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Permeation time lag and heterogeneity in adsorbed phase transport
Chemical Engineering Science 55 (2000) 3543}3551
Permeation time lag and heterogeneity in adsorbed phase transport S.W. Rutherford, D.D. Do* Department of Chemical Engineering, The University of Queensland, St. Lucia, QLD 4072, Australia Received 16 August 1999; accepted 21 December 1999
Abstract The dependence of surface di!usivity with sorbate loading for transport in a heterogeneous carbon sorbent has been investigated. Several descriptions were tested for their ability to predict the time lag for permeation measured for carbon dioxide transport at 190 K through pellets of compressed Carbolac carbon by Ash, Barrer and Pope (1963, Proceedings of the Royal Society A, 271, 1). The descriptions tested in this investigation included a parallel path model, an e!ective medium approximation and a description based on the microstructural properties of activated carbon proposed by Do (1996, Chemical Engineering Science, 51(17), 4145). All descriptions were compared with that obtained by invoking the Darken relation. It was found that only the description proposed by Do (1996) provided a suitable prediction for this system. 2000 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Many commercially available sorbent materials such as activated carbon are manufactured in order to generate a wide distribution of pore sizes within the material. Pores in the order of micron size known as macropores, are created within the material in order to allow fast transport of sorbate into or out of the porous structure. Pores in the order of nanometer size known as micropores have the function of trapping the sorbate within the micropore network. This action involves molecular interaction of the sorbate molecules with the sorbent material. Adsorption heterogeneity can arise on this nanolevel due to a distribution in size and irregular shape of the micropores. This is usually characterised by measurement of the adsorption equilibrium isotherm and energetic heterogeneity is subsequently extracted from the measurement. This has been a rich area of study for many years and a convenient compilation has been formed by Rudzinski, Steele and Zgrablich (1997). Not so well understood is the e!ect of this heterogeneity on the mobility of the sorbate within the material. Many studies of transport and the relationship with temperature have been undertaken but the e!ect of the mobility on loading is not as well understood. This e!ect is important when considering dynamic adsorption processes in which the adsorp* Corresponding author. Tel.: #61-7-3365-4154; fax: #61-7-33652789. E-mail address: [email protected] (D.D. Do).
tion and adsorbed-phase di!usion can be dominant over other transport mechanisms. A study by Kapoor and Yang (1989) attempted to relate the adsorption heterogeneity, characterised by energy distribution inside the porous matrix and extracted from the equilibrium isotherm measurement, to the adsorbed phase mobility measured along the pellet. This analysis uses a local energy distribution to characterise heterogeneity an approach which has some merit. Another study has abandoned the idea of a local energy distribution and considered an energy distribution along the pellet scale (Aharoni, 1987). A further study has considered heterogeneity to arise from a physical view of the nature of the porous adsorbent (Do, 1996). The consideration of heterogeneity and its e!ect on the adsorbed-phase process, is an important issue as signi"cant di!erences between various descriptions of the process are predicted for the mobility of the adsorbed-phase with concentration of sorbate (Do, 1996). These large di!erences make discrimination of the suitability of the models relatively simple and requires measurement of the mobility and its relationship with loading of the sorbate within the material. The actual measurement of this relationship can be undertaken by several means. Direct measurement is usually made via steady-state permeation methods. A useful parameter for characterisation of this method is the permeation time lag which is obtained from the time di!erence between the start of the permeation process and the time at which it reaches steady-state. This parameter has been used to characterise adsorption and
0009-2509/00/$ - see front matter 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 1 3 - 0
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surface di!usion processes (Rutherford & Do, 1997a). The e!ect of the inherent adsorption heterogeneity upon the time lag has not been investigated fully. It is the purpose here to characterise the in#uence of heterogeneous e!ects upon the time lag and investigate various descriptions of the adsorbed-phase di!usion process and assess their suitability in predicting carbon dioxide transport in pellets of Carbolac-activated carbon.
2. Theory The description of molecular motion of sorbate molecules through porous solids can be obtained through various means including Monte Carlo simulation and e!ective medium analysis (Zgrablich, 1997). Generally, the validity of these techniques are highly dependent upon the description of the porous solid itself and can be useful when the structure of the material is well known. For materials such as activated carbon, the structure includes ordered and disordered regions and semi-empirical descriptions are often more useful in the respect that heterogeneity can be introduced in the form of an energy distribution or use of a more phenomenological approach. In this investigation, we examine four approaches to describe adsorbed-phase transport in activated carbon: (1) use of a parallel pore model with uniform energy distribution to incorporate heterogeneity in the adsorbed-phase transport; (2) use of an e!ective medium approximation with uniform energy distribution to incorporate network e!ects in transport; (3) use of a structural model of activated carbon to incorporate heterogeneity in the adsorbed-phase transport; (4) use of the Darken relation as an approximation for transport on heterogeneous surfaces.
energies resulting in the Unilan isotherm being used to correlate the adsorption equilibria. The Unilan equation is represented as
C 1#b eQC S C " IQS ln , (1) I 2s 1#b e\QC S where s represents a heterogeneity parameter, C the IQS saturation capacity and b the a$nity constant. The S uniform distribution of energy can be further used to evaluate the e!ect of heterogeneity upon the mobility. It is well known that adsorbed-phase transport is an activated process and when the energy of activation is 50% of the heat of adsorption (denoted as a"0.5), the mobility can be expressed as (Kapoor & Yang, 1989) D eQF!1 Q "1# , (2) D eQ!eQF\ Q where h is the fractional loading (ratio of adsorbed phase concentration to saturation concentration), D is the surQ face di!usion coe$cient and D represents the limit at Q very low loading. The relationship is plotted in Fig. 1 for several values of the heterogeneity parameter s. Characteristic to this relationship is a small change in the mobility at low concentration followed by a large change with loading as saturation is approached. Fig. 1 also indicates that the surface di!usivity is twice that at the low limit when fractional loading is 0.5. When the energy of activation for adsorbed-phase transport is so great that it is equal to the heat of adsorption (denoted as a"1), the expression for the
3. Parallel path model with uniform energy distribution It is well known that adsorption of sorbate molecules occurs by energetic interaction between the micropore and the molecule itself. Due to inherent size distribution and surface properties, the micropores o!er di!ering degrees of attraction to these molecules. The variation in this level of attraction o!ered by the adsorption sites generates energetic heterogeneity which is often characterised by an energy distribution and evaluated by experimentally determining temperature dependence of adsorption equilibria. This inherent heterogenity not only a!ects the amount adsorbed but also in#uences the mobility of the adsorbed phase. Kapoor and Yang (1989) have considered the case for a uniform distribution of
Fig. 1. The dependence of the reduced surface di!usivity with loading predicted by the parallel path description proposed by Kapoor and Yang (1989). Continuous line represents the case for lower activation energy for surface di!usion (a"0.5) and the dashed line represents the high activation energy (a"1).
