Chapter 27. Gas Transport in Membranes

Membrane-based separations are becoming important in many contexts. Examples can be found in gas separations, dialysis, reverse osmosis, and so on. An additional example is the unit operation of pervaporation, which is becoming important in many applications, including the production of ethanol for biofuel applications.

For pedagogical convenience we study separately the transport of gases and liquids across the membranes in two separate chapters. This chapter introduces the analysis of transport effects in various gas transport membrane processes and the models for such systems.

Transport of a gas mixture across a membrane is done with a view of separating a gas mixture. The Fick’s law type of models are generally used to characterize the rate of transport. But many terminologies and jargon, specific to this field and often confusing, are used in the literature. We define these common terms, such as permeance, permeability, selectivity, barrer, and so on. Some complexities (nonlinear effects) encountered in membrane transport are briefly presented.

Membrane transport rate provides the local model. This is then coupled to the process model for the separation unit. The process model needs some assumption about the mixing pattern in the gas phase. These features are explained and various flow patterns are schematically shown. Finally, models for membrane permeator units are presented as examples of the coupling of local and equipment-level models. These models are useful to evaluate the performance of gas separation equipment or to do a preliminary design of the process.

The final section provides an example where a membrane separation is coupled to a reactor, an example of a reactor–separator combo. The equilibrium limitations can be overcome to some extent by the simultaneous removal of the product and some applications of membrane reactors are indicated.

Gas separation using membranes is employed in a number of industrial applications for separation of gas mixtures. A few examples are indicated here:

• Recovery of hydrogen from product streams of ammonia plants, refinery process steams, or mixtures with inert gases such as nitrogen or methane

• Enrichment of air to produce oxygen for medical or metallurgical purposes or use in oxidation processes in chemical industries

• Removal of CO₂ and/or H₂S from natural gas

Polymers in a glassy state are examples of non-porous membranes. A solution–diffusion model is used as the mechanism of transport in such membranes. The transport rates of different gases are then determined by the differences in their solubility and diffusivity, as discussed later in this section. Since diffusivity in a non-porous matrix is low, the transport rate is low, but selectivity can be high due to solubility differences between the gases in the membrane. Hence the selection of membranes is often a compromise between the rate of transport and the selectivity to a specified gas. Examples of common polymeric membranes include cellulose triacetate, polyisoprene (natural rubber), polycarbonates, polysulfones, and teflon.

Porous membranes are more commonly used for gas separation. The pore diameter must be smaller than the mean free path of gas molecules. Thus the pore size has to be on the order of 10 nm or so. Gas flux through a porous membrane is much higher (three to five orders of magnitude) than for the non-porous case, but the separation selectivity is lower since both gases become closely permeable.

Membranes can also be asymmetric. This represents a compromise between a non-porous and a porous membrane. Here a dense but thin non-porous layer is bonded to a microporous layer. The dense layer provides the selectivity while the porous layer provides increased structural strength.

Although polymeric membranes are common, in special cases materials other than polymers can be utilized. Microporous α-alumina is one example. A second example is the palladium membranes that permit transport solely of hydrogen. Hydrogen is strongly adsorbed over Pd in a dissociative manner and hence the selectivity of such membranes toward hydrogen is very high. The operating temperatures of such inorganic membranes are higher than polymeric membranes.

We can use the Fick’s law type of model for transport across the membrane itself:

where N_A is the flux of the diffusing species across the membrane. Note that the low flux mass transfer model used here and N_A is assumed to be the same as the Fick’s law flux. D_A is the diffusion coefficient of A in the membrane, which is assigned a constant value.

