water azeotrope separation by vapor permeation

water azeotrope separation by vapor permeation

Journal of Membrane Science, 68 (1992) 229-239 229 Elsevier Science Publishers B.V., Amsterdam Methods to improve flux during alcohol/water separat...

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Journal of Membrane Science, 68 (1992) 229-239


Elsevier Science Publishers B.V., Amsterdam

Methods to improve flux during alcohol/water separation by vapor permeation*


A.E. Jansen, W.F. Versteeg, B. van Engelenburg, J.H. Hanemaaijer and B.Ph. ter Meulen Netherlands Organization for Applied Scientific Research (TNO), P.O. Box lo&3700 AC Zeist (Netherlands)

(Received December 10,199O; accepted in revised form May 22,199l)


Pervaporation and vapor permeation both offer interesting alternatives to distillation for the azeotropic dehydration of alcohols. Main reasons are considerable potential energy savings and their modular, flexible system configurations. In this paper various theoretical and practical advantages of vapor permeation compared to pervaporation are discussed. Experimental results of vapor permeation are presented. It is concluded that fluxes during vapor permeation will be principally higher, mainly because the driving force is less affected by concentration polarization and decreasing temperature. This allows the reduction of membrane area to be installed. Because no heat exchangers have to be included to supply the heat of evaporation, module construction for vapor permeation systems is more simple. It is shown that flux and selectivity during vapor permeation can be controlled by pressure, superheating and membrane impregnation. Furthermore impregnation of membranes is proposed as a promising route for tailoring flux and selectivity of vapor permeation membranes to specific applications. Keywords: pervaporation;

vapor permeation;

separation of azeotropic mixtures; dehydration

Introduction Distillation still is the current technique used for the separation of alcohol-water mixtures. Some specific features of the phase diagram normally concerning these separations are the presence of an azeotrope, and of close liquid/ vapor curves between the azeotropic composition and the pure alcohol. This means that in this region extractive distillation, a high reflux

Correspondence to: A.E. Jansen, Netherlands Organization for Applied Scientific Research (TNO), P.O. Box 108, 3700 AC Zeist (Netherlands). *Paper presented at the 5th World filtration Congress, Nice, France, June 5-8,199O.


ratio and a large number of theoretical plates are needed. These facts cause distillation processes for the dehydration of alcohols to be rather complicated and highly energy-consuming. When considering the thermodynamics of distillation we can learn that no large further progress is to be expected concerning these aspects. Pervaporation and vapor permeation seem to offer promising alternatives to distillation for the dehydration of azeotropic alcohol-water mixtures. Pervanoration has been investigated now for about 15 years, and at the moment various (mostly small) plants are in operation worldwide. Substantial energy savings compared to distillation are reported [ 1,2 1.

0 1992 Elsevier Science Publishers B.V. All rights reserved.


Vapor permeation is only recently proposed as an alternative to pervaporation, and as yet no commercial plants are in operation. Both processes are based on the preferential permeation of water vapor through (water-)selective membranes, driven by a difference in partial pressure across the membrane. The main difference is the state of the feed: for pervaporation the feed is a liquid, for vapor permeation is a vapor. Four areas of application of pervaporation or vapor permeation can be distinguished: (1) addition of capacity to existing distillation equipment; (2) pre-azeotropic concentration; (3) breaking of azeotropic mixtures; (4) post-azeotropic concentration to produce pure alcohol. This paper aims at outlining several principal theoretical and practical advantages of vapor permeation compared to pervaporation for the dehydration of alcohols. Of these, especially some possibilities for tailoring flux and selectivity during vapor permeation will be illustrated with experimental results. Theory Pervaporation is evaporation across a selective membrane. As shown schematically in Fig. 1, water vapor permeates preferentially out of a liquid feed through the membrane, controlled by the driving force: a difference in partial vapor pressure between feed and permeate side. Pervaporation is a three-step process: ( 1) selective sorption in the polymer matrix; (2 ) diffusion across the membrane; (3 ) desorption at the permeate side of the (active layer of the) membrane. Flux and selectivity are normally determined by sorption and diffusion, provided that the permeate vapor pressure is low enough (established by vacuum or inert carrier gas) for desorption to occur.

