymer segments present in the sorption system and of holes. For this, one may apply the Flory method,8 based on a lattice model. In the case under consideration it leads to the formula (see refs 4 and 6 for more details)
{∑
∆S ) -Rn
∑a (νha - vpa(1 - wa))
vpa ln vpa +
[
a
ln(νha - vpa(1 - wa) + ve
3 ln(ve) + (ve-2/3 - 1) 2 2 1
]}
+C (19)
where n denotes the total number of system cells (elastic chain segments + empty holes + sorbate molecules, expressed in moles), vha is the volume fraction (in bulk volume of the sorption system) of original size holes attributed to the ath subsystem, vpa is the volume fraction of sorbate molecules in ath subsystem, ve is the volume fraction of elastic chains, and C is a constant. The last term in eq 19 represents the entropy of swelling of the sorbent, with four-functional cross linkages being assumed. Finally, the formula for the sorption isotherm in the ath subsystem attributed to holes of Rha size may be derived by differentiation (with respect to mpa) of formula 13, including eqs 14 and 19. It yields the following expression
(
[
)]
1 1/3 ve p ve + ln ) wa ln νpa + ve p0 xe 2 vpa Qa + (1 - wa) ln (vha - vpa)(1 - wa) RT ωa(1 ωa
∑b φpbωb)φhφe° 1
δ2c Va RT
∑b φpb[Eab + Eba -
∂δAh
φpBEBb] {(1 - δφh)[φeφe°δ2c Va + ∑ RT ∂ma B ∑b φpb(Qb - Ua)] + φhφe°δc2Va∑b φpbωb + 1
φpbφpBEBb} - δφh{ωa[φeφe°δc2Va ∑b ∑ RT B 1
∑b φpb(Qb - Ua)] + Qa - Ua}RT
(20)
The multiple sorption model enables us to calculate sorption isotherms for a given coal-sorbate system with different settings of its parameters. Theoretical isotherms obtained in this way confronted with empirical data provide information on possible mechanisms of sorption in coal. In particular, the factors which affect sorption capacity and need to be examined are (1) sources of coal heterogeneity and their formal representation, (2) inaccessibility of smaller submicropores for sorbate molecules, and (3) changes in coal surface area due to swelling. The model (19) involves the following parameters: (a) energetic parameters for sorbate Ua and sorbent δc; (b) parameters used in the formula for molar energy Qa of sorbate molecule, i.e., Rf, Cf, RB, ζ0, ζB, ζA, and Cp in the case of model (10) or RB, ZB and ZA if the simplified model (11) is applied; (c) number of holes mha° (with assigned size Rha) in each subsystem in the sorption system; (d) average number xew of segments in an elastic chain of coal (average volume of elastic chain relative to that of a water molecule). During examination of the dual sorption model4,5 and an earlier analysis of the multiple sorption,6 the energetic parameters δc and Ua have been treated as semiempirical quantities. In this paper we postulate that the sorption energy description can be based on theoretical values only. Hence, we take the molar evaporation energy of sorbate as the value for Ua and the value for the solubility parameter δc as a function of coal rank determined by van Krevelen.11 In this way we attempt to show that the empirical values for δc found by Hombach10 may be affected by the presence of highly dispersed pores in coals. Notice that this approach gives no possibility to influence the Flory-Huggins parameter, albeit the absorption model may be influenced directly by changes in φh°, and indirectly by shaping the function Qa(Rh). The latter is due to interactions of absorbed and adsorbed molecules, which considerably affect absorption capacity. The parameters c and d characterize the sorbent porosity and structure of the elastic phase. The number of subsystems and the size of the corresponding holes must be taken arbitrarily. To set the values for mha°, one may use the total volume of submicropores (or their volume fraction vh°) and assume a distribution function for pore size, thus reducing the number of required parameters. In ref 6, we found that a normal distribution (with the left tail truncated near zero) is adequate to represent the hole size diversity in coal. Thus only two parameters are necessary, i.e. the standard deviation σR and the average radius of the submicropre Rhav. These will be defined relative to the water molecule radius (Rhav/Rw, σR/Rw). We have distinguished 11 sorption subsystems. The first represents the elastic phase of coal (pure absorption), but further ones are for submicropores with a constant increment in size. The fraction of the sorptive phase of coal (used implicitly in the model equations) and average chain length xew, as well as the bulk characterization of pores (i.e., their volume
Hard Coal Surface Heterogeneity
Figure 4. Theoretical sorption isotherms for H2O-coal 32: O, empirical data; 1, sorption; 2, adsorption (including sorption in expanded holes); 3, pure absorption; 4, volumetric expansion related to molar volume of sorbate.
