Modelling and simulation of hydrogen permeation through supported Pd-alloy membranes with a multicomponent approach

Chemical Engineering Science 63 (2008) 2149 – 2160 www.elsevier.com/locate/ces

Modelling and simulation of hydrogen permeation through supported Pd-alloy membranes with a multicomponent approach Alessio Caravella a,b , Giuseppe Barbieri a,∗ , Enrico Drioli a,b a Institute for Membrane Technology, ITM-CNR, Via Pietro Bucci, c/o The University of Calabria, Cubo 17/C, 87030 Rende, CS, Italy b Department of Chemical Engineering and Materials, The University of Calabria, Cubo 44/A, Via Pietro Bucci, 87030 Rende, CS, Italy

Received 22 December 2006; received in revised form 11 December 2007; accepted 10 January 2008 Available online 17 January 2008

Abstract Hydrogen transport in Pd-based supported membranes was described by means of a model considering several elementary steps of the permeation process, improving what done by Ward and Dao [1999. Model of hydrogen permeation behavior in palladium membranes. Journal of Membrane Science 153 (2), 211–231] for self-supported membranes. The model includes the external mass transfer in the multicomponent gaseous phases on both sides of the membrane, described by the Stefan–Maxwell equations. The transport of the multicomponent mixture in the multilayered porous support was also considered and described by means of the dusty gas model, which takes into account Knudsen, Poiseuille and ordinary diffusion. The diffusion in the Pd-alloy layer is modeled by the irreversible thermodynamics theory, taking the hydrogen chemical potential as the driving force of the diffusion in the metallic bulk. The interfacial phenomena (adsorption, desorption, transition from Pd-based surface to Pd-based bulk and vice-versa) were described by the same expressions used by Ward and Dao (1999). Thicknesses of 1 and 10 m are considered for the Pd-alloy layer. The asymmetric support consists of five layers, each one characterized by a specific thickness and mean pore diameter. The model separates the permeation steps and consequently their influence, quantifying the relative resistances offered by each of them. Comparison with some experimental data in several conditions in the literature shows a good agreement. The developed tool is able to describe hydrogen transport through a supported Pd-based membrane, recognizing the rate-determining steps (e.g., diffusion in the metallic bulk or in the porous support) involved in the permeation. 䉷 2008 Elsevier Ltd. All rights reserved. Keywords: Hydrogen permeation; Pd-based supported membranes

1. Introduction It is well known that Pd-based membranes have the specific ability to let only hydrogen permeate. Thus, these membranes are usually employed in processes requiring the selective removal of hydrogen from the reactive zone, like dehydrogenation reactions (propane, isobutane/n-butane, ethylbenzene, etc.) or processes to produce pure hydrogen (methane or methanol steam reforming and water–gas shift reaction) for PEM fuel cells, preventing chemical equilibrium from being established. For this reason, the modelling of the hydrogen permeation through Pd-alloy membranes is very important to optimize the operating conditions of these types of process. ∗ Corresponding author. Tel.: +39 0984492029; fax: +39 0984402103.

E-mail address: [email protected] (G. Barbieri). 0009-2509/$ - see front matter 䉷 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.01.009

The relatively high costs of palladium and the necessity to increase hydrogen permeability are driving more and more efforts to fabricate ever thinner membranes, which need an opportune support for acquiring the necessary mechanical resistance. Many articles of the literature (Liang and Hughes, 2005; Dittmeyer et al., 2001; Gielens et al., 2007; Guazzone et al., 2006; Tong and Matsumura, 2004; Tong et al., 2006; Höllein et al., 2001) use semi-empirical expressions for analyzing the permeation rate through palladium alloys like the following: J = (PHn2 ,Feed − PHn2 ,Permeate ), where the original Sieverts’ exponent value (n = 0.5) is replaced with another one calculated from statistical analysis of the experimental data. However, in all these works it is not possible to recognize the relative contribution of each transport and kinetic phenomenon involved in the permeation process. This occurs because an empirical Sieverts’ exponent is a clearly inadequate parameter to

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Table 1 Permeation elementary steps of the model considered in this work and comparison with that of Ward and Dao (1999)

Membrane

This work Supported

Ward and Dao (1999) Without support

Phase compositions Streams Feed side Permeate side In multilayered porous support

Multicomponent mixture Multicomponent mixture Multicomponent mixture

Hydrogen as the unique species H2 –N2 binary mixture N.A.

