[membrane science and technology] fundamentals of inorganic membrane science and technology volume 4...

Fundamentals of Inorganic Membrane Science and Technology Edited by A.J. Burggraaf and L. Cot

�9 1996, Elsevier Science B.V. All rights reserved

Chapter 9

Transport and separation properties of membranes with g a s e s and vapours

A.J. Burggraaf

Department of Chemical Technology, Laboratory of Inorganic Materials Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

9.1 INTRODUCTION

9.1.1 Chapter Outline

Transport phenomena in porous solids have been the subject of many studies [1-6,10]. Quantitative solutions are obtained however only in a number of limiting cases of generally formulated problems or in relatively simple cases. Such a case is, e.g., the permeation of a single gas in a membrane system with a relatively simple pore architecture and under conditions when a single mechanism is predominantly operating.

Transport of mixtures is more complicated, especially in membrane systems with a more complex architecture and operated with large pressure gradients. In such cases quantitative solutions for permeation and separation efficiency (selectivity) are not available in a generally applicable form. Specific solutions have to be obtained by approximations and by combining solutions for limiting cases. The description in this chapter takes account of this situation.

First a number of important points will be summarised including a brief discussion of definitions and terminology. In subsequent sections a brief overview will be given of the most important theoretical aspects (equations) of single gas permeation and of accepted ways to combine several, simultaneously

3 3 2 9 ~ TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS

operating mechanisms in simple membrane architectures. This is followed by a brief description of those transport properties of mixtures which can be applied in membrane systems. Next, permeation and separation in real, but simple, porous membrane systems will be discussed in more detail with a focus on operational applicability. Some more complex systems (multilayered, hollow fibre) will be briefly treated. Finally, a discussion will be given of the validity of important approximations made in preceding sections, of important problems (multi-component mixtures) and opportunities, and of some interesting models (e.g., molecular sieving).

In all sections macro-, meso-, and microporous (molecular sieving) systems will be treated separately. The focus will be on the most promising systems to obtain high selectivity (separation factors) in combination with reasonable permeation values.

9.1.2 Overview of Important Points

For single gases a number of transport mechanisms exists (Sections 9.2.3.1- 9.2.3.3, 9.4). Depending on the pore diameter distribution and/or the temperature-pressure combination one of these mechanisms might be dominant. In many cases some of them act simultaneously and addition rules must be formulated and each contribution has to be "weighted" according to its own driving force. This is generally not the pressure gradient, but the gradient of the thermodynamic potential. As a consequence a thermodynamic correction factor has to be applied in diffusion or permeation equations expressed in terms of pressure or concentration. Even then appropriate descriptions cannot always be obtained (see, e.g., Section 9.4.)

In gas mixtures the permeation of components (and thus the selectivity) is only identical with that of single gases under special conditions (high temperature and low pressure). This difference is of importance in the transition region between molecular diffusion (Poiseuille flow) and Knudsen diffusion and in that of Knudsen to configurational diffusion. In multicomponent gas mixtures general descriptions make use of Stefan-Maxwell equations and e.g. the extended Dusty-Gas Model. For binary gases these more complicated models converge to Fickian type of equations and relatively easy-to-obtain solutions for permeation (and thus for ideal) separation factors.

In systems consisting of a macro-and/or mesoporous support and a meso- or microporous separation (top) layer, the permeation is a system property and the driving force for transport is distributed over the system components. In studying the permeation and separation properties of the top layer, corrections must be made on the permeation of the total system to find that of the top layer, unless it is shown that the flow resistance of the support is negligible compared to that of the top layer. Even when the permeation of the support is much larger

9 - - TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS 3 3 3

and its flow resistance therefore much smaller ~ than that of the top layer, there can be a considerable effect on the effective separation factor of the total system. This last one is usually different from the ideal separation factor of the top layer. In all cases the value of the pressure on the downstream (permeate) side of the membrane is important and should be as low as possible (back-diffusion, concentration polarisation). Relations between the effective and the ideal separation factors can be obtained in a number of cases.

Almost all physical models use simple pore geometries. Practical pore systems are, however, very complicated and contain parameters which are difficult to measure or which have a wide distribution of their characteristic parameters. The applicability of a rigorous treatment and of very refined models and physical expressions is therefore doubtful. The treatment in this chapter will make use mainly of phenomenological equations which allow description of data, data reduction and some extrapolation and which rely on experimentally determined parameter values. Gas kinetic theory and expressions based on the microscopic (atomic) level will be used only to estimate some parameter values and to predict trends.

For practical applications a combination of high selectivity and high permeation is required. As will be shown below, these two requirements are more or less contradictory and so an optimal compromise has to be sought. In this chapter a certain focus will be given to mechanisms with a large potential for high separation factors and at least reasonable permeation values. This leads to microporous systems or capillary condensation type of phenomena.

Complete membrane systems can be operated in a variety of modes with e.g. co- or counter flow of feed (high pressure side) and permeate (low pressure side) streams and with membrane modules coupled in different ways. Permea- tion and separation in these complex engineering systems will not be treated in this chapter. Heat and mass transfer limitations on the gas-membrane surfaces or interfaces can be important with high fluxes and/or strongly adsorbing gases as well as in membrane reactors. These effects will not be treated explicitly but are introduced in experimental results, e.g., by variation of sweep rates of permeated gases.

9.2 GAS TRANSPORT IN SIMPLE MEMBRANE STRUCTURES

9.2.1 Important Concepts

Transport data of membranes can be expressed in terms of flux (mol/m 2 s) or as flux normalised per unit of pressure (mol/m 2 s Pa). Following the IUPAC convention this last parameter is called permeation (note: in the literature the better term 'permeance' is frequently used). Using 'permeation' is meaningful

334 9 ~ TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

however only if there is a linear relation between flux and pressure. Despite the fact that this relation in many cases does not hold, transport data in the literature are expressed as permeation. To facilitate comparison of data the permeation can be normalised per unit of thickness and is then called permeability (tool m / m 2 s Pa). This should be done only if the thickness of the separation layer is known. In many cases only an unknown part of this layer is really active and use of the parameter permeability gives rise to large values compared with the real intrinsic ones. Therefore, in case of doubt, flux values should always be given together with the (partial) pressure of the relevant components at the high pressure (feed) and low pressure (permeate) sides of the membrane, as well as the apparent membrane thickness.

It is convenient to distinguish between permeation measurements in which the flux is measured under a known (and constant) pressure gradient and those in which the flux of a component i is driven by a concentration difference between the membrane faces under a constant and equal total pressure at both sides (Wicke--Callenbach [3]). Either of these two main methods may be performed under steady state or under transient conditions. Whether or not component fluxes and diffusivities measured with both methods give similar or different values depends on the conditions and on the type of the dominant diffusion mechanism.

An overview of the transport mechanisms in porous membranes is given in Table 9.1.

TABLE 9.1

T r a n s p o r t r eg imes in p o r o u s m e m b r a n e s

T r a n s p o r t t y p e Pore d i ame te r Select ivi ty

Viscous f low > 20 n m -

Molecu la r d i f fus ion > 10 n m -

K n u d s e n d i f fus ion 2-100 n m 1 /

Surface d i f fus ion +

Cap i l l a ry c o n d e n s a t i o n ++

Mic ropo re (config.) d i f fus ion < 1.5 n m ++

Viscous (Poiseuille) flow and molecular diffusion are non-selective. Never- theless they play an important role in the macroporous substrate(s) supporting the separation layer and can seriously affect the total flow resistance of the membrane system. Mesoporous separation layers or supports are frequently in the transient-regime between Knudsen diffusion (flow) and molecular diffusion, with large effects on the separation factor (selectivity).

9 - - TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 335

Configurational diffusion in microporous (molecular sieve) membranes will be treated separately. Here the driving force must be described in terms of a chemical potential gradient, which is coupled to partial pressure via adsorption isotherms. In cases where several mechanisms operate simultaneously, the problem of additivity arises and in real membrane systems simplifying assumptions have to made.

9.2.2 Pore Characteristics and Membrane Architecture

Porous materials have a very complex structure and morphology and many studies have been devoted to describing and characterising them [1-3,8]. Rouc- querol et al. [8] in their IUPAC report give useful advice for terminology, definitions and characterisation strategies.

Parameters which influence transport properties are porosity, pore size distribution, pore shape, interconnectivity and orientation. Indirectly particle size distribution and shape are important in the way they affect the uniformity of the pore size distribution, the pore shape and the roughness of the internal surface area.

A schematic picture of different types of pores is given in Fig. 9.1 and of main types of pore shapes in Fig. 9.2. In single crystal zeolites the pore characteristics are an intrinsic property of the crystalline lattice [3] but in zeolite membranes other pore types also occur. As can be seen from Fig. 9.1, isolated pores and dead ends do not contribute to the permeation under steady conditions. With adsorbing gases, dead end pores can contribute however in transient measurements [1,2,3]. Dead ends do also contribute to the porosity as measured by adsorption techniques but do not contribute to the effective porosity in permeation. Pore shapes are channel-like or slit-shaped. Pore constrictions are important for flow resistance, especially when capillary condensation and surface diffusion phenomena occur in systems with a relatively large internal surface area.

' ex2,

Fig. 9.1. Schematic picture of pore types in a porous solid, a: Isolated pore; b,f: dead end pores; c,d: tortuous a n d / o r rough pores (d), with constrictions (c); e: conical pore.

3 3 6 9 ~ TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

I I B

C

I - I

Fig. 9.2. Schematic representation of main types of pore structures and membrane architectures: (A) straight cylindrical pores; (B) straight asymmetric pores; (C) tortuous pore system.

A very important concept is the interconnectivity and the related tortuosity (~), as illustrated in pore d of Fig. 9.1. This parameter is used in almost all equations and will be discussed below in some detail.

Burggraaf and Keizer [9] distinguish between different main types of pore and membrane structures as shown in Fig. 9.2. These different structures are related to the way they are fabricated. There are straight and parallel pores running from one side of the membrane to the other side with a constant pore diameter or conical shaped pores (Fig. 9.2A and B respectively). The tortuosity has in this case a value of about unity. In the case of conical pores as shown in the figure the membrane is asymmetric and combines a 10w flow resistance (large pores across a considerable fraction of the membrane thickness) with a relatively large selectivity (small pores on the top side of the membrane). This structure is relatively simple and systems designed in this way are useful for model experiments.

Systems used in practice have a spongy structure (porous glass or carbon) or have the structure common in ceramic membranes. The latter have an interconnected, tortuous and randomly oriented pore network with constrictions and dead ends (Fig. 9.1) and are formed by packing of particles.

The pore structure of zeolite membranes is formed by arrays of intergrown zeolite particles or zeolite particle packings with interparticle pores filled with another material. The intracrystalline pores are a part of the crystallographic structure and are the ones which should be responsible for the selectivity.

The architecture of the ceramic membrane system is that of a multi-layer asymmetric system (Figs. 9.2C and 9.3). The separation activity is concentrated mainly in the top layer, the other parts form the supporting systems with

9 m TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 337

3

2

I

Fig. 9.3. Architecture of an asymmetric composite membrane. (1) Porous support (1-15 ~tm pores). (2) hltermediate layer(s); pore diameter dp = 100-1500 nm. (3) Mesoporous separation layer; dp = 3-100

nm. (4) Modification of 3 to microporous separation layer; dp = 0.5-2 nm.

relatively large pores to minimise the transport resistance. Hollow-fibre membranes do not need 'supports' and are thin single wall membranes. Several types of defect, such as pinholes or cracks, can exist in the morphologic structure which reduce the selectivity even when the effect on permeation is limited.

9.2.3 Single Gas Permeation in Macroporous and Mesoporous Systems

The properties of gas flow in porous media depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collisions. The Knudsen number Kn is a characteristic parameter defining different regions of this ratio. Its value is defined by Kn = ~,/dp with ;~ being the average free path length of the gas molecules and dp the characteristic pore diameter (sometimes the hydraulic pore radius is taken).

The magnitude of Kn separates three main flow regimes of gaseous diffusion (see also Table 9.1):

(a) Viscous flow: Kn <<1, ~ << dp (b) Knudsen diffusion (flow): Kn >>1, ~ >> dp (c) Transition flow: Kn = 1, ~ = dp When the pore walls strongly absorb gas molecules, surface diffusion and / or

capillary condensation accompanied by (surface) flow occurs. Usually this is the case with gases which condense rather easily at moderate temperature- pressure conditions (in any case being below their critical point) and we are dealing with 'vapour' flow.

Configurational diffusion is a separate class and occurs when the pore diameter is a factor of 1-5 larger than the molecular diameter.

9.2.3.1 Viscous Flow

When the number of intermolecular collisions is strongly dominant (Kn << 1), forced flow under a pressure or concentration gradient in a capillary can be


described by Darcy's law [14]. In capillaries with small diameter the flow is laminar and, if the gas velocity near the pore walls equals zero (no slip), the molar flux can be described by a Hagen-Poiseuil le type law:

r 2 P dP Jv = - (9.1)

81"1 RT dz

In real porous media, Eq. (9.1) must be modified to account for the number of capillaries per unit volume (porosity) and the complexities of the structure (tortuosity 1;). This leads to:

s 2 P dP Jv = - (9.2)

"r 81"1 RT dz

In the steady state the fluxes into and out of any cross section of a pore are equal. Therefore p(dP/dz) is constant and the integration of Eq. (9.2) over the thickness L of the porous m e d i u m gives the permeation:

Jv E 1-2 Fv = - ~ = ~ ~ Pm (9.3)

AP 8r1"r RTL

with the mean pressure P m = 0.5(P1 + P2) and P1 and P2 the pressure at the inlet and outlet respectively. Equation (9.3) shows that the permeat ion is proportional to"

- the (hydraulic) radius squared (r2);

- the mean pressure.

9.2.3.2 Knudsen Diffusion and the Transition Region

Kn udsen diffusion When the number of molecule to wall collisions is strongly dominant (Kn >>

1) the flow of a single gas in a long capillary under the action of a concentration pressure gradient can be described by the Knudsen equation [15]:

2 - dc Jk = - ~ v r d---z (9.4a)

2 _ 1 dP Jk = - ~ v r R--T- d---7 (9.4b)

with the thermal mean velocity of the gas molecules v given by:

v = (8RT) ~ (9.5)

In real porous media geometrical effects play an important role, similar to that discussed for viscous flow (Section 9.2.3.1.), and Eq. (9.4) must be modified by a term I~/'l~Kn.


Furthermore in the derivation of (9.4) it is assumed that after collisions with the wall the molecules are specularly reflected. This is usually not the case. The walls have a certain roughness and this causes diffuse reflections. This effect is accounted for by a factor 1/OKn; for smooth walls 0 equals unity, otherwise it is larger.

Taking these effects into account and inserting (9.5) into (9.4b) the expression for the Knudsen flow in a porous membrane is obtained:

2 ~r ( 8 ) ~ (9.6) Jk" = - 3 "~0 k ~RTM dz

Several authors introduce a shape factor [~ in (9.6) which accounts for the pore size distribution. After integration of (9.6) over the membrane thickness L the permeation is found to be:

Jk, 2~r ( 8 )0.5 Fk . . . . (9.7)

AP 31:.0 k L rcRTM

Equation (9.7) shows that the permeation of a single gas in the Knudsen regime is: - proportional to the average pore radius r; - independent of the pressure (note this is an important difference from

viscous flow); - proportional to M -~ Note that the Knudsen diffusion coefficient is obtained by introducing the

above mentioned geometrical coefficients in Eq. (9.4).

Transition flow Transition flow occurs when viscous flow and Knudsen diffusion both play

a role, that is in the region with Knudsen number values around unity. Esti- mates of the value of Kn can be made with the help of the gas kinetic expression for )~:

1 RT ~, = ~ ~ (9.8a)

~J2- ~ 2 P

with the average collision diameter, o, of different molecules c~1 and c~2 defined as:

(~1 } (~2 c ~ = ~ (9.8b)

For example for Argon (M = 40) at a pressure of I Bar and a temperature of 293 K, the value of ~, equals 6.9x10 -8 m and with d = 10 nm the value of Kn = 7. This means that in pores with d < 10 nm and under ambient conditions, Knudsen diffusion occurs; pores with a size of 100 nm however fall in the transition region.

3 4 0 9 - - TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

Most attempts at describing the Knudsen-viscous Poiseuille transition in- volve a combination of Eqs. (9.2) and (9.6). For single gases this is a good approximation and after some reorganisation and integration of Eqs. (9.2) and (9.6) and assuming a linear pressure drop across the membrane this yields the expression for the total flux Jt:

I t = - ~ ( Ar2 -P + Br) AP (9.9a) ,r L

with A = 1/(8TIRT) and B = (2/30k)~I(8/TIRTM). For further discussion, see Section 9.2.4.2 on binary gases.

The expression for F t follows from:

Ft= Jt /AP (9.9b)

This is a very useful equation which is used for several purposes (see below). It is noted that the occurrence of slip flow has been ignored so far. This point will be discussed below.

An alternative, semi-empirical, expression has been proposed by Schofield et al. [16]:

Jt = a. pb. AP (9.10)

In this equation the exponent b of the dimensionless pressure Pd (= P/Pref) is a measure for the extent of viscous flow (b = 1 for viscous flow and b = 0 for Knudsen diffusion). The reference pressure Pref is chosen as a typical or average pressure for the range of applications concerned.

Equation (9.10) can be substantiated by manipulation of (9.2) and(9.6), using the kinetic molecular expression for the gas viscosity 11:

1 - 1 P M q ( B R T I n =-~ NM v X- 2 RT ~ M (9.11)

This yields an expression for the permeation which can be approximated by Eq. (9.10) provided P and Pref do not differ by more than a factor of 3. The advantage of Eq. (9.10) as an engineering equation compared with (9.9) is that it does not require knowledge of the membrane properties, r and z but expresses the gas flux in terms of a membrane property a and of operation conditions b.

Geometric aspects Many authors have derived expressions for the transport in porous media

based on specific models, taking into account pore size distribution under different assumptions for the interconnectivity (tortuosity) of the pore network. An overview has been given by Cunningham [1], some discussions are presented by Karger and Ruthven [3] and Dullien [2]. Sometimes reasonable

9 w TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES w r r H GASES AND VAPOURS 341

predictions can be made, which require however a detailed knowledge of the pore shape as well as pore shape distribution.

The most simple definition of the tortuosity is that of the increased flow (diffusion) path length with respect to the shortest distance in the flow direction. This means 1: = Le/L w i t h L e the effective length. However other effects influencing the transport resistance play a role (e.g., roughness, interconnectivity, etc.) and are specific for the transport regime under consideration. For example the reflection coefficient 0 plays a role in the Knudsen diffusion but hardly at all in the viscous flow regime. Therefore some authors have split the tortuosity factor into a purely geometrical one (Le/L and in a part accounting for all other aspects. Other authors put all the structural complexities in a single parameter and some of them call it-torfuosity (which is incorrect). Karger and Ruthven define the tortuosity 1: with the help of:

aDp D = (9.12)

where Dp is the diffusivity for a straight cylindrical pore and D is the experimentally determined diffusion coefficient. So all complexities of the structure are hidden here in ~.

In practice it is simpler to treat the tortuosity �9 as an empirical factor and to determine it experimentally (see below). The same holds for other geometrical parameters like 0 and [~, discussed in connection with Eqs. (9.6 and (9.2). In porous pellets of packed particles a correlation of the type ~/~ = constant is frequently found [3]. The validity of this expression is not shown however for low values of the porosity (r < 0.30) and very small pore sizes. Experimental tortuosity values generally fall inthe region 2 < "r < 5, but in special cases much larger values havebeen reported. Leenaars et al. [17] reported values of "r = 6-7 for membranes consisting of a packing of plate-shaped (boehmite, gamma alumina) particles.

Experimental determination of geometric parameters of membranes The expression for the total flux Jt of a single gas through a homogeneous,

single wall membrane as given by Eq. (9.9) can be used for several purposes. For mesoporous membranes which are definitely in the Knudsen regime, the permeation plotted versus the average pressure P should give a horizontal line because F k is pressure independent. If the curve has a certain slope this points to a contribution of viscous flow and this in turn means that there are defects in the membrane in such a number and size that they cause a measurable viscous flow contribution.

A plot of F t v s P for defect-free membranes which are definitely in the transition region yields a curve with a certain slope, which intersects the permeation axis. This is shown in Fig. 9.5.


A

o

O

x f0

n

E

d) O E >,

l u

.O ca 4) E I - -

4) a.

U.

2.20

1.6S

1.10

0.SS

4 ,

m

"m

" . . . ,

H2

H�9

N2

CO

- ; .... ' ' ~' ..... i ' ' ....... ' ' I : - ~ ' ~ - - ' ' . . . . i : ' ~ ' ~ . . . .

0.5 1.0 1.5 2.0

p, Average pressure across membrane (Pa x 10 "s) (a)

Fig. 9.4. Gas permeabilifies versus average pressure at 20~ (a) and 538~ (b). Layer thickness of top layer is 3-5 ~tm. After J.C.S. Wu et al. [19].

From the slope the value of r can be calculated,while the intersection with the permeation axis (P = 0) yields the value of ~/(~.e) and so e can be calculated as well as the Knudsen contribution to the total flow. The pore size should be well defined in these cases and so the pore size distribution should be reasonably sharp. When the total porosity really is representative for all the active pores (thus, e.g., not many "dead ends" should be present), the value of the tortuosity ~ can then be calculated. Otherwise the parameter ~ is used as a fitting parameter.

Examples of this type of analysis are given by, e.g., Eichmann and Werner [18] for Nuclepore membranes with a pore diameter of 30 nm. The Knudsen permeability is given for several non-condensable gases and is reported to be 3.6x10-8/L and 1.1x10-8/L mol m/m2s Pa for H2 and N2 at room temperature respectively. Because the thickness L is not given, the actual flux obtained cannot be recalculated from their results. Wu et al. [19] and Keizer et al. [20] reported permeability data on thin gamma alumina layers with a porosity of about 0.5 and a layer thickness of 4.0 ~tm (Keizer et al. [20]) and 4.0-7.0 ~tm (Wu et al. [19]), supported by an (~-alumina supporting system. Wu did not correct for the support resistance. Some of Wu's results are shown in Fig. 9.4.

9 ~ TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 343

A

o , i . = ,

o I - =

x t~

(h t'~i"

d) O E > ,

mm =mm

.Q

O E I m

(I} CL

LI=

2 . 0

. =

1 . 5 - . =

. =

.=

.=

1 . 0 -

==

0 . 5 -

i ,r

A = L i i l l i l i I . . . . . . . . III

H2

He

N2 A - - - - - , .

I l l . . , .

C O

' ' I ' ' ' ' i .... ' "~ ' ' I +' ' ' '

0 . 5 1 .0 1 .5

p, A v e r a g e p ressure ac ross m e ~ n e ( P a x 10 4) (b)

Fig. 9.4 ( con t inued) . C a p t i o n oppos i t e .

2.0

The Knudsen permeabilifies (obtained for P = 0) at 20~ are 1.9x10 -1~ and 5.5x10 -11 mol m / m 2 s Pa for H 2 and N 2 respectively. Taking an average value of 4.0 ~tm for the thickness of the gamma alumina layer and assuming the resistance of the support ing system to be negligible, this yields values for the permeat ion F at room temperature of 5.5x10 -s and 1.1x10 -5 m o l / m 2 s Pa for H 2

and N 2 respectively. Keizer et al. [20] reported permeation data on a similarly made gamma alumina membrane, supported by a different alpha support and

\

corrected for the support resistance (see Section 9.5.2). Their results are shown in Fig 9.5. The N 2 permeation at 20~ is reported to be 4x10 -6 m o l / m 2 s Pa in reasonable agreement with the value reported by Wu et al. It must be noted that in the above-mentioned treatment the absence of surface flow is assumed.