Langmuir, Vol. 13, No. 5, 1997 1291
Figure 6. Theoretical sorption isotherms for H2O-coal 34. Denotation the same as for Figure 4.
Figure 7. Theoretical sorption isotherms for C6H6-coal 34. Denotation the same as for Figure 4. Figure 5. Theoretical sorption isotherms for C6H6-coal 32. Denotation the same as for Figure 4.
fraction vh° and specific surface area ah) can be evaluated by fitting the theoretical dual sorption isotherms for smaller sorbate molecules (water, methanol) to empirical data (see ref 4). In ref 6 we showed that the parameters found in such a way remain roughly adequate when applied to the multiple sorption model, as well as for larger sorbate molecules. In particular, the parameters of the hole size distribution can be adjusted with volume fraction vh° and specific surface area ah of holes being kept near the values found on the basis of dual sorption model. Computations have been carried out for two coals and two sorbatesswater and benzene. Molecules of water and benzene meet relatively well our model assumptions concerning sorbate molecule shape and they differ significantly in size as well as in polarity. Hence they seem to be suitable for examination of sorption mechanisms in coal. Empirical data were obtained by Z˙ yła13 using a manostat at 293 K (see Figures 4-7). For each coal both water and
benzene sorption isotherms were measured on samples taken from the same cob. The physical parameters of the systems are presented in Table 1, together with BET characterization values. The above coals (vitrine 32 and vitrine 34) differ significantly in sorption properties in spite of similar rank (compare benzene sorption isotherms in Figure 5 and Figure 7). Hence, they were selected as good examples for examination of our model capability to explain different aspects of sorption process. Coal structure parameters xew, vh°, Rhav/Rw, σRh/Rw (the same for both sorbates) were evaluated by adjusting their values found on the basis of the dual sorption model.4 The results are gathered in Table 2. Next the parameters shaping the function Qa(Rh), i.e., Rf/Rw, RB/Rw, Cf, ζ0, ζB, ζA, and Cp, were adjusted (separately for each coal-sorbate system) to reach the best fit of the theoretical sorption isotherms to empirical data. In each case Rf ) RB was assumed. For benzene we set Cp ) 1. (13) Z˙ yła M. PhD Thesis, AGH, Krako´w, 1963 (in Polish).
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Milewska-Duda and Duda
Table 1. Physical Parameters for Coal-Sorbate Systems and BET Characteristicsa coal type
%C
δC
sorbate
Va, cm3/mol
Ua/RT
δc2Va/RT
Flory-Huggins parameter
QABET/RT
vhBET
vitrine 32
82.5
25.0
vitrine 34
83.6
24.5
H 2O C6H6 H2O C6H6
18 89 18 89
15.7 12.8 15.7 12.8
4.62 4.44 27.85 21.94
3.19 1.45 3.44 1.22
-3.02 -0.64 -1.08 -1.94
0.028 0.033 0.048 0.046
a Coal type is according to polish classification. The international classification: coal 32 f coal 821 or 822 or 721 or 722 or 621 or 622 or 521 or 522; coal 34 f coal 733 or 633 or 634 or 635. QABET, molar energy of adsorption evaluated by BET method based on data shown in Figures 4-7 (to each adsorption site the volume of sorbate molecule is attributed).
