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High pressure multicomponent adsorption in porous media
Fluid Phase Equilibria 158–160 Ž1999. 565–573
High pressure multicomponent adsorption in porous media Alexander A. Shapiro ) , Erling H. Stenby Engineering Research Center IVC-SEP, Department of Chemical Engineering, Technical UniÕersity of Denmark, Building 229, DK 2800, Lyngby, Denmark Received 13 April 1998; accepted 4 January 1999
Abstract We analyze adsorption of a multicomponent mixture at high pressures on the basis of the potential theory of adsorption. The adsorbate is considered as a segregated mixture in the external field produced by a solid adsorbent. We derive an analytical equation for the thickness of a multicomponent film close to a dew point. This equation Žasymptotic adsorption equation, AAE. is a first order approximation with regard to the distance from a phase envelope. It can be applied to both normal and retrograde condensation. Different forms of the AAE make it possible to analyse phenomena occurring close to a dew point. For testing of the AAE, we compare thicknesses of the adsorbed film predicted by this equation and obtained by direct calculations. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Adsorption; Potential theory; Model; Method of calculation
1. Introduction The potential theory of adsorption, first formulated by Polanyi, is a thermodynamic concept used for modeling of gas adsorption on solid. Its applications are multilayer adsorption on homogeneous and heterogeneous surfaces, adsorption from mixtures close to their dew points and adsorption in microporous media. In the latter case a modification of the potential theory formulated by Dubinin is widely used. A specific feature of the potential theory is that the mixture is treated in a purely thermodynamic way, without detailed consideration of molecular behavior. This makes it possible to utilize the knowledge accumulated in the bulk phase thermodynamics and to automatically extend the theory onto non-trivially behaving multicomponent mixtures like hydrocarbon mixtures exhibiting retrograde behavior. On the other hand, the purely thermodynamic treatment is the origin for the limitations of the theory. It becomes inefficient in cases where specific molecular-level phenomena )
Corresponding author
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 1 4 4 - 2
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
566
have a dominant contribution into the formation of adsorbed layers. In such cases, more modern and rigorous theories, such as the density fluctuation theory, should be applied. In a modification of the potential theory developed here, we do not make a common assumption about adsorbate as a homogeneous phase, but it is considered as a segregated mixture in the external field produced by a solid adsorbent. Such a model is able to correlate experimental data of multicomponent adsorption in porous media, if only parameters of the surface potentials for the pure components are fitted, but not the parameters in an equation of state for the segregated mixture w1x. However, a generalization of the potential theory onto multicomponent mixtures encounters computational difficulties. To overcome them, we derive an analytical equation for the thickness of a multicomponent film close to a dew point. This asymptotic adsorption equation, AAE, is a first order approximation with regard to the distance from a phase envelope. It can be applied to both normal and retrograde condensation. Different forms of the AAE make it possible to analyze phenomena occurring close to a dew point. This analysis includes the phase transition from a continuous to a discontinuous form of the adsorbate and a comparison of the action of capillary forces and adsorption in porous media. For testing of the AAE, we compare thicknesses of the adsorbed film predicted by this equation and obtained by direct calculations. The comparison shows remarkably good agreement between the two theories, especially, in the region of normal condensation, where the AAE provides good estimates for the relative pressures as low as 0.1.
2. Basic equations of the potential theory Let us briefly review the fundamentals of the potential adsorption theory, following Ref. w1x. Adsorption is represented as a segregation of the mixture in the potential field emitted by the surface of the adsorbent. Each Ž ith. component of the mixture is affected by its own adsorption potential e i Ž x . depending on the distance x to the surface. An equilibrium state in the potential field is described by the system of equations for the chemical potentials m i :
m i Ž P Ž x . , z j Ž x . . y e i Ž x . s mgi ž Pg , z gj / .
Ž1.
z gi
Ž I s 1,n. and pressure Pg in the gas phase, the distributions of pressure and Given composition molar fractions at the surface are uniquely determined by the system Ž 1. . Then the surface excess G i of the ith component is found as the difference between the actual amount of this component and its amount in the absence of the adsorption forces. The molar fraction z ai of the ith component in the adsorbate is the ratio of G i to the total surface excess: n
` i
G s
H0
i
z Ž x . nŽ x .
y z gi n g
dx ,
Gs
Ý G j,
z ai s G irG .