Relative contributions of viscous and Knudsen flow; some data As will be discussed in Section 9.3, small contributions of viscous flow to the

total flow in the transition region can have a considerable effect on the selectivity in separations. Therefore some typical data are given in Table 9.2. for N 2 as

a reference gas. Note that for light gases (H2) the contribution of the viscous flow differs considerably from that given in Table 9.2.

344 9 - - TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

|

m,,i

0

�9

0 e .d

•

0 , i , a 0~ o

l 13) 0

. 0 . , , . . _

0 0

l l )

(2) "

0 100 tOO

Pressure (kPal

Fig. 9.5. Nitrogen permeation as a function of pressure for a supported 7-A1203 membrane at 20~ (1) support; (2) support + top layer; (3) top layer.

TABLE 9.2

Permeation data of macroporous supports and mesoporous layers for N2 at 20~ and an average pressure p = I bar in the transition region of viscous to Knudsen flow. The fraction of the viscous flow (b) to the total flow is given by Fr, the remainder is the Knudsen contribution (a)

Thiclaless (10-6 m) Permeation ( m o l / m 2 s Pa)

Pore radius (10-6 m)

10 1 0.1

2000 (a) 1.25x10 -'4 1.25x10 -5 1.25x10-6

(b) 2.8x10 -3 2.8x10 -5 2.8x10 -7

sum 2.92x10 -3 4.05x10 -5 1.55x10 -6

Fr 0.92 0.69 0.18

(a) 2.5x10 -4 5.1x10 -5 lx10 -5

(b) 2.8x10 -5

sum 2.8x10 -4 lx10 -5

F r 0.10 0.025

9 ~ T R A N S P O R T A N D S E P A R A T I O N PROPERTIES OF MEMBRANES W I T H GASES A N D VAPOURS 345

S lip flow The summation of viscous and Knudsen flow as given in Eq. 9.9 is not strictly

valid. In long cylindrical capillaries a minimum in the permeation has been observed at low pressure when plotting the permeation in the transition region versus the pressure [1,2]. This minimum has been described already by Knud- sen, but it has not been observed in porous media with a tortuous network, although it remains a controversial point in the literature. This minimum is caused by the occurrence of 'slip'. When the velocity of gas molecules at the wall is not zero, a slip (wall) velocity must be taken into account. This effect becomes significant when the mean free path ~ of the gas molecules is of comparable magnitude to the pore size (so in the transition region) and is negligibly small when ;~ is much smaller than the pore size. After the last collision of a gas molecule with the wall it travels a certain distance. "Wall velocity" means now the average flow velocity in the immediate vicinity of the wall, but still in the gas phase. At a distance from the wall equal to the mean free path, the gas molecules have, on average, a non-zero velocity and as the mean free path becomes an increasingly greater fraction of the capillary diameter, the wall velocity increases in significance relative to the average velocity. Starting with P = 0 and increasing the pressure, first the decrease of ~,/dp dominates (the flight length decreases) and so the flux decreases. At higher pressures intermolecular collisions increase and so does the flux.

The effect of slip flow can be treated either as an extension of a pure viscous flow or as an extension of a Knudsen flow. The simplest method is by adding an additional term (R/2~) �9 (P/RT) �9 dP/dz to Eq. (9.2), with ~ being the slip coefficient which is proportional to P.

As shown [1] the Dusty Gas Model expresses the slip flux in terms of a Knudsen diffusion. This implies that the slip flow is inversional proportional to the square root of the molecular mass and this has the interesting consequence that slip flow can contribute to segregative properties in gas mixtures.

9.2.3.3 Surface Diffusion and Capillary Condensation

Surface diffusion When the temperature of the gas is such that adsorption on pore walls is

important, experimental results show that the preceding laws for gaseous flow are no longer valid. Overviews of the subject have been given by Uhlhorn and Burggraaf [21a,b] and have been treated by many authors in detail [22-26]. The mechanism of surface flow is rather complicated and three main groups of mechanism can be distinguished [22b]:

The hydrodynamic model: In this model the adsorbed gas is considered as a liquid film, which can 'glide' along the surface under the influence of a pressure gradient.


The hopping model: This model assumes that the molecules can move over the surface by hopping over a certain distance with a certain velocity.

The random walk model: This model is based on the two-dimensional form of Fick's law and is most frequently used in the literature.

For relatively low surface concentrations, the surface flux Js for a single gas is generally described by the two-dimensional Fick law:

dCs Ds (9.13a) Is = - 1: dz

where Cs is the surface concentration (mol/m2). Expressed in terms of directly measurable parameters this gives:

J s - - P (1 - 8) (~] Dsdq dz (9.13b)

with Cs- qp(1 - 8) and p(1 - 8) the density of the porous material. To demonstrate the influence of the pore size on the magnitude of the surface flow it is considered that:

q - 0s" Sw "Csa t (9.14a)

and

s Sw = 2--= (9.14b)

pr

where 0s is the occupancy, defined as the mole fraction occupied by adsorption relative to a monolayer with sorption capacity Csat in m o l / m 2 and Sw is the surface area of the porous medium.

Substituting (9.14) into (9.13) one obtains:

2~ 2 Ds dO I s - - - Csat (9.15)

r dL

This expression shows that Js increases strongly with decreasing average pore size.

Assuming local adsorption equilibrium (adsorption processes are fast), Eq. (9.15) can be converted in terms of pressure instead of concentration using dq / dL = dq /dP .dP/dL and the expressions for the adsorption isotherm which relate q or 0s to P, e.g., for Henry's law

q - b. P (9.16a)

for Langmuir adsorption

KiP 0 i = ~ (9.16b)

1+KIP


with Ci,sa t the saturation concentration of i (mol/kg or mol /m 2) and q - 0 .Ci ,sa t

(kg/mol) = 0 .Csa t (mol/m2).Sw . Generally an Arrhenius (exponential) type of relation represents the diffu-

sion coefficient as a function of the temperature, with AQa the activation energy of diffusion. Similarly the parameters b and K (9.16) can be expressed with Arrhenius functions with Qa the (isosteric) heat of adsorption. Consequently Js is also activated with a total apparent activation energy of (Qa-AQa). For chemisorp- tion AQa has about the same value as Qa [1]. For physical adsorption the value of AQa is < (0.5--0.66)Q a. Since the surface flux is small at very low temperature as well as very high temperature there must be a maximum. The possibility of observing this maximum depends on the relative magnitudes of Qa and AQa.

Note that it is assumed so far that D~ as well as Qa are independent of 0. This is not true for larger values of 0. The value of Csa t usually decreases with increasing temperature.

Some data Recently, Bai et al [27] reported permeation and separation data of zeolite

membranes on a supporting system consisting of a thin gamma alumina layer (thickness 5 ~tm, pore diameter 5 nm) on an m-alumina substrate. The log-log plot of the (measured) permeation of the y-~ supporting system as a function of temperature for H2, Ar, SF 6 and isobutene was activated and gave linear curves with a slope of-0.66 to -0.76 depending onthe gas and conditions. For ideal Knudsen transport a slope of -0.5 is expected. Furthermore, the single gas permeation ratio of isobutene/Argon equals 2.4 (Knudsen ratio is 0.83) at room temperature and equals 2.2 at 770 K. This means that even at high temperature the transport of C4 hydrocarbons (in the Knudsen regime) is significantly increased by surface diffusion in the y-alumina layer. Uhlhorn et a1.[28] also reported surface diffusion on modified y-alumina layers with a pore diameter of 4 nm and unsupported layer thickness about 20-30 ~tm) and found that at 20~ about 30% of the total flux of CO2 through the membrane was carried by surface diffusion. Modification of the y-alumina with 2 wt% MgO strongly increased the adsorption (0 and C s a t in Eq. (9.15) increase), but this did not increase the value of Is due to the strong increase of Qa. Modification of y-alumina with 17 wt% of finely dispersed Ag increased the flux of H 2 considerably above the Knudsen level as shown in Fig. 9.6. At 25~ and P = 60 kPa the flux by surface diffusion is 2.5 times the Knudsen flux.

Increasing the H 2 pressure decreased the contribution of the surface diffusion. This is due to saturation of the adsorption (e approaches unity in Eq. (9.15)) with increasing pressure, causing the surface flow to become constant while the Knudsen flow continues to increase.

Finally, Sloot et al. [29] reported a surface flux contribution of about 40% of the total flux of SO2 in ~-alumina membranes with a pore diameter of 350 nm


12

Z g

T - 2 9 8 K

o \

O~Q,~ expiw m,mnt a~ ra rio

_ , ~ ._~ . m m ,,,,b m ~ ,,,,,, m , m u w m - , , m I I m u m m m m w u m i m ~ I w g ~ m

t l ' e ~ ~ rst~)

0 , | I ,, . . . . . . . . . . . . . .

0 100 2O0 3OO

p ~ a )

Fig. 9.6. Surface diffusion as shown by compar i son of the exper imenta l (e) and theoretical flux

ratio of H2 and N2 at 25~ on a n o n s u p p o r t e d 7-A1203 layer modi f ied wi th 17 wt% Ag. After U h l h o m

et al. [28].

and modified with impregnated y-alumina ~ in the temperature region 170-290~ and with P = 2-6 bar. This means that the membrane was in the molecular flow regime (note: Wicke--Callenbach measurements, no absolute pressure gradient) and the surface diffusion flux was combined with the flux from molecular diffusion in the gas phase.

An overview of data for different gas-membrane combinations is given by Uhlhorn [21]. It is concluded that in all treatments in the literature the surface flux is taken as an additional contribution to the gas flow and usually the total permeation is obtained as a linear combination of gas and surface permeation,

9 -- TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES wrrH GASES AND VAPOURS 349

0 ,M

0

0

surface flow

Temperature Fig. 9.7. Schematic v iew of total f low (permeation) as a function of temperature for the combinat ion

of gas and surface flow.

as derived from Eq. (9.15).This is an ad hoc assumption for which no justification is given. The generally observed trend of the total permeation of a single gas versus temperature (including surface diffusion) is given in Fig. 9.7.

Multilayer diffusion and capillary condensation An extensive analysis of data and theories describing permeation by surface

flow and capillary condensation is given by Uhlhorn [21a]. A fully satisfactory explanation of surface flow mechanisms has not been provided. Some very useful models and equations are however available and will be discussed below.

With increasing pressure and at temperatures below the critical temperature the surface coverage (occupancy) can become larger than unity. In this case the adsorbed molecules behave like a sliding film on the internal surface of the porous membrane under the action of a bi-dimensional spreading pressure related to the gas pressure. This situation is best described by a hydrodynamic model first proposed by Flood and Huber [30] and further developed by Gilliland et al. [31,32] and by Tamon and Toei [33,34]. These models cover the complete range of coverages including capillary condensation. According to Gilliland it follows that the permeation Fsm due to multilayer flow (and not too far from a monolayer coverage) is


I

,M

" i i 0

I

I

I 2 relative ~euure

Fig. 9.8. Schematic picture of the permeat ion as a function of the relative pressure in the presence of capillary condensate. After U h l h o m et al. [21]. (1) Onset of mult i layer adsorption; (2) pores are

completely filled.

2 RT Cs

= ~ ~ (9.17) Fsm ~ Cr L p

where Cr is the flow resistance, Cs the surface concentration and the constant ~t incorporating geometrical characteristics of the pore system.

With the onset of multilayer flow the measured flow strongly increases (see Figs. 9.8 and 9.10). It should be noted that in small pores the increasing thickness of the adsorbed layer decreases the effective radius of the pore for diffusion through the gas phase. This is important for the selectivity in binary mixtures.

At temperatures below the critical point of the diffusing gas, the increase of pressure first leads to multilayer adsorption until finally all pores are filled with liquid. This phenomenon is called capillary condensation and this process starts when the gas pressure P surpasses the pressure Pt given by the Kelvin equation which is for a cylindrical capillary:

RT Pt r~ s cos ~l/ V----~ In ~00-- 2 r (9.18)

where P0 is the saturated vapour pressure above a flat surface. This equation predicts that the smaller the pore radius, the lower the pressure at which capillary condensation starts, provided a good wettability of the pore surface by the condensate is present.

The general picture of flow due to capillary condensation is given in Fig. 9.8 for a narrow pore size distribution.

An important conclusion from Tamon and Toei's studies is that permeation


I ,ll I t

II I II II I I I

P 2 <P 1 < P t t 2 < t l < r

ITI P 2 < P t < P 1 t 2 < t 1 < r

P t < P 2 < P 1 t 2 < t l < r

P 2 < P t < P 1 t 2 < r < t 1

p t < p 2 < p l t 2 < r < t 1

l."r<,t,2<t i- ] Fig. 9.9. Six flow modes in the flow model of Lee and Hwang [35]. P2 and P1 are pressures at feed (high pressure) and permeate (low pressure) side of the membrane respectively; Pt is the pressure

above the meniscus; t is the thickness of the adsorbed layer.

for vapour flow is higher than for liquid flow. This implies that as soon as all pores are filled with liquid,the permeation drops (see Figs. 9.8 and 9.10). This qualitative picture was quantified by Lee and Hwang [35], based on Gillilands hydrodynamic model. In order to describe the transport, Lee and Hwang proposed six flow modes, which differ according to the site where the meniscus is formed, as illustrated by Fig. 9.9.

In this model also the decrease of the pore radius due to the formation of an adsorbed layer is incorporated. Flow I in Fig. 9.9 is the case of combined Knudsen molecular diffusion in the gas phase and multilayer (surface) flow in the adsorbed phase. In case 2, capillary condensation takes place at the upstream end of the pore (high pressure P1) but not at the downstream end (P2), and in case 3 the entire capillary is filled with condensate. The crucial point in cases 3 and 4 is that the liquid meniscus with a curved surface not only reduces the vapour pressure (Kelvin equation) but also causes a hydrostatic pressure difference across the meniscus and so causes a capillary suction pressure Pc equal to

2Gs cos Pc = - (9.19)

1"

The overall capillary pressure drop across the cylindrical pore is given by

A P c = ' ~ m l n ~ 0 - 1 n -" Wm Pm (9.20)


where Pm is the mean pressure and APg = P1-P2. Note that Eq. (9.19) predicts that the actual pressure drop is two orders of

magnitude larger than the gas pressure drop. Actually the capillary pressure Pc works on the effective pore radius, which equals ( r - t) with t the adsorbed condensate film thickness and which is assumed to be immobile; this point is discussed below. A force balance for one end of the capillary with length L and radius r then yields the effective capillary pressure Pc,eff:

2c~(r- t ) 2 COS ~ (9.21) Pc,eft = - r2

Combining (9.21) with the Kelvin equation (9.18) and using Darcy's law for liquid flow through porous media (as similarly done before for gas flow, see Eq. (9.1)) yields an expression for the gas flux Jcf3 of capillary condensate in case 3 of Fig. 9.9:

~tRT [(r-t1) 2 P1 (r- t2) 2 P21 Jcf3 - n VmL r----d--- In p--~- r--------d---- In -~o (9.22)

The other cases, which describe situations where the capillary is not completely filled and/or where the meniscus on one side is not present, follow from (9.22) by adjusting the relevant capillary pressure term and using the filled length z instead of the real length L. Note that case 6 of Fig. 9.9 obeys Eq. (9.2).

The film thickness t is estimated with the help of the BET adsorption equation (t plot) giving:

t = qBET Wt/St (9.23)

where Vt is the specific volume of the adsorbed layer, St is the specific surface area of the porous membrane and qBET is the amount of adsorbed condensate. The geometric constant ~t contains characteristic pore parameters (porosity, tortuosity) and is determined by liquid permeation experiments.

Some illustrative data A few publications have reported the permeation of capillary condensate in

inorganic, mesoporous membranes. Lee and Hwang [35], using their equations (9.22) and (9.23) found a good agreement between measured and calculated permeabilities in Vycor glass membranes for Freon at 19-41~ and a reasonable agreement for water vapour at 70~ Maxima in the permeation (or the permeability) are indeed found at relative mean vapour pressures Pm ranging from 0.6-0.8 and with permeabilifies of 20-50 times the Knudsen gas permeability.

Similar maxima have been found by Rhim and Hwang [36] for C2H 6, n-butane and CO2 in Vycor glass membranes and by Uhlhorn et al. [37] for propylene at 263 K in l-alumina membranes(pore diameter of about 3 nm) as shown in Fig. 9.10.


| 35 0

X 3O

r 25

|

pml 0

oM

~ 5

r 0 ......

0,00

~opylene ~..

desorption

~dsorption

nitrogen ; ~ A .,nlk i _ ,6 . ,= . �9 _ A A _

v v v ii v v - - - " T , v v

0,5O 1.00

Rel. pressure Fig. 9.10. Permeat ion of a suppor ted 7-alumina thin film for ni trogen and propylene at 263 K as

a function of the relative pressure of pr0pylene. After Uhlhom et al. [21,37].

Uhlhorn analysed his results in terms of the model of Lee and Hwang taking into account the slit-shaped pore geometry of ~-alumina membranes. This means that capillary condensation in the adsorption mode did not take place. The meniscus in the adsorption mode is formed due to the fact that the adsorbed layer thickness completely fills the pore width.In the desorption mode the usual description in terms of capillary condensation holds. Asaeda and Du [38] separated alcohols from water condensate by flow through a silica-alumina membrane with 3 nm pores. Sperry et al. [39] demonstrated that capillary condensation can be effective in permeation and separation with modified mesoporous 7-alumina membranes (pore diameter 4 nm) at elevated temperatures, provided the applied pressure of the condensing gas is increased. Capillary condensation of methanol was observed up to a temperature of 473 K at a partial pressure of 23 bar. This is about 0.65 Psat, so a large range of operation pressures is possible. Also here the permeability drops sharply at the onset of capillary condensation and is in this region about three times the Knudsen permeation.

A limiting factor for the maximum allowable total pressure drop across the membrane is the requirement of a stable condensate which should not be blown out of the pores. For methanol in pores of 4 nm in gamma alumina the total allowable pressure drop is 0.28 bar at 373 K and 0.05 bar at 473 K [39].

3 5 4 9 - - TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

Modified membranes and the transition to micropore diffusion Modification of mesoporous membranes can result in (i) a decreased pore

size which increases the contribution of surface diffusion, and (ii) a change in the nature of the pore surface and consequently a change in all types of interaction energies with the gas phase. Both phenomena have an effect on permeation and separation.

Abeles et al. [39a] investigated some fundamental aspects of capillary condensation and surface flow. The pore system of Vycor glass (pore radius 3.1 nm) was treated with ClsH38 molecules and in this way the pore radius was decreased to 2.1 nm. The permeation of toluene was then studied on both systems. A similar type of model was derived as used by Lee and Hwang and was investigated for case 2 in Fig 9.9. The main conclusion was that the effective viscosity of liquid toluene increases with decreasing pore size due to a rather strongly adsorbed monolayer on the pore walls. The flow of this layer is determined by a thermally activated friction parameter whose activation energy is considerably larger than that of the bulk liquid. Note that this layer should be similar to the layer assumed by Lee and Hwang. Surface diffusion is primarily due to molecules adsorbed on top of the first layer. The friction coefficient of the adsorbed layer on membranes treated with C18H38 is somewhat larger than that of the non-treated one due to a changed interaction energy. Local equilibrium is indeed obtained because the exchange rate of molecules between vapour and liquid (10 -1 mol/cm 2 s) is orders of magnitude larger than the molecular flux (10 -6 mol/cm 2 s).

Okubo et al. [40,41] treated Vycor glass membranes with tetra-ethoxysilane which was initially adsorbed and finally decomposed on the pore wall by heat treatment. The pore size was expected to be decreased by this treatment. As a result of this modification the permeation decreased and the permeation as a function of temperature increased (compared with that of the non-modified glass) for the gases He, 02, N2, Ar, H 2 and CO2 and became activated. The authors argue that surface diffusion cannot explain this result and suggest that the modified system is in the transition region of Knudsen to molecular sieving (micropore diffusion).

Rao and Sircar [42] made nanoporous carbon membranes (thickness 5 ~tm) by repeated deposition of a polymeric latex film on a mesoporous graphite substrate, followed by decomposition of the polymer to Carbon. Permeation and selectivity in separation of He, H 2 and some hydrocarbons were studied. The average pore diameter of the carbon membrane was estimated to be 0.5-0.6 nm. All hydrocarbons exhibited larger pure gas permeabilities than H2 because of their preferential adsorption on the carbon surface. The permeation was activated and followed the order H 2 < CH 4 < C2H 6 > C3I-I 8 > C4H10. For the higher hydrocarbons (>C2) the specific amount of adsorbed material increases as the molecular weight increases but the adsorption strength also increases

9 ~ TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS 3 5 5

causing a decreased mobility at the surface and resulting in a net decrease of the permeability. This is a similar phenomenon to that observed by Uhlhorn et al. [37] for CO2 transport on y-alumina modified by MgO. The transport mechanism suggested by Rao is surface diffusion, but from the size of the pores it is clear that the transport is at least in the transition region of surface diffusion to micropore diffusion.

9.2.4 Permeation in Binary Gas Mixtures in Macroporous and Mesoporous Membranes

9.2.4.1 General Considerations

Transport of components of a mixture through a porous medium is usually caused by mole fraction gradients (in isobaric systems) as well as total pressure gradients. The transport in the Knudsen regime, with mainly molecule-wall interactions, and in the continuum regime (bulk molecular diffusion) with mainly molecule-molecule interactions, are well understood. In the transition region between Knudsen and continuum diffusion interpolating models (e.g. the Bosanquet equation) are used to describe the effective diffusivity. A superimposed pressure gradient across the system forces convective motion of the components.

In order to take into account the effect of both mechanisms, more complicated models have been proposed [1,2,11,43-45]. Overviews have been given by Uhlhorn et al. [21] and more recently by Veldsink [46]. The models differ in the way the different mechanisms are combined and which coupling terms are taken into account. The most important coupling effects are the occurrence of 'drag effects' in mixtures and of momentum transfer between different species. Drag effects on molecular species a and b occur in isobaric binary mixtures a-b due to differences in molecular velocities between species a and b, which induce internal pressure differences causing a net flow of the mixture which has to be superimposed on the diffusive fluxes of a and b.

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamen- tally more correct than a description in terms of the classical Fick model, DGM/Maxwell-Stefan models yield implicit transport equations which are more difficult to solve and in many cases the explicit Fick type models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is


Feed Qf Xo, PF

Sweep , " /

/

Retentate Qr Y, P~

membrane

\ Permeate Qp /

T ,, Y, Pp

Fig. 9.11. A perfect mixing model for gas separation, x and y are mole fractions, Q is the molar flux at pressure p.

only incorporated in the later developments (extended DGM and Fick models [46]) but its implementation is not as straightforward as that of the other mechanisms.

In membrane systems, which require segregative properties e.g. in gas separation, usually large permeation in combination with a good separation factor (selectivity) is required. This is obtained by applying an external pressure gradient and a low partial pressure at the permeate (low pressure) side of the membrane (see also Section 9.3). A frequently used membrane system is schematically given in Fig. 9.11 as an example.

The gas is applied as a mixture to the retentate (high pressure) side of the membrane, the components of the mixture diffuse with different rates through the membrane under the action of a total pressure gradient and are removed at the permeate side by a sweep gas or by vacuum suction. Because the only segregative mechanisms in mesopores are Knudsen diffusion and surface diffusion/capillary condensation (see Table 9.1), viscous flow and continuum (bulk gas) diffusion should be absent in the separation layer. Only the transition state between Knudsen diffusion and continuum diffusion is allowed to some extent, but is not preferred because the selectivity is decreased. Nevertheless, continuum diffusion and viscous flow usually occur in the macroscopic pores of the support of the separation layer in asymmetric systems (see Fig. 9.2) and this can affect the separation factor. Furthermore the experimental set-up as shown in Fig. 9.11 can be used under isobaric conditions (only partial pressure differences are present) for the measurement of diffusivities in gas mixtures in so-called Wicke--Callenbach types of measurement.