Table 2. Structural Parameters of Examined Coals coal
submicropores
type
%C
xew
vh°
Rhav/Rw
σh/Rw
ah
32 34
82.5 83.6
15 15
0.04 0.10
0.62 0.81
0.25 0.24
1.26 1.07
Table 3. Parameters Evaluated by Fitting the Models to Empirical Data coal type sorbate modela QA/RT RB/Rw Cf 32
H2O
C6H6
34
H2O
C6H6
a b c d e a b c d e a b c d e a b c d e
-3.27 -3.43 -3.43 -3.27 -2.83 -0.33 -0.33 -0.33 -0.66 -3.10 -1.28 -1.28 -0.98 -0.53 -1.69 -1.98 -1.98 -3.79 -7.11 -6.21
0.60 0.60 0.60 0.75 0.63 1.06 1.06 1.09 1.06 1.06 0.88 0.90 0.88 0.95 0.78 1.31 1.31 1.31 1.31 1.31
1.3 1.0 1.3 1.3 1.3 1.1 1.0 1.1 1.0 1.2 1.5 1.0 1.5 1.7 1.5 2.2 1.0 2.2 2.2 2.1
ζ0
ζB
ζA
Cp‚ζA
0.70 0.00 0.70 0.70 0.70 0.40 0.00 0.40 0.00 0.44 0.70 0.00 0.70 0.50 0.40 0.69 0.00 0.69 0.46 0.70
0.94 0.92 0.93 0.96 0.90 0.74 0.74 0.74 0.72 0.76 0.86 0.86 0.81 0.70 0.87 0.79 0.79 0.76 0.73 0.77
0.82 1.00 0.82 0.81 0.82 0.40 0.40 0.40 0.41 0.50 0.79 1.00 0.78 0.76 0.79 0.53 0.53 0.55 0.66 0.61
1.17 1.17 1.17 1.16 1.17 0.40 0.40 0.40 0.41 0.50 1.13 1.13 1.11 1.08 1.12 0.53 0.53 0.55 0.66 0.61
a Model symbols: (a) Energetic parameter Q defined by the a formula 10. (b) Energetic parameter Qa defined by the formula 11 (simplified model: Cf ) 1; ζ0 ) 0; ZB ) ζB; ZA ) CpζA). (c) Model the same as in (a) and inaccessible holes removed: for coal 32, Rh/Rw > 0.53 (ah ) 1.23, vh° ) 0.039); for coal 34, Rh/Rw > 0.77 (ah ) 1.00, vh° ) 0.088). (d) Model the same as in (a) and hole size range limited: for coal 32, Rh/Rw > 0.67 (ah ) 1.11); for coal 34, Rh/Rw > 0.87 (ah ) 0.94). (e) Model the same as in (a) and volume fraction of submicropores vh° ) 0.07. b QA, energetic parameter Qa for pure adsorption (eq 10).
The fitting has been performed first for the model involving eq 10 (model a). The resultant values for RB/ Rw, Cf, ζ0, ζB, ζA and ξA) ζA/Cp are listed in Table 3. The best fit sorption isotherms are depicted in Figures 4-7 as consisting of absorption (curve 3) and adsorption (curve 2) isotherms. The latter includes sorption capacities of all subsystems except for that of pure absorption. The volumetric expansion (relative to the volume of the sorbate molecule) is also shown (curve 4). The difference between this curve and the absorption isotherm (3) is due to expansion of holes and so it shows the contribution of absorptive-adsorptive subprocesses to the sorption capacity. Quantitative characterization of the above components is given in Table 4 (model a) together with relative mean square fitting errors. The model does not explain the kinks observed for coal 34 near p/po ) 0.6 (Figures 6 and 7). This is probably due to a more complex pore size distribution (perhaps polymodal) than assumed in our study. Also for the watercoal 32 system, larger deviations are observed in the higher pressure range (Figure 4), which may be due to multilayer
adsorption in larger pores. Nevertheless, the fitting quality seems to be quite satisfactory and the values for the model parameters are within intuitive physical bounds. The above results allow us to conclude that the proposed model may be used to analyze sorption properties of coals with theoretical values of energetic parameters being assumed. They show that the observed significant differences in sorption capacity of individual coals can be explained as caused by the diversity of geometric properties of submicropores. During the fitting procedure, we noticed that a proper setting of the values for Rhav/Rw, σRh/Rw, RB/Rw, vh°, and ζB is of primary importance (see also ref 6). In fact, they determine the number and properties of the most attractive sites for adsorption. It was stated that adsorption in very small pores, even if relatively low, results in an unacceptable shape for the total isotherms and yields a sorption capacity much higher than empirical. Hence, the factors ζ0 and Cf that determine the sorption energy in smaller holes must ensure that the energy is high enough to make the smallest holes practically inaccessible for sorbate molecules. If this condition is met, the actual values for ζ0 and Cf are of less significance, and practically the same result can be obtained using the simplified formula (11) for Qa (ζ0 ) 0; Cf ) 1; Cp ) 1ssee formula 12 and ξA equal to 1). The results of a fit to such a model (referred to as model b) are shown in Tables 3 and 4. The fitting error compared to that of model a is acceptable (the corresponding total sorption isotherms are almost identical); hence we may conclude that the simplified model involving formulas 11 and 12 is fully adequate also for polar sorbates. In order to get more information on the role of the smallest pores in the sorption process, we also examined the model assuming the coal does not contain pores inaccessible to both sorbates. First, only the inaccessible pores