Ž2.
js1
Close to the surface the potentials may become quite strong, and the effective pressure in the adsorbate may reach several thousand atmospheres w2x. This often leads to condensation and formation of the liquid-like film on the surface. The thickness h of this film is determined from Eq. Ž1. together with the conditions of phase equilibrium:
mgi ž P Ž h . , z gj Ž h . / s m il Ž P Ž h . , z lj Ž h . . s mdi .
Ž3.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
567
Here we use the subscripts g and l to denote the values for gas and liquid mixtures. The subscript d Ž‘dew point’. is used to denote the values at phase transition, P Ž h. s Pd . The potential theory has been applied to adsorption on homogeneous surfaces, as well as in the ‘slab’ theory of multilayer adsorption on heterogeneous surfaces. Several dependencies for the adsorption potentials e i Ž x . have been suggested Ž see the analysis in Refs. w3–5x.. One of the commonest potentials is expressed by the Frenkel–Halsey–Hill Ž FHH. dependence w6,7x. A general form of this dependence is w5x
e i Ž x . s e 0i a i
al
al
rŽ b i q x . ,
e i Ž 0 . s e 0i .
Ž4.
Another application of the potential theory is adsorption in micropores. We do not consider it here, the discussion can be found elsewhere w1,3x. The system of segregation equations ŽEq. Ž1.. is not easy to solve, especially, when combined with the conditions of phase equilibria Ž Eq. Ž 3.. . The calculations become straightforward only for the case of a single-component adsorption. To overcome these difficulties, we will derive an asymptotic adsorption equation ŽAAE..
3. Derivation of the AAE The AAE may be derived on the basis of geometrical considerations. Fig. 1a represents the adsorption path and the relevant points on the P–z phase diagram, Fig. 1b in the space of the chemical potentials of the mixture. Point g on Fig. 1 corresponds to the bulk pressure Pg and composition z g s Ž z g1 , . . . , z gn .. Point D is the dew point for the composition z g ; the dew-point pressure is PD . In Fig. 1a, this point belongs to the gas branch of the phase envelope. The corresponding point on the liquid branch is L s Ž PD , z L .. In the space of chemical potentials we have L s D ŽFig. 1b.. The adsorption curve M s Ž P Ž x ., z Ž x .. determined by Eq. Ž1. starts from the point g corresponding to x s `. It intersects the phase envelope at the point d s Ž Pd , z d . and continues from l s Ž Pd , z l . from the liquid branch. In the space of chemical potentials, d s l. For a point g, which is close to the critical point, the adsorption curve can go around the phase envelope, and a smooth transition from gas to liquid is observed Žcurve MX in Fig. 1a. .
Fig. 1. Graphical representation of the adsorption curve: Ža. on the pressure-composition diagram, Žb. in the space of chemical potentials.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
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The thickness h of the adsorbed film is determined as a value of x at which M meets the phase envelope. By conventional flash calculations, we can find the dew-point pressure PD and the equilibrium liquid composition z L , which are supposed to be known. In the asymptotics considered, all the distances between the points g, d, D are assumed to be small, and we neglect the second order terms with regard to them. Let us find the vector n D orthogonal to the hypersurface of the phase transition in the space of the chemical potentials. The Gibbs–Duhem relations at constant temperature, on the ‘gas’ and on the ‘liquid’ sides of the phase transition, may be expressed as follows: n
Ý is1
i z gD
VgD
n i d mgD sd P,
Ý is1
z Li VL
d m iL s d P .
Ž5.
i Subtracting them and noticing that mgD s m iL s m iD we find:
n
Ý is1
ž
i z gD
y
VgD
z Li VL
/
d m iD s 0.
Ž6.
The last equation is interpreted as a definition for the tangent plane d m to the hypersurface of the phase transition, in the standard form of Ž n D P d m . s 0, where nD s
ž
1 z gD
y
VgD
z L1 VL
,...,
n z gD
y
VgD
z Ln VL
/
.
Ž7.
The point d, which is sought for, may be found from the vector equality dgsDgyDd We multiply this equality by n D and note that the scalar product Ž n D P Dd. is of the second order with regard to a characteristic linear size of the triangle gDd. This can be found geometrically from Fig. 1b Žconsidering the decline of the vector connecting the two points on the surface from the tangent plane.. Within the second order values, we obtain:
Ž n Ddg . s Ž n DDg. . Let us transform the last geometrical equation to the form of a thermodynamic relation. It follows from Eq. Ž 1. that the coordinates of the vector gd are ye i Ž h. . Substituting the coordinates of n D from Eq. Ž7., we obtain n
Ý is1
ž
i z gD
VgD
y
z Li VL
n
/
i
e Ž h. s
Ý is1
ž
i z gD
VgD
y
z Li VL
/ž
m iD y mgi / .