Differences in the mathematical approach External mass transfer Feed side Permeate side Multilayered porous support

Stefan–Maxwell theory Stefan–Maxwell theory Dusty gas model

N.A. Constant mass transfer coefficient N.A.

Driving force of the diffusion in the Pd-bulk

Analogies Adsorption and desorption “Surface to bulk” and “bulk to surface”

d H

C H

dPd-layer Gradient of hydrogen chemical potential along the Pd-layer

Pd-layer Difference between the hydrogen concentrations at the sides of Pd-layer

King and Wells’ (1974) equations (as Ward and Dao, 1999) Ward and Dao’s approach (1999)

describe the effects of many interacting phenomena. In fact, the increase of the driving force exponent (with respect to 0.5) can be due to the surface phenomena as well as to the porous support effects. Furthermore, the values of this exponent are estimated by means of a statistical-type analysis, usually a nonlinear regression, whose inevitable fitting errors represent another degree of uncertainty. For all these reasons, a deviation from the original Sieverts’ law (n = 0.5) may have different reasons and there is no unequivocal conclusion to be drawn from this. In order to be able to recognize the individual effect of each phenomenon occurring in hydrogen permeation through supported Pd-based membranes, it is necessary to have an adequate mathematical tool. Ward and Dao (1999) have already developed a model for hydrogen permeation in Pd membranes, considering several elementary steps. However, they did not consider the presence of any support and, furthermore, their approach to external mass transfer is of binary mixture-type. For these reasons, the aim of this work is precisely to develop a model taking into account different elementary steps, with special care paid to the multilayered porous support and the external mass transfer, both based on the multicomponent Stefan–Maxwell equations. Since the present approach is similar to that of Ward and Dao (1999), their article will be taken as a reference work, as a starting point for improving the mathematical modelling. The description of the present model and the differences and analogies with Ward and Dao (1999) are reported in Table 1. As indicated in Table 1, this model is able to describe the transport in supported membranes, which cannot be analyzed by means of the Ward and Dao’s model, since they considered self-supported ones. In the fluid phases adjacent to the membrane, multicomponent mixtures are considered, using the

Stefan–Maxwell approach to describe the mass transfer. In particular, the transport in the support is described by means of the dusty gas model (DGM), accounting for Knudsen, Poiseuille and ordinary diffusion. For transport through the Pd-alloy layer, the gradient of the hydrogen chemical potential is used as the driving force for the diffusion. On the contrary, Ward and Dao (1999) consider a single component stream on the feed side and a binary mixture on permeating side, using constant mass transfer coefficients to describe the external mass transfer. Furthermore, they took the hydrogen concentration difference at the sides of the Pd-based bulk as the driving force of the diffusion. The kind of mathematical model developed here, missing in the open literature, represents a useful tool for analysing H2 permeation through thin supported membranes in order to: • Evaluate the hydrogen profiles along the membrane top-layer, in the layers of the asymmetric porous support and in the gasphase layers adjacent to the supported membrane, describing the mass transport in a more exhaustive way. • Calculate quantitatively the influence of each step on the whole permeation process, in order to identify the ratedetermining steps as a function of the operating conditions. • Enlarge the range of applicability of the mathematical description. • Design the membrane to maximize the transmembrane flux by minimizing the resistance of each elementary step. This can be done: (a) by decreasing the Pd-based layer thickness as regards the diffusion in the Pd-layer; (b) by modifying the fluid-dynamic flow regime (as turbulent as possible) for the transport in the films; (c) by decreasing the resistance of the support (adopting a larger mean pore diameter, a higher porosity and/or a less thickness) for the transport in the support.

A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

2. Description of the model Fig. 1 shows: (a) an SEM photo of a multilayered Pd-based membrane; (b) the scheme of membrane considered in this work; (c) the mass transfer mechanisms of each elementary steps involved in the permeation. The Pd-based layer is supported by an asymmetric support characterized by several (five) layers, each one having different thicknesses and pore sizes. This study describes the transport in the orthogonal direction to the membrane surface and no axial profiles are considered. Therefore, the scheme proposed can be