were removed (Rh/Rw < 0.53 for coal 32 and Rh/Rw < 0.77 for coal 34) with the remaining ones being kept (model c). In effect, the specific surface area and volume fraction of holes were reduced (compare parameters ah and vh° in Table 2 and Table 3). Next the hole size distribution was changed radically by taking a lower bound for Rh: Rh/Rw > 0.67 for coal 32 and Rh/Rw > 0.87 for coal 34, with the volume fraction of holes being held fixed (model d). Notice that the bounds are higher than the average values used for model a (see Table 2) and they reduce significantly the specific surface area of the coals (see the explanation below Table 3). The parameter settings giving the best fit to models c and d are listed in Table 3, and the corresponding theoretical isotherms are characterized in Table 4. As seen in Table 4, both models a and c are acceptable. It means that free surface, which in fact influences the sorption capacity, may be due to the presence of small inaccessible holes or to a poorer fit of the shape of the larger pores to the sorbate molecule (compare parameter ζB for the models a and c in Table 3). Model d, which distorts the structure of the accessible pores, is noticeably worse. This suggests that the size
Hard Coal Surface Heterogeneity
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Table 4. Characterization of Theoretical Sorption Isotherms coal type
sorbate
modela
% contribution of absorption at p/po ) 0.8
vhads b
% linear expansion at p/po ) 0.8
fitting error related to mp (0.8)
32
H2O
a b c d e a b c d e a b c d e a b c d e
18.0 21.0 18.7 20.6 12.3 73.1 73.0 75.9 76.4 56.8 10.1 13.3 12.1 10.9 13.5 48.0 47.4 56.2 53.1 60.6
0.055 0.053 0.055 0.053 0.059 0.018 0.018 0.016 0.016 0.029 0.048 0.046 0.047 0.048 0.046 0.043 0.044 0.036 0.039 0.033
1.12 1.24 1.09 0.94 0.77 2.00 2.00 2.00 2.03 1.85 0.24 0.27 0.26 0.20 0.36 2.17 2.10 2.25 2.17 2.39
0.029 0.023 0.031 0.037 0.037 0.011 0.011 0.016 0.009 0.025 0.015 0.019 0.013 0.032 0.017 0.023 0.030 0.024 0.029 0.041
C6H6
34
H2O
C6H6
a
Model symbols the same as those given in Table 3. b vhads, volume fraction of adsorbed molecules at p/po ) 0.8.
and energy of the most attractive adsorption sites are relatively well detectable. Similar conclusions may be drawn from examination of model e, which assumes the same volume fraction of holes for both coals (vh° ) 0.07) (increased for coal 32 and decreased for coal 34). To fit this model to empirical data, changes in the amount of adsorptive holes are compensated by suitable changes in their energy (see Table 3). This also changes the absorption isotherms, so the fitting is rather poor (see Table 4). The multiple sorption model shows that the adsorption process (filling of submicropores) proceeds mainly in submicropores of relatively narrow size range. The properties of such a fraction of submicropores play a dominant role in affecting both the adsorption and absorption capacity. In order to reach a theoretical sorption capacity close to empirical data, it is necessary to assume that a fraction of the coal surface does not contact sorbate molecules. The results in Tables 3 and 4 give insight into the uncertainty of the parameters describing the energetic heterogeneity of the coal surface and the possible dispersion of the linear expansion predicted by the well-fitted models. As seen in Table 3, the parameters corresponding to models a, b, and c for the same coal are fairly close. The resultant values for the quantities QA and vhads may be compared to those obtained from the BET method (Table 1). They reveal that the BET characterization is inadequate from a physical viewpoint. In fact, the linearized BET equation fits empirical data shown in Figures 4-7 only for p/po < 0.2, and the parameters QABET and vhBET are very sensitive to changes in the pressure interval taken for fitting. The degree of filling of submicropores (see values for vh° and for vhads) is different for different systems. In particular, vhads may exceed vh° due to expansion of holes. The molar energy of sorption computed according to eq 14 for models a-e, is plotted against p/po in Figures 8-11. The energy of free surface expansion for model a is also shown to give insight into the contribution of this phenomenon. The sorption energy is significantly higher than that predicted by the BET method. Notice that, for a given sorption system, the curves corresponding to different models are close if the corresponding sorption isotherms are well fitted to empirical data. It means, that the multiple sorption model enables us to evaluate the sorption energy reliably, in spite of the uncertain
Figure 8. Molar sorption energy for H2O-coal 32 corresponding to the models (a, b, c, d, e): - - -, molar energy of free surface expansion (model a).