Isobaric applications in the continuum regime, making use of molecular bulk diffusion and /or some viscous flow are found in catalytic membrane reactors. The membrane is used here as an intermediating wall or as a system of micro- reactors [29,46]. For this reason some attention will be paid to the general description of mass transport, which will also be used in Sections 9.4 and 9.5.


The treatment in Section 9.2.4 will first start with some simple limiting cases (Knudsen diffusion and viscous flow in mixtures), followed by a comparison of an extended Fick model with the DGM model derived equations for binary gas mixtures. Subsequently a treatment will be given of a direct application to membrane separation of a set of equations derived from the model of Present and Bethune by Wu et al. [18] and by Eichmann and Werner [19].

9.2.4.2 Knudsen Diffusion

For Knudsen diffusion collisions between particles are negligible and molecules of different species move entirely independent of each other under the action of their own concentration (or partial pressure) gradient. There is no fundamental difference between flow and diffusion. The resulting expression for the total flux Jk,t of a mixture with component fluxes Jk,1 and Jk,2 is

Jk, t = Jk,1 + lk,2 (9.24a)

where Jk,1 and Jk,2 are related by

/k,__}_l = X l (9.24b) Jk,2 1 - Xl

where xl is the mole fraction of component 1. The expression for Jk,2 and Jk,2 is given by Eq. (9.6).

If the mean molecular weight <M> is defined by

<M> -~ = Xl' Mi ~ + (1 - Xl)" M2 ~ (9.25)

then the Knudsen permeation of the mixture is obtained by inserting <M> -~ from (9.25) for (M) -~ from (9.7). Equation (9.24b) in combination with (9.6) predicts for a non-isobaric and equimolecular mixture (xl = x2 = 0.5) that the ratio Jk,1/Jk,2 is proportional to sqrtM2/M1. This is the ideal permselectivity of the mixture.

9.2.4.3 Viscous Flow and the Transition Region

Viscous flow The viscous flow of a binary mixture which is fully in the continuum regime

does not affect the concentration of both gases and relation (9.2) applies for the mixture as for a single gas with the mixture viscosity rl(x) of the mixture with a constant mole fraction x.

The viscous flow Jv~ of each species i of the mixture equals the total flow Jv,t multiplied by the mole fraction x (proportional to the partial pressure p):

Jv,i = xi. Jv,t (9.26)

and Jv~ given by Eq. (9.2).


The transition region of Knudsen and continuum diffusion or viscous]low Two important cases must be considered: (i) non-isobaric, and (ii) isobaric

situations. The non-isobaric situation will first be discussed.

- Estimate of magnitude of different contributions: According to Eq. (9.9a) the viscous flow increases with r 2 and with P, while

the Knudsen diffusion increases with r and is independent of pressure. This means that the contribution of the viscous flow to the total permeat ion increases with r and p.

Using relations (9.3) and (9.4b) or (9.7) it can easily be shown that in a first approximation the total permeation F can be writ ten as:

( 3 F ' P / ( 1 - ~ ) F = F k . l + l - - 6 . r l . v =Fk" 1+ .A (9.27a)

Using the gas kinetic relations between r, rl, v and ;~ we find A = n K~ and so:

( / f = f k �9 1 + ~ (9.27b) 16 �9 K n

Equation (9.27b) is useful to estimate the contribution of viscous flow to the total permeation.

For argon at 1 bar and 293 K it is found that with r = 10 nm (K n = 7), 98% is Knudsen diffusion, with r = 1 ~tm (Kn = 0.07), 67% is viscous flow and 33% is Knudsen diffusion. So with larger pores and higher pressure in non-isobaric systems viscous flow is the dominant contribution and molecular diffusion can be assumed to be negligible. Note that in this treatment m o m e n t u m transfer is ignored.

- The extended Fick model: An extended Fick type of equation is used by Veldsink [46] to incorporate this

momentum transfer. The total flux Ji of component i can be written as a superpo- sition of the total pressure driven viscous flow on the diffusional flow component.

5(xiP) Bo 5P) 1 D e ~ + ~ x ~ P (9.28)

J i - - RT 5z 11

where Bo is the permeabili ty coefficient, xiP the partial pressure and D e is the effective diffusion coefficient of i in the mixture. The term 'effective' indicates that geometric effects of the pore structure are incorporated in D e and Bo (with D e = TI/'cD ~ with D ~ the expression for a cylindrical pore).

In the transition region the transport resistances are assumed to be in series as expressed by the Bosanquet equation:

1 1 1 ~ = ~ + ~ (9.29) Die, j Diem Diek


where e e D i,m and D i,k a re the effective diffusion coefficients for continuum and Knudsen diffusion respectively. Di,k is given by the gas kinetic expression. So it follows:

4 ,N/8RT e (9.30) D i,k ----- T, -3 Kn ~ M i

If the mixture continuum diffusivity is unknown it can be estimated using Blanc's law:

t l

De = ~ 1 y__, Died �9 xj (9.31a) z,m 1 - - X i j=l, j~l

For binary diffusion (9.31a) reduces to

d In p _ Dll (Fick) (9.31b) Diem = D12 d In c i

Dij (here D12) is the diffusion coefficient of the pair i-j. It can be experimentally measured by Wicke-Callenbach type (isobaric) measurements (see Sections 9.2.4.3 and 9.4.2,3) or calculated with the help of the first order approximation Chapman-Enskog relation [1,4] which is written as

.N/,r3 +__ M___2.1 L M 2 j (9.32)

D 1 2 - 0 . 0 0 2 6 2 P (~12 ~'~12

where oh2 is the collision diameter (taken as the arithmetical mean of the individual component diameters), ~'~12 is the first order collision integral, which is tabulated by e.g. Hirschfelder [4] and which is a function of the temperature. P is the pressure in atm and D12 is obtained in c m 2 s -1.

The Dusty Gas Model (DGM) In the DGM model as presented by Mason and Malinauskas [11a] all the

different contributions to the transport are taken into account. The wall of the porous medium is considered as a very heavy component and so contributes to the momentum transfer. The model is schematically represented in Fig. 9.12 for a binary mixture (in analogy with an electrical network). As can be seen from this figure, the flux contributions by Knudsen diffusion Jk, i and of molecular (continuum) diffusion of the mixture Jm,12 are in series and so are coupled. The total flux of component i (i = 1,2) due to these contributions is Ji, km" Note that Jk, i = Jm,12. The contribution of the viscous flow Jv,i and of the surface diffusion Js,i are parallel with Ji, km and so are considered independent of each other (no coupling terms, e.g. no transport interaction between gas phase and surface diffusion).

360 9 -- TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

J m,12 I =''"

,i.

L I

L_

P Jr

J v,i J,.,

Fig. 9.12. Schematic representat ion of the Dusty Gas Model. Ji is the molar flux of component i; k = knudsen, m = molecular diffusion, v = viscous flow, s = surface diffusion.

The flux expression for a single species i in a multi-component mixture with n components according to the DGM model results in

1"/

~_~ xi ' J j - x.j. ' Ji Ji 1 ~)xi xi l BoP ) SP j=l, j,i Pi Dfj - p . Di,---~k - RT 8z + PRT ('11 Di,--~k + 1_ ~ (9 .33)

with Diek- g/T,.Di~ given by (9.30) and D,~ = 8/t.D~ given by (9.32)or directly measured. B0 is the permeability coefficient for a porous medium (m 2) and it can be obtained from the slope of the curve obtained by plotting the permeation F (in the transition region) versus the average pressure, as discussed in Section 9.2.3.2.

For multi-component mixtures the flux ]i as described by (9.33) can only be obtained in implicit form. For binary mixtures (9.33) can be solved directly in explicit form.

- Comparison of DGM and extended Fick models, some data: A comparison of DGM and the extended Fick model for the transition region

has been made by Veldsink et al. [46] and is illustrated by many transport data and applied to describe transport in a macro-porous membrane reactor. Their main conclusion is that for ternary mixtures the use of the DGM model is necessary and predicts the transport of a gas mixture within a few percent (5%). For binary gases usually the extended Fick model is sufficient, but with an overall pressure over the membrane the accuracy is less than that obtained by use of the DGM. A further discussion will be given in Section 9.7.


TABLE 9.3

Relative importance of molecular (continuum), Knudsen diffusion and Poiseuille flow for air at 20~ in a straight cylindrical pore (after Karger and Rutven [3])

Dpoiseuille p Dm r Dk D DPoisuille Dtotal (atm) (cm2/s) (cm) (cm2/s) (cm2/s) (cm2/s) (cm2/s) Dtotal

1.0 0.2

10 0.02

10-6 0.03 0.027 0.0007 0.027 0.026 10 -5 0.3 0.121 0.07 0.19 0.37 10-4 3.0 0.19 7.0 7.2 0.97

10 -6 0.03 0.012 0.007 0.019 0.37

10 -5 0.3 0.019 0.7 0.719 0.97

10 -4 3.0 0.020 70 70 1.0

The relative importance of different transport contributions in a porous structure is given in Table 9.3 which shows that the contribution of Poiseulle (viscous) flow becomes important in larger pores (range 0.1-0.3 ~tm). At high pressure (10 bar) the Poiseuille flow is already important in pores with a radius of 10 nm.

- The ex tended P - D model:

Present and De Bethune [48] were the first to develop a model (P-D model) including diffusion, intermolecular momentum transfer and viscous flow. Based on the P-D model, Eickmann and Werner [18] incorporated two parameters (n k and [5) i n t h e P-D equations to account for geometric and reflection characteristics of a real membrane. This extended P-D model is very successful to describe the effect of a variety of parameters on permeation and separation [18] and will also be used in Section 9.3. Note that surface diffusion is not incorporated in the model.

The flux of component i in a binary mixture is given by:

Ji g ' [ ~176 d(x .P) ~f~ dP , dP] = L 1 + B'--------P d-----~- + 1 + B ' P d z + x A P--~z (9.34)

with the mol fractions for components 1 and 2 (i = 1 or 2) given by x and l-x, respectively. The terms in (9.34) describe the Knudsen diffusion (1st term), momentum transfer (2nd term) and viscous flow, respectively.

The different coefficients in (9.34) are described below:

8r[ ~ g = - ~ 2 k T M (9.34a)


g '= nkr I r 2 ~g (9.34b)

Mathematically nk[-] accounts for the porosity (r and the tortuosity (z) for gas permeation dominated by Knudsen diffusion (see Eq. (9.16)). ~[-] is used to correct for behaviour deviating from the ideal Knudsen behaviour, e.g., due to reflection conditions deviating from elastic specular collisions with the pore wall.

3 r A = ~ (9.34c)

16rlv2

and

A A' = -- (9.34d)

B=8r qrckT q Mx 1 3---ff " - - -2--M " M1 + M2 P D12

(9.34e)

B' = B. [3 (9.34f)

= ~ / M 1 + (1 - x ) ~4M 2 (9.34g)

MIM2 M + = (9.34h)

M 1 + M 2

with M2 > M1

(9.34i)

o0: (9.34j)

- x / f 8kT / 34k

D12 in (9.34e) can be calculated from (9.32) or directly measured. Equation 9.34 is used by Eichmann and by Wu et al [19] to s tudy separation

in porous media and this will be discussed in Section 9.3. Wu et al. [19] used (9.34) for single gas permeation (see Fig. 9.4a,b) to obtain values of n k and ~ in an asymmetric membralox membrane consisting of a top layer of T-A1203 (thickness 3-4 gm, pore radius r = 4(-7.8) nm) supported on an o~-A1203 support.


Gases studied were He, N 2, H 2 and CO in the temperature range 20-815~ and pressure range 3-38 bar.

The pore diameter calculated from the measurements is not the same for all gases. The same holds for the values of n k which vary from 2.43x10 -12 m -2 (H2)

to 4.3x10 -16 m -2 (N2). The values of [3 (representing reflection conditions of molecules after colliding with the pore wall) decrease with increasing temperature for all four gases and strongly different values are found for the different gases. Especially the ~ values formed for CO are much lower: 0.27 (20~ and 0.06 (T = 538~ compared with that for N 2 w i t h ~ = 0.40 (20~ and 0.024 (815~ respectively. This unexpected behaviour of CO may be attributed to the interaction of CO with the aluminium oxide surface. The small value of [~ explains the much lower permeation of CO compared with the theoretical Knudsen diffusion in the membrane (for the other gases there is a good agreement) (see Fig. 9.4a,b).

It should be noted that surface diffusion of CO is possible, but is probably negligible because the permeation is decreased (with respect to expectations based on Knudsen diffusion) instead of increased (if surface diffusion is important).

- Determination of effective diffusion coefficients: The effective diffusion coefficient, and so the permeation of a component in

a mixture, can be determined with the so-called Wicke-Callenbach cell [7]. The cell has a similar design to that given in Fig. 9.11 but in this Wicke-Callenbach type of measurement there is no total pressure difference across the membrane (isobaric). The feed is in this case gas a, the permeate in Fig. 9.11 is replaced by an incoming flow of gas b (countercurrent configuration). Gases a and b diffuse through the membrane (counter diffusion) with fluxes Ja and Jb, and so the retentate (Fig. 9.11) is now a flow of gases a+db, the outgoing stream ('sweep' in Fig. 9.11) is b+da. In the measurement of D a, the volume flow ~v,d of the gas mixture b+da in the bottom compartment (d) and the concentration Ca,e in Qba,d are measured; this gives the mol fraction Ja,e.

In the equilibrium state using a mass balance over the cell and using the DGM expression for a binary gas (under isobaric conditions) it can be described that [49]

2 r P d ' Ya,d " Tcell 1 - 2ya , d A P

"a- "- (I)v'd Pcell Ya,cell T d = D e a + Ke a (9.35) �9 " ' Ya, cell ' Pcell

This equation takes into account that usually P, y and T are measured not in the cell but at a different site in the measuring equipment. A plot of the left-hand side of Eq. (9.35) v e r s u s AP/Pcell yields the effective flow factor Ke~ from the slope of the curve. The value of De,a can be calculated from the intersection of the curve at AP/Pcell = 0 because the mol fraction Ya,d is known.


9.3 SEPARATION OF BINARY MIXTURES IN SIMPLE M E S O P O R O U S M E M B R A N E S

9.3.1 Important Concepts

The separation of gas mixtures in practice can be performed in a variety of modes e.g. counter- or concurrent flows and cross flow (dead end mode) with different conditions concerning the variation of pressure and concentration on the feed and permeate chambers and along the membrane surface. Examples are discussed by e.g. Eichmann and Wemer [18]. The most simple experimental set-up, suitable to define some important parameters, is given in Fig. 9.11. Assumptions made here are well mixed flow on both permeate and retentate streams (this means constant concentration), no pressure drop throughout the permeate and retentate sides respectively and ideal gas behaviour. These assumptions hold usually in small modules with not too large membrane permeation. In large modules (long tubes, capillaries or large plates) with high membrane fluxes other conditions prevail. This will be discussed in Section 9.5.

The parameter to describe the separation efficiency for a binary mixture is the separation factor (x which is a measure of the enrichment of a gas component after it has passed the membrane.

(x= y . 1 - x (9.36) 1 - y x

with x and y the mol fractions of feed and permeate respectively. For a given mixture, 0~ is influenced by the membrane and the process specific parameters.

In mesoporous membranes the most effective separation mechanism outside the capillary condensation region is Knudsen diffusion. In this case the ideal separation factor 0~* equals the square root of the ratio of masses:

0~* = ~ M 2 / M 1 with M 2 > M1 (9.37)

In general c~* is not equal to ~ due to back diffusion, caused by non-zero pressure at the permeate side, or to contributions of non-separative mechanisms to the total flow and concentration polarisation on feed or the permeate side. Also the presence of surface diffusion influences the ideal separation factor.

Back diffusion due to a non-zero value of the pressure at the permeate side is a very general phenomenon to decrease the value of (x. The permeant gases at the permeate side of the membrane are removed by pumping or by a sweep gas. In the last case the total pressure is usually relatively large, but the partial pressure of the permeant is low. Using a sweep gas makes the mixture effec- tively a ternary system and ignoring the effect of the sweep gas (as is frequently done) is not always allowable as will be discussed in Sections 9.4 and 9.5.


If the pressure at the downstream (permeate) side is in the transition or continuum regime and is not negligible, there is a back-diffusional flux into the membrane decreasing the value of c~. Equation 9.38 gives the effect of back diffusion on the actual separation factor [23,24].

(1 - Pr) (a* - 1 ) a = 1 + (9.38)

1 + Pr(1 - y ) (0~*- 1)

where Pr is the ratio of permeate pressure divided by the feed pressure. It is obvious from (9.38) that the permeate pressure directly after the separation layer should be kept low. This is in principle possible with single wall, symmetric membranes. With asymmetric (supported) membranes the support represents always a certain flow resistance and this means that the actual, or partial, pressure of the interface between separation layer and support is larger than the pressure at the permeate side of the support. This implies that the flow resistance of the support should be as small as possible to minimise back-diffusional effects.

The separation factor a as determined from gas mixtures is generally not the same as the permselectivity which is defined as the ratio of the permeation of the single gases at a given membrane thickness. They are similar only when all interactions between the different phases and between gases and the pore wall can be neglected, e.g., in the Knudsen region and at high temperature (surface diffusion negligible).

9.3.2. Separation in the Knudsen and Transition Regions

As discussed above, the ideal separation factor (x" in the case of pure Knud- sen diffusion is given by Eq. (9.37) and is equal to the permselectivity provided that surface diffusion is not present (high temperature). As can be seen from (9.37) the highest ideal separation factors are obtained for mixtures of light and heavy gases. Back-diffusion effects are taken into account by Eq. (9.38) to give the real separation factor.

The support can have a considerable influence on the separation factor of the membrane consisting of separation layer and support when its flow resistance is not negligible and the gases in the support pores are in the transition or viscous flow regime [20]. This point will be discussed in Section 9.5.

In the transition region intermolecular momentum transfer decreases the separation factor considerably. The effects of the pressure ratio Pr, with feed pressure as a parameter of temperature of pore size and of concentration, are analysed by Wu et al. [18] and by Eichmann and Werner [19].

Wu et al. used Eq. (9.34) to simulate the permeate composition and separation factor for H2/N2, H2/Co and H e / O 2 gas mixtures and compared them with experimental results obtained on a Membralox asymmetric membrane system,

366 9 w TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

4 .00

3 . S 0 .

3.00

2.S0

3 . O 0

8. I . s o

1 . 0 0

0.S0 -

0.(

II ~ _ I _ ] I I I I II I I l i l l I I Il l i

. . " . ' ~ _ - ~ II ILl l a b I In _ ; . -__] . . . . | I I I I

1 I d e a l sepmidon

2

m I . . . . . m m i m m m - m m m m , m m

nomuaaon J �9 " " ' I ""'" " �9 I �9 i , " I ' I I I I I ' l �9 �9 I

lO 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0

Pressure Ratio, Pr Fig. 9.13. Feed pressure effect on separation of H2-N2 mixtures at T = 538~ feed H2X0 = 0.5, stage

cut = 0.01, pore diameter 5.6 run. Feed pressures (1) I atm; (2) 7 atm; and (3) 34 atm. After Wu et al.

[19].

whose characteristics are described in Section 9.2.4.3. Correction for the support resistance was not applied.

The simulation is generally in good agreement with the experimental results generated for a wide range of operation conditions (20-815~ P = 1-34 atm, P~ = 0.1-0.8, stage cut 0.01-0.36). Deviations between predicted and experimental mole fractions are within 10%, with a consistent overestimate of the light component in the permeate.

The effect of the pressure ratio Pr of permeate and feed and feed pressure on the c~ value of a H2/N2 membrane is given in Fig. 9.13 for T = 538~ and feed pressures ranging from 1-34 arm using a separation layer of 5.6 nm.

As is shown in Fig. 9.13 for a given pressure ratio, the higher the feed pressure, the lower the separation factor. At all pressures (1-34 atm) the separation factor decreases continuously with P~ (0.10--0.70). At P~ = 0.70 allsepara- tion factors converge to a value of 1.5. Note that even at the lowest pressure (1 atm) and lowest value of P~ = 0.10 the value of (~ = 3.20 which is considerably smaller than the ideal value (~* = 3.70) as given by Eq. (9.37). So even a small amount of non-Knudsen contribution to the total flow in a pore considerably influences the separation.

At higher temperature the separation factor increases because the mean free path increases and consequently less momentum loss is expected for H2. The effect is stronger at lower Pr value, and at Pr = 0.10 and P = 7 arm the values of


for H 2 / N 2 are 2.50 and 3.00 at 20~ and 815~ respectively for the same membrane as used in Fig. 9.13.

As the H 2 concentration in the discussed H 2 / N 2 mixture decreases the separation factor also decreases under selected operating conditions. When the partial pressure (concentration) of H2 decreases the number of H 2 to N2 collisions increase relative to that of the H 2 to H 2 collisions and consequently more H 2 momentum is lost at low H 2 concentration and the separation efficiency decreases. The effect is weak however compared to that of pressure and temperature, because collisions with the pore wall are much more frequent compared with intermolecular collisions.

Finally the 'stage cut' Sc = Qp/Qb (Qp,f = feed and permeate flow respectively, see Fig. 9.11) is important. At high stage cut the driving force for gas separation in terms of a partial pressure difference is reduced to maintain the material balance. At low Pr the effect of Sc is largest and the lower the value of Sc the larger the separation. For H 2 / N 2 and the conditions given for Fig. 9.13 with Pr

= 0.01 and P = 7 atm the values of ~ are 1.90 and 2.90 at Sc = 0.4 and 0.01 respectively (ideal separation is 3.70).

The effect of the pressure, temperature and pore radius on the separation factor is investigated also by Eichmann and Werner [19] using Eq .(9.34) with a constant and experimentally determined value of ~ for all gas membrane combinations, in contrast to Wu et al. Who fitted the value of ~ for each gas membrane combination. Figure 9.14 shows the effect of the pressure ratio Pr for different mean pressure levels P (assuming a linear pressure drop in the membrane) on the separation factor of a N 2 / C O 2 mixture (ideal separation factor equals 1.25) in a membrane with pore radius Rp = 0.03 ~tm.

In contrast to the situations given in Fig. 9.13, maxima can be seen which shift to larger Pr values with higher pressures. Similar curves are obtained for different pore radii as shown in Fig. 9.15, where the maxima become smaller and shift to larger Pr values with increasing pore radius.

The maximum is caused by a viscous flow contribution in the relatively large pores (0.015-0.12 ~tm) considered here. In the rising part of the curves the (non-separative) viscous flow contribution decreases with increasing Pr (smaller pressure difference). The contribution of the viscous flow decreases with decreasing pore radius and with small enough pores the maximum vanishes and continuously decreasing curves are obtained which exhibit greatly reduced pressure dependency. This is shown in Fig. 9.16 with similar shapes to those of Fig. 9.13. The results of Fig. 9.16 are obtained on membranes of y-A1203 with a pore radius of 2.5 nm as prepared by Leenaars and Burggraaf [17b].

In conclusion it can be said that the key operating parameters to approach the ideal Knudsen separation factor (determined by mass ratio) in mesoporous membranes are: small pore diameters; low pressure ratio, adjusted to produce maximum separation; relatively low pressure level; and high process temperature.


1.15

=2x10 s

1.10

C 0

0

1.05

1.00

Ix10 s

0 0.2 0.4 0.6 0.8 1.0

PR : I~ I p . Fig. 9.14. Influence of pressure ratio Pr on the separation factor of N2/CO2 mixtures. Pore radius

is 0.03 l~n. After Eickmann and Wemer [18].

To increase the separation factor above the ideal Knudsen separation factor requires contribution of surface diffusion and /o r capillary condensation or the presence of micropore systems.