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used to describe the permeation, e.g., in a continuous stirred system or in each abscissa of tubular systems. The whole permeation mechanism has been divided into the following elementary steps: On the feed side: 1. Mass transfer in the fluid phase of the film just near the interface. 2. Adsorption: H2Feed ↔ 2H(Pd-surfaceFeed ) ⇒ Hydrogen decomposition in atoms on the Pd-alloy surface. Inside the Pd-alloy layer: 3. Surface to bulk: H(Pd-surfaceFeed ) ↔ H(Pd-bulk Feed ) ⇒ Atomic hydrogen transition from Pd-alloy surface to Pdalloy bulk. 4. Diffusion: H(Pd-bulk Feed ) → H(Pd-bulk Permeate ) ⇒ Transport in the Pd-alloy bulk. 5. Bulk to surface: H(Pd-bulk Permeate ) ↔ H(Pd-surfacePermeate ) ⇒ Atomic hydrogen transition from Pd-alloy bulk to Pdalloy surface. On the permeation side: 6. Desorption: 2H(Pd-surfacePermeate ) ↔ H2Permeate ⇒ Hydrogen atomic recombination on the Pd-alloy surface. 7. Mass transfer in the fluid phase of the multilayered porous support. 8. Mass transfer in the fluid phase of the film just near the interface. As regards the adsorption, desorption, surface to bulk and bulk to surface steps, virtual thicknesses are introduced in the graphical representation in order to visualize their location, since these phenomena are interfacial and take place on surfaces. Thus, the lines connecting the initial and final pressure values of these virtual domains are not profiles, because only the boundary points have a physical meaning. Furthermore, in this analysis the surface coverage and the atomic hydrogen concentration were expressed in terms of equivalent pressure, in order to represent homogeneous quantities and compare the H2 profile inside the selective layer to those in the fluid phases (films and porous support). 2.1. Mass transfer in the fluid phases on feed and permeate side The transport of the gas mixtures is modelled by using the Stefan–Maxwell multicomponent theory. In the simplest case of a binary mixture and for slow mass transfer velocity, the partial pressure of a component at a solid interface can be related to its flux by the following expression (Bird et al., 1960):

Fig. 1. (a) Example of a multilayered Pd-based membrane (adapted from Höllein and co-workers (2001). Reprinted from “Catalysis Today, 67(1–3), Höllein V., Thornton M., Quicker P., Dittmeyer R., Preparation and characterization of palladium composite membranes for hydrogen removal in hydrocarbon dehydrogenation membrane reactors, 33–42, Copyright (2001)”, with permission from Elsevier. (b) Scheme of the multilayered membrane considered in this work. (c) Mass transfer mechanisms of all the elementary steps involved in the permeation process.

N1 =

1 NTotal Surface Kc(P1Bulk − P1Surface ) + P . RT PTotal 1

(1)

The quantities of the other components are known by the difference, since the total pressure is constant. In the film theory, the mass transfer coefficient is related to the diffusivity and to

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the transfer film thickness in the following way: Sh =

where

Kcl = g(Re)Scq D12

(2)

where g(Re) is a function of Reynolds number, q is a real number and l is a characteristic dimension of the mass transfer equipment, which in this case can be represented by the equivalent diameters in the feed and permeate side. In the more complex case of multicomponent mixtures, Toor (1964) and Stewart and Prober (1964) proposed the same approach, based on the Stefan–Maxwell transport theory, by transforming some dimensionless scalar group into dimensionless matricial group. Using this theory, the expression for the fluxes is (Krishna and Taylor, 1993) 1 NTotal  Surface N = P , {Kc} · {Z} · (P Bulk −P Surface ) + RT PTotal

(3)

“ →”

where all the vectors and matrices, indicated with and “{}”, have a dimension of (n − 1) and (n − 1) × (n − 1), respectively, since the profiles of one of the components can be calculated by the others. {Kc} is the matrix of the mass transfer coefficients and {Z} is that of correction factors for taking into account the high transfer rates. Using the film theory, {Kc} is expressed by:

{} =

RT N Total {Kc}−1 . PTotal

(9)

In these calculations, the expression for {Z} was trunk at second order, that is {Z} {I } −

1 2

{} +

1 12

{}2 .

(10)

Finally, once chosen the semi-empirical correlation, the following expression is obtained: {Kc}dEquivalent = r g(Re){D}1−q ,

(11)

where the matrix power is calculated by the approximation of Alopaeus and Nordèn (1999). The vector of the fluxes is N = [N1 0 · · · 0],

with a dimension of (n − 1)

(12)

because only hydrogen can pass through the membrane. In this case, the total flux is NTotal =

Species N

Ni = N1 = N1Surface .