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
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mobility becomes
D 2s eQF!1 Q "1# . (3) D eQ!e\Q eQ!eQF\ Q Characteristic to this description is a smaller change in the mobility at low loading, a result of the fact that a larger energetic barrier must be overcome. Once again, at high loadings a sharp increase in the mobility results. The description proposed by Kapoor and Yang (1989) has merit in the fact that a distribution of micropore size represented by an energy distribution is included, but the media is modelled as a series of transport paths each with a characteristic energy and each alligned parallel to one another. As a result, once a molecule has chosen its path it cannot interact with any other paths and experiences a constant mobility throughout. This approach has been termed a `parallel patha description. For most porous solids, in particular activated carbon, there is often much variation in mobility along the path of transport. E!ective medium analysis can account for this variation and its description is expected to be a truer representation of transport in many porous materials.
Fig. 2. The dependence of the reduced surface di!usivity with loading predicted by the e!ective medium approximation discussed by Kapoor and Yang (1990). Continuous line represents the case for lower activation energy for surface di!usion (a"0.5) and the dashed line represents the high activation energy (a"1).
adsorption (a"0.5), we have (Kapoor & Yang, 1990): 4. E4ective medium analysis with uniform energy distribution
D (b C Q" S . D tan\((b eQC)!tan\((b e\QC) Q S S
E!ective medium analysis provides an alternative description of transport in porous media in the respect that the solid is not viewed as allowing discrete and independent transport paths as in the parallel pore description, but is viewed as a network in which a molecule can be free to travel along a path of randomly changing mobility. The analysis is made by examining a representative element assumed to be randomly distributed in an equivalent homogeneous media. It is possible to undertake this analysis in one and two dimensions as is discussed in Kapoor and Yang (1990). One-dimensional analysis allows for a molecule to choose its discrete path through the solid and the path o!ers randomly changing mobility such that the e!ective mobility is limited by the least mobile patch it encounters. The two-dimensional case additionally allows the molecule to randomly change paths. For the one-dimensional case in which the adsorbate experiences high activation energy for transport (a"1), we have (Kapoor & Yang, 1990)
Characteristic to this relationship is a slow increase at low loading similar to that shown in the parallel path description and this is indicated in Fig. 2. The limitation of this analysis when dealing with carbon-based materials lies in the assumption that the media is random in nature. For highly amorphous carbons whose structure is disordered this analysis may be appropriate however for those carbons whose structure approaches that of highly ordered graphite, this analysis would be less applicable. An analysis which attempts to incorporate such structural factors into the description of adsorbed-phase transport in activated carbon has been proposed by Do (1996).
D eQF!1 Q" . (4) h[eQ!eQF\] D Q Fig. 2 shows the nature of this relationship and Kapoor and Yang (1990) have shown that steady changes of large magnitude are possible at low loadings. In the one-dimensional case in which the adsorbate experiences an activation energy equal to half that of the heat of
(5)
5. Description of Do (1996) The inherent structure of activated carbon must be addressed when considering adsorbed-phase transport within such materials. The microstructure is of concern because it is within the micropores that the adsorption/di!usion takes place. Various techniques including X-ray studies show that the microdomain consists of graphite-like units in the order of 10 nm embedded within disordered amorphous carbon. Thermal history of a carbon-based material determines the relative amount of graphite like and amorphous material in the
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microstructure. Low-temperature chars have high amounts of amorphous carbon in which graphite microdomains are cross-linked. Low density is characteristic to these non-graphitising carbons and this issue is discussed in a review by Foley (1995). In contrast, high-temperature carbons are more graphite like and hence have densities approaching that of graphite. An example is Carbolac carbon studied by Ash, Barrer and Pope (1963). It has been proposed that surface di!usion in activated carbon is highly dependent upon structural di!erences between the crystalline and amorphous regions of the carbonaceous solid (Do, 1996). Adsorption heterogeneity is accounted for in this description by utilising the Toth isotherm given as bC C R IQR , C " I [1#(b C)R] R R
(6)
where t represents the heterogeneity. This relation is often used in describing equilibria in activated carbon (Valenzuela & Myers, 1989). The relationship between surface di!usivity and loading developed by Do (1996) is given as 1 D" , Q f#d[1!(C /C )R]> R I IQR
(7)
where the parameters f and d are related to microstructural properties of the activated carbon sample. It has been shown that the parameter f, can be very small depending upon the nature of the sorbate/sorbent interaction (Do, 1996) in which case the following simpli"ed form of the relationship results: D 1 Q" . D (1!hR) R> Q
(8)
Characteristic to this relationship is a large change in the mobility with loading even at low concentrations as is shown in Fig. 2 for several values of the heterogeneity parameter t. The magnitude of this change is also large as is shown in this "gure.
6. Approximation of Darken (1948) It has been argued that the true driving force for di!usion of the adsorbed phase should be based on a thermodynamic quantity, the chemical potential. This approach was originally used by Darken (1948) in the study of interdi!usion of metal alloys and has been extended to the study of di!usion of the adsorbed phase. This relation has been subsequently used by Ruthven (1992) in the analysis of micropore di!usion of oxygen and nitrogen in carbon molecular sieve. The relationship
Fig. 3. The dependence of the reduced surface di!usivity with loading predicted by the description proposed by Do (1996).
can be represented as D d ln(C) Q" (9) d ln(C ) D I Q and necessarily requires evaluation of the isotherm relationship in order to obtain an explicit form for the surface mobility. Using the Toth isotherm, this relationship becomes D 1 Q" . (10) D (1!hR) Q The Darken relation was shown by Ruthven (1992) to provide a useful correlation for transport in pores of size in the order of molecular dimensions in which steric hindrance for entry into the micropore network plays an important role. In the larger pores of activated carbon, molecular interaction with the surface dominates and the nature of surface heterogeneity is important. For this reason the Darken correlation is often considered to be an approximation for heterogeneous surfaces of activated carbon (Zgrablich, 1997), However, using the Toth isotherm, adsorption heterogeneity has been indirectly introduced into the description in the form of the isotherm parameter t. Fig. 3 shows the anticipated dependence of the mobility upon loading for a number of values for the isotherm heterogeneity parameter. It can be seen that this relationship predicts a slow increase in the mobility at low loading followed by a sharper rise at higher loadings.