Here C_A0 and C_AL are the concentration on the membrane side of the interface. This concentration is usually not directly known. What is known is the concentration in the fluid phase on either side of the membrane. The concentration in the fluid is related to the concentration in the membrane by an equilibrium relationship, Thus if C_A0,G is the concentration in the gas at x = 0 then the concentration in the membrane C_A0 is given as

where K_A is an equilibrium solubility constant, also called the partition coefficient. A similar equation holds at x = L. Hence

Using these equilibrium relations in Equation 27.1, we obtain the flux in terms of the concentration driving force based on the fluid concentrations:

Hence the overall transport rate depends on the product of D and K (as well as the thickness L) and not merely on the value of the diffusion coefficient. This composite parameter DK is defined as the permeability of the membrane. This has units of m²/s, similar to that of diffusivity, but it includes the partition coefficient as well.

The driving force can also be based on partial pressure units and a permeability P_M,A can be defined by the following equation:

where Δp_A is the partial pressure difference across the membrane. The permeability defined in this manner has units of mole – m/m² s Pa this case. The two are related by

PM=DKRgT

one barrer = 10^–10 cm³ STP. cm/cm² s Hg – cm

The conversion factor to traditional SI units is as follows:

one barrer = 3.348 × 10^–16mole – m/m² s Pa

The barrer unit is named after early work by Barrer (1951, 1984) in this area. One reason for the persistence of this unit is that the permeability for many membranes is on the order of 1 to 10 barrer and hence the values can be easily remembered. Typical values are shown in Table 27.1 from more extensive data provided by Seader et al. (2011).

Table 27.1 Permeability in Barrer of Common Gases in Typical Membranes at 25 °C

	H2	O2	N2	CO2
Polyethylene	9.8	2.93	0.97	12.66
Polymethacrylate	–	1.18	0.23	5.05
Polyvinlchloride	1.73	0.046	0.12	0.16
Butyl rubber	7.24	1.30	0.32	5.19

The transport rate across the membrane can thus be described (in terms of the fluid phase concentration difference as the driving force) as

Note that the permeance has the same units as the mass transfer coefficient (m/s) and is hence a measure of transport efficiency of a membrane. The driving force is also defined in this equation in terms of the gas phase concentration difference. The superscript (C) on permeance is used to indicate this.

The permeance of a membrane P is a composite parameter that depends on the diffusion coefficient of the membrane itself, the solubility or the partition coefficient of the solute in the membrane, and finally the thickness of the membrane itself. Thus the values of P may vary from one type of membrane to another and cannot be correlated in a theoretical sense unless all three effects can be quantified separately. However in practice the permeance is often used to describe the transport rate and in membrane module analysis as a matter of convenience. Often the permeance values are directly reported in the literature, since they can be directly calculated from the measured rate data.

The partial pressure difference is often used in Equation 27.7 instead of the concentration difference. The flux is given as

where P_A (with no superscript) is now based on the partial pressure difference. The permeance has units of mol/m².s.Pa (or mol/m².s.atm if the pressure is expressed in atm). The units are the same as k_G, the gas-side mass transfer coefficient used in absorption studies. We will use this definition and the partial pressure difference as the driving force for local flux calculation in permeator scale modeling in Section 27.3.

The effect of temperature is to increase permeance. An increase in temperature causes an increase in diffusivity while the effect of solubility may act in either direction. Overall, a modest increase in permeance with temperature is observed.

S12=P1P2

or

S12=K1K2⋅D1D2

Thus selectivity depends on both the solubility ratio and the diffusivity ratio.

This equation is called Sievert’s law and describes hydrogen transport in Pd-based membranes. The square root dependency arises due to the law of mass action: the concentration of H atoms (the diffusing species) is proportional to the square root of H₂ concentration.

• Constant diffusion and constant Henry’s law (K): This is the simplest case and the model in subsection 27.1.2 is applicable. Diffusion of permanent gases in elastomers and many harder polymers are often described by such a model.

• Diffusivity as a function of position: Membranes prepared from composite materials of two or more layers show different values of D for each layer.

• Concentration-dependent diffusivity D but constant K: This phenomena is exhibited by gases with critical temperatures near the ambient to 200° C, for example, C₄ hydrocarbons in rubbery membranes.