(13)

i=1

In the calculation, the membrane curvature is neglected. In order to solve the problem, that is, to find the partial pressures (4) of all the components at the interface between the support and the mass transfer film, an iterative procedure was used. A value In analogy to the scalar case, the Sherwood matrix can for Film is fixed and the following equation is solved by a be expressed by semi-empirical correlations (Krishna and Newton–Raphson method:        Surface   Surface  = 0,    (14) −P f = P  Stefan.Maxwell  Stefan.Maxwell   

{Sh} · {D} {Kc} = . dEquivalent

Differential equations

Taylor, 1993): {Sh} = g(Re){Sc}q .

(5)

The function of Reynolds number continues to be scalar, while Schmidt number becomes a matrix, given by {Sc} = {D}

−1

.

(6)

Matrix {D} is the matrix of the multicomponent diffusivities, calculated in according with the Stefan–Maxwell theory (Krishna and Taylor, 1993), where the index “n” marks the nth species of the mixture: {D} = {B}−1 ,  Bij = −xi

Multicomponent mass transfer

  where P Surface  Stefan.Maxwell

where Bii =

1 1 − Dij Din

 .

xi + Din

Species N

k=1 k=i

xk , Dik

Differential equations

partial pressures calculated by integrating the Stefan–Maxwell equations along the mass transfer thickness of the fluid phases: Species

N Pi N j − P j N i dPi , = dl CDij

i = 1 . . . (N Species − 1).

(15)

j =1

The semi-empirical correlation used in this work is the following (Cussler, 1997): for Re > 2100 (turbulent flow) ⇒ g(Re) = 0.026Req , q = 0.80. 2.2. Dissociative adsorption and recombinative desorption

(7)

In accordance with the film theory, the expression for the matrix {Z} is (Krishna and Taylor, 1993) {Z} = {}(e{} − {I })−1 ,

is the vector of interfacial

(8)

Analogously to Ward and Dao (1999), the adsorption and desorption rates were described by the expressions introduced for the first time by King and Wells (1974) for the adsorption of nitrogen on tungsten and used by Behm and co-workers (1980) for hydrogen adsorption on palladium. Hydrogen adsorption on palladium surfaces involves a bi-atomic dissociation, while the

A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

desorption involves a superficial recombination of two hydrogen atoms to form one molecule. The adsorption rate is rAdsorption = 2S(),

(16)

where: = 

PHSurface 2 2MH2 RT

.

(17)

 is the molecular bombardment rate and S() is the sticking coefficient, depending on the surface coverage , and PHSurface 2 is the hydrogen partial pressure in the fluid just near the interface fluid-solid. S() can be modelled by the quasi-chemical equilibrium approximation of King and Wells (1974): S0 

S() =

, 1 −1 oo

1 + Kw oo = 1 −  −

1+



2(1 − ) 1 − 4(1 − )[1 − e(w/RT ) ]

,

(18)

where S0 is supposed to be equal to unity. The same procedure leads to the following expressions for the desorption rate: 0 e(−2EDesorption /RT ) NS2 HH rDesorption = 2KDesorption

(19)

with HH =  −

1+



2(1 − ) 1 − 4(1 − )[1 − e(w/RT ) ]

.

(20)

2.3. Transitions “surface to Pd-bulk” and “Pd-bulk to surface” As indicated by Ward and Dao (1999), the “surface to bulk” transition rate is 0 (− rSurface to bulk = NS Nb KSurface to bulk e

ESurface to bulk RT

)

(1 − ), (21)

where is the atomic hydrogen concentration just near the Pdsurface. By analogy, the “bulk to surface” transition rate is rBulk

to surface

0 = NS Nb KBulk

to surface e

(−

EBulk to surface ) RT

(1 − ). (22)

The two kinetic constants are functions of the temperature and the surface coverage, as shown by the following expressions (Ward and Dao, 1999): ⎧ j 0 0 ⎪ = , KBulk ⎪ to surface ⎪ ⎪ 3Nb (1 − ) ⎪ ⎪ ⎪ 0 0.25 ⎪ KBulk 1 −  HH S0 ⎪ to surface (, T )T ⎨ K0 , Surface to bulk =  S() 10.154 ⎪ ⎪ ESurface to bulk = EBulk to surface ⎪ ⎪ ⎪ ⎪ EAdsorption − ESurface to bulk ⎪ ⎪ . ⎪ ⎩ + 2