7. Time lag Experimental determination of the nature of the adsorbed-phase #ux is often undertaken using the permeation
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
experiment because it can provide a direct means of measurement and allows evaluation of gas-phase transport also. A parameter often used to characterise the permeation measurement is the time lag obtained from the time di!erence between the start of permeation and the point at which the process reaches a steady state. In order to evaluate the time lag we must examine the mass balance which usually involves transport with adsorption and local equilibrium and di!usion of the adsorbed phase occurring simultaneously with the gas phase. This is represented as
e#(1!e)
*C *C * *C I " H , *C *t *x *x
(11)
where the mobility function H is de"ned as *C I H"eD #(1!e)D . Q *C
(12)
and where C represents the concentration in the gas phase (mole per unit volume gas phase), C represents the I concentration in the adsorbed phase (mole per unit volume adsorbed phase) and e represents the volume fraction of void available for gas-phase #ow. The measure of the gas-phase mobility is the e!ective di!usivity D . While the gas-phase di!usion mechaC nism is one of Knudsen di!usion, the pore di!usivity remains constant. It is usually evaluated through measurement of non-adsorbing helium permeation (Ash et al., 1963). When a constant upstream concentration (C ) is applied and the downstream concentration is kept to a value negligible in comparison to the upstream, the permeation process will reach a steady state after some time delay. The time delay is known as the `time laga and can be obtained in integral form by manipulating the mass balance equation by the method of Frisch (1957) ¸A (eC#(1!e)C )H(u)(A H(w) dw) du I S t " . (13) (A H(u) du) This is the general form for the time lag as a function of the upstream pressure chosen for measurement. The speci"c form of this relation is dependent upon the isotherm equation. Using the Unilan equation (Eq. (1)) the time lag assumes the following form:
a"0.5, we have (1!e) eQ!e\Q b C S IQS H? (C)"eD # C 2s 1#b eQC 1#b e\QC S S ;D (1#b C) Q S and for a"1 (Eq. (3)) we have (1!e) eQ!e\Q b C S IQS H?(C)"eD # C 2s 1#b eQC 1#b e\QC S S 2sb C S ;D 1# . Q eQ!e\Q
(15)
(16)
Using the e!ective medium approximation in one dimension considered by Kapoor and Yang (1990) (Eq. (5)) for a"0.5, we have (1!e) eQ!e\Q b C S IQS H? (C)"eD # C 2s 1#b eQC 1#b e\QC S S (b C S ;D Q tan\((b eQC)!tan\((b e\QC) S S (17)
and for a"1 (Eq. (4)) we have (1!e) eQ!e\Q b C S IQS H?(C)"eD # C 2s 1#b eQC 1#b e\QC S S b D C S Q ; . (18) ln((1#b eQC)/(1#b e\QC)) S S Using the Toth isotherm equation (Eq. (6)) the time lag assumes the following form:
(1!e)b C u R IQR H(u)(A H(w) dw) du ¸A eu# S [1#(b u)R] R R , t " (A H(u) du)
(19)
where, for the description proposed by Do (1996) the mobility parameter H takes the form (1!e)b C R IQR H (C)"eD # (20) C f [1#(b C)R]> R#b R and when the Darken relation is used for the description of the adsorbed-phase mobility (Eq. (10)) the mobility parameter H assumes the form (Fig. 4) (1!e)b C D R IQR Q . H (C)"eD # C [1#(b C)R] R R
¸A [eu#[(1!e)C /2s]ln((1#b eQu)/(1#b e\Qu))]H(u)(A H(w)dw) du IQS S S S t " , ( H(u) du) where the parameter H represents the mobility and as such is dependent upon the chosen description for adsorbed phase transport. Using the parallel path description considered by Kapoor and Yang (1989) (Eq. (2)) for
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(21)
(14)
The experimental determination of the time lag is undertaken at a constant upstream pressure for a number of pressures in order to establish a relationship between the time lag and the pressure at which it is measured. After
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Fig. 4. The dependence of the reduced surface di!usivity with loading predicted by the Darken relation.
Fig. 5. The adsorption equilibrium isotherm for carbon dioxide at 190 K on Carbolac (Taken from Ash et al., 1963). The "t of the Unilan and Toth isotherm equations are also shown.
this relationship has been established experimentally and the isotherm relation evaluated, "tting of the abovementioned equations for time lag can be undertaken to verify the validity or otherwise of the description used for the adsorbed-phase transport. Evaluation of time lag equations must be undertaken numerically after non-dimensionalisation.
well as permeation rate were made for various upstream pressures. By measurement of helium permeation rate, the gas-phase #ux was evaluated and it was shown that the adsorbed-phase #ux in their material dominated that of the gas phase. Hence, the results are useful for analysis of the surface di!usion process. Previous studies have shown that this time lag and its dependence upon the upstream pressure can reveal information about the mobility of the adsorbed phase as is shown in the study by Rutherford and Do (1997a). However, it was assumed in that study that the adsorption and surface di!usion process was una!ected by heterogeneity and as a result the Dubinin}Radushkevich (DR) isotherm was used to "t the isotherm data and Darken relation used to describe surface di!usion. The isotherm data for carbon dioxide upon this Carbolac material at 190 K is shown in Fig. 5. The "t of the Unilan and Toth isotherm equations are also included in this plot. The "t for both equations are very good and the parameter values are shown in Table 2. The values obtained compare well with those presented in Valenzuela and Myers (1989) for a number of carbons in which the heterogeneity parameter for the Toth isotherm t, was found to be in the order of 0.35 to 0.45 and the corresponding parameter of the unilan isotherm s, was found to be in the order of 3 to 4. The values obtained for Carbolac are within these bounds.
8. Comparison with experimental data The permeation experiment has been widely utilised as a means for characterisation particularly with carbon-based materials (Rutherford & Do, 1997b). A broad understanding of the adsorbed-phase #ow phenomena and the dependence of this phenomena upon structural properties of the porous medium has not yet been achieved. For homogeneous surfaces there are several interpretations now used to describe adsorbed-phase #ow (Kapoor, Yang & Wong, 1989). In the study of Ash et al. (1963), carbon dioxide permeation was used to investigate mass transfer processes in Carbolac carbon pellets formed from compression. The nature of the macro and microstructures are discussed by Ash et al. (1963) and the details of the pellet used in their study is shown in Table 1. Measurement of the equilibrium isotherm as
Table 1 Properties of Carbolac pellet used by Ash et al. (1963) Material
Surface area (m/cc)
Porosity
Helium density (g/cm)
Cross-sectional area (cm)
Pellet length (cm)
Carbolac
785
0.50
2.12
0.07
0.91
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551 Table 2 Adsorption equilibrium isotherm parameters for "t of Toth and Unilan equations Unilan
Toth
C "0.0216 mol/cm IQS b "3.0;10 cm/mol S s"3.38
C "0.0271 mol/cm IQR b "3.23;10 cm/mol R t"0.37
From the magnitude of the heterogeneity parameters in both isotherm equations it is obvious that this material is highly heterogeneous with respect to the adsorbed phase. Whether this inherent heterogeneity in#uences the mobility of the adsorbed phase can be determined by analysis of the time lag for permeation measured by Ash et al. (1963). The time lag as a function of the upstream pressure is shown in Fig. 6 for measurement at 190 K. Against this measured data, we have "tted the time lag for the descriptions discussed earlier: (1) Unilan isotherm equation with the parallel path description for the adsorbed-phase mobility obtained using the uniform energy distribution and developed by Kapoor and Yang (1989). The time lag for the case of high activation energy (a"1) is calculated from Eqs. (14) and (16) by numerical integration. Likewise for a"0.5, the time lag is calculated from numerical solution of Eqs. (14) and (15). (2) Toth isotherm equation with the heterogeneous description of Do (1996) for the adsorbed-phase mobility. The time lag is calculated for this case from Eqs. (19) and (20) by numerical integration. (3) Toth isotherm equation using the description of Darken (1948) for the adsorbed-phase mobility. The time lag is calculated for this case from Eqs. (19) and (21) also by numerical integration. (4) Unilan isotherm with the e!ective medium approximation given by numerical solution of Eqs. (14) and (17) when a"0.5 and Eqs. (14) and (18) when a"1. Fig. 6a shows the "t of the data using the description of Darken which was used in a previous study with the DR isotherm (Rutherford & Do, 1997a). In the present study, the Toth isotherm which provides a better "t of the equilibrium data is used with the Darken relation. One parameter is adjusted for the "tting procedure, that being D . The Q "gure shows that the "t is poor and this serves to reinforce the view that the Darken relation only provides an approximation for heterogeneous surfaces. Fig. 6b shows the "t of the data using a uniform energy distribution and parallel path model proposed by Kapoor and Yang (1989) for high (a"1) and low (a"0.5) activation energy. The "t is obtained by optimisation adjusting one parameter, D . It can be seen that the "t is poor Q for both high and low activation energy. Fig. 6c shows the
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"t of an e!ective medium approximation description proposed by Kapoor and Yang (1990) for high (a"1) and low (a"0.5) activation energy. This prediction is slightly better than Darken or parallel path description but is still poor indicating that incorporation of network e!ects (based on the assumption that the porous media consists of a randomly distributed microstructure) may be of limited validity for application with highly ordered graphitic carbon such as Carbolac (Biscoe & Warren, 1942). The analysis of adsorbed-phase motion according to Do (1996), results in two unknown parameters to be evaluated: f, b. It has been shown by Do (1996) that di!usion of ethane in Ajax-activated carbon the parameter f is small. For the "tting procedure used in this investigation, we have initially assigned the value of f"0 and adjusted the value of b for the best "t shown in Fig. 6d. The "t is good in this instance and is far better than that obtained by the other descriptions considered in this study. All four descriptions of the adsorbed-phase mobility contained one parameter for adjustment and both Unilan and Toth equations had an equally accurate "t to the isotherm data. Consequently, the failure of the Darken relation, parallel path model and e!ective medium approximation must be the result of an inherent failure in the description of mass transfer. It should be noted that, common to the unsuccessful descriptions is the parent HIO model which characterises homogeneous surface transport (Kapoor & Yang, 1990). Under homogeneous conditions, the successful description of Do (1996) is the only one that does not obey the HIO model and this is possibly the reason for its better "t.