Ž8.
The last equation will be referred to as a general or a potential form of the asymptotic adsorption equation ŽAAE. for the value of h. To use it, conventional thermodynamic calculations to determine the dew point pressure PD for a given composition z g are first performed. In these calculations, the composition z L is found. Then the liquid volume V L at the pressure PD is evaluated. Finally, Eq. Ž8. is solved as a transcendent equation for h. Due to approximations in its derivation, Eq. Ž 8. is valid asymptotically, close to a dew point of the mixture. A degree of proximity to the dew point at which the equation gives a reasonable
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
569
approximation will be studied numerically later on. However, it is clear beforehand that the accuracy of the AAE is high if the phase transition surface in the chemical potential space is close to planar, at least, in a vicinity of the points D and d. On the other hand, Eq. Ž 8. loses its validity close to the critical point.
4. Different forms of the AAE The expression in the right hand side of the AAE Ž Eq. Ž 8.. may be substituted by different asymptotically equivalent expressions, depending on a purpose of the study. This is performed, first, by expanding m iD y mgi around the point D: i m iD y mgi s yVgD D P y Ez gD m iD D z g ,
D P s Pg y P D ,
D z g s z g I z gD .
Ž9.
Further, since D belongs to the boundary of the phase transition, for the corresponding infinitesimal changes D z g , D z L we have Ez gD m iD D z g s Ez L m iD D z L .
Ž 10.
Also, we will use the Gibbs–Duhem relations in the form of n
n
i Ez Ý zgD
m iD s gD
is1
Ý z Li Ez
L
m iD s 0
Ž 11 .
is1
Substituting Eq. Ž 9. to Eq. Ž11. into the right hand side of Eq. Ž8., we reduce it to the form of i z gD
n
Ý is1
ž
y
VgD
z Li VL
/
e i Ž h. s
ž
VgL VL
/
y 1 D P s Pd
ž
VgL VL
/
y 1 Ž x y 1. .
Ž 12.
Here we introduce the relative pressure x s PgrPd and the thermodynamic parameter n
VgL s
Ý z Li VgDi is1
which will be called the mixed Õolume. This parameter was introduced and discussed previously in connection with the problem of multicomponent capillary condensation w8x. When adsorption in meso- and macroporous media close to a dew point is studied, one of the problems is to distinguish it from the capillary condensation. In w8x it was shown that capillary pressure under multicomponent capillary condensation obeys an asymptotic equation called the modified Kelvin equation Ž MKE.: Pc s P D
ž
VgL VL
/
ln x y x q 1 .
Ž 13.
The right-hand side of the MKE is equal within second-order terms to the right-hand side of the linear AAE ŽEq. Ž12... Hence, the general AAE may be reduced to the form of n
Ý is1
ž
i z gD
VgD
y
z Li VL
/
e i Ž h . s PD
ž
VgL VL
/
ln x y x q 1 s Pc .
Ž 14.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
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Use of Eqs. Ž13. and Ž14. makes it possible to study simultaneously the effects of capillary forces and adsorption in macroporous media. For simplicity, let us assume that all the pores are cylindrical and all the surface potentials are proportional: e i Ž x . s d ie Ž x ., which is valid, for example, for the FHH potentials Ž Eq. Ž 4.. with different e 0i but the same a and a . The Kelvin radius rc of a capillary which may be filled at given thermodynamic conditions is found on the basis of Eq. Ž13. and of the Laplace equation. Comparison of Eqs. Ž 13. and Ž14. leads to a simple relation between rc and the thickness h of the adsorbed film n
rc s gre Ž h . ,
g s y2 sr Ý d i is1
ž
i z gD
VgD
y
z Li VL
/
where s is the surface tension.
5. Numerical testing of the AAE We analyzed surface adsorption of methane Ž 1. –n-butane Ž2. at 277.6 K. Thermodynamic calculations were performed on the basis of the SRK EoS with Peneloux volume corrections, and on ` the basis of the modified FHH potentials Ž Eq. Ž 4.. . The values of e 0i for these potentials were taken from the original Halsey equation: e 0i s 5RT. Both a i and b i are of the order of the effective thickness of a monolayer Žsee discussion in Refs. w5,9x for details. . These thicknesses were reported for a few substances Ž for nitrogen, if a cubic packing is assumed, it is 0.4 nm. . As a basic variant, the adsorption potentials were chosen with a1 s b 1 s 0.4 nm and a 1 s 3 for methane, and with a1 s 0.4 nm, b 1 s 0.3 nm and a 2 s 2.5 for butane. Fig. 2 represents the dependence of the thickness h on the relative pressure, for the mixtures corresponding to Pd s 10 bar Ž86.2% of methane. and 20 bar Ž 92% of methane. . These dew point pressures lie in the region of normal condensation, x - 1. The calculations based on the exact equations ŽEqs. Ž1. and Ž 3.. were compared with predictions of the general AAE Ž Eq. Ž 8.. . A
Fig. 2. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4. 1— P D s 20 bar, 2— P D s10 bar.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
571
Fig. 3. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4 at P D s 20 bar. 1—exact calculationss general AAE, 2—‘capillary’ AAE, 3—linear AAE.