9.3.3 Separation with Surface Diffusion and Capillary Condensation

The permeation of gases in membranes due to surface diffusion and capillary condensation has been discussed in Section 9.2.3.3. together with some illustrative data. The total flux of a single gas is usually calculated as the sum of the flux by surface diffusion and the flux through the gas phase. As shown the surface flux can contribute considerably to the total flux (increased by factor 2-3 of gas diffusional flux), especially with smaller and uniform pore sizes (compare Eqs. (9.9a) and (9.15). With decreasing pore size the flux through the bulk gas decreases while the surface diffusional flux increases. With very small pore diameter (< 2-3 nm) the effective diameter for bulk gas transport is less than the geometric pore diameter due to the thickness of the absorbed layer which


I , �9 I I I . . . . . . . .

i .

0 o 0

C o

....

,.mlw

o

0 Q. o ul

1.15

1.10

1.0S

0.03

(106

�9 0 015 pm

1.00

0 0.20.t . 06 (18 1.0 P,,

Fig. 9.15. Influence of pore radius r onthe separation factor of N2/CO2 mixture at a pressure of 2 bar. After Eickmann and Wemer [18].

decreases the space available for the gas phase. With gas mixtures this means that the bulk gas phase diffusion of a non absorbing molecule is decreased by absorption of an adsorbing molecular species in the mixture resulting in an increase of the separation factor. This is especially the case with lower temperatures of a few hundred degrees and intermediate pressures which give rise to partial blocking by capillary condensation. Some illustrative examples and special phenomena will be discussed below.

Separation by surface diffusion With gas mixtures, enhancement of the separation factor can be obtained by

preferential sorption of mobile species of one of the components of the gas mixture. Adsorption does not always lead to enhanced separation. In a mixture of light non-adsorbing molecules and heavy molecules, the heavy molecules move slower than the lighter ones but in many cases are preferentially adsorbed. Con- sequently the flux of the heavier molecules is better enhanced by surface diffusion and the separation factor decreases. This occurs, e.g., in CH4/CO 2


I

1.20

O u 1.15 0

- 1 2 a,10 s D 1.10- O, 0

6,2,~os./.z a , @

1.05

0 0.2 0.~, 06 0.8 1.0

Pn = p, I P, Fig. 9.16. Influence of pressure level on the separation factor of a N2/CO2 mixture. Pore radius is 2.5

nm. After Eickmann and Wemer [18].

mixtures in Vycor glass membranes. With two adsorbing molecular species, competi t ion for the adsorpt ion sites may exist and sorption isotherms for single gas species are no longer valid.

Uhlhorn et al [28] reported for a H2/N 2 mixture a separation factor of about 9 compared to the Knudsen value of 3.74. As shown in Fig. 9.17 the ratio of the H 2 flux over that of the N 2 flux decreases from 9 at a pressure of 50 kPa to 5 at 200 kPa. This result is obtained on ~'-A1203 membranes (thickness 100 ~tm, pore diameter 2.5-4.0 nm) impregnated with 17 wt% (finely dispersed) Ag. The increase of the H 2 flux is obtained by the Ag impregnation. Probably the decrease of the separation factor is caused by a decreasing contribution of the surface diffusion to the total flux with increasing pressure due to saturation of the adsorption.

Keizer et al [20] found a similar phenomenon for CO2/N 2 separation (with C - O ~ 2 as the fastest diffusing species) on non-modified ~'-A1203 membranes (0~ = 1.5-2.0 at 240 K, c( = 0.8 (Knudsen value) at 360 K, pressure I bar). In order to


12

Z 9

e~e--.~....e �9 l i o r e ~ r A i ratm

~ . . . . . . _ . . A _ _ _ 1 . , . ~ , , , , ~ . . . ~ , ~ . - , ~ . , , , ~ , , , , ~

0 100 2O0 30O

Fig. 9.17. Experimental (o) and theoretical flux ratio of H2 and N2 at 25~ on a nonsupported 7-Al203 layer modified with 17 wt% silver, measured in counter-diffusion configuration. After Uhlhom et

al. [28].

enhance the surface contribution the 7-A103 membrane was modified with 2.2 wt% MgO [28,20]. The result was a decrease of the separation factor to 1.0 due to the formation of strongly bonded, immobile CO2 species, the total concentration of adsorbed CO2 remaining constant. As shown by Eq. (9.15) this results indeed in a lower CO2 surface flux.

Separation by multilayer diffusion and capillary condensation (see also Section 9.2.3.3)

Brief overviews are given by Keizer et al. [50] and Sperry et al. [39] and these show that very high separation factors in combination with large permeation can be obtained in cases of mixtures of an easily condensable gas (vapour) and a difficult (non)-condensable gas which has a low solubility in the condensed phase. Pore blocking by capillary condensation takes place at 0.5--0.8 of the saturated vapour pressure (depending on pore size) and is preceded by multilayer diffu-


A

X 3o a.

2o 0 E C 0

m

0 E 0

q l "

r

"E 0 E to

c 0

m

E Q Q.

~ii 5 - , . . . . . . . . . . . . . .

A prol:)ytene .~

cleS~:~Dhon

.15 ~ a d s ~ p t i o n

10

s r . t rogen

0.00 (xs0 t .00

Rel. prestmm

2 0 - ' . . . . ' . . . .

C aclsorptton

( ~ ~ ,

o~oo

+ c ~ m c x p t | ~

p, o o y J ~ / " ~.

ru t rooen

~L p ~

30 - ' . . . . . -

B

" Oesor

O . . . . . . .

0.00 0.50 1.00

R~a. ~ m

9O

i m

o ~ 60

J l 30

D

O - ' '

ooo oa0 ,~x~

c l e ~ p t ~ n

iior

Fig, 9.18. Permeation and separation factors of supported y-alumina thin film for nitrogen and propylene at 263 K (A,B). Propylene is the preferentially permeating component; dashed line gives the relative pressure at which the maximum in the permeation of Fig. 9.18a. occurs. (C) and (D) as

(A) and (B) but for a supported film modified with MgO. After Uhlhom et al. [37].

sion and an increased flux of the condensable gas and an increased separation factor. Uhlhom et al. [37] reported separation factors of c~ up to 27 for propene/N2 (60:40) mixtures at 263 K (with propene the fastest permeating species). Note that the Knudsen factor is 0.8 and the permselectivity (ratio of single gas phase fluxes) amounts 7.4. As shown in Fig. 9.18, the region with the highest separation factors coincides with the maxima in the permeation curves which in turn are determined by the blocking of pores by adsorbate (capillary condensation).

The permeation of propylene at the maximum amounts 30x10 -6 mol /m 2 s Pa. A further improvement of the separation factor is obtained by modification of the ~/-A120 3 membrane with the reservoir method [51]. The membrane pores are filled up to 85% of the pore volume with MgO. This process enhances the value of 0c to 85 with a corresponding decrease of both propylene and N2 permeation values to 15x10 -6 mol /m 2 s Pa for propene (equivalent to 300 Nm3/m 2 day bar.

9 ~ T R A N S P O R T A N D SEPARATION PROPERTIES OF MEMBRANES W I T H GASES A N D VAPOURS 373

The shape of the curves (steeper, shift of desorption branch to lower relative pressures) indicates a narrower pore size distribution with smaller average pore size (below 3 nm) and less defects for the modified membrane.

Sperry et al. [39] reported capillary condensation up to 473 K in methanol/H2 mixtures for certain pressure ranges. They used a similar type of membrane as used by Uhlhorn, but treated with NaOH to poison the surface for chemical dehydration reactions.

Using the Wicke-Callenbach method (no absolute pressure drop) the highest value of o~ equals 680 (methanol being the faster permeating species) and is obtained at 373 K and 2.2 bar methanol pressure, with a methanol permeability of 51x10 -6 cm3(STP) cm/cm 2 s cmHg. At higher temperatures the maximum obtainable values of both {x and permeability decrease and {x = 110 (with methanol permeability is 4.2x10 -6 mo l /m 2 s Pa) at 473 K. (Note: 1 cmB(STP) cm/cm 2 s cmHg is equivalent to 3.12x10 -6 mol m / m 2 s Pa). Capillary condensation takes place at Pr = 0.60. This is considerably lower than predicted by the Kelvin equation (9.18) for pores with a diameter of 4 nm.

Separations with a pressure drop must be carried out with pressure drops smaller than 0.25-0.28 at T < 448 K or 0.05 bar at 473 K due to blow-out of the condensate under these conditions.

The observed flow rates in the capillarycondensation regime are larger than those obtained for Knudsen diffusion at lower pressures. Together with the results reported by Sperry e t al., the conclusion is that separation by capillary condensation yields a combination of large separation factor and high permeation even at increased temperature provided the appropriate temperature- pressure, pore size combination is chosen. A disadvantage is the sensitivity of the process for pressure changes (blow-out phenomena).

Finally, Asaeda and co-workers [52,53,64] reported separation results using membranes which are modified in such a way that pore sizes below the mesopore range (<2 nm) are obtained; no definitive pore characteristics are given however.

A type of pore blocking by one of the components occurs but whether this is capillary condensation is not certain. Asaeda and Du [38] reported values up to o~ > 100 for water-light-alcohol mixtures at 70-90~ in alumina-silica membranes. The water permeability is dependent on its concentration in the mixture. At atmospheric pressure and 20% water a typical water permeation value is 1.3X10 -2 m -2 s -1 (= 20 1 H 2 0 (liquid) m -2 day-l) . Azeotropic points can be bypassed in this way with an alcohol concentration much higher than the azeotropic concentration.

Similar results are given for mixtures of water and organic acids (acetic, propionic, acrylic) by Kitao and Asaeda [52] for rather thick (10 ~tm) silica membranes supported by 7-A1203 and made in a multi step process (up to 15 layers on top of each other). A permeation mechanism and a model for the pore

374 9 ~ TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS

structure is proposed by Kitao et al [53]. The pore shape is assumed to be conical, changing from rather wide on the support side to very small at the surface. Here the 'neck' diameter is suggested to be 0.4 nm. Equations for the (preferentially) permeating water flux are given. Near the surface an additional resistance to the flow builds up due to osmotic effects caused by rejection of organic molecules at the pore entrance.

9.4 PERMEATION A N D SEPARATION IN MICROPOROUS MEMBRANES

9.4.1 Introduction and Important Concepts

Existing ceramic, mesoporous membranes (with a 4 nm pore diameter) perform most gas separations according to Knudsen diffusion. The obtainable separation factors (Section 9.3.2.) are usually not economical for most gas separations and provide incremental but limited conversion enhancement in catalytic membrane reactor applications. Capillary condensation and preceding surface flow yield economically interesting separation factors but this mechanism is limited to easily condensable gases and is limited to rather low pressure drops due to stability problems (Sections 9.2.3. and 9.3.3.).

To enhance the separation factor the average pore diameter should be decreased considerably. According to Eqs. (9.9a) and (9.15) the contribution to the total gas flux of the gas (Knudsen) diffusion decreases and at the same time that of surface flow (diffusion) increases with decreasing pore radius. In recent years modification of existing membranes for improving their separation efficiency has been actively pursued especially by attempts to decrease the pore size of membranes. This resulted in different types of microporous membranes. Ac- cording to IUPAC convention these are porous systems with a pore diameter below 2 nm. In the literature the name 'microporous' is frequently misused and this should be avoided.

An overview of microporous membrane types is given in Table 9.4. The oldest microporous membranes are based on carbon and are reported by Koresh and Softer in a series of papers from 1980 to 1987 (see overviews in Refs. [6,42]). They are made by pyrolysis of a suitable polymer (hollow fibre) as reviewed by Burggraaf and Keizer [9]. More recently Rao and Sircar [42] developed a new technique. A macroporous graphite sheet was coated with a suitable polymer (latex) which was pyrolysed subsequently. This process was repeated 4-5 times and resulted in a total carbon layer thickness of 2.5 ~tm with an average pore diameter between 0.5 and 0.6 nm. The membrane has interesting properties (see Section 9.4.3).

Finally, very recently Linkov and Sanderson et al. [55] modified and improved the method reported by Koresh and Softer and produced flat sheets as well as hollow-fibre systems.


TABLE 9.4

Microporous membrane types

Type Ref.

.

2. 3. 3.1 3.2 4.

Carbon hollow fibre, film on (C) support Porous silica glass (Vycor) Amorphous silica based systems Sol-gel techniques C.V.D. Zeolite films on supports (alumina, steel)

42,54,55 56

21,57-63,64 65-68 69-79

Mesoporous glass (Vycor type) can be produced by a combined heat-treatment and leaching procedure [9]. Modification of this process can lead to microporous hollow-fibre systems with interesting properties as discussed by Shelekhin, Ma et al [56]. For further discussion see Sections 9.4.2 to 9.4.4.

The most promising results from the viewpoint of a combination of large separation factors and reasonable-to-large flux values are reported for supported silica based systems.

Burggraaf and co-workers reported in a series of publications [21,57,63] the sol-gel, two-step synthesis of silica and silica-titania films supported by a composite membrane of mesoporous 7-A1203 and macro-porous (x-al203. The film has a thickness of 50-100 nm and is situated for about 50% within the mesopores of the y-A1203 and for the rest on top of it. The pore diameters are around 0.5 nm. A combination of large separation factors and large fluxes was reported for several gas combinations [60,61] (See sections 9.4.2-9.4.4). As described by de Lange et al., the precursor sol consist of a polymeric silica solution with low fractal dimension [59,62] and the support quality (roughness) is important to obtain defect-free membranes [59,60]. Asaeda et al. [64] produced a microporous film directly in a macroporous c~-A1203 support with a 15-step coating process starting with colloidal silica solution and ending with a polymeric silica solution. This means that a mesoporous intermediate silica layer was first produced. The final top layer was said to have a pore diameter < I nm. The system shows very interesting (isomer) separation properties (see Sections 9.4.3-9.4.4).

In a series of papers (1989-1994) Gavalas and co-workers reported the synthesis of silica films in porous Vycor glass substrates with chemical vapour deposition (CVD) techniques [65]. A similar technique was used by Heung et al. [66]. The separation factors reported by Gavalas and by Heung are very high but the fluxes are low. In fact the silica layers are non-porous (no interconnected pore network). Wu et al. [67,68] improved the method used by Gavalas using a


composite support of 0~-A1203 with a 3-5 ~tm thick 7-A1203 in the top region of the 0~-A1203. A 1.5-3 ~tm thick silica film was deposited in the 4 nm pores of the 7-A1203.

The smallest obtained effective pore diameter in the silica plugs was estimated to be -- 0.5 nm. A combination of large separation factors and reasonable fluxes was reported for H2/N 2 and H2-isobutane mixtures (see Section 9.4.3).

Zeolite membranes form the most recent branch of the inorganic membrane field. It is only very recently that well characterised and properly described real microporous zeolite membranes have been reported [69,72-78,88,89].

Geus et al. [69,70] and Bakker et al. [70] described the synthesis of 50 ~tm thick silicalite (MFI) membranes on porous stainless steel supports; Vroon et al. synthesised 3 ~tm thick silicalite membranes on o~-A1203 supports [72-74]. These membranes consist of very small crystals (100-200 nm).

Jia and Noble and co-workers et al. reported a 10 ~tm thick silicalite membrane on a composite support of c~-A120 3 [27,77]. Finally, Xiang and Ma [76] partially filled the pores of a microporous (~-alumina support with ZSM5 crystals. All the authors used an in situ hydrothermal crystallisation method to grow directly polycrystalline zeolite layers.

The layers reported by Jia et al. and by Xiang and Ma contain a relatively large number of defects, in contrast to that of Geus/Bakker and Vroon, but nevertheless show interesting separation and flux properties provided that good condensable gases are present (e.g. methanol, xylenes).

The microstructure of the layers plays an important role as shown by Vroon et al. [72,74] as well as does the support (compare clay with stainless steel) as shown by Geus et al. [69,75]. Examples of properties will be discussed in Section 9.4.3.

Zeolite membranes on porous support with good to reasonable quality has been reported so far only for silicalite and (related) ZSM5 systems. In the literature since 1985 a number of other systems are reported including a series of patents. They are reviewed by Geus [69] and Vroon [72] and briefly by Matsukata et al. [78] and Burggraaf [79]. This older literature concerns either membrane systems which are not real (but very defective) membrane systems but sometimes have interesting properties for membrane reactors or concern single crystal work or very fragile non-supported membranes on which important fundamental studies have been performed.

In the first category belong the pioneering work of Suzuki (patents 1985, 1987) and of I.M. Lachmann (patent 1989) yielding NaA/CaA and X or Y or mordenite zeolites. Unsupported ZSM5 layers were prepared by Haag and Tsikoyannis (1992) and Sano (1991/1992). Work on single crystals of NaX and silicalite were reported by Wernick and Osterhuber (1985) and Geus [69] respectively. For literature references see cited overview papers.

It is not the place here to treat structural characteristics of zeolites. Neverthe- less a very brief summary with a focus on silicalite/ZSM5 systems is necessary


as a background for permeation/separation studies in Section 9.4.3. For details the reader is referred to books of e.g. Breck [80], Meier [81] and van Bekkum et al. [82]. Zeolites can be represented by the empirical formula [80]

M 2 /nO" al 203" xSi O2" yH 20

in which n is the cation (M) valence, x/2 is the Si/A1 ratio (equal to or larger than two). The cations M are present to balance the negative charge introduced in the crystalline framework by the substitution of Si 4+ by A13+. These cations can be exchanged (in exchange reactions). The aluminium-rich zeolites are hydrophilic (high affinity for water), the silica-rich zeolites are hydrophobic (small affinity for water) and /or organophilic. Also the thermal stability increases with increasing Si/A1 ratio.

The crystalline framework consists of a three-dimensional network of SiO 4 and A104 tetrahedra, linked to eachother by sharing the oxygen atoms. The framework structures contain channels of voids interconnected by ring openings. These channels can be isolated from each other (one-dimensional) or are interconnected by ring openings and form two or three-dimensional network structures.

More then 85 different framework structures are known [81]. Silica-rich zeolites are ZSM5 with a Si/A1 ratio of 11/1000 and silicalite (Si/A1 > 1000). Both have a similar structure (i.e. MFI type) but ZSM5 contains some cations and is more hydrophilic. The structure of MFI-type zeolites is given in Fig. 9.19. The structure has two sets of intersecting channels (10-membered oxygen rings, see Fig. 9.19b), one set consisting of straight channels with ring openings of 0.52x0.57 nm, the other set consists of sinusoidal channels of 0.53x0.56 nm (Fig. 9.19a). At the intersection points cavities are formed with a size of about 0.9 nm. The lattice of ZSM5 is stable up to 1175 K; that of silicalite to a somewhat higher temperature. Both zeolites have a good stability in strongly acidic environ- ments, are relatively easy to prepare and have a low affinity for water, which is important for (gas) separation properties.

In recent years zeolites with very large pores (supercages) and ring openings up to 0.6xl.32 nm (cloverite) have been synthesised.

9.4.2 Phenomenological Description of Single Gas Permeation

The theory of transport in microporous solids is complex and involves many aspects and steps. Although many aspects has been treated separately (e.g., sorption, diffusion, simulation studies, mechanisms, etc.) there are no coherent descriptions of permeation and separation in microporous membranes cover- ing all the important aspects. In this chapter an attempt is made to introduce such a description. It is useful to give a qualitative picture first (Section 9.4.2.1).


(.) (b)

Fig. 9.19. Schematic picture of zeolite MFI structure.

This will show that a quantitative description involving all the complexities in simple microporous membranes is not available (if possible). However a number of boundary cases can be described quantitatively, as in Section 9.4.2.2, and trends in more complex situations can be predicted in combination with the qualitative pictures based on mechanistic considerations.

9.4.2.1 Qualitative Description of Gas Permeation

As discussed in Section 9.4.1, the contribution of Knudsen diffusion to the total flux decreases with decreasing pore radius of the membrane material. Initially the selectivity of binary mixtures of gases is constant and equal to the Knudsen value.

Lin et al. [67] reported in the region between pore diameters of 3.0-2.0 nm small negative deviations for H e / N 2 mixtures, but with pore diameters < 2.0 nm a strong increase occurs to values above the Knudsen value. This is a typical phenomenon for microporous systems together with the onset of activated gas permeation.

As will be shown, it is useful to distinguish microporous membranes in systems with relatively large, intermediate and small pores. This is discussed by de Lange and Burggraaf et al. [59,63] and is schematically shown in Fig. 9.20. Note that here the location of the minima and the shape of the potential as a function of z is given schematically and is not exact.

Simulation results yielding pictures as given for region c2 are reported by Petropoulos and Petrou [83]. For mesopores the minimum in the potential curves is equal to the (isosteric) adsorption heat at 'free' surfaces with respect


r 2- -

0

II <--2 N

4-

deft , B , ........ ,

1!1 - ' nt si i

a b l b 2 c1 c 2

A Z

r /OA.- 0.9 1.086 1.239 2 3

Fig. 9.20. The relative potential U = U(Z)/UA* of molecule A as a function of the distance Z from the pore centre for several values of the relative (dimensionless) pore radius r~ r~A for three different regions (a-c). UA* is the absolute value of the (Lennard Jones) potential on a free surface, OA the

molecular diameter. After de Lange et al. [59,63].

to the ambient a tmosphere (horizontal line in Fig. 9.20, the effects of curvature are neglected here).

In the central region of the pore the gas molecules move freely (Knudsen). At the boundary of meso- and micropore region the potential fields of both walls start to overlap and the potential curve shows two minima separated by a m a x i m u m (region c2). This max imum decreases (larger negative values) wi th decreasing pore diameter. This means that molecules are adsorbed more strongly. For relatively wide pores the enhancement of the sorpfion energy wi th respect to a free surface is significant and is calculated by Everett et al. to be 30-50% for R/r~ = 3 and cylindrical pores [84]. The m a x i m u m enhancement is by a factor of 3.39 at R/r~ = 1.086 (region bl).

With decreasing pore size the desorpfion energy from the wall to the gas phase within the pores (the max imum in the curves in Fig. 9.20) becomes smaller but remains positive. This implies that the molecules in the central part of the pore behave in a Knudsen-like manner (i.e. no intermolecular collision) and can pass each other (region q, upper part of Fig. 9.20) but nevertheless are not 'free' and follow curved trajectories (see Ref. [83] and Sections 9.4.3.1-2). In this region c we can speak of a surface flow enhanced micropore difhasion (SEMP). Because in surface diffusion the activation energy is a fraction of the adsorption heat (see


Section 9.2.3.3) this implies that the transport in the SEMP regime is activated. With further decreasing pore size, a single minimum is finally obtained at R

= 1.24 G (region b2). When the effective pore diameter def t < (~A + (~B (region bl in Fig. 9.20) the molecules in cylindrical pores cannot pass each other (except e.g. at wider intersections or channels) and the molecules interact strongly during diffusion. Finally in region a the pore diameter is about equal to the molecular diameter and molecules have increasing difficulties to enter the pore (less negative sorption heat) and relatively large molecules cannot enter the pore at all. This is the size exclusion region with mixtures.

In regions b and c we cannot speak any longer of 'bulk' gas phase in the pore and the diffusion shows strong similarities with solid state diffusion (solution- diffusion models).

From the discussion so far it follows that sorption as well as diffusion play a role. Their relative importance depends on such sorbent material characteristics as pore size, sorption strength, and gas properties such as molecular size and shape, concentration, etc.

A theoretical model describing gas transport in microporous (single crystal zeolite) membranes in regions a and bl of Fig. 9.20 was proposed by Barrer [85] and is schematically given in Fig. 9.21.

Essentially the flux J consist of two parallel and additive components F1 and F 2. The flux F1. J c o m e s directly from the gas phase to the first sites 01 in the micropore via the pore entrance. The flux F2"J consist of two parallel fractions f2"J andfl-J, each consisting of several sequential steps, the first being an adsorption step. Infl.J adsorption takes place at sector 00 at the external surface near the pore entrance. Jumps from 00 -~ 01 may require the passage of an energy barrier.

The flux f2"J involves first adsorption at the external surface at sites 00~urf followed by surface diffusion from sites 00~urftO sites 00 and subsequently from 00 ~ 01. The sites 01 are occupied to a degree (concentration) depending on the sorpfion isotherm.