description of coal surface properties, provided that it yields sorption isotherms close to empirical data. In order to illustrate the sensitivity of the sorption isotherms to changes in coal properties (represented in the multiple sorption model), we have computed theoretical sorption isotherms based on settings found for model a with different modifications. The results obtained for coal 34 are presented in Figures 12 and 13. Curves 1 and 6 show that changes in the structure of adsorptive holes essentially affect the sorption capacity. In particular, the effect of the number of adsorption sites is important because of the adsorption-absorption interactions (a decrease in the volume fraction of the submicropores reduces the sorption of water but increases the sorption of benzenessee curve 6 in Figures 12 and 13). The curves 2-5 illustrate the sensitivity of sorption isotherms to changes in the parameters RB/Rw, ζ0, ζB, and ζA that determine the energy of the adsorption sites. These parameters describe the heterogeneity of the coal elastic phase itself. In order to expose their role in the multiple sorption model, we have computed the sorption isotherms assuming that RB/Rw ) 0, Cf ) 1, ζ0 ) 1, ζB ) 1, and ζA
1294 Langmuir, Vol. 13, No. 5, 1997
Milewska-Duda and Duda
Figure 9. Molar sorption energy for C6H6-coal 32 corresponding to the models (a, b, c, d, e): - - -, molar energy of free surface expansion (model a).
Figure 11. Molar sorption energy for C6H6-coal 34 corresponding to the models (a, b, c, d, e): - - -, molar energy of free surface expansion (model a).
Figure 10. Molar sorption energy for H2O-coal 34 corresponding to the models (a, b, c, d, e): - - -, molar energy of free surface expansion (model a).
Figure 12. Theoretical sorption isotherms for H2O-coal 34. The parameters are the same as for model (a) with the following modifications: 1, Rh/Rw > 0.87 (like in the model d); 2, RB/Rw increased by 0.02 (RB/Rw ) 0.9); 3, ζ0 increased by 0.1 (ζ0 ) 0.8); 4, ζB increased by 0.02 (ζB ) 0.88); 5, ζA increased by 0.02 (ζA ) 0.81); 6, vh° decreased by 0.03 (vh° ) 0.07 like in model e); 7, δAh ) 0 (expansion of free surface neglected); 8, with no correction (ζ0 ) 1; ζB ) 1; ζA ) 1; RB ) 0); 9, with no correction and Rh/Rw > 0.87; O, empirical data.
) 1 (disregarding any peculiarities of the surface of the submicropores). The results are shown in curves 8 and 9 and give evidence that differences in sorption energies in holes of different size cannot be explained simply by the different expansion ratios required for adsorption. Such a model predicts a sorption capacity much higher than empirical, and the shape of the isotherms does not agree with that observed for coals. Proper correction is of vital importance not only for smaller holes but also for larger ones (compare the curves 8 and 9). Let us recall the problem of coal swelling raised at the beginning of this section. Its formal description is based on formula 18, where regular expansion is assumed. The swelling of coal is certainly much more complicated. One observes anisotropic expansion together with contraction1 of coal samples. Nevertheless, it is likely to be due to changes in the volume of larger pores which have no effect
on sorption isotherms. It can be shown theoretically14 that nonmonotonic expansion of submicropores may imply a shape of sorption isotherms which is not observed in practice. Hence, formula 18 seems to be the only reasonable description of this complicated phenomenon. Eventually the value taken for the constant factor (now equal to 2/3) is up for discussion. The simplest test is to assume δAh ) 0 in eqs 18 and 20, thus neglecting the expansion effect. The resultant isotherms are depicted in Figures 12 and 13 as curve 7. They show that the effect of free surface changes on sorption capacity can be very important but depends strongly on molar volume of sorbatessee eq (14) Milewska-Duda, J. Appl. Chem. 1988, 32, 71 (in Polish).