remarkably good agreement of the exact and approximate values was established. In the whole region PD - 20 bar, and for the values of x up to 0.1 Žwhich is by no means ‘asymptotic’., the error did not exceed 1%. The error is lower at higher relative pressures and at lower dew-point pressures Žso that it is only 0.4% at x s 0.1 and P D s 10 bar.. Note that h depends much stronger on the relative pressures than on the dew-point ones. Comparison of the different forms of the AAE at PD s 20 bar is shown on Fig. 3. Both the ‘capillary’ Ž Eq. Ž14.. and the linear Ž Eq. Ž 12.. forms overestimate the values of h. The ‘capillary’ form is in a relatively good agreement with the exact calculations, although the general form Ž Eq. Ž 8.. is much superior. The relative errors for the AAE Ž Eq. Ž 14.. are, respectively, 0.7% at x s 0.9, 2.6% at x s 0.75 and 6% at x s 0.5. The linear form of the AAE, however, is far from being precise. The relative error is 3.6% already at x s 0.9. Figs. 4 and 5 represent similar results for the retrograde condensation, for the dew-point pressures of 80 and 100 bar. The general form of the AAE is again superior in this region. However, the relative
Fig. 4. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4. 1— P D s80 bar, 2— P D s100 bar.
572
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
Fig. 5. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4 at P D s80 bar. 1—exact calculationss general AAE, 2—‘capillary’ AAE, 3—linear AAE.
errors are much higher. At x s 1.2, the errors of the general AAE vary from 1.7% Ž PD s 100 bar. to 2.4% Ž PD s 80 bar. . Other forms of the AAE are closer to the exact dependence in this region than in the region of normal condensation. At P D s 100 bar, disappearance of the phase transition is observed at x ) 1.21. At higher relative pressures the adsorption curve goes around the phase envelope, as the curve MX on Fig. 1. However, all the forms of the AAE predict a non-zero thickness of the adsorbed layer for these relative pressures. Thus, either form of the AAE must be used with care in the near-critical region. In conclusion, on the basis of comparative calculations, the general form of the AAE is recommended for the calculation of the thickness of the adsorbed film. The ‘capillary’ form, although less precise, can be used for comparison of the action of capillary forces and adsorption. The linear form can be applied if the mixture is very close to its dew point. List of symbols a, b, a d h n n P r V x z g G e
parameters in the surface potentials proportionality constant in surface potentials thickness of the adsorbed layer molar density normal Žto the surface of phase transition. pressure radius of a capillary molar volume distance to the surface molar fraction surface parameter surface excess surface potential
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
m s x
chemical potential surface tension relative pressure
Subscripts a c d, D g l, L
adsorbed capillary dew point gas liquid
Superscripts i, j
number of a component
573
References w1x w2x w3x w4x w5x w6x w7x w8x w9x
A.A. Shapiro, E.H. Stenby, J. Colloid Interface Sci. 201 Ž1998. 146–157. A.V. Neimark, J. Colloid Interface Sci. 165 Ž1994. 91–96. W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, AP Academic Press, London, 1992. B. Rangarajan, C.T. Lira, R. Subramanian, AIChE J. 41 Ž1995. 838–845. A.W. Adamson, Physical Chemistry of Surfaces 3rd edn., Wiley-Interscience, New York, 1976. T.L. Hill, Advances in Catalysis, Vol. IV, in: W.G. Frankenburg, E.K. Rideal, V.I. Komarewsky ŽEds.., Academic Press, New York, 1952. J. Frenkel, Kinetic Theory of Liquids, The Clarendon Press, Oxford, 1946. A.A. Shapiro, E.H. Stenby, J. Fluid Phase Equilibria 134 Ž1997. 87–101. T. Allen, Particle Size Measurement, Chapman & Hall, London, 1991.