9. Implication of model 5t As discussed earlier, the microstructure of activated carbon materials is viewed as being comprised of discrete microcrystalline units joined by disordered regions crosslinking these crystalline units. It is believed that this microstructure provides a continuous path for transport of the adsorbed phase through the material. This is discussed by Do (1996) and according to the description proposed, analysis of the relationship between loading and mobility of the adsorbed phase may allow evaluation of the structure of this material. According to Do (1996), the value of the parameter b evaluated from the relationship can be related to the structure of the graphitic units within the material. It has been proposed that ¸ bC b" % R IQR , ¸ D I N
(22)
where the ratio ¸ /¸ represents the length scale of the % I disordered cross-linked region to the length scale of the graphitic region within the carbon microstructure. Under this analysis, the ratio ¸ /¸ is in the order of 5% for the % I material studied in this investigation. This would imply
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Fig. 6. (a) The measured time lag for several upstream pressures and the "t of the description of Darken (Eqs. (19) and (21)). The surface di!usivity at low loading obtained from the "t is 273D "4.5;10\ cm/; (b) the measured time lag for several upstream pressures and the "t of the parallel path Q description of Kapoor and Yang (1989). The broken line represents high activation energy (a"1) obtained from the solution of Eqs. (14) and (16). The continuous line represents the case when a"0.5 obtained from the solution of Eqs. (14) and (15). The surface di!usivity at low loading obtained from the "t is D "1.8;10\ cm/s; (c) the measured time lag for several upstream pressures and the "t of the e!ective medium approximation of Kapoor Q and Yang (1990). The broken line represents high activation energy (a"1) obtained from the solution of Eqs. (14) and (18). The continuous line represents the case when a"0.5 obtained from the solution of Eqs. (14) and (17); (d) the measured time lag for several upstream pressure and the "t of the description of Do (1996) (Eqs. (19) and (20)). The parameter extracted from this "t is b"5.7;10 s/cm.
from our kinetic measurement, that the amount of graphitic crystallites is much greater than that of the disordered amorphous region, in fact about 95}5% ratio. The helium density of Carbolac measured by Ash et al. (1963) is 2.13 g/cm which is 95% of the density of graphite (2.26 g/cm, Foley, 1995). This measurement implies a 95}5% ratio of graphitic to amorphous carbon within the microstructure, a result which agrees well with that obtained from dynamic adsorption measurement. This corroboration by helium density analysis coupled with the good "t of the time lag relationship with upstream pressure shown in Fig. 6d, provides good support for the description of Do (1996). Further-
more, the value of the ratio ¸ /¸ for Carbolac is much less % I than that obtained under analysis of Ajax-activated carbon in which the ratio is in the order of 1 (Do, 1996). This di!erence points to fundamental structural di!erences between the two materials and as discussed earlier implies that Ajax-activated carbon has a higher proportion of amorphous carbon characteristic of low temperature chars (Foley, 1995). The magnitude of the change in the predicted surface di!usivity using the description of Do (1996) for carbon dioxide permeation, is plotted in Fig. 7. High values of the reduced surface di!usivity (D /D ), characteristic to this Q Q description are noted.
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
D C D Q D Q ¸ P s t t <
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e!ective di!usivity of the gas phase surface di!usivity adsorbed-phase di!usivity at zero loading length of the pellet pressure heterogeneity parameter of unilan isotherm heterogeneity parameter of Toth isotherm time lag volume of vessel in which measurement takes place
Greek letters h e
surface coverage porosity
Subscripts Fig. 7. The predicted dependence of the reduced surface di!usivity upon adsorbed-phase concentration.
0
upstream
Acknowledgements 10. Conclusion In this investigation, analysis of the permeation time lag for carbon dioxide transport indicates that adsorbed phase heterogeneity has a signi"cant in#uence upon the dynamics of mass transfer. Use of a parallel pore model with a uniform energy distribution to characterise heterogeneity in transport of adsorbed carbon dioxide is shown to provide an inadequate description of the mass transfer. Use of the e!ective medium approximation and Darken relation also provide inadequate descriptions. However, the physical model proposed by Do (1996) based on structural properties of activated carbon, provides a useful description of the adsorbed phase #ow of carbon dioxide under heterogeneous transport conditions.
Notation a A b C C IQ C d, f
ratio of the activation energy for surface di!usion to the heat of adsorption cross-sectional area of medium adsorption a$nity concentration of di!using species saturation capacity upstream concentration parameters characterising the adsorbed-phase mobility in activated carbon
This project is supported by the Australian Research Council. References Aharoni, C. (1987). A.I.Ch.E.Journal, 33(2), 303. Ash, R., Barrer, R. M., & Pope, C. G. (1963). Proceedings of the Royal Society A, 271, 1. Biscoe, J., & Warren, B. (1942). Journal of Applied Physics, 13, 364}371. Darken, L. (1948). Transactions of AIME, 174, 184. Do, D. D. (1996). Chemical Engineering Science, 51(17), 4145. Foley, H. C. (1995). Microporous Material, 4, 407}433. Frisch, H. L. (1957). Journal of Physical Chemistry, 61, 93. Kapoor, A., Yang, R. T., & Wong, C. (1989). Catalysis Reviews in Science and Engineering, 31, 129. Kapoor, A., & Yang, R. T. (1989). A.I.Ch.E. J., 35(10), 1735. Kapoor, A., & Yang, R. T. (1990). Chemical Engineering Science, 45(11), 3261. Rudzinski, W., Steele, W., & Zgrablich, G. (1997). Equilibria and dynamics of gas adsorption on heterogeneous solid surfaces. Amsterdam: Elsevier Science. Rutherford, S. W., & Do, D. D. (1997a). Chemical Engineering Science, 52(5), 703. Rutherford, S. W., & Do, D. D. (1997b). Journal of Adsorption, 3, 283. Ruthven, D. M. (1992). Chemical Engineering Science, 47(17/18), 4305}4308. Valenzuela, D. P., & Myers, A. L. (1989). Adsorption equilibrium handbook. Englewood cli!s, NJ: Prentice-Hall. Zgrablich, G. (1997). In W. Rudzinski, W. Steele, & G. Zgrablich, Equilibria and dynamics of gas adsorption on heterogeneous solid surfaces. Amsterdam: Elsevier Science.