A.E. Jansen et al/J. Membrane Sci. 68 (1992) 229-239 alcoholMater (liquid)

F waler

Fig. 1. Principle of pervaporation.

Vapor permeation alcohol/water


i water Fig. 2. Principle of vapor permeation.

The main disadvantage of pervaporation is the requirement to supply the heat of evaporation AH,, necessary to evaporate the permeate through the membrane. Because this supply cannot occur in place, but only between (arrays of) membrane modules, the liquid retentate cools down, leading to lower vapor pressure and thus decreased driving force. This results in lower permeate fluxes and to overcome this effect more membrane area and/or more heat exchangers have to be installed. The main advantage of pervaporation compared to extractive distillation is the low energy requirement. This is mainly due to the fact that the ‘reflux’ during pervaporation is zero, meaning that the heat of evaporation has only to be supplied once. Reported energy savings amount to 62 to 92% [1,2]. Vaporpermeation differs from pervaporation mainly because the feed is supplied as a vapor (Fig. 2). It is membrane process aiming at the

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

same applications as pervaporation but act as a membrane-gas separation process where no temperature drops in the feed flow can occur. This is a major advantage of vapor permeation compared to pervaporation, resulting in higher permeate fluxes and much less complicated system design. In both pervaporation and vapor permeation separation is based on sorption, diffusion and desorption. These processes are governed by the driving force, which usually is the difference in chemical potential. This implies a linear function between the water flux through the membrane and the activity, which again is a function of temperature, pressure and composition. At very low water activity at the permeate side of the membrane and neglecting non-ideal behaviour of water in the vapor phase (this assumption can lead to errors in the order of percents for vapor permeation), the flux during azeotropic dehydration is given by eqn. (1). Ji = LiPXi


where Ji is the permeate flux of component i ( kg-mm2-hr-’ ), Li is the permeability coefficient ( kg-m-2-hr-‘-Pa-1), P is the vapor pressure of retentate (Pa), and Xi is the mole fraction in feed (- ) . Temperature influences several terms in the above equation: l because sorption, diffusion and desorption are temperature-dependent, the permeability coefficient is an ( Arrhenius-like ) function of temperature; l in order to operate at elevated retentate pressure, in vapor permeation the temperature has to be raised to prevent condensation; l vapor permeation is usually operated at a temperature above the dew point of the mixture. This superheating decreases the flux by decreasing the sorption, as will be discussed later. The major difference between pervaporation and vapor permeation, which is the state of the


feed stream, has consequences for all three parameters Xi, P and Li in eqn. ( 1) : (1) Xi. During pervaporation permeate fluxes are such (up to 5 kg-m-2-hr-‘) that serious mass transfer limitations may occur. This is caused by concentration polarization, characterized by much lower water concentrations at the membrane surface compared to the bulk liquid. Besides by the flux, this concentration polarization is governed by the retentate hydrodynamics, by the diffusion coefficients of the components to be separated and by the selectivity of the membrane. This effect causes flux decreases during pervaporation of about 10% up to several factors, depending on the type of application and on process conditions. During vapor permeation the effect of concentration polarization is rather small; mass transfer coefficients will be approximately 10 times higher than during comparable pervaporation processes. This is due to the fact that the increase in diffusion coefficients in the vapor feed (about 500 to 1000 times) is much higher than the decrease in density (about 50 to 100 times, or even less when operating at elevated pressure). In this respect the ‘real’ driving force during vapor permeation will be higher than during pervaporation. (2) P. For both pervaporation and vapor permeation the water transport through the membrane increases linearly with vapor pressure. However, during pervaporation the temperature drop, caused by the withdrawal of heat of evaporation, lowers the vapor pressure (following an Antoine function): a temperature drop of 10 K in a pervaporation module causes approximately 35% decrease in vapor pressure, resulting in a loss of driving force (and flux) per module between 15 and 20%. During vapor permeation principally no drop in temperature takes place, apart from a possible LIT of at most a few tenths of K due to a Joule-Thomson volume enlargement effect.