High pressure multicomponent adsorption in porous media Alexander A. Shapiro ) , Erling H. Stenby Engineering Research Center IVC-SEP, Department of Chemical Engineering, Technical UniÕersity of Denmark, Building 229, DK 2800, Lyngby, Denmark Received 13 April 1998; accepted 4 January 1999
Abstract We analyze adsorption of a multicomponent mixture at high pressures on the basis of the potential theory of adsorption. The adsorbate is considered as a segregated mixture in the external field produced by a solid adsorbent. We derive an analytical equation for the thickness of a multicomponent film close to a dew point. This equation Žasymptotic adsorption equation, AAE. is a first order approximation with regard to the distance from a phase envelope. It can be applied to both normal and retrograde condensation. Different forms of the AAE make it possible to analyse phenomena occurring close to a dew point. For testing of the AAE, we compare thicknesses of the adsorbed film predicted by this equation and obtained by direct calculations. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Adsorption; Potential theory; Model; Method of calculation
1. Introduction The potential theory of adsorption, first formulated by Polanyi, is a thermodynamic concept used for modeling of gas adsorption on solid. Its applications are multilayer adsorption on homogeneous and heterogeneous surfaces, adsorption from mixtures close to their dew points and adsorption in microporous media. In the latter case a modification of the potential theory formulated by Dubinin is widely used. A specific feature of the potential theory is that the mixture is treated in a purely thermodynamic way, without detailed consideration of molecular behavior. This makes it possible to utilize the knowledge accumulated in the bulk phase thermodynamics and to automatically extend the theory onto non-trivially behaving multicomponent mixtures like hydrocarbon mixtures exhibiting retrograde behavior. On the other hand, the purely thermodynamic treatment is the origin for the limitations of the theory. It becomes inefficient in cases where specific molecular-level phenomena )
Corresponding author
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 1 4 4 - 2
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
566
have a dominant contribution into the formation of adsorbed layers. In such cases, more modern and rigorous theories, such as the density fluctuation theory, should be applied. In a modification of the potential theory developed here, we do not make a common assumption about adsorbate as a homogeneous phase, but it is considered as a segregated mixture in the external field produced by a solid adsorbent. Such a model is able to correlate experimental data of multicomponent adsorption in porous media, if only parameters of the surface potentials for the pure components are fitted, but not the parameters in an equation of state for the segregated mixture w1x. However, a generalization of the potential theory onto multicomponent mixtures encounters computational difficulties. To overcome them, we derive an analytical equation for the thickness of a multicomponent film close to a dew point. This asymptotic adsorption equation, AAE, is a first order approximation with regard to the distance from a phase envelope. It can be applied to both normal and retrograde condensation. Different forms of the AAE make it possible to analyze phenomena occurring close to a dew point. This analysis includes the phase transition from a continuous to a discontinuous form of the adsorbate and a comparison of the action of capillary forces and adsorption in porous media. For testing of the AAE, we compare thicknesses of the adsorbed film predicted by this equation and obtained by direct calculations. The comparison shows remarkably good agreement between the two theories, especially, in the region of normal condensation, where the AAE provides good estimates for the relative pressures as low as 0.1.
2. Basic equations of the potential theory Let us briefly review the fundamentals of the potential adsorption theory, following Ref. w1x. Adsorption is represented as a segregation of the mixture in the potential field emitted by the surface of the adsorbent. Each Ž ith. component of the mixture is affected by its own adsorption potential e i Ž x . depending on the distance x to the surface. An equilibrium state in the potential field is described by the system of equations for the chemical potentials m i :
m i Ž P Ž x . , z j Ž x . . y e i Ž x . s mgi ž Pg , z gj / .
Ž1.
z gi
Ž I s 1,n. and pressure Pg in the gas phase, the distributions of pressure and Given composition molar fractions at the surface are uniquely determined by the system Ž 1. . Then the surface excess G i of the ith component is found as the difference between the actual amount of this component and its amount in the absence of the adsorption forces. The molar fraction z ai of the ith component in the adsorbate is the ratio of G i to the total surface excess: n
` i
G s
H0
i
z Ž x . nŽ x .
y z gi n g
dx ,
Gs
Ý G j,
z ai s G irG .
Ž2.
js1
Close to the surface the potentials may become quite strong, and the effective pressure in the adsorbate may reach several thousand atmospheres w2x. This often leads to condensation and formation of the liquid-like film on the surface. The thickness h of this film is determined from Eq. Ž1. together with the conditions of phase equilibrium:
mgi ž P Ž h . , z gj Ž h . / s m il Ž P Ž h . , z lj Ž h . . s mdi .