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2.4. Diffusion through the Pd-bulk The diffusion of the atomic hydrogen in the Pd-bulk is described by the irreversible thermodynamics theory, for which the atomic hydrogen flux can be given as JH = −Nb DH H

j , jl

(23)

where l is the generic abscissa spacing along the Pd-thickness. The thermodynamic factor H takes into account the deviation from the ideal case ( >1). When the hydrogen concentration is high, H can be expressed as H = 1 +

j ln H , j

(24)

where H is the activity coefficient, which is related to the chemical potential as   j ln( H ) RT j ln H jH = RT = 1+ j j j RT jH = H ⇒ H = . (25) RT j The hydrogen chemical potential in the palladium lattice can be expressed by   0 (26) + 2 WHH − VH , H = H + RT ln 1− where 0H is the chemical potential at the standard state (infinite dilution, → 0). The last term takes into account the crystalline reticule deformation due to the stress caused by the atomic hydrogen passage. This deformation (elastic) becomes ever stronger as the hydrogen concentration increases, being negligible for dilute systems ( >1). The term  can be expressed as follows (Adrover et al., 2003): j 2YVH Nb j =− . jl 3 jl

(27)

From the previous expressions, the atomic hydrogen flux is j JH = − Nb DH H jl 

2YV2H Nb j 1 2WHH = − N b DH + + , 3RT 1− RT jl

(28)

where the diffusion coefficient of hydrogen in palladium, DH can be described as an activated process (Cussler, 1997): 0 (−EDiffusion /RT ) e . DH = D H

(29)

If the flux expression is integrated along the palladium thickness up to a generic abscissa by supposing a constant mean value for VH (see Table 4), this relation is obtained:      Nb Y V2H 1 − 1 WHH N b DH 2 2 ln + JH = ( 1 − ) . l 1− RT 3RT (30)

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A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

which for the entire palladium thickness becomes:    1 − 1 Nb D H JH = Pd-layer ln 1 − 2 

  Nb Y V2H WHH 2 2 + + ( 1 − 2 ) . RT 3RT

3. Operating conditions and numerical values used in the simulation

(31)

For dilute systems ( >1) is H 1 and the flux expression can be simplified in this way: JH − DH Nb

1 − 2 . Pd-layer

(32)

It can be demonstrated that when the diffusion through the Pdbulk is the unique rate-determining step and the other steps can be considered at the equilibrium, the latter relation leads to a theoretical demonstration of the Sieverts’ law. 2.5. Transport through the porous support The transport through the multilayered porous support is described by means of the DGM (Krishna, 1987). The mathematical expression used to analyze each layer is the following: n 

xj Ni − x i N j

Maxwell.Stefan j =1 CTotal Dij ,effective

=−

+

Ni Knudsen CTotal Di,effective

B0 xi xi V i ∇P , ∇T ,P i − ∇P − xi Knudsen RT RT Di,effective

(33)

where the factor B0 is calculated considering the support pore as cylindrical (Poiseuille flow relationship): B0 =

2 dpore

32

.

Since the components are considered as ideal gases, the previous expression can be transformed into n 

xj Ni − x i N j

Maxwell.Stefan j =1 CTotal Dij ,effective

+

Ni

The following tables report the operating conditions (Table 2), the geometrical parameters (Table 3) and physical and kinetic constants (Table 4) used in the simulation. The geometry of a Pd-alloy membrane (Table 3) is typical and common of many supported membranes analyzed in the open literature. 4. Results and discussion Fig. 2 shows the transmembrane hydrogen partial pressure profiles calculated at different temperatures for a Pd-alloy thickness of 1 m. At 300 ◦ C, the transport in the film on the feed side can be neglected with respect to the other permeation steps. The pressure drop in the adsorption and desorption steps is low but not negligible, while the surface to bulk and bulk to surface steps are very rapid (pressure drop flat in all the operating conditions considered). The greatest resistance is concentrated in the Pd-alloy layer and in the multilayered porous support, where the layers 5, 4, 2 and 1 provide the majority of resistance. The importance of the support is more and more important at a higher temperature (∼ 500 ◦ C). Thus, even the presence of the thinnest support layers have a certain influence on the permeation and cannot be neglected in the analysis. At a higher temperature, the rate of the surface phenomena increases and their slopes decrease, whereas the transport resistance is concentrated in the Pd-alloy layer and in the support. In this case, the increase of the pressure drop in the support—especially in the first layers—is relevant and is due to the Knudsen mechanism, which exceeds the other two diffusional mechanism considered in the support (Poiseuille and ordinary diffusion). Table 3 Geometrical data of the Pd-alloy supported membrane considered in the simulation Pd-alloy layer thickness, m = {1.100} Multilayered porous support Layer