In all cases the steps at the surface are followed by micropore diffusion in the pore channels from sites 01 t o the other side of the membrane. Here desorpfion takes place directly to the gas phase or via desorpfion from the pore to the external surface and than to the gas phase. Equations for the several steps and the total flux have been derived and will be discussed in Section 9.4.2.2. Some important conclusions can be drawn however from this qualitative picture:

(i) Direct entrance from the gas phase (FI-J) is important for relatively small, spherically shaped molecules and will then dominate the flux for weakly adsorbing molecules or at high temperature. For larger, branched molecules (e.g. hydrocarbons) direct entrance is unlikely and an adsorption step at the external surface is necessary.

(ii) The adsorption step is important at lower temperature (increasing concentration). With increasing temperature the diffusion rate becomes more im-


G A S P H A S E

f2.J fl .J

A

F1 .J 1 +F2"19

A~ 6n.,,rt

PORE Fig. 9.21. Schematic model of gas permeation in microporous membranes. Flux F consist of fractions

F1 (direct from gas phase to site 01) and F2 (=fl +f2) (via adsorption at the external surface).

portant, while the concentration decreases (adsorption isotherm). So maxima in the flux can be expected at a certain temperature given the pressure.

(iii) With mixtures (gases A and B) competitive adsorption can take place and a strongly adsorbing gas can exclude a weakly or non-adsorbing one from being sorbed. This will severely affect the flux component F 2 and may enhance the selectivity (sorption selectivity) depending on the differences in mobility of A and B (diffusion selectivity).

(iv) In the size exclusion range adsorption of a large, non-penetrating molecule at the external surface (at sites O0) can block or strongly hinder the flux of the smaller, penetrating component. This decreases flux and selectivity.

(v) At larger concentrations (occupancy degree) within the pores of components A and B, strong interactions will take place and in regions a and bl of Fig. 9.20 selectivity by mobility differences vanish and only selectivity by sorption remains.

As will be shown later all these phenomena have been observed. The consequence is that the permeation behaviour in mixtures depends strongly on the character Of the mixture and it is necessary to distinguish several categories in terms of combinations of weakly (W) and strongly (S) adsorbing gases as shown in Table 9.5 [72,74]. The quantitative description of permeation and separation in terms of operational equations is today only reported in a few limiting cases mainly characterised by relatively low concentrations (Henry regime, initial part of Langmuir type regime) or by size exclusion at higher temperature where adsorption at the external surface is not important and we have essentially permeation of a single gas species.

Finally should be stressed that firm conclusions on the magnitude of permeation and separation factors are only possible after appropriate control of the defect level of the membranes (non- micropore/Knudsen contributions to the


TABLE 9.5

Overview of categories of (binary) gas mixtures in terms of separation regimes as a function of relative concentrations and mobilities

Regime Code Concentration on Concentration in Mobility in the the external surface the zeolite pore zeolite pore

I W-W A:low; B:low A:low; B:low A < B

IIa W-S A:low; B:high A:low; B:high A < B IIb W-S A:low; B:high A:low; B:high A > B IIIa S-S A:high; B:high A:high; B:high A = B IIIb S-S A:high; B:high A:high; B:high A > B or B > A IV S-S (SE) A:high; B:high A:high; B:low* A > B, A >> B V A:low; B:low A:low; B:low* A > B VI A:high; B:high A:low; B:low* A > B

w and S are weakly or strongly adsorbing components respectively of mixtures: W-W, etc. SE is size exclusion. *B is large molecule.

flow). The best way to perform this is determinat ion of permeat ion values and ~ separat ion factors in binary mixtures of gases consisting of small, weakly adsorbing and very large molecules at high temperature. This is the size exclusion regime under exclusion of strong adsorpt ion on the external surface of the larger molecule.

9.4.2.2 Quantitative Description of Gas Permeation and Separation

Single gas permeation The equat ions given below are derived for single wall or unsuppor t ed

membranes under similar conditions to those given in Section 9.2.4 and Fig. 9.11. These are homogeneous and uniform concentrations (well mixed) and pressures on the feed and permeate sides of the membrane and near equilib- r ium between concentrations in the bulk gas phase and in the membrane surface. As discussed in Section 9.4.2.1, small and large micropores should be dis t inguished. This t reatment will start wi th a general descript ion which is appl ied to small micropores. Subsequent ly the consequences for larger micropores will be treated.

Under isothermal conditions it follows from irreversible the rmodynamics [1-3] for the flux Ji in a mixture of k components:

Ji = - ~_~ Lik V ~tk (9 .39 )

k

9 n TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 383 _

So the real driving force is the sum of the gradients of the chemical potentials as is also implicit in the general Maxwell-Stefan formulation [87-89].

For a single gas this reduces to:

3 In P dqi (9.40a) J i - - ~tDo, i 0 ha qi dz

Here the term 31nP/31nq is the so-called thermodynamic factor (hereafter called F), Do(q) is the corrected or intrinsic diffusion constant and Ix is a correction term (see notes below). With q = qsat'0, Eq. (9.40a) becomes:

d0i Ji-- ~tqsat, i D 0 ( 0 ) F d---~- (9.40b)

Notes on Eqs. (9.40a,b): (1) When qsat~ is expressed in mo l /kg the density (kg /m 3) enters the nominator of Eqs. (9.40a,b) and ~t = p. (2) If the zeolite is supported with a support having porosity ~, the effective surface area of the zeolite available for transport is ~.m2/m 2 and the term ~ enters the nominator of Eqs. (9.40a,b). If the flux is measured on a supported system and one wants to calculate the intrinsic zeolite properties, ~ enters the denominator of Eqs. (9.40a,b). (3) The term Do(q).blnP/blnq is identical to the Fick diffusion coefficient DF, while the intrinsic diffusion coefficient Do(q) is identical to the Max- well-Stefan diffusion coefficient Dms.

The thermodynamic factor F corrects for differences in activities (chemical potentials) of different gases which can exist with similar concentration gradients. It is similar to the factor that has been described in solid state diffusion by Darken and is sometimes named after him.

Equations (9.40a,b) can be integrated over the thickness L of the membrane to yield expressions for the flux of specimen i:

qp

Ji dz = - ~ , D 0 (q) F dq (9.41)

qf

with q - qf (feed) at z = 0; q = qp (permeate) at z = 1. Note that qp and qf are steady-state concentrations which are not necessarily

equal to the equilibrium concentrations. Equation (9.41) can be integrated under a number of different boundary conditions which will be treated below.

The Langmuir and Henry adsorption regions In many cases single gas adsorption in zeolites can be adequately describe d

by a Langmuir-type adsorption isotherm as given in Section 9.2.2.3.:

KiP O~ = ~ (9.16b)

I+KiP


with Ki the equilibrium Langmuir adsorption constant (Pa-1). Inserting (9.16b) in the expression for F yields the equation for the thermo-

dynamic factor in the case of Langmuir adsorption:

1 F - (9.42a)

1 - 0

At higher values of 0 small deviations from the Langmuir isotherm are corrected in (9.42a) by introduction of an empirical constant k [86b]"

k F - (6.42b)

1 - 0

Substitution of Eqs. (9.16b) and (9.42a) in (9.41) and integration, assuming D is not dependent on q, yields an explicit equation for the single gas flux in the Langmuir regime in terms of sorption and diffusion parameters:

Do, i" qsat,i In qsat, i - qf, i (9.43a) J i - L qsat, i - qp, i

1 + KPf, i _ ~ D o , i qsat, i In (9.43b) L 1 + KPp, i

Equation (9.43a) can easily be converted in terms of occupancies by dividing numerator and denominator of the In term by qsatd. Note again that Do,i is the intrinsic diffusion coefficient and that DFick = D0/(1 - 0 ) and so DF increases strongly when e assumes larger values.

At low occupancy we are in the Henry regime and Eqs. (9.16b) and (9.43b) can be simplified because KiP < 1:

Ji = ~ D ~ " qsat'i " K L (Pf, i - Pp, i) (9.44)

The temperature dependency of Ji can be introduced using a van 't Hoff-type relation for K and an Arrhenius relation for D:

i: 0iexp

Do, i = D~,i exp - (9.46)

where Ed is the activation energy for diffusion in the micropores. Insertion of (9.45) and (9.46) in (9.44) yields the temperature dependency of

the flux in the Henry regime of a supported zeolite (~t ~ 1):


Ji = -~ D~,i Ko, i qsat, i exp (Ed, i - Qa, i)

RT (Pf- Pp) (9.47)

Here D~,i is the pre-exponential coefficient of the intrinsic diffusion coefficient. Note that K'qsat is the Henry constant b as given in Eq. (9.16a) (mol kg -1Pa -1

or mol m -3 Pa -1) which can be directly determined from experiments without separate knowledge of the value of qsat. Equation (9.47) shows that the flux Ji is activated with an apparent activation energy (Ed~- Qa~) which is determined directly from permeation experiments. Since both parameters are positive quantifies, positive as well as negative values can be expected and the flux can be increase as well as decrease with temperature depending on the relative values of Ed~ and Qa~. Equation (9.47) has been used by several authors to describe, analyse and/or simulate permeation and diffusion in silica [59,63,92] and in zeolite membranes [69,72,75]. 1

- Some limiting cases and discussion: At high concentration (high 0, low temperature, relatively large pressure), but

within the Langmuir regime, KP >> I and with (9.16b) and (9.43b) or (9.41) one finds

d ln pi li = - ~t . Do, i(q) �9 qsat, i dz (9.48a)

and

bt P Li ]i = -~ Do, i qsat.i i n /~p,i (9.48b)

and with (9.45) and (9.46) assuming qsat~ is independent of F:

~ t , ln Pf, i E(_~I = ~ exp - (9.48c) Ji ~ D~ qsat'iln Pp, i

Equation (9.48c) shows that at high values of 0 (low temperature) the apparent activation energy of the permeation equals that of the diffusivity provided that intra-crystalline diffusion is still the controlling mechanism.

Outside the Henry region calculation of the permeation from adsorption and diffusion data requires knowledge of the value of qsat~" Especially for weakly adsorbing gases the value is not always known nor can be easily determined from experiments. As discussed by Kapteyn et al. [88] the value of qsat can be estimated from the molar volume which is obtained from extrapolation of the liquid state [90] or from volume filling theory [91]. Some results will be discussed below (binary gas permeation). In the Henry regime separate values of qsat are not necessary as discussed above and the product K'qsat~ = b (Henry coef.)

1 Ed~ can be larger than Qa,i because molecules can penetrate pores directly without preceding adsorption.


can be directly measured and used in the permeation equation. To obtain Eqs. (9.43), (9.47) and (9.48) it was necessary to assume that the

intrinsic diffusion coefficient D o should be independent of the concentration (occupancy). This is only correct when there are no intermolecular interactions, so for lower values of the occupancy 0. An extensive discussion has been given by Xiao and Wei [86]. Based on model calculations and analysis of experimental results, they showed that Do is approximately constant until 0 -- 0.5-0.6 and then starts to change in a way depending on the value and the character of the interaction energy. This is expressed by a parameter W = A E / R T being a non-dimensional energy change from the non interacting to the intermolecular interacting situation. Analysing their model results it is obvious that for W = 2 and until 0 = 0.85 the relation between DFick and Do is almost similar to that obtained from the Langmuir adsorption type of isotherm. This is equivalent to an occupancy independent D o until large values of 0. For 0 _< 0.5 this holds for all values of W _< 3.

9.4.2.3 Permeation and Separation in Binary (Ternary) Gas Mixtures

For multicomponent gas mixtures the generalised Maxwell-Stefan (GMS) equations should be used. Krishna [87b] derived an expression for the flux of specimen Ji:

n

Oi V~t i = ~,, RT

j=l j~i

- l i O/Ji Oi J/ + (9.49) qsat, i Di,j qsat, i " D i

The first term at the right-hand side represents the friction due to the counter- exchange of adsorbed molecules, while the second term represents the friction with the zeolite. Note that this equation shows a strong similarity with the GMS equation (9.33) for gaseous diffusion.

For a two-component gas mixture, Eq. (9.49) reduces to a set of three equations [89]"

J1 J2 A 1 A 2 0 ln(1 - 01 - 02) + - + - (9.50)

~t " qsat,1 " D1 ~ " qsat,2 " D2 D1 D2 0z

001 A1

3z D1

302 A2

~)z D2

01 + q- 02 ~ - - 01 ~ (9.51a) D12 D12

A(~_~ 1 A2) A1 A2 - 02 + ~22 - 02 ~D12 + 01 ~D12 (9.51b)

and A1 and A2 defined by (9.51).

9 -- TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 387

For small por~s with a pore diameter of about the same size as the molecular diameter (situation b 2 in Fig. 9.20) counter-exchange will not take place and terms with D12 a re absent. This is the so-called single file diffusion for which can be written [89]"

D1 { ~01 C)02 l J1 - - ~tqsat,/(1 - 01 - 02) (1 - 02) -~z + 01 --~-z j (9.52a)

{ l D2 (1 - 0~) + 02 (9.52b) 12-"--~tqsat,2 1 - 01 -- 0 2 ~ Z - ~ z J

The second component affects the diffusion of the first component in two ways: (i) by occupying a number of sites which now are not any longer available for component 1 (term (1 - 02) in denominator of D), and (ii) by an entrainment contribution due to the gradient of component 2, which can be either positive or negative depending on conditions.

If single file diffusion prevails the single component permeation data (which determine Di) and single component adsorption data are sufficient to predict the binary fluxes via Eq. (9.52).

The values for the concentration qi or occupancy 0i must be determined from the competitive Langmuir adsorption isotherm:

qi Ki Pi 0i = ~ = (9.53)

qsat, i n

1 + E KiPi i=1

In most cases these adsorption data from gas mixtures are not available and estimates must be made by insertion of single gas data (K and Qa values) in (9.53) and setting

/l

0i = 1 (9.53a) i=1

- Permeation in large micropores: The situation in large micropores is schematically represented in situation C1

and C2 of Fig. 9.20 and as discussed in Section 9.4.2.1. A part of the molecules move rather "free" in the gas phase (central part of pores) with a free length of diffusion in the Knudsen range [Xiao, 86] but with curved path trajectories [Petropoulos, 83], an increased potential with respect to the gas phase outside the sample and a small activation energy.

Shelekhin et al. [92] have modeled this situation while in the transition region Xiao [86] describes the total micropore diffusion coefficient D t as:


-1 (9.54) Dtl= DK 1 + Dconf

where Dconf is the configurational diffusion coefficient in small micropores. Shelekhin developed his model for a single wall (Vycor type) membrane

with pore diameter of approximately 1.5 nm. The large pores in the tortuous network are interconnected by pore openings(windows) with a smaller diameter in the range 0.5-0.6 nm.

As discussed in Section 9.4.2.1. and Fig. 9.20 there are two possible states for the diffusant molecules inside relatively large micropores (1.0 < dp < 2.0 nm). In the SEMP model a certain fraction of the gas molecules move through the "gas phase" in the pore, while the remainder reside on the pore walls. According to Shelekhin [92] the total gas concentration CT inside the membrane is determined as:

CT = Cg + Ca (9.55)

where Cg and Ca are the gas phase and adsorbed phase concentrations respectively in mole /m 3 (membrane).

The permeation is described by Eq. (9.40a) for both phases and the total permeation is taken as the sum of both gas (in the pore) and surface flow.

An expression for the adsorbed concentration is obtained with the help of the Dubinin-Radushkevitch adsorption isotherm for microporous materials:

W p p (9.56a) C R = Vm

where pp is the density of the porous medium (kg/m3), Vm is the adsorbate molar volume (m3/mol) and W the adsorbed volume given by:

I W = W0 exp A (9.56b)

where W0 is the limiting adsorption volume, [31 is an empirical factor (affinity coefficient), e is the adsorption potential e = RT ln(po/p) which is equal to the free energy to remove one mole of adsorbate molecules from the surface to the gas phase and W is the adsorbate volume at temperature T and relative pressure P/P0. Note that the meaning of W and W0 are equal to that of q and qsat in (9.40) in the case that CT = C~ (see below).

The diffusion process is described now by a single diffusion coefficient D:

D = ~ v v (9.57a)

v = v g. v E exp - (9.57b)

where v is the molecular mean velocity, ~ is the molecular mean free path, Vg is


the probability that the gas molecule jumps in the desired direction (a geometrical parameter) and VE is the probability that the molecule has sufficient kinetic energy to surmount the energy barrier AE.

Assuming all the obstructions in the way of diffusing molecules are rigid, expressions for Vg can be derived. For the case discussed this yields"

1 a 2 2 (9.58) Vg=3 dp

where dn is the diameter of the window betweenthe large pores with diameter dp.

Expression (9.58) accounts to some extent for shape selectivity which occurs with non-spherical molecules.

The thermodynamic factor F in Eq. (9.40) can be calculated now with Eq. (9.56) and yields for the adsorbed phase:

~2 P0 F =-~- (RT) 2 In ~ (9.58a)

P

with A an empiricalfactor. Substituting (9.57b) in (9.57a), using the gas kinetic expression for v, taking s

=dp yields expression for D in both gas and surface phases:

(8az~ 0"5 /AEad,/ DO,ad s =Vgdp ~/t M ) exp - ~ aT ) (9.59a)

(8RT/~ (kEgas I Do,gas =Vgdp (=M) exp -~ RT- (9.59b)

where Do is the corrected (intrinsic) diffusion coefficient which is related to the Fickian diffusion coefficient by DF = D0.F (see Eqs. (9.40b) and (9.58a)).

The total transmembrane flux is obtained now by summing up the expressions obtained from (9.40) for bulk gas and surface flow:

dCads D0,ga~ ) ap l = - g DO,ad s F - - ~ p + RT L (9.60)

with g = 0/~ and assuming D is independent of the occupancy (concentration). Note that the form of (9.60) resembles that of (9.47) for small pores.

Shelekhin defines a parameter Tiso which defines the temperature where the amount of gas adsorbed on the pore walls equals that in the gas phase (iso concentration point).


At T > Tiso + AT and C T ~ C g the first term in Eq. (9.60) can be ignored. For T < Tis o - AT and C T ~ C a the last term in Eq. (9.60) is negligible. From adsorption isotherms the value of T~o was determined for a number of gases. For highly adsorbable gases like CO2, the magnitude of T~so may be as high as 160~ At p = 5 bar, and in the investigated membrane T~o equals -20, 30, 70 and 160~ for N2, 02, CH4 and CO2 respectively.

Adsorption is negligible for He and H 2 at temperatures above ambient. With decreasing pore size the contribution of the gas phase decreases and

that of the surface flow increases and interaction between gas molecules and pore wall increases (see Section 9.4.2.1.). This is the case in the small windows between the larger pores. This transition situation from Knudsen-like to configurational diffusion has been modelled by Xiao and Wei [86] for zeolite systems. The activation energy of the gas phase is calculated using Lennard- Jones potentials for the interaction energy, the activation energy for the adsorbed molecules is determined as a difference between the potentials in the pores and in the necks. The ratio of molecular diameter om and pore diameter dm at which the transition takes place depends on molecular shape and zeolite pore characteristics but is situated in the region 0.6 < ~m/dm < 1.

A maximum permeability coefficient was estimated by Shelekhin using Eq. (9.60) assuming porosity r and tortuosity �9 values equal to 0.3 and 2 respectively, a pore diameter of 1.5 nm and a micropore volume of 0.11-0.13 m3/g. For gases with T >> T~s~ and so in the regime where bulk gas diffusivity with AEg is dominant, the permeability is strongly dependent on the magnitude of AE. Permeability values for He at T = 90 K are estimated to be 5000 and 9000 Barrer for z~E = 10 and 6 kJ/mol respectively (note: for AE = 0 (Knudsen) this value is 3500 Barrer). With a membrane thickness of 30 ~tm, estimated permeation values for He are 5x10 -~ and 10 -7 mol m-2s -1Pa respectively.

Hassan et al. [95] using porous Vycor glass with a pore diameter of about 0.8 nm reported a separation factor ~ equal to 11.5 for O 2 / N 2 at 298 K and of 0~ = 4.6 at 423 K which values are about 20% larger than the perm selectivities. This is due to competitive adsorption in which the relatively strongly adsorbing component (02) saturates the surface and blocks the transport of the weakly adsorbing component (N2). Similar results are reported for C O 2 / C H 4 mixtures (~ - 186-122 in the same temperature range). This explanation seems qualita- tively in accordance with sorption data of Shelekhin [92] giving a sorption of 2 c m 3 / c m 3 membrane for 02 which is a factor 100 larger than that of N 2 at 30~

- Diffusion coefficients and kinetic information: The simplest way to obtain kinetic information is to perform permeation

measurements under transient conditions with a non-adsorbing gas in a Wicke-Callenbach experiment [3]. In this case the total amount of permeant qt that has passed through the membrane as a function of time is given by


qt 1 De' t 2 (-1) n De" t L" c-----~ = 6 g 2 ~2 ~ /,/2 exp _//2/i;2 g 2

n=l

(9.61a)

which for t --4 ~ approaches the asymptote:

OeC0 / '2 / qt= L t - - ~ e (9.61b)

which yields a straight line with intercept (time lag) equal to L2/6De on the time axis. A similar result is obtained by plotting any quantity which is directly proportional to qt.

Here Co = c(z = 0,t) and D e is the effective diffusion coefficient (porosity and tortuosity effects are incorporated in De). If the upstream (high) pressure is constant and much larger than the downstream (low) pressure, the slope of the asymptote will correspond to the steady state and so it is possible to determine the diffusivity under both steady state and transient conditions from a single permeation experiment. With a narrow and unimodal pore size distribution both methods yield reasonable consistent values. Large discrepancies point to strong microstructural effects (bimodal broad distribution, many dead ends, many defects).

9.4.3.4 Illustrative Examples of Permeation and Separation with Microporous Membranes

Usually membranes investigated in literature do not have the simple architecture assumed in the preceding theoretical treatment. This requires a number of corrections or modified equations before data of the separating layers can be compared and analysed. This problem is treated in Section 9.5 but results will be used in this section.

(a) Large micropores Shelekhin et al. [92, 56] reported some interesting results for Vycor type of

hollow-fibre membranes (for membrane characteristics see Section 9.4.2.2). The theory of permeation of hollow fibre systems will be treated in Section 9.5.

- Pressure dependence of permeation: For He, H2, 02 and N2 a linear dependence of the transmembrane flux on the

pressure gradient across the membrane was observed. So the permeation is constant and independent of pressure as expected for Knudsen diffusion and sorbed gases in the Henry regime (and accordingly to the sum of both mechanisms).


It,

in I

m

I e l . a

g t -

eq o

120

U

T ' I ~ ~

O ~ T-3,0 ~

3"heory

imi

0 4 8 12 16 Pressure, s tm

Fig. 9.22. Pressure dependency of the permeabil i ty of CO2. After Shelekhin et al. [92].

For C O 2 , which is a highly adsorbable gas, the permeability (Barrer) as a function of pressure at T = 30 and 100~ is given in Fig. 9.22. At 100~ there is a weak maximum above which the permeability slightly decreases with increasing pressure, at 30~ there is a continuous decrease. The two curves could be described with Eqs. (9.60) using Eqs. (9.59) and (9.56) with values for AEads and AEgas of 21 kJ/mol and 10 kJ/mole, respectively. These values were obtained from a best fit of the curves to the experimental results (note: AE=E in Fig. 9.22).

The maximum was explained with Eq. (9.60) considering a pressure independent bulk gas term (second term in (9.60)), while the first term for highly adsorbable gases may initially increase or decrease and then decrease with increasing pressure.