• Variable D and variable K: Gases with high critical temperatures, organic vapors, and so on, are examples of systems showing this pattern.

• Time-dependent effect, that is, D(C_A, t): Here a time- and history-dependent diffusion phenomena is observed. These are seen in polymers with longer relaxation times, for example, organic vapor in ethyl cellulose.

Further discussion of these complex effects is not considered here. The main point is for you to be aware of these phenomena in practical applications and use specific modifications in the model as needed. The models for all the preceding cases can be set up using the methodology described in Chapter 6 for variable diffusivity problems. Numerical computation is usually needed to integrate the local model across the membrane thickness.

The gas translation model can be viewed as an extension of the Knudsen diffusion concept. In small micropores the pore diameter is smaller than the molecular mean free path and hence diffusion is through the Knudsen mechanism. Note that the Knudsen diffusion coefficient in a cylindrical straight pore (Section 7.5.1) is given as

DK,A=13dp8RgTπMA

where d_p is the diameter of the pore. The term within the square root is the mean free path of the diffusing gas.

The corresponding permeability is given by using the porosity and a tortousity factor:

PA=(∊τL)DKA

The gas translation model is a modification of the above relationship, which was based on the Knudsen diffusion concepts. In this model, the permeate diffusion is assumed to be restricted from the potential field of the membrane wall and a correction factor, p_i, is applied to the this formula. The permeabiity equation is then multiplied by the following factor:

The factor p_i represents the probability factor, which represents the fraction of molecules that have the energy to overcome the activation barrier; this is modeled by an Arrhenius type of equation:

pi=p0exp(−EiRgT)

p₀ is the pre-exponential factor and E_i is the activation energy barrier for transport through the pores.

The concepts behind the modified gas translation model are shown in Figure 27.3. Since the center of the permeating molecules cannot approach the pore wall, the effective cross-sectional area is modeled as a circle with a diameter equal to the effective diffusion length d_p – d_A, as shown in this figure.

d_A is the effective molecular diameter of species A. The ratio of this area to the physical area is used as the pre-exponential factor: hence the pre-exponential factor, p₀, is modeled as

p0=(dp−dA)2dp2

Introducing this into Equation 27.10, the permeability can be written as

This provides a useful expression to correlate data on the various gases in terms of structural parameters and an activation energy parameter.

The model also provides a way to assess the selectivity parameter for two gases based on the gas properties. Taking the ratio of the permeability for two gases, the selectivity parameter is given by

S12=M2M1⋅(dp−d1)3(dp−d2)3

d₁ and d₂ are the molecular diameters of species 1 and 2 respectively. This equation indicates a dependency on the molecular size as well in addition to the molecular weight. The Knudsen model would have predicted only the effect of molecular weight. The second difference is in the temperature effect. The classical Knudsen predicts a square root relation while the gas translation model predicts an additional exponential effect. Thus the temperature dependency is

PA∝Texp(−EARgT)

Basic aspects and useful relations common to all four models are first presented. A binary feed consisting of A and B is considered.

The following notation is used here: x is the mole fraction of A in the retentate and y is the mole fraction of A in the permeate (i.e., species subscript A is dropped.) These are also local values at any position in the bulk gas.

where P₁ is the pressure on the membrane side and P₂ is the pressure on the pemeate side. PA is the permeance of species A.

Similarly for B the flux is given as

The pressure ratio parameter and selectivity parameter are introduced to simplify the above equations.

The selectivity parameter is defined as

S=PAPB

The pressure ratio parameter is defined as

P*=P2P1

With these definitions, the fluxes are given as

y′=JAJA+JB

Using the fluxes the following relation can be shown for the local permeate composition, y^′:

This equation provides the local composition of A crossing the membrane and it can then be used to model the performance of a membrane separator. The material balance equations are to be coupled with these flux relations. The material balances are different for different flow patterns as shown in Figure 27.4. Hence the effect of flow patterns needs to be accounted for; these are studied next.