Thickness (m)

Mean pore diameter (nm)

Porosity Tortuosity (dimensionless) (dimensionless)

1a 2 3 4 5 Total

1 50 100 500 850

1.5 mm

5 100 1000 10,000 50,000 –

0.5 0.5 0.5 0.5 0.5 –

Knudsen CTotal Di,effective

B0 xi V i ∇P − xi = −∇xi − ∇P Knudsen RT Di,effective

(34)

which is the final relation used in the simulation. The calculations of the physical properties for the fluid phases are reported in Appendix A.

a In

1.2 1.2 1.2 1.2 1.2 –

contact with the Pd-alloy surface.

Table 2 Operating conditions considered in the simulation Side

Pressure (kPa) H2

N2

O2

Feed 600 75 75 Permeate 100 80 20 Temperatures (◦ C) = {200, 250, 300, 350, 400, 500}

CO2

H2 O

CO

Total

50 –

50 –

25 –

875 200

Reynolds number

Total flow rate, mmol/s

4600–2500 6400–3200

100 20

A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

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Table 4 Physical and kinetic constants of the model

EAdsorption (J/mol) (Behm et al., 1980) EDesorption (J/mol) (Behm et al., 1980) EBulk to surface (J/mol) (Ward and Dao, 1999) EDiffusion (J/mol) (Holleck, 1970) VH (m3 /mol) (Adrover et al., 2003) Y (Pa) (Adrover et al., 2003) j 0 (s−1 ) (Ward and Dao, 1999) Ns (molPd /m2 ) (Ward and Dao, 1999)

83,680 41,840 22,175 22,175 1.77 E − 06 1.84 E + 11 2.30 E + 13 2.80 E − 05

Kw (dimensionless) (Ward and Dao, 1999) 0 KDesorption (s−1 ) (Behm et al., 1980) ESurface to bulk (J/mol) (Ward and Dao, 1999) 0 (m 2 /s) (Holleck, 1970) DH WHH (J/mol) (Ward and Dao, 1999) w (J/mol) (Ward and Dao, 1999) S0 (dimensionless) (Behm et al., 1980) Nb (molPd /m2 ) (Behm et al., 1980)

0.05 4.80 E + 17 16,736 2.90 E − 07 −23, 400 2,092 1 1.13 E + 05

Resistances, %

Permeate Side

Diffusion

50 Porous Support

1

Desorption Adsorption

Porous Support

Film

Bulk to Surface Desorption

Pd-alloy Bulk

Adsorption

Surface to Bulk

200

Film

200

4

T Ref = 350°C

Bulk

Feed Side

400

...................

Layer 5

Layer 1

250

Bulk

H2 Partial Pressure, kPa

500 C 300

0

5

100

Supported Membrane

600

H2 Flux

Fig. 2. Calculated hydrogen partial pressure profiles through the membrane and in the fluid phases adjacent the membrane for different temperatures. The lengths of the steps are not to scale. Pd-layer = 1 m. See Table 2 for the other operating conditions.

The mass transfer in the external films is rapid and provides an appreciable effect only on the feed side film. At lower temperatures (200.250 ◦ C), the influence of the surface phenomena (adsorption and desorption) is relevant and cannot be neglected, e.g., 200 ◦ C, at which the profile in the Pd-alloy layer and support are almost flat. Fig. 3 shows the relative resistances of each step and the normalized overall resistance of the whole membrane determined in the same conditions as Fig. 2. The normalized overall resistance is calculated with respect to the overall resistance at 350 ◦ C (see Appendix B for its definition), since this is a typical temperature at which this type of membrane is used. In this type of figure, the rate-determining steps can be recognized by their greater resistances. In particular, the importance of the porous support is clear, which provides a relevant increasing contribution from 250 ◦ C ca. onwards. This occurs since the transport in the pores is a non-activated process, which is not favored by the temperature increase. At 300.320 ◦ C the situation is quite complex, because there are four steps controlling the process. In this case, it is not possible to recognize only one rate-determining step and the value of the driving force exponent (n) in the modified Sieverts’ law cannot provide the relative influence of each step without any other information. Thus, the diffusion in the Pd-based layer is the controlling step if and only if n is equal to 0.5, whilst it is generally false that another value different from 0.5 is a unique identifiable