However, due to the higher retentate volume flow during vapor permeation the pressure drop at the retentate side cannot be neglected, and this causes a decrease in vapor pressure (and flux). By proper module design and operation at elevated pressure this pressure drop can be minimized. (3) Li. The permeability coefficient during pervaporation and vapor permeation principally depends on the type of membrane used, and thus is not affected at all by the phase of the feed. However, for vapor permeation this opens an indirect way to improve fluxes by increasing the permeability by impregnating the membrane with hygroscopic salts. These salts lower the water vapor pressure in the membrane, thereby increase sorption and so improve permeability and flux. On this impregnation technique a patent is pending [ 31. The technique cannot be used in pervaporation because of leaching out of the salts into the liquid feed. When the salts are leached out is is expected that the original membrane characteristics return. In additional to the desired impregnation effects some not wanted other effects can occur. ( 1) Swelling of the top layer. This swelling will

Fig. 3. Test equipment for measurements at elevated pressure.

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

cause higher diffusion and leads also to higher fluxes, however, accompagned by a stronger decrease in selectivity. This effect can be deminished by crosslinking of the active layer of the membrane. (2) Lower diffusitivity. Higher salt contents in a toplayer influence the diffusivity, especially when no swelling occurs. Lower fluxes combined with high selectivities are expected. (3) Plasticizing: an effect caused by higher water contents leads to irreversible change in membrane structure. An irreversible change in membrane performance (higher fluxes and slow selectivities) can be expected. Considering all three parameters ( Li, P, Xi) discussed above it is demonstrated that by applying a vapor feed instead of a liquid, and thus by applying vapor permeation instead of pervaporation, it is principally possible to obtain a better flux performance under comparable process conditions. Experimental Several flat sheet membranes (homogeneous or asymmetric) were purchased or manufactured. Special test equipment was designed and

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239


Fig. 4. Vapor permeation test equipment.

constructed to characterize flat sheet membranes and small hollow fiber modules at conditions (pressure, temperature) occurring in distillation processels. Figures 3 and 4 show the experimental set-up. Attention has been given to accurate control of process parameters: l

l l l

( 20.05 vapor feed flow: O-8 kg-hr-l kg-hr-‘) retentate pressure: 100-500 kPa ( -t 0.1 kPa) temperature: 295-425 K ( + 0.5 K) permeate pressure: 0.1-100 kPa (+ 0.1kPa)

Test conditions and a proper design of the flat-sheet-membrane test cells enabled the membrane characterization to be carried out a negligible concentration polarization and minimized change in retentate vapor composition. The permeate side pressure was kept at 0.1 kPa. The test equipment was designed to operate under fully continuous conditions (24 hr-d-l), because it can take several hours to reach a steady state [4]. The main parameters obtained during characterization are selectivity and flux. Fluxes were calculated from the

weighed amount of liquid condensed during certain time intervals in the permeate condensers. Compositions of feed, retentate and permeate were determined by refractometry and densimetry. Permeabilities for water and alcohol were calculated as component flux per unit of driving force; the driving force being expressed as the difference in partial vapor pressure between retentate and permeate. The selectivity of the membranes is defined by the separation factor ( CZ):

X, cy= (100-X,)


(100-X,) x,


where X, is the alcohol content in retentate (wt.% ), and X,, is the alcohol content in permeate (wt.% ). It should be noted that a! is a very sensitive function of X, and X, at small X, and high X, values, so at cy> 1000 small differences in selectivity result in large changes in CLFor practical use in breaking azeotropes an (I!higher than 50 is required.


A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

Results and discussion Characterization of membrane materials Several commercial membranes for reverse osmosis and pervaporation, and some other polymers made into homogeneous films have been screened with respect to their ability to dehydrate alcohols by vapor permeation. As is shown in Table 1, most membrane materials tested have sufficient selectivity. Regarding both selectivity and flux, best results were obtained with a poly(viny1 alcohol) -composite membrane which is commercially available for pervaporation, and with a cellulose membrane, produced by alkaline hydrolysis of a commercial asymmetric cellulose acetate membrane. Effect of concentration and pressure on flux and selectivity The effect of feed water content and of increased pressure was investigated for ethanol dehydration using poly (vinyl alcohol) (PVA) composite membranes. The effects on flux and selectivity are shown in Figs. 5 and 6. Effect of concentration With decreasing water content in the retentate, flux decreases and selectivity increases.