Ž3.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
567
Here we use the subscripts g and l to denote the values for gas and liquid mixtures. The subscript d Ž‘dew point’. is used to denote the values at phase transition, P Ž h. s Pd . The potential theory has been applied to adsorption on homogeneous surfaces, as well as in the ‘slab’ theory of multilayer adsorption on heterogeneous surfaces. Several dependencies for the adsorption potentials e i Ž x . have been suggested Ž see the analysis in Refs. w3–5x.. One of the commonest potentials is expressed by the Frenkel–Halsey–Hill Ž FHH. dependence w6,7x. A general form of this dependence is w5x
e i Ž x . s e 0i a i
al
al
rŽ b i q x . ,
e i Ž 0 . s e 0i .
Ž4.
Another application of the potential theory is adsorption in micropores. We do not consider it here, the discussion can be found elsewhere w1,3x. The system of segregation equations ŽEq. Ž1.. is not easy to solve, especially, when combined with the conditions of phase equilibria Ž Eq. Ž 3.. . The calculations become straightforward only for the case of a single-component adsorption. To overcome these difficulties, we will derive an asymptotic adsorption equation ŽAAE..
3. Derivation of the AAE The AAE may be derived on the basis of geometrical considerations. Fig. 1a represents the adsorption path and the relevant points on the P–z phase diagram, Fig. 1b in the space of the chemical potentials of the mixture. Point g on Fig. 1 corresponds to the bulk pressure Pg and composition z g s Ž z g1 , . . . , z gn .. Point D is the dew point for the composition z g ; the dew-point pressure is PD . In Fig. 1a, this point belongs to the gas branch of the phase envelope. The corresponding point on the liquid branch is L s Ž PD , z L .. In the space of chemical potentials we have L s D ŽFig. 1b.. The adsorption curve M s Ž P Ž x ., z Ž x .. determined by Eq. Ž1. starts from the point g corresponding to x s `. It intersects the phase envelope at the point d s Ž Pd , z d . and continues from l s Ž Pd , z l . from the liquid branch. In the space of chemical potentials, d s l. For a point g, which is close to the critical point, the adsorption curve can go around the phase envelope, and a smooth transition from gas to liquid is observed Žcurve MX in Fig. 1a. .
Fig. 1. Graphical representation of the adsorption curve: Ža. on the pressure-composition diagram, Žb. in the space of chemical potentials.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
568
The thickness h of the adsorbed film is determined as a value of x at which M meets the phase envelope. By conventional flash calculations, we can find the dew-point pressure PD and the equilibrium liquid composition z L , which are supposed to be known. In the asymptotics considered, all the distances between the points g, d, D are assumed to be small, and we neglect the second order terms with regard to them. Let us find the vector n D orthogonal to the hypersurface of the phase transition in the space of the chemical potentials. The Gibbs–Duhem relations at constant temperature, on the ‘gas’ and on the ‘liquid’ sides of the phase transition, may be expressed as follows: n
Ý is1
i z gD
VgD
n i d mgD sd P,
Ý is1
z Li VL
d m iL s d P .
Ž5.
i Subtracting them and noticing that mgD s m iL s m iD we find:
n
Ý is1
ž
i z gD
y
VgD
z Li VL
/
d m iD s 0.
Ž6.
The last equation is interpreted as a definition for the tangent plane d m to the hypersurface of the phase transition, in the standard form of Ž n D P d m . s 0, where nD s
ž
1 z gD
y
VgD
z L1 VL
,...,
n z gD
y
VgD
z Ln VL
/
.
Ž7.
The point d, which is sought for, may be found from the vector equality dgsDgyDd We multiply this equality by n D and note that the scalar product Ž n D P Dd. is of the second order with regard to a characteristic linear size of the triangle gDd. This can be found geometrically from Fig. 1b Žconsidering the decline of the vector connecting the two points on the surface from the tangent plane.. Within the second order values, we obtain:
Ž n Ddg . s Ž n DDg. . Let us transform the last geometrical equation to the form of a thermodynamic relation. It follows from Eq. Ž 1. that the coordinates of the vector gd are ye i Ž h. . Substituting the coordinates of n D from Eq. Ž7., we obtain n
Ý is1
ž
i z gD
VgD
y
z Li VL
n
/
i
e Ž h. s
Ý is1
ž
i z gD
VgD
y
z Li VL
/ž
m iD y mgi / .
Ž8.