Permeation time lag and heterogeneity in adsorbed phase transport S.W. Rutherford, D.D. Do* Department of Chemical Engineering, The University of Queensland, St. Lucia, QLD 4072, Australia Received 16 August 1999; accepted 21 December 1999
Abstract The dependence of surface di!usivity with sorbate loading for transport in a heterogeneous carbon sorbent has been investigated. Several descriptions were tested for their ability to predict the time lag for permeation measured for carbon dioxide transport at 190 K through pellets of compressed Carbolac carbon by Ash, Barrer and Pope (1963, Proceedings of the Royal Society A, 271, 1). The descriptions tested in this investigation included a parallel path model, an e!ective medium approximation and a description based on the microstructural properties of activated carbon proposed by Do (1996, Chemical Engineering Science, 51(17), 4145). All descriptions were compared with that obtained by invoking the Darken relation. It was found that only the description proposed by Do (1996) provided a suitable prediction for this system. 2000 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Many commercially available sorbent materials such as activated carbon are manufactured in order to generate a wide distribution of pore sizes within the material. Pores in the order of micron size known as macropores, are created within the material in order to allow fast transport of sorbate into or out of the porous structure. Pores in the order of nanometer size known as micropores have the function of trapping the sorbate within the micropore network. This action involves molecular interaction of the sorbate molecules with the sorbent material. Adsorption heterogeneity can arise on this nanolevel due to a distribution in size and irregular shape of the micropores. This is usually characterised by measurement of the adsorption equilibrium isotherm and energetic heterogeneity is subsequently extracted from the measurement. This has been a rich area of study for many years and a convenient compilation has been formed by Rudzinski, Steele and Zgrablich (1997). Not so well understood is the e!ect of this heterogeneity on the mobility of the sorbate within the material. Many studies of transport and the relationship with temperature have been undertaken but the e!ect of the mobility on loading is not as well understood. This e!ect is important when considering dynamic adsorption processes in which the adsorp* Corresponding author. Tel.: #61-7-3365-4154; fax: #61-7-33652789. E-mail address: [email protected] (D.D. Do).
tion and adsorbed-phase di!usion can be dominant over other transport mechanisms. A study by Kapoor and Yang (1989) attempted to relate the adsorption heterogeneity, characterised by energy distribution inside the porous matrix and extracted from the equilibrium isotherm measurement, to the adsorbed phase mobility measured along the pellet. This analysis uses a local energy distribution to characterise heterogeneity an approach which has some merit. Another study has abandoned the idea of a local energy distribution and considered an energy distribution along the pellet scale (Aharoni, 1987). A further study has considered heterogeneity to arise from a physical view of the nature of the porous adsorbent (Do, 1996). The consideration of heterogeneity and its e!ect on the adsorbed-phase process, is an important issue as signi"cant di!erences between various descriptions of the process are predicted for the mobility of the adsorbed-phase with concentration of sorbate (Do, 1996). These large di!erences make discrimination of the suitability of the models relatively simple and requires measurement of the mobility and its relationship with loading of the sorbate within the material. The actual measurement of this relationship can be undertaken by several means. Direct measurement is usually made via steady-state permeation methods. A useful parameter for characterisation of this method is the permeation time lag which is obtained from the time di!erence between the start of the permeation process and the time at which it reaches steady-state. This parameter has been used to characterise adsorption and
0009-2509/00/$ - see front matter 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 1 3 - 0
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S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
surface di!usion processes (Rutherford & Do, 1997a). The e!ect of the inherent adsorption heterogeneity upon the time lag has not been investigated fully. It is the purpose here to characterise the in#uence of heterogeneous e!ects upon the time lag and investigate various descriptions of the adsorbed-phase di!usion process and assess their suitability in predicting carbon dioxide transport in pellets of Carbolac-activated carbon.
2. Theory The description of molecular motion of sorbate molecules through porous solids can be obtained through various means including Monte Carlo simulation and e!ective medium analysis (Zgrablich, 1997). Generally, the validity of these techniques are highly dependent upon the description of the porous solid itself and can be useful when the structure of the material is well known. For materials such as activated carbon, the structure includes ordered and disordered regions and semi-empirical descriptions are often more useful in the respect that heterogeneity can be introduced in the form of an energy distribution or use of a more phenomenological approach. In this investigation, we examine four approaches to describe adsorbed-phase transport in activated carbon: (1) use of a parallel pore model with uniform energy distribution to incorporate heterogeneity in the adsorbed-phase transport; (2) use of an e!ective medium approximation with uniform energy distribution to incorporate network e!ects in transport; (3) use of a structural model of activated carbon to incorporate heterogeneity in the adsorbed-phase transport; (4) use of the Darken relation as an approximation for transport on heterogeneous surfaces.
energies resulting in the Unilan isotherm being used to correlate the adsorption equilibria. The Unilan equation is represented as
C 1#b eQC S C " IQS ln , (1) I 2s 1#b e\QC S where s represents a heterogeneity parameter, C the IQS saturation capacity and b the a$nity constant. The S uniform distribution of energy can be further used to evaluate the e!ect of heterogeneity upon the mobility. It is well known that adsorbed-phase transport is an activated process and when the energy of activation is 50% of the heat of adsorption (denoted as a"0.5), the mobility can be expressed as (Kapoor & Yang, 1989) D eQF!1 Q "1# , (2) D eQ!eQF\ Q where h is the fractional loading (ratio of adsorbed phase concentration to saturation concentration), D is the surQ face di!usion coe$cient and D represents the limit at Q very low loading. The relationship is plotted in Fig. 1 for several values of the heterogeneity parameter s. Characteristic to this relationship is a small change in the mobility at low concentration followed by a large change with loading as saturation is approached. Fig. 1 also indicates that the surface di!usivity is twice that at the low limit when fractional loading is 0.5. When the energy of activation for adsorbed-phase transport is so great that it is equal to the heat of adsorption (denoted as a"1), the expression for the
3. Parallel path model with uniform energy distribution It is well known that adsorption of sorbate molecules occurs by energetic interaction between the micropore and the molecule itself. Due to inherent size distribution and surface properties, the micropores o!er di!ering degrees of attraction to these molecules. The variation in this level of attraction o!ered by the adsorption sites generates energetic heterogeneity which is often characterised by an energy distribution and evaluated by experimentally determining temperature dependence of adsorption equilibria. This inherent heterogenity not only a!ects the amount adsorbed but also in#uences the mobility of the adsorbed phase. Kapoor and Yang (1989) have considered the case for a uniform distribution of
Fig. 1. The dependence of the reduced surface di!usivity with loading predicted by the parallel path description proposed by Kapoor and Yang (1989). Continuous line represents the case for lower activation energy for surface di!usion (a"0.5) and the dashed line represents the high activation energy (a"1).