A decreasing water content in the retentate decreases the driving force for permeation, so one expects a linear relationship with the flux (eqn. 1). The fact that at high water concentration the flux decreases slightly more than proportional can be explained by a somewhat higher permeability coefficient Li, caused by swelling of the PVA toplayer [ 41. The separation factor at isobaric and isothermic conditions more or less doubles in value when the ethanol concentration changes from 90 to 99%. This is caused by a different behaviour of the permeability of the pure components, as is shown in Figs. 7 and 8. Water permeability only slightly decreases at decreasing water content in the retentate (Fig. 7). In contrast to this, ethanol permeability shows a clear decrease (Fig. 8), resulting in increased selectivity. This can be understood from the difference in sorption behaviour of water and ethanol. It can be derived from water sorption isotherms [5] that the interaction energy of the PVA/ water system is similar to water/water. For ethanol, however, sorption in water is much higher than in PVA. So it can’ be expected that a decreasing water content of the retentate (which implies decreasing water content in the membrane) leads to increased selectivity.


Fluxandseparation factor for the dehydration of EtOH/HxO mixture using several flat sheet membranes. Results obtained at atmospheric pressure Membrane material

Polysulphonate Polysulphonamid FT-30 Permasep Poly(viny1 alcohol) Cellulose Hydrolysed C.A.

Thickness (pm)

Feed (wt.% EtOH)

t (“C)

Flux (kg-m-2-hr-‘)

Separation factor

30 30 20 25 -

99.7 96.0 96.0 96.0 96.0 94.0 96.0

84 84 89 86 82 86 86

0.006 0.016 3 0.02 0.26 0.04 0.21

1630 450 12 500 15000 180 1390


A.E. Jansen et al. jJ. Membrane Sci. 68 (1992) 229-239 ethanol permeability mg/(m*.h.Pa)

A& A

500 kPa (126OC)


400 kPa (11 ST)


300 kPa (109YJ


200 kPa (SPC)

f + +

A + + 0







retentate composition (wt % ethanol)

Fig. 5. Relationship between the flux through the membrane and the ethanol content of the feed at five different pressure levels. Membrane: PVA-composite. Superheating: 1 K.



retentate composltion/(wt%





Fig. 7. Relationship between water permeability and the ethanol content of the feed at five different pressure levels. Membrane: PVA-composite. Superheating: 1 K. water ~ermeabllity mg/(m h Pa)

separation factor

500 kPa (126.X)


400 kPa (116°C) 300 kPa (1 OSOC)


200 kPa (97Tz) 115 kPa (62°C) ,&*






1001 90



I 94

I 96


I 98


I 100

retentate composition (wl % ethanol)

Fig. 6. Relationship between the separation factor and the ethanol content of the feed at five different pressure levels. Membrane: PVA-composite. Superheating: 1 K.

Effect of pressure According to eqn. (1)) a linear relationship is expected between pressure and permeate flux. As can be derived from Fig. 5, the observed flux increase with pressure is somewhat more than proportional. Also, the permeability (Figs. 7 and 8) shows an increase with pressure, while


92 r&Nate







Fig. 8. Relationship between the ethanol permeability and the ethanol content of the feed at five different pressure levels. Membrane: PVA-composite. Superheating: 1 K.

already being corrected for the pressure increase by dividing by driving force! This phenomenon can be understood from the fact that increasing the pressure at constant superheating means increasing the temperature too. By increasing the temperature, the permeability coefficient Li increases because the increase in diffusivity through the membranes overacts the decrease in sorption. The separation factor clearly decreases with

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

increasing pressure (Fig. 6). This is caused by a much larger increase in ethanol permeability compared to water permeability (Figs. 7 and 8 ) . This can also be explained by the fact that at higher pressure (and temperature) the diffusivity plays a role of increasing importance in the transport mechanism, and the impact of sorption difference decreases. The lowest separation factor observed for the PVA-composite membrane was 540 at 500 kPa and 93 wt.% ethanol in the retentate. Process consideration show that a separation factor above 50 is sufficient for efficient breaking of azeotropic alcohol-water mixtures; above this level, increase of flux is normally more important than increase of selectivity.

when the sorption behaviour of water in PVA is considered. A temperature increase at saturated conditions causes an increase in permeability, as is shown above, because diffusivity increases and sorption only slightly decreases. Sorption, however, is not only a function of the temperature, but also a strong function of the relative pressure: the ratio of partial to saturated pressure. Characteristic for the sorption isotherm of water in PVA is the sharp decline at slightly unsaturated conditions [6]. This means that a small degree of superheating already causes a large decrease in sorption; much larger than the decrease in diffusivity. Net result is a decrease in permeability caused by superheating.