Before we show these details of permeator performance calculations, it is worth noting two limiting cases of Equation 27.16. The first is when P* equal to zero. This will give the maximum local mole fraction in permeate:

ymax′=Sx1+(S−1)x

The second limiting case is when P* = 1. The value of y^′ is now zero since there is no driving force for diffusion when the pressures are equal on both sides. Separation is now possible only by adding a third component on the permeate side, which is called the sweep gas. This is used in some applications to improve separation. However, the sweep gas dilutes the permeate gas and hence may not be suitable in some applications.

There is also a minimum value of P* that is independent of the value of S. This arises due to the requirement that the partial pressure of A in the permeate does not exceed that in the feed:

P₁x_f ≥ P₂y_max

Since y_max can not exceed one, the operating pressure ratio has to be less than the following minimum value:

P* ≤ x_f

For example if the feed is 40% A, the pressure ratio has to be less than 0.4 in order to get a nearly pure A (in the permeate stream) under ideal operating conditions. This is similar to the minimum reflux concept in distillation operation. Here we have a minimum retentate side pressure criteria.

Gas permeator models are considered next. We examine the simplest case (backmixed-backmixed) first.

The following notation is used in this section: F_f is the molar flow rate of the feed with x_f the mole fraction of A in the feed. x is the exit mole fraction in the retentate; y is the exit mole fraction on the permeate side. Due to the assumption of complete mixing on the permeate side, the mole fractions x and y are also representative of the mole fractions in the permeater itself.

It is customary to use a cut parameter, denoted as Θ. This is defined as the fraction of the feed that becomes the permeate:

Θ=FpFf

Here F_p is the permeate molar flow rate.

The mass balance for A gives

This can be expressed using the cut parameter and rearranged to give

This has the status of an operating line since it is based on mass balance and relates x to y. A second relation between these two is given by transport considerations, which are shown next.

The mole fraction in the retentate side y is now equal to the local mole fraction y^′ since there is no other flow in the retentate (no sweep gas). Hence Equation 27.16 relates y and x. This is reproduced below with the left-hand side changed to y now:

Equations 27.18 and 27.19 can be solved simultaneously to find the exit mole fractions for a given feed mole fraction with the cut as a parameter. Additional parameters to be specified are the selectivity ratio S and the pressure ratio P*. Since the cut varies from 0 to 1, a design type of plot can be constructed for the permeate mole fraction as a function of the cut. An illustrative plot of membrane performance is shown in Figure 27.6.

The model is useful for laboratory studies to interpret data and to evaluate permeability values as these are often operated such that the back-mxied assumption is reasonably valid.

The transfer area needed can then be calculated from the following equation:

A=FfΘyPAP1(x−P*y)

This will provide a conservative estimate since the driving force will be larger in other flow patterns. The flow pattern examined next is the counterflow of the two streams.

Here the membranes are made in tubes or fibers and tied together to form a bundle. The fiber bundle is put into a shell where the feed stream is contacted. The species diffuse into the fiber and the permeate is collected at the open end as shown in the figure. The flow is usually countercurrent since it provides a better degree of separation compared to cocurrent flow. A model for performance analysis is presented in this section.

A differential model is called for since the mole fractions vary along the tube length. However, if the differential model is discretized (e.g., by an implicit Euler scheme) an equivalent stagewise representation is obtained and can be used to simulate the process. A schematic of this is shown in Figure 27.8. The model equations are the species balance and the total flow balance for the permeate and the feed side; these are presented next. The permeater is divided into N stages with stage 1 starting at the closed end of the system. Each stage is assumed to provide an area ΔA_m for mass transfer.

The following notation is used: L is the molar flow rate on the feed side and V is the flow rate on the permeate side. The subscript will be used to denote the stream leaving a stage. The stream entering a stage will have the subscript of the stage from where it came. This notation is similar to that used in distillation. The balance equations using this notation are presented next.