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
3545
mobility becomes
D 2s eQF!1 Q "1# . (3) D eQ!e\Q eQ!eQF\ Q Characteristic to this description is a smaller change in the mobility at low loading, a result of the fact that a larger energetic barrier must be overcome. Once again, at high loadings a sharp increase in the mobility results. The description proposed by Kapoor and Yang (1989) has merit in the fact that a distribution of micropore size represented by an energy distribution is included, but the media is modelled as a series of transport paths each with a characteristic energy and each alligned parallel to one another. As a result, once a molecule has chosen its path it cannot interact with any other paths and experiences a constant mobility throughout. This approach has been termed a `parallel patha description. For most porous solids, in particular activated carbon, there is often much variation in mobility along the path of transport. E!ective medium analysis can account for this variation and its description is expected to be a truer representation of transport in many porous materials.
Fig. 2. The dependence of the reduced surface di!usivity with loading predicted by the e!ective medium approximation discussed by Kapoor and Yang (1990). Continuous line represents the case for lower activation energy for surface di!usion (a"0.5) and the dashed line represents the high activation energy (a"1).
adsorption (a"0.5), we have (Kapoor & Yang, 1990): 4. E4ective medium analysis with uniform energy distribution
D (b C Q" S . D tan\((b eQC)!tan\((b e\QC) Q S S
E!ective medium analysis provides an alternative description of transport in porous media in the respect that the solid is not viewed as allowing discrete and independent transport paths as in the parallel pore description, but is viewed as a network in which a molecule can be free to travel along a path of randomly changing mobility. The analysis is made by examining a representative element assumed to be randomly distributed in an equivalent homogeneous media. It is possible to undertake this analysis in one and two dimensions as is discussed in Kapoor and Yang (1990). One-dimensional analysis allows for a molecule to choose its discrete path through the solid and the path o!ers randomly changing mobility such that the e!ective mobility is limited by the least mobile patch it encounters. The two-dimensional case additionally allows the molecule to randomly change paths. For the one-dimensional case in which the adsorbate experiences high activation energy for transport (a"1), we have (Kapoor & Yang, 1990)
Characteristic to this relationship is a slow increase at low loading similar to that shown in the parallel path description and this is indicated in Fig. 2. The limitation of this analysis when dealing with carbon-based materials lies in the assumption that the media is random in nature. For highly amorphous carbons whose structure is disordered this analysis may be appropriate however for those carbons whose structure approaches that of highly ordered graphite, this analysis would be less applicable. An analysis which attempts to incorporate such structural factors into the description of adsorbed-phase transport in activated carbon has been proposed by Do (1996).
D eQF!1 Q" . (4) h[eQ!eQF\] D Q Fig. 2 shows the nature of this relationship and Kapoor and Yang (1990) have shown that steady changes of large magnitude are possible at low loadings. In the one-dimensional case in which the adsorbate experiences an activation energy equal to half that of the heat of
(5)
5. Description of Do (1996) The inherent structure of activated carbon must be addressed when considering adsorbed-phase transport within such materials. The microstructure is of concern because it is within the micropores that the adsorption/di!usion takes place. Various techniques including X-ray studies show that the microdomain consists of graphite-like units in the order of 10 nm embedded within disordered amorphous carbon. Thermal history of a carbon-based material determines the relative amount of graphite like and amorphous material in the
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S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
microstructure. Low-temperature chars have high amounts of amorphous carbon in which graphite microdomains are cross-linked. Low density is characteristic to these non-graphitising carbons and this issue is discussed in a review by Foley (1995). In contrast, high-temperature carbons are more graphite like and hence have densities approaching that of graphite. An example is Carbolac carbon studied by Ash, Barrer and Pope (1963). It has been proposed that surface di!usion in activated carbon is highly dependent upon structural di!erences between the crystalline and amorphous regions of the carbonaceous solid (Do, 1996). Adsorption heterogeneity is accounted for in this description by utilising the Toth isotherm given as bC C R IQR , C " I [1#(b C)R] R R
(6)
where t represents the heterogeneity. This relation is often used in describing equilibria in activated carbon (Valenzuela & Myers, 1989). The relationship between surface di!usivity and loading developed by Do (1996) is given as 1 D" , Q f#d[1!(C /C )R]> R I IQR
(7)
where the parameters f and d are related to microstructural properties of the activated carbon sample. It has been shown that the parameter f, can be very small depending upon the nature of the sorbate/sorbent interaction (Do, 1996) in which case the following simpli"ed form of the relationship results: D 1 Q" . D (1!hR) R> Q
(8)
Characteristic to this relationship is a large change in the mobility with loading even at low concentrations as is shown in Fig. 2 for several values of the heterogeneity parameter t. The magnitude of this change is also large as is shown in this "gure.
6. Approximation of Darken (1948) It has been argued that the true driving force for di!usion of the adsorbed phase should be based on a thermodynamic quantity, the chemical potential. This approach was originally used by Darken (1948) in the study of interdi!usion of metal alloys and has been extended to the study of di!usion of the adsorbed phase. This relation has been subsequently used by Ruthven (1992) in the analysis of micropore di!usion of oxygen and nitrogen in carbon molecular sieve. The relationship
Fig. 3. The dependence of the reduced surface di!usivity with loading predicted by the description proposed by Do (1996).
can be represented as D d ln(C) Q" (9) d ln(C ) D I Q and necessarily requires evaluation of the isotherm relationship in order to obtain an explicit form for the surface mobility. Using the Toth isotherm, this relationship becomes D 1 Q" . (10) D (1!hR) Q The Darken relation was shown by Ruthven (1992) to provide a useful correlation for transport in pores of size in the order of molecular dimensions in which steric hindrance for entry into the micropore network plays an important role. In the larger pores of activated carbon, molecular interaction with the surface dominates and the nature of surface heterogeneity is important. For this reason the Darken correlation is often considered to be an approximation for heterogeneous surfaces of activated carbon (Zgrablich, 1997), However, using the Toth isotherm, adsorption heterogeneity has been indirectly introduced into the description in the form of the isotherm parameter t. Fig. 3 shows the anticipated dependence of the mobility upon loading for a number of values for the isotherm heterogeneity parameter. It can be seen that this relationship predicts a slow increase in the mobility at low loading followed by a sharper rise at higher loadings.