- Temperature dependence of permeation: For He the theoretically predicted permeability (Barrer) using Eq. (9.60)

exhibits a maximum as a function of temperature for AEgas- 4 kJ/mole. Note that the adsorbed gas phase is hardly present here. For larger values of AEgas


TABLE 9.6

Activation energies of diffusion and molecular kinetic diameters for different gases in microporous silica and zeolite membranes

Gas (~m Eperm Ed Eperm Ed (nm) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol)

He 0.26 H 2 0.289 CO2 0.33 02 0.346 N 2 0.364 CH 4 0.38 C~-I 8 0.43 n-C4H8 iso-C4H8 0.5 benzene 0.585

22.5 (b) 0.52 (a)

3.68 (b) 9.9 (a) 13.0 (b) 10.6 (a) 13.0 (b) 18.2 (a) 23.4 (b) 28.7 (a)

30 [88,89]

15-21 [63] 21 [63] --10 32 [63]

(a) Ref. [92]; (b) Ref. [56].

there is a cont inuous increase of the permeabi l i ty . No discuss ion was g iven of the occurrence of this m a x i m u m .

Such a cont inuous (non-linear) increase of the pe rmeabi l i ty as fiT) was observed indeed for CO2, O~- N2 and CH4 in the t e m p e r a t u r e reg ion of 300- 520 K. The act ivat ion energies of the pe rmeabi l i ty were obta ined wi th a non-l in- ear leas t -squares fit to the exp. curves and are given in Table 9.6 toge ther w i th the theoret ical calculated ones. These act ivat ion energies reflect the act ivat ion ene rgy of diffusion E d in the membrane . This table i l lustrates that, g iven the d iamete r of the pore there is an increase of the va lue of E d w i th increas ing kinetic (molecular) d iameter . The reverse t rend is found in the pe rmeab i l i ty values.

Theoret ical ly it is p red ic ted that the va lue of B = D.(M/T)l /2should converge to a single va lue for all gases for T ~ co.

A plot of l i terature da ta of B values as af(T) for a large n u m b e r of gases yie lds va lues of 1.1x10 -4 for the Vycor m e m b r a n e and 2.2x10 -4 for ZSM5 (Shelekhin [92]) in good ag reemen t wi th theory.

- Selectivity: Based on pe rmeab i l i ty data, permselect ivi t ies (selectivity factor a = FA/FB)

were calculated for a n u m b e r of pairs of gases A-B. At 30~ some typical va lues

are a -- 4190 for H2/CH4, (z = 2.5 for H 2 / C O 2 and a = 1675 for C O 2 / C H 4. All selectivity factors decrease wi th t empera tu re e.g. at 250~ ~ = 62 for C O 2 / C H 4.


(b) Small micropores

- Silica membranes (permeation and separation): Silica microporous membranes combining high separation factors and high

permeation values were first reported by Uhlhorn et al. [28,58] and were further developed and analysed by de Lange et al. [59-63]. More recently silica membranes made by a CVD process with similar qualities were reported by Lin et al. [67] and by Wu et al. [68].

The membranes, synthesised by Uhlhorn and by de Lange et al., were formed from polymeric silica solutions in an ultra-thin layer of about 100 nm thick partly on top (50 nm), partly within (50 nm) the pores of the y-A1203 support (pore diameter ~- 4 nm, thickness 3-8 nm) which was in turn supported by an ~-alumina support (disc) with a pore diameter of about 0.2 ~tm. The characteristics of the silica layer depend strongly on details of the synthesis procedure and a high quality supporting system is required (with low roughness and no or few defects) to obtain good quality membranes. The pore diameters were in the range 0.4-0.5 nm.

Discussion of permeation and separation requires some characteristic parameter for the membrane quality. As shown below the apparent activation energy for H 2 permeation gives a good correlation with the separation factor and is used as a measure of quality. Furthermore, the total measured permeation has to be corrected for influence of the support to obtain permeation and activation energy values characteristic for the silica layer (see also Section 9.5).

The experimental permeation results could be consistently described using Eqs. (9.43b) and (9.47) for Langmuir and Henry sorption respectively as shown by de Lange in a full analysis of sorption, permeation and separation results of five different gases [63]. This description requires knowledge of adsorption isotherms which could be measured only on unsupported membranes. To use these data for calculation of the permeation of supported membranes requires the assumption of equal pore characteristics in both cases. As discussed by de Lange et al. this is probably not correct in the case of silica layers. Based on sorption data a microporosity of about 30% and a pore size distribution with a peak at 0.5 nm is found. Analysis of permeation data point to a pore diameter of -- 0.4 nm and a considerably smaller porosity. Table 9.7 summarises the sorption data. H 2 and C H 4 have relatively low (isosteric) adsorption h e a t s (qSt) while CO2 and isobutane strongly adsorb.

Henry behaviour in the pressure range up to 125 kPa exist at temperatures larger than the limiting temperature Zl imi t ,Henry given in Table 9.7. At ambient temperature (323 K) C H 4 showed Henry behaviour up to 8 bar while H 2 exhibited Henry behaviour to at least 15 bar. CO2 exhibited Langmuir behaviour at I bar (323 K).


TABLE 9.7

Henry constants (b), isosteric heats of adsorption qst and lower limiting temperature for sorption behaviour Zlimifl-Ienry for CH4, H2, CO2 and isobutane in microporous silica at P < 125 kPa. After de Lange et al. [59,63]

Gas (--~) CO2 CH4 H2 iso-C4Hlo

qSt (kJ / mol) 22.3 10.3 (~) Zlimit~ Henry (K) (-~) 348 T (K) b (mol kg -1 Pa -1) ($) ~)

77

194

273 3.2x10 -5

303 7.8x10 -6

323 5.9x10 -6 -

348 3.2x10 --6 -

373 2.1x10 -6 -

473 4.8x10 -7 -

6.1 22.9

273 194

0.43 1.5x10 -4

4.2x10 -6 3.7x10 -7

2.3x10 --6 1.8x10 -7 w

1.7x10 -4

2.9• -5

2.4x10 -5

9.4• -6

TABLE 9.8

Typical values of permeation and activation energies of microporous silica membranes. Phigh ~ 3 bar. After de Lange et al. [59,63]

Permeation (10 -7 tool m -2 8 -1 Pa -1)

Apparent Eact* (kJ mo1-1)

Gas 50~ (H2) 200~ 28~ (CO2)

H2 4.1 (4.5) 21.7 (52.7) CO2 2.3 (3.0) 6.8 (32.3)

14.9 (21.7)

6.1 (14.9)

*Values between brackets corrected for support influence.

The sur face cove rage (0) for CO 2 w a s m a x i m u m 20% at 273 K a n d 125 kPa

a n d the isosteric hea t w a s prac t ica l ly i n d e p e n d e n t of coverage . This resu l t

ind ica te tha t for all o ther gases in the p r e s s u r e r ange up to -- I ba r cove rage w a s

also low. C o n s e q u e n t l y , Eq. (9.47) can be u s e d to descr ibe the p e r m e a t i o n

resul ts . Typica l p e r m e a t i o n resul ts are g iven in Table 9.8.

The p e r m e a t i o n va lues for H2, CH4 a n d CO2 at T > Tiso,Henry w e r e app rox i -

m a t e l y i n d e p e n d e n t of p r e s s u r e (as expec ted for H e n r y behav iou r ) a n d in-

3 9 6 9 m TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

creased with temperature from about 4.5-20x10 -7 mol /m 2 s Pa (425-473 K) for H 2 and 2.3-7x10 -7 mol/m2 s Pa (273-473 K) for CO2 [63]. For membranes with lower quality the increase is less pronounced due to the smaller apparent activation energy.

Typical values for iso-butane are 0,6-0.35x10 -7 mol /m 2 s Pa at 50~ and 200~ respectively. Note that the permeation in this case decrease with increasing temperature. Similar conclusions were drawn by Wu et al. [68] who reported apparent activation energies in the range 11-20 kJ/mol for considerably lower H 2 permeation values.

Table 9.8 shows that in the case where the flow resistance is not negligible, corrections should be applied on the total permeation value of the system to obtain the true permeation values of the silica separation layer. Consequently the true values of the activation energy may also differ considerably compared to the apparent ones. (See further Section 9.5).

The conclusion of de Lange et al. [61] is that the activation energy of permeation of H2 exhibit a good correlation with the quality of the membranes (permeation, separation factor) and high quality membranes should have an apparent activation energy of at least 10 kJ/mol.

Sometimes a weak maximum in the permeation of CO2 as a function of the feed pressure of a similar type as reported by Shelekhin et al. [92] has been observed by de Lange [61] also for small micropores.

Separation factors (defined by Eq. (9.36)) obtained from mixtures are usually smaller than permselectivities obtained from the ratio of single gas permeation (see qualitative discussion in Section 9.4.2.1.). At hightemperature and lower concentrations the mixture separation approaches the permselectivities which in turn tend to approach the Knudsen value at high enough temperatures.

Typical values for some gas mixtures in combination with permeation data (in the mixture) for different silica membrane systems are given in Tables 9.9a and 9.9b respectively. Several interesting conclusions can be drawn from Tables 9.8 and 9.9. The synthesis method and related membrane quality strongly determines the obtainable combination of permeation and separation values (as characterised by E~ct,H2). High quality membranes have activation energies for permeation (after correction for support influences) in the range 15-22 kJ/mol for H 2 and 10-15 kJ/mol for CO2 with typical permeation values at 200~ of 20x10 -7 for H2 and 5x10 -7 mol /m 2 s Pa for CO2 respectively. The permeation value of isobutane at 200~ is very small which indicates a pore size close to that of the kinetic diameter of i-butane and the absence of (larger) defects. Separa- tion factors are in the order of 20-30 for H2/CH4 and 150-200 for H2/isobutane. Lower quality membranes (lower values of Eapp,H2 tend to give larger permeation and smaller separation values for non-adsorbing gases. For strongly adsorbing gases (i-butane, CO2) even with moderate quality membranes ( E a p p , H 2 =

5-10 kJ/mol) good separation factors can be obtained up to about 200~


TABLE 9.9a

Separation factors (defined by Eq. (9.36)) for some gas mixture-silica membrane combinations

Membrane Gas mixture Separation factor a at T (~

~-25 50 100 150 200 >_250

Eapp

k J / m o l

A13Sil a C O 2 / C H 4 48 65 28 7-(10)

A13Sil a H2/C3I~ 13 62 156 270 7-(10)

a. A13Si 2 H2/CO 2 1.7 2.5 4.5 5.5 6.6 5-(7)

b. H2/N2 15-36

A13Sil H2/Ct-I4 2 3 5 8 11 12 5-(7)

A13Si2 H2/CH4 2 3 5 10 18 30-40 7-(9)

A13Si2Ti H2/CH4 12 50 150 200 165 - 12-(16)

AllSi2 H2/CH4 =9 10 11 11 11 -

AllSil H2-iC4H8 - 80 105 110 110 110 -

AllSi2 H2-iC4H8 80 130 170 180 170 -

Values are taken from de Lange et al. [59-63], unless otherwise referred. aTaken from Uhlhom et al. [58]. bTaken from Shelekhin et al. [56]. Membrane code: AlxSiy with x and y are number of A1203 and SiO2 layers respectively. Eapp is the apparent activation energy of permeation for H2. Figure in parenthesis is corrected for support influence.

TABLE 9.9b

Permeation values of some gas mixtures in different silica membranes

Membrane Gas mixture Permeation F at T (~

50 100 150 200

A13Sil a CO2/CH4 4 (CO2) 10

A13Si2 H2/CH 4 3.7 (H2) 1.8 (CH4)

A13Si2 H2/CH4 1.8 (H2) 50 (H2)

A13SilTi H2/CH4 2 (H2) 3 (H2) 4 (H2)

Membrane code: see Table 9.9a. Permeation given in 10-7mol/m 2 s Pa.

For non-adsorbing or weakly adsorbing gases (H2, CH4, N2, 02) the permeation increases with temperature (for high quality membranes). This is in accordance with data of Wu et al. [68] who reported increasing permeation values (H2) for membranes with lower quality (characterised by positive values of Eap p for

398 9 - - TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES w r r H GASES AND VAPOURS

N2). Permeation values of Wu at 600~ are in the range 0.03-1.0x10 -7 m o l / m 2 s Pa for H 2 (for N 2 a factor of 20-70 lower) with the highest value for the lower quality membrane (Eact,H2-~ 11 kJ/mol).

As will be discussed in Section 9.5 the different values cannot be compared directly because of the strong influence of pressure conditions and support effects.

- Silica membranes (diffusion data): Equation (9.47) can be used to calculate the activation energy for "intracrys-

talline" micropore diffusion Ed, / of specimen i provided sufficient sorption data are available. The value of Ed, / follows from:

Edd = Eapp- Qa,i (9.62)

where Qa,/is the isosteric heat of adsorption and Eapp is the measured apparent activation energy of permeation (Eq. (9.47)) after correction for support influences.

With typical values of Eap p equal to 15 kJ/mol (H2) and 10 kJ/mol (CO2) and typical values of Qa equal to 6 (H2) and 23 kJ/mol (CO2) [63] the resulting calculated activation energies of the intra channel (micropore) diffusion are about 21 kJ/mol for H2 and 32 kJ/mol CO2 [59,93]. This is in accordance with the expectation that larger molecules will have a larger activation energy for diffusion than smaller ones [92,82].

Equation (9.47) is also used by de Lange [63] to calculate the value of the diffusion coefficient of several gases in silica membranes.

The term ~t in (9.47) takes the form ~t = p(1 - ~)/~ with the skeletal density of silica p = 2.2 k g / m 3, the silica porosity ~ = 0.4 and the membranes thickness L = 100 nm. Taking all the sorption terms together in the Henry constant b (which can be directly measured) and substitution in (9.47) yields:

J = 3.3 x 101SD x b (9.63)

Typical values of b and D for a range of membranes are given in Table 9.10 together with some other parameter values. The range in D values reflects differences in membrane quality, the smallest D values being formed in high quality membranes. The diffusion coefficients become smaller in the same order as the kinetic molecular diameter (see Table 9.6) increase. The large differences in the D values indicate that the pore diameter is of the order of the molecular diameters (0.4~.5 nm).

The differences in D values are much larger than the differences in permeation values and indicate the effect of the sorption term even for weakly adsorbing gases (compare H 2 and CH4).

The absolute magnitude of the diffusion coefficient is rather uncertain, because all uncertainties concerning the value of ~t are reflected by the D values.


TABLE 9.10

Typical values for the diffusion coefficient of different gases in silica membranes at room temperature after de Lange et al. [63]

Gas b (289 K) D (,298 K) Ed,i qst (mol /kg Pa) (m ' / s ) (kJ/mol) (kJ/mol)

H2 1.4x10 -7 5.25x10 -11 13-21 --6

CO2 1.4x10 -5 6-9x10 -13 --30-33 =23

CH4 1.6x10 -6 5-15x10 -13 --10

iC4H10 7.3x10 -5 3x10 -14 ---23

The value of the porosity is taken from adsorption measurements on unsupported silica membranes and probably the porosity of supported silica membranes is considerably smaller and the calculated D values give a lower limit.

A comparison with zeolite data and effects of surface reactions will be discussed below.

- Z e o l i t e m e m b r a n e s :

Permeation and separation data on well defined, high quality zeolite membranes are only reported for MFI (ZSM-5, silicalite) zeolites grown in situ directly from the precursor solution on top of a substrate. The experimental single gas permeation results could be in a number of cases consistently described using Eqs. (9.43)-(9.48) for the Langmuir and Henry regimes.

Geus et al. [70] give a detailed description of the synthesis of a MFI layer with a thickness of about 50 ~tm on top of a porous steel support. Vroon et al. [72,74] synthesised thin (2-6 ~tm) MFI layers on a (x-A120 3 support and varied the crystallite size (0.1-0.4 ~tm) in the layers by varying the synthesis temperature and using a very high pH (-~ 12.5). Both groups of authors investigated the quality of their membranes. Both groups of authors measured a very small flux of gas molecules which are much too large to pass the pores of the MFI structure indicating that some larger pores were present in the layer. The measured fluxes for iso-octane (or of 2-2-di-methylpentane) were more than 5 orders of magnitude smaller than that of C H 4 indicating a good membrane quality. This conclusion is supported by the large observed separation factors for e.g. H2/butane, CH4/butane and n-butane/i-butane (see below). Vroon had to apply two silicalite layers on top of each other in order to obtain this good quality.

Typical single gas permeation data for relatively thick MFI membranes are given in Fig. 9.23 [71]. At 673 K all the gases show a linear dependence on the (feed) pressure (Henry behaviour) as is the case at 300 K for the noble gases and for C H 4 , whereas butane and ethane exhibit saturation at low and higher


22

20

18

~ " 16 W

N

s ~4 . , . .m

0 E ~2 E ~'~ 10

ft. e

(a)

~" , .butane* 100 A

0 20 40 60 80 I O0

10

O. 8 -~

@ W

T 0 E

6 0

5 .E 0

4 W W o

3

O. t

Partial pressure in feed (kPa) (a)

Fig. 9.23. Steady-state flux and permeate pressure as a function of partial feed pressure for different gases at 300 K (a) and at 673 K (b). After Bakker et al. [71]. (0) neon, (+) argon, (V) krypton, ( �9 ) methane,

(A) ethane, ( I ) n-butane, (&) isobutane, (O) CFC-12.

pressure (30 and 80 kPa) respectively. The permeation increases with temperature for all gases except krypton and CH4 which were almost independent of temperature.

The maximum observed permeation values (673 K) of noble gases and CH 4 are about the same and correspond with a permeation of 1.6-2.3x10 -7 m o l / m 2 s Pa. Permeation values of 1.2 and 0.9x10 -7 m o l / m 2 s Pa are found for n- and i-butane respectively.

Similar results concerning the trends in the permeation values as a function of pressure are reported by Vroon et al. [72,73,74] for CH4, ethane, propane and butane. The absolute values of the permeation reported by Vroon et al. for these


22

20

18

A 16 (/)

E ~4 0 E 12 E

10 x

~.. 8

(b)

I I

0 20 40 60 80 ~00

1 0

O. 8 ..~

Q)

7 m

E 6

S .E

4 ~

ID 3 ~" t2.

2 "~.

Partial pressure in feed (kPa) (b)

Fig. 9.23 (continued). Caption opposite.

MFI membranes on (x-A120 3 supports showed for CH 4 a decrease of the permeat ion from lx10 -7 (298 K) to 0.6x10 -7 m o l / m 2 s Pa (473 K) with increasing temperature.

Plots of the flux of butane (Fig. 9.24), propane and ethane versus temperature exhibit a (weak) maximum which values shifts from 440 K for n-butane to 350 K for ethane at 100 kPa. This maximum depends on the (partial) pressure of the gas (e.g. for n-butane at 8 kPa pressure the maximum is situated at about 390 K). Similar maxima are found [72,74] in the curves of H 2 and CO2 vs temperature as shown in Fig. 9.25 and are also reported by Kapteyn for n-butane [88,89].

The absolute values found for CH4, CO 2 and n-butane can be compared with that obtained by Bakker [71], Geus [75] and Kapteyn et al. [88]. At 473 K the values obtained by Vroon are lower by a factor of about 2.5 compared to that of


15

A qll I

E ,/.

m m

0 E E

X m u

10

0 0 0

0 0 O

0 o 0 o

0 0

0 0 @ 0 0 0 0 O O0 o

vvVVVVVVVv v vvvvVVVVYV

v v V t ~ v v v v V V

i , I , I 0 273 373 473

Temperature (K) Fig. 9.24. Comparison of the methane and n-butane flux measured by the dead-end method (O,V)

and the Wicke-Kallenback method (o,v). After Z. Vroon [72-74].

20

15 o O ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 ,..

10r oO - v v v v v v v v v v v v l ~1 v v v V V V

F + "l" "1" "1" + + + + + + + + ,1" + + / +++t

0 50 100 150 200

Temperature (~

Fig. 9.25. Flux of hydrogen (O), helium (V), carbon dioxide (+) and sulphur hexafluoride A as a function of temperature at a feed pressure of 100 kPa. After Vroon et al. [72-74].


Bakker. This means that the permeation values corrected for thickness differences (permeability) of the thin membranes with small crystallites on a support with relatively narrow pores (0.2 ~tm) are much lower than of the thicker layers on a steel support. This point will be discussed later.

Kapteyn et al. [88,89] and Vroon et al. [72,74] could model and describe their single gas permeation measurements for C H 4 and n-butane rather well with Eqs. (9.41)-(9.48) taking the thermodynamic factor in the Langmuir regime from adsorption measurements.

Using Eqs. (9.40) and (9.43), Kapteyn showed that the corrected (intrinsic) value of the diffusion constant Do at 300 K of n-butane is independent of pressure (up to 1 bar) and Do (300 K) equals 0.4x10 -6 c m 2 / s (contrary to the Fickian diffusion constant which is strongly increasing with increasing 0). The maximum in the flux versus temperature for n-butane could be correlated with a strong decrease of the occupancy 0 at higher temperatures. Occupancies at p = 0.5 bar vary almost linearly from 0 = 0.8 at 350 K to 0 = 0.2 at 450 K and the maximum in the cu~e is situated at 0 -- 0.4. Initially the change in 0 is less. So at lower temperature the diffusion coefficient increases more rapidly than the concentration (occupancy) decreases, at higher temperature the reverse is true and this give rise to the observed maximum asflT). A similar result is reported by Vroon et al. [72,74] who calculated the flux of C H 4 and n-butane using also Eqs. (9.40) and (9.43) and using diffusion constants taken from literature [94] and measured on twinned single crystals by the membrane method. The sorption data for methane taken from literature agree within 20% from data obtained by Vroon, for n-butane not sufficient literature data are available and measured data (gravimetric method) are used. The set of data used in the calculations is given in Table 9.11 and the calculation results in Figs. 9.26 and 9.27.

TABLE 9.11

Henry constants and saturation concentrations obtained from the gravimetric sorption measurements on silicalite particles and diffusion constants obtained by the membrane method of methane and n-butane. After Vroon et al. [72-74]

Gas Temperature Henry constant Saturation Diffusion coefficient (K) (mol Pa -1 m -3) concentration (m 2 s -1)

(mol m -3)

Methane 298 8.6x10 -3 - 0.7x10 -1~ 323 5.4x10 -3 - 1.0x10 -1~

n-Butane 298 17.5 2.2x10 3 1.0xl0 -12

323 4.0 1.8x103 2.2x10 -12 348 1.0 1.4><103 4.3x10 -12


A

E v -

I

t~

o E E

v

x i F �9 _~ A u. 1 �9

@

III

I @

0 2 0 40 60 80 1 0 0

Feed pressure (kPa) Fig. 9.26. Calculated n-butane flux as a function of the feed pressure at 298 K (I), 323 (II) and 348 (1II) K. Permeate pressure is 0 kPa. Curve A is experimentally obtained. After Vroon et al. [72-74].

A

E T w

0 E E

10

8

4

X

n

u. 2

I I I

0 1 2 3 4 5

Permeate pressure (kPa) Fig. 9.27. Calculated methane (I) and n-butane (II) fluxes at a feed pressure of 100 kPa and 298 K as function of the permeate pressure. After Vroon et al. [72-74]. Figures in Curve II have been multiplied

by a factor of 3.

9 - - TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS 405

The activation energy for intra-crystalline diffusion for n-butane is 30 kJ mo1-1 [72,74,89]. For the isosteric heat of adsorption values of =38 kJ mo1-1 are reported by Kapteyn and by Vroon, which value is considerably lower than other values (=50 kJ mo1-1) reported in literature. For CH4 a good agreement between calculated and measured fluxes is obtained, for n-butane the agreement is reasonable to bad at low pressure and good at higher pressure. A difficult problem is the value of the saturation concentration. In many cases no reliable experimental data are known and theoretical estimates have to be made usually under the questionable assumption that qs is independent of temperature. For n-butane the theoretical sorpfion capacity of MFI/silicalite equals about 2.1x10 -3 mol g-1 (equivalent to 12 molecules per unit cell).