The last equation will be referred to as a general or a potential form of the asymptotic adsorption equation ŽAAE. for the value of h. To use it, conventional thermodynamic calculations to determine the dew point pressure PD for a given composition z g are first performed. In these calculations, the composition z L is found. Then the liquid volume V L at the pressure PD is evaluated. Finally, Eq. Ž8. is solved as a transcendent equation for h. Due to approximations in its derivation, Eq. Ž 8. is valid asymptotically, close to a dew point of the mixture. A degree of proximity to the dew point at which the equation gives a reasonable
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
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approximation will be studied numerically later on. However, it is clear beforehand that the accuracy of the AAE is high if the phase transition surface in the chemical potential space is close to planar, at least, in a vicinity of the points D and d. On the other hand, Eq. Ž 8. loses its validity close to the critical point.
4. Different forms of the AAE The expression in the right hand side of the AAE Ž Eq. Ž 8.. may be substituted by different asymptotically equivalent expressions, depending on a purpose of the study. This is performed, first, by expanding m iD y mgi around the point D: i m iD y mgi s yVgD D P y Ez gD m iD D z g ,
D P s Pg y P D ,
D z g s z g I z gD .
Ž9.
Further, since D belongs to the boundary of the phase transition, for the corresponding infinitesimal changes D z g , D z L we have Ez gD m iD D z g s Ez L m iD D z L .
Ž 10.
Also, we will use the Gibbs–Duhem relations in the form of n
n
i Ez Ý zgD
m iD s gD
is1
Ý z Li Ez
L
m iD s 0
Ž 11 .
is1
Substituting Eq. Ž 9. to Eq. Ž11. into the right hand side of Eq. Ž8., we reduce it to the form of i z gD
n
Ý is1
ž
y
VgD
z Li VL
/
e i Ž h. s
ž
VgL VL
/
y 1 D P s Pd
ž
VgL VL
/
y 1 Ž x y 1. .
Ž 12.
Here we introduce the relative pressure x s PgrPd and the thermodynamic parameter n
VgL s
Ý z Li VgDi is1
which will be called the mixed Õolume. This parameter was introduced and discussed previously in connection with the problem of multicomponent capillary condensation w8x. When adsorption in meso- and macroporous media close to a dew point is studied, one of the problems is to distinguish it from the capillary condensation. In w8x it was shown that capillary pressure under multicomponent capillary condensation obeys an asymptotic equation called the modified Kelvin equation Ž MKE.: Pc s P D
ž
VgL VL
/
ln x y x q 1 .
Ž 13.
The right-hand side of the MKE is equal within second-order terms to the right-hand side of the linear AAE ŽEq. Ž12... Hence, the general AAE may be reduced to the form of n
Ý is1
ž
i z gD
VgD
y
z Li VL
/
e i Ž h . s PD
ž
VgL VL
/
ln x y x q 1 s Pc .
Ž 14.
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Use of Eqs. Ž13. and Ž14. makes it possible to study simultaneously the effects of capillary forces and adsorption in macroporous media. For simplicity, let us assume that all the pores are cylindrical and all the surface potentials are proportional: e i Ž x . s d ie Ž x ., which is valid, for example, for the FHH potentials Ž Eq. Ž 4.. with different e 0i but the same a and a . The Kelvin radius rc of a capillary which may be filled at given thermodynamic conditions is found on the basis of Eq. Ž13. and of the Laplace equation. Comparison of Eqs. Ž 13. and Ž14. leads to a simple relation between rc and the thickness h of the adsorbed film n
rc s gre Ž h . ,
g s y2 sr Ý d i is1
ž
i z gD
VgD
y
z Li VL
/
where s is the surface tension.
5. Numerical testing of the AAE We analyzed surface adsorption of methane Ž 1. –n-butane Ž2. at 277.6 K. Thermodynamic calculations were performed on the basis of the SRK EoS with Peneloux volume corrections, and on ` the basis of the modified FHH potentials Ž Eq. Ž 4.. . The values of e 0i for these potentials were taken from the original Halsey equation: e 0i s 5RT. Both a i and b i are of the order of the effective thickness of a monolayer Žsee discussion in Refs. w5,9x for details. . These thicknesses were reported for a few substances Ž for nitrogen, if a cubic packing is assumed, it is 0.4 nm. . As a basic variant, the adsorption potentials were chosen with a1 s b 1 s 0.4 nm and a 1 s 3 for methane, and with a1 s 0.4 nm, b 1 s 0.3 nm and a 2 s 2.5 for butane. Fig. 2 represents the dependence of the thickness h on the relative pressure, for the mixtures corresponding to Pd s 10 bar Ž86.2% of methane. and 20 bar Ž 92% of methane. . These dew point pressures lie in the region of normal condensation, x - 1. The calculations based on the exact equations ŽEqs. Ž1. and Ž 3.. were compared with predictions of the general AAE Ž Eq. Ž 8.. . A
Fig. 2. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4. 1— P D s 20 bar, 2— P D s10 bar.