7. Time lag Experimental determination of the nature of the adsorbed-phase #ux is often undertaken using the permeation
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
experiment because it can provide a direct means of measurement and allows evaluation of gas-phase transport also. A parameter often used to characterise the permeation measurement is the time lag obtained from the time di!erence between the start of permeation and the point at which the process reaches a steady state. In order to evaluate the time lag we must examine the mass balance which usually involves transport with adsorption and local equilibrium and di!usion of the adsorbed phase occurring simultaneously with the gas phase. This is represented as
e#(1!e)
*C *C * *C I " H , *C *t *x *x
(11)
where the mobility function H is de"ned as *C I H"eD #(1!e)D . Q *C
(12)
and where C represents the concentration in the gas phase (mole per unit volume gas phase), C represents the I concentration in the adsorbed phase (mole per unit volume adsorbed phase) and e represents the volume fraction of void available for gas-phase #ow. The measure of the gas-phase mobility is the e!ective di!usivity D . While the gas-phase di!usion mechaC nism is one of Knudsen di!usion, the pore di!usivity remains constant. It is usually evaluated through measurement of non-adsorbing helium permeation (Ash et al., 1963). When a constant upstream concentration (C ) is applied and the downstream concentration is kept to a value negligible in comparison to the upstream, the permeation process will reach a steady state after some time delay. The time delay is known as the `time laga and can be obtained in integral form by manipulating the mass balance equation by the method of Frisch (1957) ¸A (eC#(1!e)C )H(u)(A H(w) dw) du I S t " . (13) (A H(u) du) This is the general form for the time lag as a function of the upstream pressure chosen for measurement. The speci"c form of this relation is dependent upon the isotherm equation. Using the Unilan equation (Eq. (1)) the time lag assumes the following form:
a"0.5, we have (1!e) eQ!e\Q b C S IQS H? (C)"eD # C 2s 1#b eQC 1#b e\QC S S ;D (1#b C) Q S and for a"1 (Eq. (3)) we have (1!e) eQ!e\Q b C S IQS H?(C)"eD # C 2s 1#b eQC 1#b e\QC S S 2sb C S ;D 1# . Q eQ!e\Q
(15)
(16)
Using the e!ective medium approximation in one dimension considered by Kapoor and Yang (1990) (Eq. (5)) for a"0.5, we have (1!e) eQ!e\Q b C S IQS H? (C)"eD # C 2s 1#b eQC 1#b e\QC S S (b C S ;D Q tan\((b eQC)!tan\((b e\QC) S S (17)
and for a"1 (Eq. (4)) we have (1!e) eQ!e\Q b C S IQS H?(C)"eD # C 2s 1#b eQC 1#b e\QC S S b D C S Q ; . (18) ln((1#b eQC)/(1#b e\QC)) S S Using the Toth isotherm equation (Eq. (6)) the time lag assumes the following form:
(1!e)b C u R IQR H(u)(A H(w) dw) du ¸A eu# S [1#(b u)R] R R , t " (A H(u) du)
(19)
where, for the description proposed by Do (1996) the mobility parameter H takes the form (1!e)b C R IQR H (C)"eD # (20) C f [1#(b C)R]> R#b R and when the Darken relation is used for the description of the adsorbed-phase mobility (Eq. (10)) the mobility parameter H assumes the form (Fig. 4) (1!e)b C D R IQR Q . H (C)"eD # C [1#(b C)R] R R
¸A [eu#[(1!e)C /2s]ln((1#b eQu)/(1#b e\Qu))]H(u)(A H(w)dw) du IQS S S S t " , ( H(u) du) where the parameter H represents the mobility and as such is dependent upon the chosen description for adsorbed phase transport. Using the parallel path description considered by Kapoor and Yang (1989) (Eq. (2)) for
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(21)
(14)
The experimental determination of the time lag is undertaken at a constant upstream pressure for a number of pressures in order to establish a relationship between the time lag and the pressure at which it is measured. After
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Fig. 4. The dependence of the reduced surface di!usivity with loading predicted by the Darken relation.
Fig. 5. The adsorption equilibrium isotherm for carbon dioxide at 190 K on Carbolac (Taken from Ash et al., 1963). The "t of the Unilan and Toth isotherm equations are also shown.
this relationship has been established experimentally and the isotherm relation evaluated, "tting of the abovementioned equations for time lag can be undertaken to verify the validity or otherwise of the description used for the adsorbed-phase transport. Evaluation of time lag equations must be undertaken numerically after non-dimensionalisation.
well as permeation rate were made for various upstream pressures. By measurement of helium permeation rate, the gas-phase #ux was evaluated and it was shown that the adsorbed-phase #ux in their material dominated that of the gas phase. Hence, the results are useful for analysis of the surface di!usion process. Previous studies have shown that this time lag and its dependence upon the upstream pressure can reveal information about the mobility of the adsorbed phase as is shown in the study by Rutherford and Do (1997a). However, it was assumed in that study that the adsorption and surface di!usion process was una!ected by heterogeneity and as a result the Dubinin}Radushkevich (DR) isotherm was used to "t the isotherm data and Darken relation used to describe surface di!usion. The isotherm data for carbon dioxide upon this Carbolac material at 190 K is shown in Fig. 5. The "t of the Unilan and Toth isotherm equations are also included in this plot. The "t for both equations are very good and the parameter values are shown in Table 2. The values obtained compare well with those presented in Valenzuela and Myers (1989) for a number of carbons in which the heterogeneity parameter for the Toth isotherm t, was found to be in the order of 0.35 to 0.45 and the corresponding parameter of the unilan isotherm s, was found to be in the order of 3 to 4. The values obtained for Carbolac are within these bounds.