The species balance on the feed side can be derived as

L_k+1x_k+1 – L_kx_k = (J_A + J_B)y^′_kΔA_m

The total balance on the feed side is

L_k+1 – L_k = (J_A + J_B)ΔA_m

The following equation holds for the species balance on the permeate side:

V_k–1yk–1 – V_kyk = –(J_A + J_B)y^′_kΔA_m

Finally, the total balance is needed for the permeate side:

V_k–1 – V_k = –(J_A + J_B)ΔA_m

y^′_k and x_k are related by Equation 27.16.

A simple solution procedure is as follows. The computations are usually started at the closed end since the permeate conditions can be specified at this end. The retentate flow and mole fraction are assumed at this end. This is stage one. Hence L₁ corresponds to the specified (or guessed) retentate flow and x₁ is the mole fraction at the exit. For the permeate, V₀ and y₀ are zero in the absence of the sweep gas. All the quantities for starting the calculation for stage one are therefore known and the composition and flow leaving stage one can now be calculated. These are then used for the next stage and we can move across the system to end up in stage N.

What value should be assigned to the number of stages N? This depends on whether it is a simulation or design problem. If it is a simulation problem the total area for mass transfer A_m is known. We can discretize this to any number of stages. The larger N is, the more accurate the results (with more computations) since the differential model is now closely approached. The parameter ΔA_m has to be selected as A_m/N and will depend on the value of N chosen. The calculations are done for N stages and provide the values for the inlet feed conditions and also the exit permeate flow and mole fraction for the chosen exit retentate conditions. Using many different exit conditions, a design chart (a parametric plot) can be prepared.

For a design problem a value of ΔA_m is chosen as an incremental area for simulation. The stagewise calculations are continued until the inlet feed mole fraction is reached. This provides the number of stages (N) needed to achieve the separation. The required membrane area is then calculated as N ΔA_m.

The analysis of the cocurrent flow pattern shown in Figure 27.4 involves merely switching the flow arrangement in Figure 27.8 and is not presented here.

A stagewise model is useful and is shown in Figure 27.10.

The permeate from each stage is assumed to flow into the center pipe. The model is merely a simplification of the countercurrent flow. The V terms in the permeate balance are set as zero. Stage k contributes a permeate molar flow of (ΔV)_k, which is equal to J_A + J_B from that stage. The composition of this stream is the same as the local composition y^′. Hence y_k = y^′_k is used in the feed species balance equations. The total permeate flow is the sum of the flow from the N stages and the composition is the weighted average:

yexit=∑kyk′ΔVk∑kΔVk

A schematic of a membrane reactor with product removal is presented in Figure 27.11. One widely studied example is the steam reforming of methane to produce synthesis gas:

CH₄ + H₂O ⇌ 3H₂ + CO

The reaction is endothermic and equilibrium limited. The water gas shift reaction also occurs in parallel:

CO + H₂O ⇌ H₂ + CO₂

This reaction is mildly exothermic. The simultaneous removal of hydrogen shifts the equilibrium to the right. Also it is possible to operate the reactor at somewhat lower temperature.

A second example is the cracking of cyclohexane to benzene:

C₆H₁₂ ⇌ C₆H₆ + 3H₂

Again equilibrium limitations can be overcome by the use of a membrane reactor. Hydrogen-selective membranes that operate at high temperatures are used in these processes.

Another potential application is in partial oxidation. Here one of the reactants (oxygen) is introduced into the reactor by transport across a membrane. The membrane provides a means of supplying oxygen to the system at a controlled rate. The concentration of oxygen can be adjusted (usually to a low value) so that the secondary oxidation reactions are suppressed. A schematic of a membrane reactor with reactant dosing is shown in Figure 27.12.