The Delft group (Kapteyn, Bakker) reported for their large MFI crystals experimental values of qs in the range of 1.6-210x10 -3 mol g-i, whereas Vroon et al. for small crystals (0.1-0.4 ~tm) reported qs = 1.2x10-3 mol g-1. Obviously the sorpfion capacity depends on synthesis conditions and/or crystal size.

For C H 4 a theoretical estimate of qs = 4x10-3 mol g-1 is reported by Kapteyn et al. [89], whereas the qs values decrease with increasing molecular weight in the C1--C 4 series.

Figure 9.27 shows a decrease by a factor of 2 of the calculated flux values, when at a constant feed pressure of 100 kPa the permeate pressure changes from I to 10 kPa. This illustrates the large influence of the permeate pressure on the flux. In the supported membranes under consideration the driving force may be decreased due to a pressure gradient in the support (decreased occupancy at interface) and/or equilibrium conditions may not exist at the interface between support and zeolite due to a slow desorption process and/or due to insufficient removal of the permeate. Bai et al. [27] observed that a pressure drop across the membrane leads to higher selectivities than obtained from W-C type of measurements using a sweep gas to remove the permeant. Comparison of the flux values obtained by Bakker/Kapteyn and by Vroon et al. show comparable values after correction for differences in support porosity. This is unexpected because the layer thickness of the membranes made by Vroon et al. is about 3 ~tm, a factor 17 smaller than that prepared by Bakker/Kapteyn. The reason for this discrepancy is not known and may be originated in a difference of the effective and nominal thickness, e.g., by a strongly porous part of the thick layers. Another reason might be effects of crystal size and number of grain boundaries and of synthesis conditions (lower crystallinity at lower synthesis temperatures). These points have been investigated by Vroon et al. [72,74]. The results are tabulated in Table 9.12 for the methane flux in membranes with different crystal sizes and thickness. As shown in the table the membrane flux corrected for the layer thickness (J.L) decrease with increasing synthesis temperature (and so with decreasing particle size). A direct correlation with bulk properties due to particle size is however not present because the flux is


TABLE 9.12

Methane flux of typical Mb1 membranes with different micro structures. After Vroon et al. [72-74]

Membrane L r A J** J.L*** J.r*** J.A*** code* (~tm) (nm) (L/r)

P100 3.5 275 13 13 46 3.6 2.7 P20 3.5 350 10 10 35 3.5 1.2 P150 5 550 9 7 35 3.8 0.6 P180 5.5 700 8 5 28 3.5 0.4

*PlOO, P20, P150 and P180 are membranes grown in situ in two steps at 373, 393, 423 and 453 K, respectively. **Methane flux (mmol m -2 s -1) at a feed pressure of 100 kPa and 298 K. ***J-L is the flux corrected for the layer thickness, J.r is the flux corrected for the particle size, J.A is the flux taken into account the munber of interfaces perpendicular to the support surface.

independent of the crystal size itself (product J.r is constant). So obviously the interfaces (grain boundaries) between zeolite particles play a role and an increase of the number of grain boundaries results in an increase of the flux (product J.A is largest for largest A value). The membranes show large separation factors including size exclusion for large molecules, and this exclude cont inuous and rapid t ranspor t pa thways wi th large dimensions. A model is deve loped therefore of grain boundar ies containing small microvoids and slits of a size similar to the pore size which p romote the flux. This does not explain however the observed difference be tween flux values of Vroon and of Kapteyn/Bakker .

- Separation and permeation with binary gas mixtures in zeolite membranes: Separat ion results wi th suppor ted MFI membranes of good quality are

repor ted by the Delft group (Kapteyn, Geus, Bakker [69-71,89]) wi th membranes on porous steel suppor ts and by the Twente group (Burggraaf, Vroon et al. [72,73]) wi th membranes on porous 0~-alumina suppor t and of m e d i u m (defect quality) by, e.g., the groups of Noble and coworkers [27,77,97] on y-alumina supports , of Ma and coworkers [76,96]. Some other groups report results wi thout sufficient indication of membrane quali ty [64,75,98].

It is interesting to note that membranes wi th m e d i u m quality ~ as indicated by relatively low separat ion factors for non- or weakly adsorbing gases ~ can have good separat ion factors for mixtures wi th a strongly adsorbing, easily condensing component (e.g., H2-methanol). Obviously the defects are blocked by filling the defect wi th the easily condensing component (capillary condensation type of blocking [98]). This makes it clear that we must dis t inguish the several categories of gas mixtures as ment ioned in Table 9.5 wi th the main categories of mixtures of (i) weakly ( W ) - weakly (W) adsorbing gases ii) weakly ( W ) - strongly (S) adsorbing and iii) strongly (S) - strongly (S) adsorbing gases.


For a combination of two weakly adsorbing gases the separation factors approach the permselectivity values but are usually somewhat lower whereas the permeance values in the mixture are somewhat decreased with respect to single gas permeances. This is precisely what can be expected for supported membranes with little interactions of molecules within the pores. Interesting results are described by Kapteyn [89] and by Krishna and van der Broek [100] for isomer separation in the range of intermediately adsorbing (between S and W) gases (ethane/ethene, propane/propene) using the Maxwell-Stefan approach (Eqs. (9.50) and (9.51)) and the competitive Langmuir adsorption model (Eq. (9.53)). Krishna et al. also simulated the behaviour of H2/n-butane mixtures. The exchange contributions were assumed to be negligible (single file diffusion) and so diffusivity data of the single gas permeation experiments were used and Eqs. (9.50) and (9.51) were numerically integrated.

The best description was obtained with diffusivity values of ethane which are decreased and those of ethene slightly increased with respect to single gas measurements, while for propane/propene the diffusivifies became equal. In this latter case high occupancies (98%) were present and the faster component is decelerated by the slower one. The alkanes in the mixture exhibit higher (measured) fluxes compared with their unsaturated analogues and some selectivity over the corresponding alkanes exist. At 292 K and for 50 kPa: 50 kPa mixtures the selectivity factors are 1.9 (ethane/ethene with a flux value of 17x10 -3 mol m -2 s -1 for ethane) and 1.3 (propane/propene with a flux of 5x10 -3 mol m -2 s -1 for propane).

Typical curves for a combination of weakly and strongly adsorbing gases (S-W) are given in Fig. 9.29 (after Kapteyn [99]) and Fig. 9.28 (after Vroon [72,74]). A similar result to that shown in Fig. 9.29 is also observed for H 2 / C O 2

mixtures [71]. Typical in all the cases is the occurrence of a maximum in the permeation curve of the S component in the mixture as is also observed in their single gas permeation curves. As shown in Fig. 9.28 the permeation of the S component (n-butane) is only slightly lower than the single gas permeation. The W component (H2) is drastically decreased and, in contrast to the single gas permeation, increases with increasing temperature. When the temperature increases sufficiently, the permeation values in the mixture first become equal and finally cross each other (see Fig. 9.29), with the W component becoming faster permeating. So we have a conversion of the selectivity factor ~ > I for the S /W mixture at lower temperature to values ~ < 1 at high temperature (note a > I for the W/S combination is equivalent to ~ < 1 for the S/W). This can be explained by preferential sorpfion of the strongly adsorbing component which excludes (or decreases) the concentration of the S component. With increasing temperature the concentration of the S component decreases much more strongly than that of the W component, the "blocking" effect decreases and finally vanishes and at high temperature the mixture starts to behave in a similar way to a mixture of two W components.


6

A

t

I

o E E

N l

I la,

4

2

V v

V

0 �9 �9 , , , , i , , , i

Temlmrature (K) Fig. 9.28. Permeation and separation behaviour of a mixture of 50 EPa C1-~ ( �9 and 50 EPa n-butane (V) as a function of temperature of a MFI membrane. Single gas permeation values are added: CH4

(O), n-butane V). After Vroon et al. [72-74].

25

- '" 2o

?

E ~s , , , , , =

O E E 10

X

ft. s

f

n-Butane

/ "0 e ~

H y d r o g e n / �9 , ~ & , , i

300 400 500 600

Temperature (K) Fig. 9.29. Separa t ion behav iou r of a H 2 / n - b u t a n e mixture (1:1) as a funct ion of t empera tu re of a MFI

(silicalite) m e m b r a n e at 100 kPa. After K a p t e y n et al. [99].

A quant i ta t ive t r ea tmen t of this complex behav iour is not yet publ i shed . The case of a mix ture of two S componen t s is even more complex and general qual i ta t ive descr ip t ions have not yet been publ ished. Examples of p e r m e a t i o n


A (N I

E T I

m

o E E

x =l

m u,,,

10-'

10"

10-*

V T

V V V V V V' V V

V

V A

A A A A

of equipment 10"

273 373 473

Temperature (K) Fig. 9.30. Separation and permeation behaviour of a mixture of 0.31 kPa paraxylene (V) and 0.26 kPa o-xylene (A) as a function of temperature. Single gas permeation data are also given: 0.62 kPa px (A) and 0.52 kPa ox (V). The total pressure was 100 kPa, the balance being He. After Vroon et al.

[72-74].

and separation results of these combinations are reported by Vroon et al. [72,74] for n-butane/i-butane, benzene/cyclohexane, methane or hexane/2,2-dimethyl- benzene and p/o-xylene mixtures.

The separation behaviour of a p / o xylene mixture is given in Fig. 9.30. The permeation of the paraxylene is much larger than that of the o-xylene at higher temperature, the last one has a permeation which is at the detection limit of the equipment used. The molecule has a diameter which is larger than that of the pore diameter of the MFI and so we have here an example of separation by size exclusion. The flux of p-xylene shows a weak maximum as a fiT) and consequently the separation factor does the same with a peak value of c~ = 100 at ~400 K under the given conditions. The separation factors and the permselectivities are equal as expected for the size exclusion mechanism.

Xiang and Ma [76] reported a value of ~ = !5 for p/meta-xylene separation with a flux of 35 ml m -2 h -1 (=--4.3x10-7mol m -2 s -1) for the m-xylene at room temperature.

An even more straightforward example of size exclusion is exhibited by the mixture of n-hexane and 2-2 dimethylbutane where the flux of the hexane is three to four orders of magnitude larger than that of the 2-2 dimethylbutane up to the highest temperature measured (473 K) and the flux of the 2-2 dimethylbutane


is of a similar magnitude of that of o-xylene Vroon [72,74]. Obviously a small number of small defects accounts for the remaining flux of the large molecule.

Finally it should be noted that isomers like n-i-butane [73,89] and cis-butane/ 2-trans-butanes [76] exhibit different permeances indicating the shape selective properties of the zeolite membranes.

- Di f fus ion in zeolite membranes:

Diffusion data can be obtained by a wide variety of different techniques which yield diffusion data which, for the same material, can differ by more than 4 orders of magnitude. So it is outside the scope of this paper to give a full discussion of diffusion data. Some relevant aspects for membrane permeation will be mentioned. A comparison of diffusivities of n-butane in silicalite obtained by different techniques is given by Kapteyn et al. [88]. Compared with his experimental results obtained from steady-state permeation measurement using Eqs. (9.50) and (9.51), values obtained by single crystal (membrane) measurements [94] are too low by more than two orders of magnitude. These single crystal data however reproduce reasonably the permeation results of Vroon et al. [72,74] as discussed in the preceding sections. The diffusion data of Kapteyn et al. agree well with diffusion data obtained by frequency response (FR) and square wave (SW) methods. Kapteyn argues that the diffusivity of n-hexane in silicalite is not influenced by the fact that the crystals in the membrane are intergrown and assumes that the same holds for n-butane. The intrinsic (corrected, Maxwell-Stefan) diffusion coefficient Do of n-butane in silicalite can be described by an Arrhenius equation with a pre-exponenfial coefficient D~ -- 0.053 c m 2 s -1 and an activation energy for diffusion E d = 29.8 kJ mo1-1. At 300 K this gives a value of Do = 4x10 -7 cm 2 s -1.

It should be noted that the absence of effects due to intergrown particles does not mean that grain boundaries do not play a role, as has been shown by Vroon [72,74] and discussed in the preceding section.

Vroon et al. report values obtained by transient measurements on their silicalite membranes using Eq. (9.61b) and find a good agreement with values obtained from steady-state membrane measurements. Values obtained from transient measurements in sorption experiments on powdered material are two orders of magnitude smaller.

Geus et al. [75] reports diffusion data at 21 and 145~ for H2, N2, CH4, C O 2

and CF2C12 in silicalite membranes on a clay support which are obtained with the similar transient permeation technique as used above by Vroon. The diffusion coefficients for methane are about two orders of magnitude smaller than those obtained by PF-NMR methods. Usually this last technique gives relatively large diffusion coefficient values, which in the case of n-butane are of the same order of magnitude as reported for FR techniques and membrane techniques as reported by Kapteyn.


Geus ascribes his low values to the influence of the support which has a low porosity. Indeed, uncertainties in geometric aspects of the separation layer and the membrane system affect the value of the measured diffusion coefficients.

The conclusion so far must be that synthesis and sample preparation techniques play an important role. Diffusion data to be used in permeation experiments should be measured on membranes with techniques which reflect as closely as possible the transport phenomena during permeation. This also minimises heat effects due to adsorpfion/desorption which play an important role in diffusion experiments based on large crystals, but is of minor importance in membrane experiments [101].

9.4.4 Surface Effects on Permeation in Microporous Membranes

In the preceding discussion it was assumed that the transfer from molecules from the gas phase to the solid (porous) membrane was not the rate-determining step in the permeation. This assumption will be evaluated in this section because in oxygen permeation of dense oxidic membranes surface reactions become clearly rate determining for several groups of materials (see also Chapter 10).

For the best permeating dense materials (perovskites) with relatively large exchange coefficients, surface reactions become rate determining with membrane thicknesses in the range 0.3-1.0 mm corresponding with flux values in the range 0.4-4.0x10 -6 mol m -2 s -1 (corresponding to a permeation value of 0.4- 4.0x10 -7 mol m -2 s -1 Pa -1 with pressures of I bar and about zero at the feed and permeate side respectively). This high oxygen permeation is comparable with or somewhat lower than many of the permeation values for microporous membranes.

De Lange [63] used a gas kinetic expression to estimate the total number of molecules Zwa n colliding per second with the walls of a volume:

1 N _ Zwall -" 4 V v (9.64)

where V is the molar volume, v the mean molecular velocity and N / V is the number of molecules per unit volume.

At 1 atm and 300 K the calculated value of Zwall is approximately 1.8 mol c m -2 s -1. A typical hydrogen flux through the microporous membranes is 10x10 -6 mol c m -2 s -1 (calculated from a permeation of 1 0 X 1 0 -7 mol m -2 s - 1 P a -1 at a pressure difference of I bar).

Not every collision leads to penetration of the molecule into the membrane. This is expressed by the sticking factor t as defined by Eq. (9.65)"

Ra= t [ P 1 E('R-T-/ (2~MRT)I/2 exp - (9.65)

4 1 2 9 a TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

The sticking factor gives the ratio of the number of activated collisions divided by the total number of collisions, whereas Ra in Eq. (9.65) gives the rate of adsorption (in mol cm -2 s-l), with an activation energy Ears for adsorption at the external surface, the other parameters having their usual meaning.

According to Turkdogan [5] the maximum value of R a in Eq.(9.65) is obtained by setting t equal to unity and zero activation (adsorption) energy (Ea = 0). Equation (9.65) then transforms to the classical Hertz-Knudsen equation for the number of moles striking a unit surface area per unit time Rmax:

Rmax = P (9.66a) 2 ~ ( M R T ) 1/2

and with p given in atmosphere:

Rmax - 44.3 p (9.66b) - (MT) I /2

with Rmax in mol cm -2 s -1.

Note that Eqs. (9.66b) and (9.64) will produce figures of similar orders of magnitude and that Eq. (9.66a) also gives the maximum rate of vaporisation from a non-contaminated surface at low pressures.

For microporous membranes only the porous part of the surface (~) is available for penetration; the solid is assumed not to accept molecules. For small molecules hitting the surface under not too low angles it is reasonable to assume a low value of the activation energy for pore penetration (this is process F1 i n Fig. 9.21). A pessimistic estimate for microporous silica membranes using values of ~ = 0.01 and t = 0.01 yields at 300 K and I atm a collisional flux (of H2) which is at least one order of magnitude larger than the permeation (flux) values found by de Lange et al. [63].

The conclusion is that for relatively small molecules (H2, CO2, etc.), permeation in microporous (silica) membranes is not limited by surface reactions and direct penetration in the pores is the dominant mechanism in a wide range of temperature and pressure conditions [63]. This conclusion does not hold for large non-spherical molecules. Here sorption is necessary, the sticking coefficient becomes very important and surface reactions probably will limit the permeation as soon as bulk permeation becomes appreciable. To the knowledge of the present author, no investigations of this phenomenon in microporous membranes have yet been reported.

In dense, non-porous membranes, surface limitations to oxygen permeation are a common phenomenon as can be understood from the very low adsorption levels and large activation energies on the dense membrane materials (see Chapter 10). For hydrogen permeation in dense metal membranes estimates have been made by Govind [105].


The implication of the theoretical considerations given above is that the permeation can be increased in cases of low adsorption and sticking coefficients by application of a mesoporous top layer with better sorption properties on top of the microporous membranes. Selective sorption should then also lead to an enhanced separation factor (see Eq. (9.71)). Indications for this effect are reported for dense membranes by Deng et al. [106] and for microporous silica membranes by Nair [107].

9.5 PERMEATION A N D SEPARATION IN MORE COMPLICATED SYSTEMS

Real membrane systems to be used in practice usually do not have the simple architecture assumed in the preceding quantitative treatments (single-wall, non-supported) nor do they fulfil basic boundary conditions, i.e. well mixed gas mixtures, homogeneous gas compositions and pressure (no gradient) across the membrane length (flow direction of feed/permeate). In those cases the aerody- namic conditions of the feed and permeate flow, the precise design and the type of permeate removal (sweep gas, vacuum suction) are important.

In the case of supported membranes the effect of the support has always to be evaluated, and if not negligible, corrections for support effects should be applied even with simple membrane architectures.

A full description of permeation and separation in practical systems is out of the scope of this paper. Two important cases will be treated for illustration because of their importance for laboratory experiments.

9.5.1 Hollow Fibres

In the case of hollow fibres, or long cylindrical tubes, the pressure drop across the membrane length is not negligible. In the case of hollow fibres with a characteristic ratio of length-to-inside-diameter of 104 this pressure drop is very large and the gas densities at inlet and. outlet differ considerably. Then the gas flow is a compressible flow.

Shelekhin et al. [56] derived a set of three expressions to describe the permeation of single gases through a micro porous hollow fibre (Vycor type) which, in the general case, should be solved numerically. In the special case of a relatively low permeable gas, the pressure drop along the fibre becomes again negligible and the permeation Fp (mol m -2 s -1Pa -1) can be calculated directly:

�9 T . r i In (r o/ri) (9.67) Fp = S(po - P3)

with (I) T the transmembrane flow rate, ro and ri the outer and inner radii of the fibre, S the membrane surface area (m 2) and P0 and P3 the inlet pressure and the pressure on the permeate side of the membrane, respectively.


The pressure distribution along the fibre was expressed as P/Po and was calculated with the complete set of equations. Values of P/Po across the complete fibre length (so Po/Pl with Pl is outlet pressure) for He at 30 and 250~ are equal to 0.8 and 0.5 respectively and use of Eq. (9.67) gives wrong results. This is illustrated by a comparison of the permeation versus temperature curves of He which give a maximum if Eq. (9.67) is used but give a continuously decreasing function when the pressure drop is taken into account.

In the case of gas mixtures the gas composition of both feed and permeate flows changes along the membrane length. There is a difference in behaviour between co-current and counter-current flow of feed and permeate streams. A brief description for separation in a single stage module with ideal mixing and of a coupling of modules to form cascades or membrane rectification units is given by Eichmann and Werner [18]. It is illustrated that the concentration at the permeate side changes across the membrane length. An implicit expression to calculate the concentration of a binary mixture at the outlet of the membrane system as a function of the inlet concentration is given and so the separation factor can be calculated. This equation gives a good description of the actual behaviour of gases as illustrated e.g. for H2/CO 2 mixtures. The effect of several important parameters (e.g. average pressure, feed or permeate pressure, feed or permeate pressure at outlet, temperature) is illustrated and the necessity to select an optimum set of parameters, given economical boundary conditions, is shown. An extensive treatment of this type of problem is given by Sengupta and Sirkar [114]

9.5.2 Multilayered, Asymmetric Supported Systems

The use of supports in asymmetric, supported membranes introduces a number of complications in the interpretation of permeation and separation data as well as in the optimalisation of membrane systems. If the flow resistance of the support is not negligible, there is a pressure drop across the support. This implies that the pressure and so the occupancy at the interface of separation layer and support is different from the (directly accessible) pressure at the support surface, usually the permeate side. Consequently, the driving force for permeation through the separation layer is different from the total driving force across the membrane system. In cases where one wants to calculate or compare transport properties of the separation layer material, it is necessary to correct for this effect (for illustration see below).

Expressions to calculate the pressure Pint at the interface of top layer and substrate and thus to calculate the pressure drop across the top layer only are originally derived by Uhlhorn et al. [21] and further developed and used by Lin et al. [103,104] and de Lange et al. [59,60]. More recently Uchytil [102] used and refined this method for different cases. De Lange [60] gives an illustration of the


calculation on a typical, supported mesoporous y-A1203 membrane. It is assumed that the flow resistance in single gas permeation experiments is a series combination of the flow resistances of support and top layer, respectively, where the permeation is a reciprocal resistance. The support permeation can be expressed as (see also Section 9.2.3.):

F0,s = gs + ~ls" Pav (9.68a)

where gs expresses the Knudsen component, [3s- Pay expresses the Poiseuille (Laminar) flow component and Pav is the average pressure (Phigh- P!o:w). For gs and as gas kinetic expressions can be given [103].

A similar expression to Eq. (9.68) usually fits very well the behaviour of the membrane (F0,m: support + top layer) so:

F0,m = g m + ~ m " Pav (9.68b)

The values of gs and [3s are calculated from measured permeation data for non-adsorbable gases (He, Ar, H2) using Eq. (9.68a). The permeation or permeability properties of the top layer are calculated now by subtracting the permeation data of the support only from the measured permeation data of the membrane using the series model. Note that Ph (high pressure) and P1 are measured at the interfaces of gas/ top layer and gas (permeate side) / support respectively.

The pressure Pi at the interface of top layer and support can be calculated by

1 -gs + 2 + [3s. P~ + 2gs" P1 + 2 (9.69) Pi= ~s

where ~ is the flow rate (tool or m 3 s -1) and A the membrane surface area. The support permeation for the actual experiment is given by:

F~ - A(Pi- P1) (9.70)

The theoretical validity of Eq. (9.68b) is discussed by Lin et al. [104] and it is shown that this equation is a special simplified case of a more general, but very complicated expression which strictly holds for the case that ~m/gm = ~s/gs. Uchytill [102] also devotes an extended discussion to this problem. Typical examples of the value of Pi and of the magnitude of the corrections are given in the cited literature.

For ~/-A1203 top layers (thickness -- 4 ktm, pore diameter 4 nm) on an ct-A1203 support (thickness -- 2 ram, pore diameter -- 0.2 ktm) Uhlhorn [21] reports a value of Pi = 55 kPa with Ph = 80 kPa and P1 = 7 kPa and H 2 as the permeating gas. This means that in this case only 30% of the total pressure drop is across the 7-A1203 top layer; the remainder is across the support. De Lange et al. [60] applied a

416 9 --TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS

microporous silica layer on top of supporting system similar to that used by Uhlhorn. The silica layer has a thickness of about 100 nm and a pore diameter of 0.4-0.5 nm. At 50~ the relative pressure drop across the support is about 4%, while at 250~ this is 15% (with the same gas flow �9 of 7.1x10 -6 mol s -1) and the main pressure drop being across the silica top layers. At 250~ the value of Ph = 0.71 bar and of Pi = 0.12 bar. So in this case the correction increases with increasing temperature and with increasing total permeation values due to a decreasing contribution to the permeation of the silica top layer.