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
571
Fig. 3. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4 at P D s 20 bar. 1—exact calculationss general AAE, 2—‘capillary’ AAE, 3—linear AAE.
remarkably good agreement of the exact and approximate values was established. In the whole region PD - 20 bar, and for the values of x up to 0.1 Žwhich is by no means ‘asymptotic’., the error did not exceed 1%. The error is lower at higher relative pressures and at lower dew-point pressures Žso that it is only 0.4% at x s 0.1 and P D s 10 bar.. Note that h depends much stronger on the relative pressures than on the dew-point ones. Comparison of the different forms of the AAE at PD s 20 bar is shown on Fig. 3. Both the ‘capillary’ Ž Eq. Ž14.. and the linear Ž Eq. Ž 12.. forms overestimate the values of h. The ‘capillary’ form is in a relatively good agreement with the exact calculations, although the general form Ž Eq. Ž 8.. is much superior. The relative errors for the AAE Ž Eq. Ž 14.. are, respectively, 0.7% at x s 0.9, 2.6% at x s 0.75 and 6% at x s 0.5. The linear form of the AAE, however, is far from being precise. The relative error is 3.6% already at x s 0.9. Figs. 4 and 5 represent similar results for the retrograde condensation, for the dew-point pressures of 80 and 100 bar. The general form of the AAE is again superior in this region. However, the relative
Fig. 4. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4. 1— P D s80 bar, 2— P D s100 bar.
572
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
Fig. 5. Thickness of the adsorbed layer vs. relative pressure for the mixture C1– nC4 at P D s80 bar. 1—exact calculationss general AAE, 2—‘capillary’ AAE, 3—linear AAE.
errors are much higher. At x s 1.2, the errors of the general AAE vary from 1.7% Ž PD s 100 bar. to 2.4% Ž PD s 80 bar. . Other forms of the AAE are closer to the exact dependence in this region than in the region of normal condensation. At P D s 100 bar, disappearance of the phase transition is observed at x ) 1.21. At higher relative pressures the adsorption curve goes around the phase envelope, as the curve MX on Fig. 1. However, all the forms of the AAE predict a non-zero thickness of the adsorbed layer for these relative pressures. Thus, either form of the AAE must be used with care in the near-critical region. In conclusion, on the basis of comparative calculations, the general form of the AAE is recommended for the calculation of the thickness of the adsorbed film. The ‘capillary’ form, although less precise, can be used for comparison of the action of capillary forces and adsorption. The linear form can be applied if the mixture is very close to its dew point. List of symbols a, b, a d h n n P r V x z g G e
parameters in the surface potentials proportionality constant in surface potentials thickness of the adsorbed layer molar density normal Žto the surface of phase transition. pressure radius of a capillary molar volume distance to the surface molar fraction surface parameter surface excess surface potential
A.A. Shapiro, E.H. Stenbyr Fluid Phase Equilibria 158–160 (1999) 565–573
m s x
chemical potential surface tension relative pressure
Subscripts a c d, D g l, L
adsorbed capillary dew point gas liquid
Superscripts i, j
number of a component
573
References w1x w2x w3x w4x w5x w6x w7x w8x w9x
A.A. Shapiro, E.H. Stenby, J. Colloid Interface Sci. 201 Ž1998. 146–157. A.V. Neimark, J. Colloid Interface Sci. 165 Ž1994. 91–96. W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, AP Academic Press, London, 1992. B. Rangarajan, C.T. Lira, R. Subramanian, AIChE J. 41 Ž1995. 838–845. A.W. Adamson, Physical Chemistry of Surfaces 3rd edn., Wiley-Interscience, New York, 1976. T.L. Hill, Advances in Catalysis, Vol. IV, in: W.G. Frankenburg, E.K. Rideal, V.I. Komarewsky ŽEds.., Academic Press, New York, 1952. J. Frenkel, Kinetic Theory of Liquids, The Clarendon Press, Oxford, 1946. A.A. Shapiro, E.H. Stenby, J. Fluid Phase Equilibria 134 Ž1997. 87–101. T. Allen, Particle Size Measurement, Chapman & Hall, London, 1991.