27.2 What is meant by a sweep? When is it used?

27.3 What is a barrer?

27.4 What is the effect of temperature on the permeability?

27.5 State Sievert’s law. Where is it used?

27.6 State factors that could lead to nonlinear models for membrane transport.

27.7 What are the assumptions in the dual mode transport model in polymeric membranes?

27.8 What additional effects are included in the gas translation model compared to the simple Knudsen diffusion model?

27.9 State four common flow patterns used in membrane separation modeling.

27.10 Define the cut parameter.

27.11 What is the value used for the permeate composition in the cross-flow arrangement?

27.12 State some merits of reactive membranes.

27.13 What is the dosing membrane reactor?

27.14 What is the separating membrane reactor?

27.2 Relative rate based on permeance values. A nitrogen (20%) and methane (80%) mixture is fed to a membrane at a rate of 1000 kmol/hour. The feed pressure is 550 kPa and the permeate is 100 kPa. The permeance values are 50,000 for nitrogen and 10,000 barrer/cm for nitrogen and methane, respectively. Find the flux of each component across the membrane.

27.3 Gas translation model. Estimate the selectivity of hydrogen (d_A = 0.25 nm) to nitrogen (0.4 nm) in a membrane with a pore diameter of 0.6 nm. Use the activation energy parameters reported in the paper by Nagasawa et al. (2014) of 4.90 kJ/mole for hydrogen and 7.53 for nitrogen. Compare the values with the selectivity predicted by using the classical Knudsen model. The temperature is 200 °C.

27.4 Permeabilty from differential data. Consider a case where air is separated at a feed rate of 20 L/min with P₁ = 3 atm and P₂ = 2.6 atm gauge. A 3 L/min STP of permeate with 40% O₂ was obtained. The retentate has 17% O₂, which is close to the inlet value. Hence an average value for mole fraction can be used for oxygen. Calculate the permeance of oxygen and the selectivity of the membrane for oxygen. A trial and error calculation is needed since the local mole fraction at the permeate side at the inlet is not known. This needs a selectivity value according to Equation 27.16.

27.5 Single-stage separation. Oxygen is separated from air in a membrane that has a selectivity of 8 for oxygen. Determine the maximum separation that can be achieved in a single stage unit. If 60% of oxygen is recovered, find the approximate permeate composition.

27.6 Hydrogen separation. A gas containing 70% hydrogen and the rest methane is separated into a nearly pure hydrogen stream in a counterflow membrane permeator that has a selectivty of 100 for hydrogen. The pressure ratio employed is 0.2. Find the fraction of hydrogen recovered in the permeate if the permeate has a purity of 96% for hydrogen and the cut. Model the separator as a countercurrent flow with ten stages.

27.7 Helium separation from natural gas. Helium has a mole fraction of 0.82% in a natrual gas and a high permeance with a selectivity factor of 200. Find the helium recovery for different cut values. Assume both the permeate and the retentate are close to backmixed flow in the membrane.

27.8 CO₂ separation from natural gas. Separation of CO₂ from natural gas in a hybrid fixed-site carrier type of membrane has been studied by He et al. (2014). Their data indicate a permeance value of 0.2 m³ STP/m² hour bar for CO₂ and a selectivity in the range of 30. The feed contains 10% CO₂ and the retentate should have less than 3% CO₂. A countercurrent unit is used and the pressure ratio is 0.3. Suggest a design by specifying a feed cut. Find the permeate composition.

27.9 Recovery of VOC from air. VOC can be separated by using highly selective membrane and Baker et al. (1987) examines this potential. A membrane with permeability of 20,000 barrer for VOC (acetone) and 4 barrer for air was used with a membrane thickness of 2 μm. The feed was 0.5% acetone at atmospheric pressure while a pressure of 3 cm Hg was used in the permeate side. The flow rate was 0.2 m³/min in a spiral wound type of arrangement. Find the membrane area needed if the retentate has 0.05% mole of acetone and the permeate has 5% mole of acetone.