Effect of superheating During vapor permeation some degree of superheating is usually applied and necessary to prevent condensation. As shown in Fig. 9, vapor permeation fluxes can be increased very clearly by decreasing the temperature to levels close to the dew point of the vapor mixture. Selectivity seems not to be affected by superheating to a large extent. The effect of superheating can be understood

Effect of membrane impregnation An interesting feature of vapor permeation is the possibility to impregnate the membranes with flux or selectivity promoting substances, which cannot leach out because no liquid is in contact with the membrane. Figure 10 shows that by impregnation of homogeneous cellulose films (thickness 45 pm), featuring high selectivity but low flux, fluxes can be raised considerably while selectivity decreases. The effect depends very much on the

ermeateflux Pkg.m-?h-‘]

permeate llux [kg.m-*.h-l]



0.01’ 01


10 100 degree of superheating (K)

Fig. 9. Effect of superheating of the vapor mixture on the flux during dehydration of ethanol. Membrane: PVAcomposite.






separation factor

Fig. 10. Effect of impregnation of cellulose membranes with several electrolytes on flux and selectivity during vapor permeation of an azeotropic IPA-water mixture.


A.E. Jansen et al/J. Membrane Sci. 68 (1992) 229-239

type of impregnant: salts with the highest interaction with water (CsF and LiBr) show the largest increase in flux. Although results presented in Fig. 10 were not completely stable because of degeneration of the membrane material (especially caused by LiBr ), they illustrate the ability of membrane impregnation to vary functional properties of vapor permeation membranes across a wide area of fluxes and selectivities. Another illustrative example is given in Fig. 11. Here results are shown for the dehydration of ethanol using a non-impregnated (standard) and a CsF-impregnated PVA-composite membrane. It was observed that at each feed pressure fluxes broadly doubled in value by impregnation, while selectivities decreased to still acceptable levels. During the total experimental time of about 700 hr membrane performance did not change, and after leaching out CsF the original (non-impregnated) flux and selectivity could be restored. Repeated impregnation with CsF shows a remarkable effect. In Fig. 12 this is illustrated by the separate permeate fluxes of water and

ethanol using the same type of PVA composite membrane as in Figs. $9 and 11. At the zero-point on the x-axis we find the original membrane performance (flux and selectivity using no impregnation). After impregnation the membrane has a low water activity (water vapor pressure< 100 Pa). When measurements begin the membrane swells due to high water sorption under influence of the water







Fig. 12. Effect of repeated impregnations of a PVA-composite membrane with CsF on water and ethanol fluxes during vapor permeation of ethanol (95 wt.% ) . Superheating: 1 K. Permeare


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12 500kPa

’ 1

H20 EtOH total


\ 0.8.











I 105

0 z-

103 104 -separation factor(a)

Fig. 11. Effect of impregnation of a PVA-composite membrane with CsF on flux and selectivity during vapor permeation of ethanol (95 wt.%) at different pressure levels. Superheating: 1 K. n CsF impregnated 0 non impregnated

Fig. 13. Effect of repeated impregnations of a PVA composite membrane (another type) with CsF. on water and ethanol fluxes during vapor permeation of ethanol (95 wt.%). Superheating: 1 K.


Fig. 14. Effect of impregnation of a PVA composite membrane with different TPAB amounts on permeate fluxes and selectivity Superheating: 1 K.