8. Comparison with experimental data The permeation experiment has been widely utilised as a means for characterisation particularly with carbon-based materials (Rutherford & Do, 1997b). A broad understanding of the adsorbed-phase #ow phenomena and the dependence of this phenomena upon structural properties of the porous medium has not yet been achieved. For homogeneous surfaces there are several interpretations now used to describe adsorbed-phase #ow (Kapoor, Yang & Wong, 1989). In the study of Ash et al. (1963), carbon dioxide permeation was used to investigate mass transfer processes in Carbolac carbon pellets formed from compression. The nature of the macro and microstructures are discussed by Ash et al. (1963) and the details of the pellet used in their study is shown in Table 1. Measurement of the equilibrium isotherm as
Table 1 Properties of Carbolac pellet used by Ash et al. (1963) Material
Surface area (m/cc)
Porosity
Helium density (g/cm)
Cross-sectional area (cm)
Pellet length (cm)
Carbolac
785
0.50
2.12
0.07
0.91
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551 Table 2 Adsorption equilibrium isotherm parameters for "t of Toth and Unilan equations Unilan
Toth
C "0.0216 mol/cm IQS b "3.0;10 cm/mol S s"3.38
C "0.0271 mol/cm IQR b "3.23;10 cm/mol R t"0.37
From the magnitude of the heterogeneity parameters in both isotherm equations it is obvious that this material is highly heterogeneous with respect to the adsorbed phase. Whether this inherent heterogeneity in#uences the mobility of the adsorbed phase can be determined by analysis of the time lag for permeation measured by Ash et al. (1963). The time lag as a function of the upstream pressure is shown in Fig. 6 for measurement at 190 K. Against this measured data, we have "tted the time lag for the descriptions discussed earlier: (1) Unilan isotherm equation with the parallel path description for the adsorbed-phase mobility obtained using the uniform energy distribution and developed by Kapoor and Yang (1989). The time lag for the case of high activation energy (a"1) is calculated from Eqs. (14) and (16) by numerical integration. Likewise for a"0.5, the time lag is calculated from numerical solution of Eqs. (14) and (15). (2) Toth isotherm equation with the heterogeneous description of Do (1996) for the adsorbed-phase mobility. The time lag is calculated for this case from Eqs. (19) and (20) by numerical integration. (3) Toth isotherm equation using the description of Darken (1948) for the adsorbed-phase mobility. The time lag is calculated for this case from Eqs. (19) and (21) also by numerical integration. (4) Unilan isotherm with the e!ective medium approximation given by numerical solution of Eqs. (14) and (17) when a"0.5 and Eqs. (14) and (18) when a"1. Fig. 6a shows the "t of the data using the description of Darken which was used in a previous study with the DR isotherm (Rutherford & Do, 1997a). In the present study, the Toth isotherm which provides a better "t of the equilibrium data is used with the Darken relation. One parameter is adjusted for the "tting procedure, that being D . The Q "gure shows that the "t is poor and this serves to reinforce the view that the Darken relation only provides an approximation for heterogeneous surfaces. Fig. 6b shows the "t of the data using a uniform energy distribution and parallel path model proposed by Kapoor and Yang (1989) for high (a"1) and low (a"0.5) activation energy. The "t is obtained by optimisation adjusting one parameter, D . It can be seen that the "t is poor Q for both high and low activation energy. Fig. 6c shows the
3549
"t of an e!ective medium approximation description proposed by Kapoor and Yang (1990) for high (a"1) and low (a"0.5) activation energy. This prediction is slightly better than Darken or parallel path description but is still poor indicating that incorporation of network e!ects (based on the assumption that the porous media consists of a randomly distributed microstructure) may be of limited validity for application with highly ordered graphitic carbon such as Carbolac (Biscoe & Warren, 1942). The analysis of adsorbed-phase motion according to Do (1996), results in two unknown parameters to be evaluated: f, b. It has been shown by Do (1996) that di!usion of ethane in Ajax-activated carbon the parameter f is small. For the "tting procedure used in this investigation, we have initially assigned the value of f"0 and adjusted the value of b for the best "t shown in Fig. 6d. The "t is good in this instance and is far better than that obtained by the other descriptions considered in this study. All four descriptions of the adsorbed-phase mobility contained one parameter for adjustment and both Unilan and Toth equations had an equally accurate "t to the isotherm data. Consequently, the failure of the Darken relation, parallel path model and e!ective medium approximation must be the result of an inherent failure in the description of mass transfer. It should be noted that, common to the unsuccessful descriptions is the parent HIO model which characterises homogeneous surface transport (Kapoor & Yang, 1990). Under homogeneous conditions, the successful description of Do (1996) is the only one that does not obey the HIO model and this is possibly the reason for its better "t.
9. Implication of model 5t As discussed earlier, the microstructure of activated carbon materials is viewed as being comprised of discrete microcrystalline units joined by disordered regions crosslinking these crystalline units. It is believed that this microstructure provides a continuous path for transport of the adsorbed phase through the material. This is discussed by Do (1996) and according to the description proposed, analysis of the relationship between loading and mobility of the adsorbed phase may allow evaluation of the structure of this material. According to Do (1996), the value of the parameter b evaluated from the relationship can be related to the structure of the graphitic units within the material. It has been proposed that ¸ bC b" % R IQR , ¸ D I N
(22)
where the ratio ¸ /¸ represents the length scale of the % I disordered cross-linked region to the length scale of the graphitic region within the carbon microstructure. Under this analysis, the ratio ¸ /¸ is in the order of 5% for the % I material studied in this investigation. This would imply
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S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
Fig. 6. (a) The measured time lag for several upstream pressures and the "t of the description of Darken (Eqs. (19) and (21)). The surface di!usivity at low loading obtained from the "t is 273D "4.5;10\ cm/; (b) the measured time lag for several upstream pressures and the "t of the parallel path Q description of Kapoor and Yang (1989). The broken line represents high activation energy (a"1) obtained from the solution of Eqs. (14) and (16). The continuous line represents the case when a"0.5 obtained from the solution of Eqs. (14) and (15). The surface di!usivity at low loading obtained from the "t is D "1.8;10\ cm/s; (c) the measured time lag for several upstream pressures and the "t of the e!ective medium approximation of Kapoor Q and Yang (1990). The broken line represents high activation energy (a"1) obtained from the solution of Eqs. (14) and (18). The continuous line represents the case when a"0.5 obtained from the solution of Eqs. (14) and (17); (d) the measured time lag for several upstream pressure and the "t of the description of Do (1996) (Eqs. (19) and (20)). The parameter extracted from this "t is b"5.7;10 s/cm.
from our kinetic measurement, that the amount of graphitic crystallites is much greater than that of the disordered amorphous region, in fact about 95}5% ratio. The helium density of Carbolac measured by Ash et al. (1963) is 2.13 g/cm which is 95% of the density of graphite (2.26 g/cm, Foley, 1995). This measurement implies a 95}5% ratio of graphitic to amorphous carbon within the microstructure, a result which agrees well with that obtained from dynamic adsorption measurement. This corroboration by helium density analysis coupled with the good "t of the time lag relationship with upstream pressure shown in Fig. 6d, provides good support for the description of Do (1996). Further-
more, the value of the ratio ¸ /¸ for Carbolac is much less % I than that obtained under analysis of Ajax-activated carbon in which the ratio is in the order of 1 (Do, 1996). This di!erence points to fundamental structural di!erences between the two materials and as discussed earlier implies that Ajax-activated carbon has a higher proportion of amorphous carbon characteristic of low temperature chars (Foley, 1995). The magnitude of the change in the predicted surface di!usivity using the description of Do (1996) for carbon dioxide permeation, is plotted in Fig. 7. High values of the reduced surface di!usivity (D /D ), characteristic to this Q Q description are noted.
S.W. Rutherford, D.D. Do / Chemical Engineering Science 55 (2000) 3543}3551
D C D Q D Q ¸ P s t t <
3551
e!ective di!usivity of the gas phase surface di!usivity adsorbed-phase di!usivity at zero loading length of the pellet pressure heterogeneity parameter of unilan isotherm heterogeneity parameter of Toth isotherm time lag volume of vessel in which measurement takes place
Greek letters h e
surface coverage porosity
Subscripts Fig. 7. The predicted dependence of the reduced surface di!usivity upon adsorbed-phase concentration.
0
upstream
Acknowledgements 10. Conclusion In this investigation, analysis of the permeation time lag for carbon dioxide transport indicates that adsorbed phase heterogeneity has a signi"cant in#uence upon the dynamics of mass transfer. Use of a parallel pore model with a uniform energy distribution to characterise heterogeneity in transport of adsorbed carbon dioxide is shown to provide an inadequate description of the mass transfer. Use of the e!ective medium approximation and Darken relation also provide inadequate descriptions. However, the physical model proposed by Do (1996) based on structural properties of activated carbon, provides a useful description of the adsorbed phase #ow of carbon dioxide under heterogeneous transport conditions.
Notation a A b C C IQ C d, f
ratio of the activation energy for surface di!usion to the heat of adsorption cross-sectional area of medium adsorption a$nity concentration of di!using species saturation capacity upstream concentration parameters characterising the adsorbed-phase mobility in activated carbon
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