Hard Coal Surface Heterogeneity
Figure 13. Theoretical sorption izotherms for C6H6-coal 34. The parameters are the same as for model a with the following modifications: 1, Rh/Rw > 0.87 (like in the model d); 2, RB/Rw increased by 0.034 RB/Rw ) 1.344); 3, ζ0 increased by 0.1 (ζ0 ) 0.79); 4, ζB increased by 0.02 (ζB ) 0.81); 5, ζA increased by 0.02 (ζA ) 0.55); 6, vh° decreased by 0.03 (vh° ) 0.07 like in model e); 7, δAh ) 0 (expansion of free surface neglected); 8, with no correction (ζ0 ) 1; ζB ) 1; ζA ) 1; RB ) 0); 9, with no correction and Rh/Rw > 0.87 O, empirical data.
20. This phenomenon is negligible for water, while for benzene it plays a crucial role. At p/po ) 0.8, curve 7 in Figure 7 reaches 4.1 mmol/g, where the real sorption is 0.7 mmol/g. For the benzene-coal 32 system the corresponding values are 1.8 and 0.55 mmol/g, respectively. Bearing in mind the possible uncertainty of formula 18, one may expect that the multiple sorption model is more reliable for sorbates of smaller molecules so that such substances should be used as probing sorbates in examinations of coal surface properties. On the other hand, the above result suggests that the amount of gases deposed in natural coal beds may be much higher than the sorption capacity observed in laboratories at the same pressure. Relaxation of coal matter just after mining may lead to enlarging of its internal surface (creation of free surface) and so to explosive desorption of gases contained in them. This is observed in coal mines as squealer phenomena. The multiple sorption model may be used to evaluate the scale of such a danger by simulation of the sorption capacity of particular gases with coal expansion properly reduced. Summary and Conclusions Sorption in coal consists in the placement of sorbate molecules among polymer chains of the coal elastic phase (absorption) and the filling of pores dispersed in coal matter (adsorption). The sorption capacity is strongly affected by the structure and the quantity of highly dispersed pores of molecular size (submicropores) referred to as holes. Larger pores are of less importance. The energetic heterogeneity of the coal surface is due to differences in size of particular submicropores, the irregularity of their shape, and the differences in local cohesion energy of the coal elastic phase. For polar sorbates, the presence of active groups on the surfaces of the larger pores is also of importance in significantly decreasing the local sorption energy. The model proposed in this paper expresses the local energy of sorption as a function of hole size (eqs 10 and
Langmuir, Vol. 13, No. 5, 1997 1295
11) and enables us to analyze the sorption process as consisting of a number of interactive subprocesses, each attributed to holes of assigned original size. Absorption is viewed as creation and filling of new holes (for which the original size is equal to zero). Such a description of sorption energy (eqs 14-18), together with a proper expression for the sorption system entropy (eq 19), leads to the multiple sorption model (eq 20). This involves a number of parameters that determine the hole size distribution and energy of particular sorption sites. The model, with the parameters being set arbitrarily, makes it possible to compute theoretical sorption isotherms. By fitting them to empirical data one may find the most appropriate settings for the parameters. On the basis of an examination of the model properties we have stated the following: (a) In formulas 10 and 11 that define the energetic parameter Qa, one may use the theoretical values for the cohesion energy of coal and sorbate (from the solubility parameter of coal determined according to van Krevelen curve and the evaporation energy for sorbate) to determine the Flory-Huggins parameter. (b) The energetic parameter Qa for adsorption must be properly chosen with respect to the energetic heterogeneity of the coal surface. This is done by a correction of cohesion and adhesion energies, which can be expressed as a function of hole size. (c) Values for parameters for the energy of smaller holes should imply the inaccessibility of the smallest pores. (d) The dispersion of the remaining parameters evaluated by this fitting procedure is relatively small. (e) The correction factors for adhesion and cohesion energies (eqs 6 and 9) may be replaced by the simplified factor Z(Rh,Ra) (eq 12) which corrects the adhesion energy only. (f) The model gives a reliable evaluation of the total sorption energy, provided that it yields an sorption isotherm close to empirical data. The multiple sorption model has a clear physical meaning; hence it can be used as a theoretical tool for examination of sorption mechanisms in coal. In particular, based on the computation results discussed in the paper, the following conclusions on sorption properties of coals may be drawn: 1. Significant differences in sorption capacity of individual coals (as observed in practice) can be explained as caused by the diversity of geometric properties of submicropores. 2. The adsorption process (filling of submicropores) proceeds mainly in submicropores of relatively narrow size range, being close to that of a water molecule. The properties of this fraction of the submicropores play a dominant role in affecting both the adsorption and absorption capacity. Hence they may be evaluated relatively well using the proposed model. 3. Very small pores (if they do occur) do not contribute to the sorption capacity (they remain empty within the entire pressure range). Because of this fact, properties of the smallest pores (size distribution, shape characteristics) cannot be reliably determined from sorption data. 4. Changes in free surface of coal due to swelling of coal matter may have a significant effect on the sorption process, but this depends strongly on the molar volume of sorbate. For water the effect is negligible, while for benzene it is of crucial importance (surface expansion causes a radical decrease in sorption capacity). This may explain the squealer phenomenon observed in coal mines.