Effects of the support on separation The low pressure PI is measured and can be manipulated only at the interface

gas(permeate)/support. This implies that when the support resistance is not negligible the value of Pi at the interface of support and top layer can be considerably larger than P1. Especially important is the case with strongly adsorbing gases where even a small increase of Pi can lead to a large increase of the occupancy at this interface and consequently to strong effects on the relative permeation contributions (separation) in gas mixtures. According to Eq. (9.38) this also means that the real separation factor of the top layer is decreased with respect to the ideal separation factor by back diffusion from the support (see Sections 9.3.1 and 9.3.2). This becomes especially serious when the conditions are such that the support is in the viscous flow or in transition regime from viscous flow to Knudsen flow. (This means that the support has no or hardly any separation properties itself.)

Even relatively small amounts of non-Knudsen contributions in the diffusive transport (which hardly affects the permeation) can decrease the separation factor considerably (see Eqs. (9.38) and (9.34)). This implies that to obtain maximum separation factors the support resistance should be as small as possible and vacuum suction is preferred above use of a sweep gas to remove the permeate (from the permeate side).

If the conditions are such that the mesoporous support is in the Knudsen regime, and so has some separation properties, the separation factor can be enhanced when the feed is applied from the support side. In this case the gas composition at the interface between support and separation layer is enriched already somewhat. This effect is reported by Keizer et al. [20] and could be described by the empirical relation:

Or = O~supp- O~oplayer (9.71)

9.6 OVERVIEW OF IMPORTANT RESULTS

In this section a brief overview will be given of the most important results of permeation and separation. It is not the intention to give a complete review of

9 ~ TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS 4 1 7

all available literature but merely to illustrate the state of the art, to show possibilities and to compare results with porous systems with competing dense membranes.

9.6.1 Introductory Remarks

Permeation and separation data reported in the literature are difficult to compare directly. This is due to the variety of parameters which influence the absolute value of permeation and separation data and which are usually badly described and sometimes cannot even be adequately described. As is shown in the preceding sections the pressure conditions and the flow dynamics (aerody- namic conditions) play a very important role. These pressure conditions are not always adequately described and data describing the external flow conditions do not directly reflect flow conditions in the membrane (model design and /or membrane architecture playing a role).

Flux data in mol (or m 3) per unit of time and surface area are the preferred data. To obtain data reduction and to make comparison easier permeation (permeance) data are usually given. One should realise however that this is only meaningful if the flux is a linear function of pressure (difference), so in the Henry region. Permeation data given as permeation (permeance) must be accompanied by information concerning the validity region (pressure boundaries) and the form of the pressure dependency. In the latter case this leads generally to a dimension of mol m -2 s -1 Pa -x with 0 < x < 1.

A membrane material with a high permeation which is valid only in a small pressure range and which "saturates" at low pressure is inferior compared with a membrane material with lower permeation which is valid in a wide pressure range.

Data given in the form of permeability (mol m / m 2 s Pa) are usually meaningful only in symmetric membranes (single, homogeneous wall, non-supported).

In asymmetric supported membranes the use of permeability data can give rise to much confusion and erroneous conclusions for several reasons. In most cases the layer thickness is not precisely known and usually it is not known whether this layer is homogeneous or has property gradients (e.g. a "skin" and a more porous part). In many cases the material of the layer penetrates the support to some extent and so it is not possible to separate properties of separation layer and support without giving account of the interface effect. Finally, even if all these complications can be avoided, a comparison based on separation layer properties expressed in terms of permeabilities can give a completely wrong impression of the practical possibilities (as done in e.g. Ref. [109]). This is illustrated by comparison of hydrogen permeabilities of ultra-thin silica layers (see Tables 9.14-9.16) with other materials such as zeolites and metals. The "intrinsic" material properties of these silica layers are not impressive;


nevertheless membranes give the highest permeation values reported in literature in combination with good separation properties. This is due to a "technology factor", i.e., the possibility to make them extremely thin. This cannot be obtained so far with other materials. Comparison per unit of thickness ~ which is the essence of p e r m e a b i l i t y - gives the impression that equal thicknesses for all materials can actually be obtained, which is not the case. Furthermore, limitations imposed by surface reactions becoming rate limiting at a given thickness are not taken into account.

9.6.2 Typical Permeation and Separation Data for Porous Membranes

Most of the data are taken from an overview of Burggraaf [108] which has been updated with results reported later. Some typical results obtained with capillary condensation a n d / o r surface diffusion as transport mechanism are given in Table 9.13. A discussion of these data is given in Section 9.2.3.3. As is shown, interesting combinations of high to very high separation factors with reasonable to good flux values can be obtained.

Typical results for supported microporous silica membranes are given in Table 9.14 and are part ly discussed in Section 9.4.3.

The data given by de Lange (see Table 9.14) are all in the Henry regime and the permeation of H 2 and CO2 is in the range of 10-25 and 3-6x10 -2 mol m -2 s -1 bar -1

TABLE 9.13

Some typical results with capillary condensation and surface diffusion in meso- and macroporous membranes

Membrane Thickness Pore dia- Separation Permeation Temp. (~ Ref. (~Jm) meter factor (mol m -2 S -1 Press. (bar)

(nm) Pa -1)

T-A1203 modi- 10 3--4 H2/N2:8 H2:35x10-6 230 fied with Ag 0.1 (H2) T-A1203 4 3 C3H6/N2: 26 C3H6:30x10-6 -10 T-A1203 m o d i - C3H6/N2: 84 C3H6:1.6x10 -6 -10 fied with MgO T-A1203 5--8 4 Methanol/H2: Methanol: 100

680 =20x10 -6 2.2 Methanol/H2: Methanol: 200 110 =1.6x10 -6 23 Methanol/H2: Methanol: 100 >1000 5.6x10 -6 7.7

T-A1203 4-5 + silicate 10 0.5

Ulhorn et al. [28] Ulhom [37] Ulhorn et al. [37] Sperry et al. [39] Sperry et al. [39] Bai/Jai Noble [27]

Permeation is measured at relative pressure p/po where pore is filled (capillary condensation). p0 is condensation (saturation pressure) of free liquid.


TABLE 9.14

Overview of typical flux and separation data of supported microporous silica membranes made by different processes

Material / gas o~ Flux Temp. Pressure / (10 -2 tool m -2 s -1) (~ thickness

SiO2 (microporous) de Lange [59- on 3'-(x-A1203 support 63]

d=100 nm

H2/CO 2 4-10 H2: 10-25 200 I bar CO2: 3-6 (-~ small

H2/CH 4 30--40 CH4: 0.9-2.5 200 "

H2/n / i -bu tane 160 i-but: 0.3--0.6 250 "

O2/N 2 2-4 02: 2-6 200 "

O2/N2 11.5 ? 25 Hassan [95] 4.5 ? 150 ?

H2/N2

H2/i-butane

15.7 H2: 1 620 Wu [68] 4.2 H2: 0.3 340

243 H2: 2.7 300 Pfeed: i-but: --0.010 0.3-0.8

40 i-but: --0.1 300 Pperm: small

C3H6/C3H8 75 21

C3H 8 (single)

C3H 6 (single)

1.6 (C3H6) 35 Asaeda [64] 3.2 (C3I-I6) 150 d < 1 ~tm

pfeed: 6 bar

1.5 50-100 Pf = 2 bar

2.5 50-100 Pperm = 1 bar

respectively in combination with reasonable separation factors. High pressure data up to 20 bar for H 2 (de Lange et al.) indicate the possibility of very high permeation values.

H 2 / N 2 mixtures are investigated by Kim and Gavalas [65] using Vycor glass supports with silica deposited partly in and partly on top of the support. At 500~ they report a separation value of c~ equal to 1000 and a permeation of 3.6x10 -3 mol m -2 s -1 bar -1 for H2. More recently Wu et al. [68] improved this method and at 600~ reported (x = 12-16 with a permeation of 1x10 -2 mol m -2

s -1 bar -1 which is about one order of magnitude smaller than that reported by de Lange. This is partly explained by the rather thick plugs (2.0-2.5 ~tm) of silica, completely deposited within the support pores.

Interesting results are reported by Hassan et al. [95] using hollow fibre silica with an estimated pore size of 1.3 nm. For O 2 / N 2 the separation factor c~ = 11.5 at 298 K and c~ = 4.6 at 423 K. Permeation data are not given.


De Lange (Table 9.14) reports a value of (~ = 2 at 473 K with a permeation of 2-6x10-2 mol m -2 s -1 bar -1, which is reasonable in agreement with Hassan's data considering the strong decrease of c~ with increasing temperature.

These results indicate that interesting combinations of flux and separation factor in air separation can be obtained with silica membranes. A similar conclusion can be drawn for separation of saturated-unsaturated hydrocarbons as shown by Asaeda et al. [64] for propane-propene (see Table 9.14). In this case permeation values cannot be calculated from the flux values due to non-linear behaviour of the flux as function of pressure.

Finally Rao and Sircar [42] report data for microporous carbon layers (thickness 2.0-2.5 ~tm) deposited on carbon supports. For C4H10/H2 mixtures, 0~ equals 94 (at 295 K) which is much larger than the permselectivity (for explanation see Sections 9.4.2 and 9.4.3). The permeability for C4H10 is reported to be 112 Barrer. The author of this chapter recalculated this value to a permeation of 1.4x10 -3 mol m -2 s -1 bar -1. It is, however, very questionable to assume a linear

TABLE 9.15a

Overview and separation data of typical supported microporous zeolite (MFI) membranes

Gas o~ Flux Temp. Pressure/ (10 -2 tool m -2 s -1) (~ thickness

50 kPa" 50 kPa

H2/CO 2 1-2 H2: 2-3 200-350 d=50 ~tm (Bakker) CO2: 1.7 stainless support

1.25 CO2: 0.1 200 d=3.0 ~tm (Vroon) H2: 0.12 A1203 support

CO2/H 2 10 H2: 0.18 25 d=50 ~tm (Bakker) CO2: 1.8 [71]

14 CO2: 0.15--0.5 25 d=3.0 ~tm

H2/CH 4 low CH4: ? 200 Bakker [71]

1.9 CH4: 0.5-0.7 200 Vroon [72-74]

H2/n-butane 2.5 n-butane: 1.0 350 Bakker

1.0 n-butane: 1.5 200 Bakker

n-butane/H2 >100 n-butane: 0.5 25 Bakker

n/ i -butane 50 n-butane: 0.2 25 Vroon

27 ? 25 Bakker

O2/N2 1 02: 0.5-0.35 25-200 Vroon

p / o xylene 1 p-xylene 25 Vroon [72-74] =60 3.5x10 --6 100-150 0.36:0.26 kPa 25 3.5x10 -6 200 0.36:0.26 kPa

Benzene / cyclohex. 5 2.6x10 -7 25 Vroon 4.5 18x10 -7 200 4.6:4.6 kPa

(benzene) 4.6:4.6 kPa


TABLE 19.15b

Overview of typical flux and separation data of supported zeolite (MFI) membranes

Material 0~ Flux Temp. Pressure / (10 -2 tool m -2 s -1) (~ thickness

p /m xylene (triisopropyl 15 benzene)

35 ml m -2 h* (m-xylene) 25

n-butane - 5.5** 25 i-butane - 4.5 25

Xiang [76]: Pfeed" 17 bar Pperm: 10 Torr ratio 1:1:1 Pfeed' I bar Pperrn' zero

Jia [27,97]: CH3OH/CH4 190 1.7 (CH3OH) 100 Ptotal:1100 kPa

0.001 (CI-~ Pfeed (CH3OH): 440-220

CH3OH/CH 4 29 1.8 (CH3OH) 100 Ptotal:1500 kPa 0.007 (CH4) Pfeed (CH3OH):

400-165 kPa n/i-butane 3-6 ? ? ?

thickness: 10.0 ~tm

*Not clear whether this is ml liquid or gas. **Calculated from given permeance assuming linear relation with partial pressure (=40x10 -2.

re la t ionship be tween the flux and the p ressure for bu tane at 295 K. In other mic ropo rous sys tems such as silica and zeolites this is not the case. In ano ther

pape r Rao and Sircar [Gas Separation and Purification (1993) 279-284] r epor t ed

for C O 2 / H 2 mixtures at 296 K, c~ = 5 (Phigh = 2.36 bar) to 0~ = 20 (Phigh = 3.7 bar) w i th a permeabi l i ty of 1200 Barrer in both cases. This indicates an increase in the separa t ion factor w i th pressure , whe reas the p e r m e a t i o n remains constant .

Typical results for zeolite (MFS) m e m b r a n e s are collected in Table 9.15 and pa r t ly d iscussed in Section 9.4.3. As is s h o w n in the table the separa t ion factor of mix tures of w e a k l y and s t rongly adsorb ing gases (see Section 9.4.3) shows a convers ion as a funct ion of t empera tu re .

In teres t ing separa t ion values can be obta ined for C O 2 / H 2 and n - b u t a n e / H 2 mix tures at low tempera ture . These are in the same range as those ob ta ined in carbon membranes ; for the flux values a s imilar conclusion holds.

I somer separa t ion is d e m o n s t r a t e d by several au thors (see Table 9.15). Good separa t ion factors (27-50) are repor ted for mix tures of n- and i sobutane

by Vroon et al. [72,74] and Bakker et al. [71] with, however , modes t flux values. Separa t ion of para- f rom or tho-xylene is r epor ted by Vroon et al. [72,74] w i th c~ equals 60 in the t empe ra tu r e range of 100-150~ and (x = 25 at 200~ and a flux of 3.5x10 -6 m o l / m -2 s q for the fastest pe rmea t i ng p-xylene (100-150~ wi th h o w e v e r a ve ry small dr iv ing force.


The partial pressure at feed side (high pressure side) is only 0.36 kPa. Using higher partial pressures and increasing the temperature might bring the flux in the range 10-3-10-4 mol /m -1 s -1. Xiang and Ma [76] reported results for mixtures of para- and meta xylene. At room temperature the value of (x equals 15 with a flux for m-xylene of 35 c m 3 m -2 h -1. Assuming that the permeating gas volume is expressed a s c m 3 gas under standard conditions (this is not defined) this permeation value corresponds to -- 8x10 -s mol m -2 s -1 for m-xylene and with 3x10 -7 mol m -2 s -1 for p-xylene which is one order of magnitude smaller compared to values reported by Vroon et al.

9.6.3 Comparison of Permeation and Separation Data of Porous and Dense Membranes

Typical data for dense membranes are collected in Table 9.16. A full discussion of these data is outside the scope of this chapter. Using permeation values the reader should be aware of the fact that the pressure dependence of the flux is usually strongly non-linear, but takes the form of a power law with values for the exponent around 0.5. This makes direct comparison on the basis of permeance or permeability not meaningful. Furthermore, the permeation value is limited by surface reactions with a critical thickness varying between 0.1 and 2 mm depending on material and condition.

Finally, dense (i.e. non-porous) membranes permeate 02 o r H 2 o n l y and so are important only in applications where these gases play a role such as in air

TABLE 9.16a

Compar ison of typical flux data of microporous and dense m e m b r a n e s

Hydrogen Permeation Temp. (mol m -2 s -1) (~

SiO2 amorphous silica 6-20x10 -2 25-250 AP = 1 bar (1---~0) (measured)

Calculated >300x10 -2

Zeolite (silicalite) on steel 1-3x10 -2 100--400

Zeolite (silicalite) on 0.5-0.85x10 -2 a lumina

Pd resp. P d / A g films on 3.0-4.5x10 -2 a lumina

Pd film 0.1x10 -2

Pd film within pores of 10-40x10 -2

0r 0c > 1000 H2/N2

25-250

400-900

100

300

AP> 15bar

Bakker et al. [71] Thickness 50 ~tm

Vroon et al. [72-74] thickness 3-4 ~tm AP = 1 atm. (1~0)

Armor [115]: AP = 2 bar; H2 thickness: 4.5; resp. 22 ~tm

Nagamoto [116] AP = I bar H 2

Yan /Morooka [113]; AP = 1 bar H 2 thickness 2 pm


TABLE 9.16b

Comparison of typical flux data of microporous and dense membranes

Oxygen Flux Temp. (mol m -2 s -1) (~

La0.6Sr0.4Co3_~ 4.0x10 -2 900

La0.3Sr0.?Co3_8 0.3-0.4x10 -2 900

Y0.05BaCo0.9bO3.6 0.4x10 -2 900

La0.2Sr0.8Fe0.6Co0.403-8 0.2x10 -2 850

ZY-Pd (40 vol%) 0.1-0.2x10 -2 0.2-0.5x10 -3

0.6x10 -2

Teraok~ [110]

v. Doom/Bouwmeester [119] Thickness I mm air vs. 10 -2 bar

Brinkman et al. [118] air vs. 10 -2 bar

Balachandral [117] Thickness 0.25-1.2 mm air vs. CH4/H2 (4:1)

1100 Chen et al. [112] 900 air vs. 10 -2 bar

Thickness 0.5 mm 1100 air vs. CO/CO2

BiEr-Au (40 vol%) 0.68x10 -3 850 Chen et al. [112] BiEr-Ag (40 vol%) 0.17x10 -2 850 Thickness 1-1.5 mm BiEr-Ag (40 vol%) 0.85x10-3 750 air vs. --2x10 -2bar

BiY-Ag (35 vol%) 1.0x10 -2 750 Shen et al. [111] Thickness 90 ~tm air vs 6x10 -5 bar

si02 microporous film 2.0-5.0x10 -2 35-200 de Lange [59--63] on alumina thickness 100 nm

AP = I bar

s e p a r a t i o n a n d d e h y d r o g e n a t i o n or par t i a l o x i d a t i o n reac t ions in m e m b r a n e

reactors . As is s h o w n in Tab le 9.16 a n d b y c o m p a r i s o n of Table 9.16 w i t h Table 9.14

the v a l u e s of ob t a inab l e s e p a r a t i o n fac tors of m i c r o p o r o u s m e m b r a n e s is m u c h

l o w e r t h a n those o b t a i n e d w i t h d e n s e m e m b r a n e s (wh ich s h o u l d be inf in i te in

the case of c o m p l e t e l y defec t - f ree d e n s e m e m b r a n e s ) .

V e r y r ecen t ly o x y g e n p e r m e a t i o n va lue s r e p o r t e d b y Shen et al. [111], C h e n

et al. [112] a n d T e r a o k a et al. [110] s h o w tha t the ob t a inab l e f lux v a l u e s at high

4 2 4 9 - - TRANSPORT A N D SEPARATION PROPERTIES OF MEMBRANES WITH GASES A N D VAPOURS

temperature (>600~ are at least a factor of 5-10 lower than those obtainable with microporous membranes at ambient or somewhat increased temperature (200~

For hydrogen a similar situation exists, except for the results reported by Yan and Morooka [113]. In this case the flux data are comparable with those obtained by de Lange et al. but with c~ > 1000 for H2/N2 mixtures.

9.7 CONCLUSIONS AND EVALUATION

A general description of gas transport properties of inorganic membranes with complex architecture and for multicomponent gas mixtures is not yet available. Quantitative descriptions based on phenomenological (thermodynamic) equations and/or microscopic models can be given in a number of limiting cases like single gases or binary gas mixtures and single wall, unsupported membranes or small plate shaped, asymmetric supported ceramic membranes. In the latter case the support properties are important and must be taken into account in the description of the membrane system and of the separating top layer.

In mesoporous membranes the maximum obtainable separation factor for non-condensable gases is limited to the Knudsen separation factor. For adsorbing gases below their critical point, surface flow can play an important role and high values of the permeation and of the separation factor can be obtained in some cases up to temperatures of 300~

In the case of macro- and mesoporous supports their flow resistance should be as small as possible. If the transport resistance is not negligible corrections must be applied in the study of the separation properties of the separating layers. It is shown that even small pressure gradients across the support can cause a considerable decrease of the permeation and of the separation factor of the top layer, especially in the case of adsorbing gases. The absolute value of the permeate pressure is important in addition to the pressure ratio of feed and permeate streams. Increasing support resistance causes an increase of the permeate pressure on the interface between support and separation (top) layer in the case of supported membranes.

High separation factors can be obtained with microporous membranes with a pore diameter smaller than 2 nm and are realised with carbon, silica and zeolite membrane systems. The description of these systems is still in its infancy.

In some cases reasonable agreement is obtained between calculated and measured permeation and separation properties. Permeation values of a single gas and of that gas in a mixture are generally different and so the separation factor of binary mixtures and the permselectivity (ratio of single gas permeation values) is also different.


The permeation of a gas is strongly affected by the sorption properties of the combination of gas and membrane and by the ratio of the molecular diameter of the gas molecule and the pore diameter. Mixtures must be classified on the basis of these two properties and the transport properties of these classes differ considerably.

The highest separation factors are obtained in the case of: (i) mixtures of strongly (S) and weakly (W) adsorbing gases at intermediate

temperature and pressure values, and (ii) the size exclusion regime; here one of the gases in the mixture has a

molecular diameter which is larger than the pore diameter. Typical values for permeation and separation factors of microporous mem-

branes are given in Tables 9.14 and 9.15. A comparison is also made with dense membranes in Table 9.16.

LIST OF SYMBOLS

A A* b B0 B*

sat Ci

C

dp

D D* Ed ;Co: F F p

gs J

k K Kn L M Mx,M* FI k

Surface area (m 2) Coefficient in Eq. (9.34). Subscript o: per mol. absorbed Henry constant (mol/kg Pa) Permeation coefficient (m) Coefficient in Eq. (9.34). Subscript o: per mol. absorbed Saturation concentration in material (mol/kg or mol / m 3) Concentration. Subscripts: s, surface; sat, saturated Pore diameter Diffusion coefficient (m2/s) Pre-exponential coeff, in Arrhenius equation Activation energy for diffusion Coefficient in Eq. (9.34) Permeation (mol/m2 s Pa) Permeability (mol m / m 2 s Pa) Fitting parameter in Eq. (9.65) Molar flux (mol/m 2 s). Subscripts" v, viscous; k, Knudsen; c, capillary condensation Correction term in Eq. (9.41b) Langmuir constant (Pa -1) Knudsen number Thickness (m) Molecular mass or molecular weight (kg/mol) Eq. (9.34g-h)) Fitting parameter in Eq. (9.34a) (m -2)


P P Pd Pr q Q Qa AQa Y

Ra

R Sc

t U V

Vm x

Z

Zwall

Partial pressure (Pa) Total pressure (Pa) Dimensionless pressure: P/Pref Ratio Plow/Phigh Amount absorbed gas (mol/kg) or (mol/m 3) Molar flow (tool/s) Heat of absorbtion (kJ/mol) Activation energy for surface diffusion (kJ/mol) Pore radius (m) or particle radius (m) Rate of adsorption Eq. (9.65) Gas constant (8.314 J/mol K) Stage cut (Qp/Qf) Sticking factor, defined by Eq. (9.65) Potential energy (kJ/mol). Subscript r: relative Molecular velocit~ (m/s) Molar volume (m/mol) Mol. fraction Distance coordinate (m) Number of molecular collisions with the walls of a volume ( c m - 3 S -1 )

Greek letters

F

0

Ok

V

Gs

Separation factor. Subscript 0: ideal separation factor (Eqs. (9.36) or (9.38)) Fitting parameter in Eq. (9.34b) (-) Fitting parameter in Eq. (9.68) Affinity coefficient in Eq. (9.56) (J/mol) Thermodynamic factor (-), defined in Eq. (9.40) Porosity (-) Dynamic viscosity (Pa s) Occupancy (c/cs) (-) Reflection factor in Eqs. (9.6) and (9.9) Molecular mean free path length (m) Geometric constant of pore structure (-) Jump probability, Eq. (9.57) Collision diamete r (m 2) Surface tension (J / m 2) Tortuosity (-) Volume flow (m3/s) or mol. flow (mol/s) Contact angle (-)


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