activity in the feed stream (ca. 15 kPa). This swelling causes an increase of the membrane flux and decrease of selectivity in time as is illustrated in Fig. 12 by the vertical arrows. A steady state is reached within 10 hr. When regarding the line through the points of the arrows, we find that the steady state water and ethanol-fluxes increase sharply at the first impregnation combined with still a fairly good selectivity (a! = 87). Repeated impregnation shows a further small increase of water flux and a decrease in ethanol flux. This effect can be explained by a combination of two effects: ( 1) increase of water sorption and a smaller increase of EtOH sorption; (2) decrease of both diffusion coefficients caused by the high salt content in the membrane. As an overall result the water permeability will be higher while the ethanol permeability will be lower. The upper line ends at the right side of the figure in a lower point on the vertical axes. This point represents the membrane characteristics after 5 repeated impregnations and increased superheating: 10 K. The fluxes decrease as expected and the selectivity restores to a higher level. Another membrane, characterized by a more crosslinked PVA toplayer, shows after repeated impregnation a water flux increase from 0.105 to 1.07 kg-m-‘-hr-’ (Fig. 13). The smaller ar-

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

rows, compared to the arrows in Fig. 12, can be explained by less swelling of this membrane due to the higher degree of crosslinking. This also explains the preservation of high selectivity ((x > 1000) when impregnating this membrane. In Fig. 14 the type I PVA composite membrane, impregnated with tetrapropylammoniumbromide (TPAB ), a hygroscopic salt with organic character, shows a dramatic ethanol flux increase (> 1000) with only moderate water flux increase. Moving to the left each point presents another increased amount of TPAB in the top layer. After leaching out the TPAB the original flux and selectivity of the membrane were restored each time, indicating that these effects are fully reversible. The nature of these spectacular effects are now point of study. Expected is a combination of higher ethanol sorption and increased sorption. These high fluxes with low selectivities are not wanted for dehydration applications, but indicate possibilities for influencing permeabilities of organic vapors permeabilities, which allows new separation applications. Conclusions Vapor permeation offers several attractive advantages over pervaporation for the dehydration of alcohols: l Module construction is rather simple, because no heat of evaporation has to be supplied. This also allows for simple process configurations and easy upscaling. l Although fluxes under ideal conditions will be identical for both pervaporation and vapor permeation using a certain membrane, the practical flux during vapor permeation will be considerably higher; this is because the driving force is less affected by concentration polarization and decreasing partial pressure difference. l Fluxes during vapor permeation can be in-

A.E. Jansen et al./J. Membrane Sci. 68 (1992) 229-239

creased considerably by increasing pressure and decreasing superheating. Impregnation of vapor permeation membranes offer several additional advantages to vapor permeation: Reversible flux increases up to 10 times are achieved, while the selectivity is still high ((.w>1000). Increasing effects on flux and selectivity are found after repeated impregnation. Complete reversibility on flux and selectivity appears after leaching out the salt. Tailoring vapor permeation membranes/ modules to specific separation applications seems possible by membrane impregnation. l








References 1 P.O. Cogat, Dehydration of ethanol, pervaporation compared with azeotropic distillation, in: R. Bakish (Ed.), Proc. of Third Int. Conf. on Pervaporation Processes in the Chem. Ind., Nancy, France, September 19-


22,1988, Bakish Materials Corporation, Englewood, NJ, 1988, pp. 305-316. J.G.A. Bitter, Evaluation of membrane technology for separation of dilute alcohol solutions, in: R. Bakish (Ed.), Proc. of Third Int. Conf. on Pervaporation Processes in the Chem. Ind., Nancy, France, September 1922,1988, Bakish Materials Corporation, Englewood, NJ, 1988, pp. 476-485. H.F. van Wijk and A.E. Jansen, Methodandmembrane for the removal of water vapor from a gas/vapor mixture by means of vapor permeation, U.S. Pat. 4, 913, 907, April 3,199O. U. Meyer-Blumenroth and R. Rautenbach, Dampf Permeation organischer und organisch-wlsriger Gemische, Preprints Aachener Membran Kolloquium, Aachen, Germany, March 16-18, 1987, GVC-VDI, Dusseldorf, 1987, pp. 252-266. D. Hauser, B. Schmittecker and R.N. Lichtenhaler, Experimental investigations of sorption and diffusion in polymeric materials, in: R. Bakish (Ed. ), Proc. of Third Int. Conf. on Pervaporation Processes in the Chemical Industry, Nancy, France, September 19-22,1988, Bakish Materials Corporation, Englewood, NJ, 1988, pp. 78-88. S.J. Gregg and K.S.W. Sing, Absorption, Surface Area and Porosity, Academic Press, London and New York, 1967, pp. 93-95.