1296 Langmuir, Vol. 13, No. 5, 1997
Nomenclature Ah, total surface area of pores relative to that of the sorbate molecule (in moles) ah, specific surface area of coal relative to that of a water molecule a, b, indices used to count sorption subsystems attributed to holes of different size Cf, a constant in eq 6 (Cf > 1) Cp, a parameter expressing the effect of specific interactions (Cp ) 1 for nonpolar sorbates and Cp > 1 for polar sorbate) Eab, energetic constant defined by eq 15 ea, eb, ec, molar energy attributed to stages (a, b, c), respectively (see eq 1) f(Rh), correction factor for cohesion energy (eq 6) mp, total sorption capacity mpa, amount of sorbate in the ath subsystem of a sorption system (in moles) N, Avogadro number n, the number of system cells (in moles) Qa, molar energy of sorbate molecule in a considered site (hole) R, gas constant Ra, effective radius of sorbate molecule Rh, effective radius of a submicropore (hole) Rhav, average radius of submicropores (parameter of hole size distribution function) RB, a characteristic hole size where ζ reaches its characteristic value ζB Rf, a characteristic size of hole such that for larger holes the cohesion energy may be taken the same as the average value for the elastic phase (Uh ) δc2) (eq 6) Rw, radius of water molecule xe, average length of cross-linked chains related to volume of sorbate molecule Ua, energetic parameter for sorbate (molar evaporation energy) Uh, cohesion energy of coal s , an energetic parameter Uha
Milewska-Duda and Duda Va, molar volume of sorbate vc, volume fraction of sorbent in the system vh° volume fraction of holes in dry coal vha volume fraction (in bulk volume of the sorption system) of original size holes attributed to ath subsystem vpa volume fraction of sorbate molecules sorbed in the ath subsystem ve volume fraction of elastic chains wa volume expansion ratio of holes attributed to ath subsystem xew, average volume of elastic chain in coal related to that of water molecule Z(Rh,Ra), simplified correction factor for adhesion energy (eq 12) ZA, a characteristic value of Z for pure adsorption (eq 12) ZB, a characteristic value of Z for holes of RB size (eq 12) δc, solubility parameter of coal determined according to the van Krevelen curve δAh, δφh, changes in surface area Ah of pores and in its fraction φh due to sorption σh, standard deviation of hole size distribution function ζ(Rh,Ra), correction factor for adhesion energy (eq 9) ζA, a characteristic value of ζ for pure adsorption (eq 9) ζB, a characteristic value of ζ for holes of size RB (eq 9) ζ0, a value of ζ for infinitesimally small holes (eq 9) ξa, fraction of effective hole-sorbate contact surface area in ath subsystem ηab, a parameter representing the effectiveness of sorbatesorbate energetic interactions φpb, surface fraction of sorbate molecules placed in ath subsystem holes φh, surface fraction of holes in the sorption system φh°, surface fraction of holes in dry coal ωa, surface expansion ratio of holes attributed to ath subsystem
Acknowledgment. The work was supported by KBN (Warsaw, Poland), Project No. 2P 303 010 07. LA951049Z