0

0 200

300 400 Temperature, °C

Normalized Overall Resistance, -

Data taken from the open literature (Ward and Dao, 1999; Tong et al., 2006; Krishna, 1987)

500

Fig. 3. Relative resistance of each elementary step and normalized overall resistance (see Appendix B) as a function of temperature. Pd-layer = 1 m. The non-reported steps do not provide any appreciable resistance. See Table 2 for the other operating conditions.

permeation step, especially in the case of very thin Pd-based layer. At high temperature (> 350 ◦ C ca.), the diffusion in the Pd-layer and in the porous support are the rate-determining steps, whereas in the middle, all of the four reported phenomena are relevant. The normalized overall resistance decreases very weakly from 350 ◦ C onwards, since there is a shift from the controlling steps characterized by high activation energies (the surface phenomena) to non-activated ones (transport in the support) or ones having lower activation energy (the diffusion in the Pd-layer). At low temperature (< 250 ◦ C ca.) the surface phenomena control the process with a small contribution from the diffusion in the Pd-layer and less from the support. The contributions of the other permeation steps (the transitions “surface to bulk” and “bulk to surface” and the mass transfer in the fluid phases at both sides of the membrane) are fully negligible. When a higher Pd-layer thickness (10 m) is considered (Fig. 4), a very different situation is found. In fact, at higher temperatures, the diffusion in the palladium influences the process much more than at 1 m), because the resistance provided by the Pd-alloy layer is greater. For the same reason, the influence of the diffusion in the Pd-alloy cannot be neglected neither at low temperatures (200.250 ◦ C). In this case, the influence of the film on the feed side is negligible even at high temperatures, and the pressure profiles

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A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

Temperature, °C

H2 Flux

5 Diffusion

Resistances, %

4

50 Adsorption

TRef = 350°C

Desorption

Porous Support

1

0

0 200

300

400

Normalized Overall Resistance, -

Fig. 4. Calculated hydrogen partial pressure profiles through the membrane and in the fluid phases adjacent to the membrane for different temperatures. The lengths of the steps are not in scale. Pd-layer = 10 m. See Table 2 for the other operating conditions.

100

400

300

200

100

500

Temperature, °C Fig. 5. Relative resistance of each elementary step and normalized overall resistance (see Appendix B) as function of temperature. Pd-layer = 10 m. The non-reported steps do not provide any appreciable resistance. See Table 2 for the other operating conditions.

in the support do not show such a great slope as before. This effect is due to the decreased permeating flux and to the relative position of the resistances with respect to the flux direction. These conditions correspond to a situation closer to the range of applicability of the Sieverts’ law. As done in the previous case, the relative resistances of each elementary steps and the normalized overall resistance are evaluated (Fig. 5). The steps not reported in the plot are negligible. The decreasing rate of the normalized overall resistance is larger than before. The relative resistance of the support is much lower than before in the whole temperature range considered, arriving at a value of 20% ca. at 350 ◦ C, against 80% ca. of that of the Pd-based layer. Thus, the influence of the support is less relevant than before, but cannot be neglected. Only whenever greater Pd-based layer thicknesses are considered, the influence of the support can be ignored. Anyway, the high-performance membranes are those characterized by a very thin Pd-based layer and, thus, they represent the objects of our study.

105

Feed Film Limited

104

Support Limited Diffusion Limited 1 µm Diffusion Limited Ad 10 µm so rp De tion so Li m rp ite tio d n Li m ite Overall fluxes d

103 102

350°C

H2 Permeating Flux, mmol/m2s

Porous Support

Permeate Side

...................

Bulk

Pd-alloy Bulk

Adsorption

Surface to Bulk

Film

200

0

200

Feed Side

400

600 106

Film

250

Layer 5

300

Desorption Layer 1

Bulk to Surface

600

Bulk

H2 Partial Pressure, kPa

Supported Membrane 500°C

10

at 1 and 10 µm

1 0.1 1

2

3

1000/T, K-1 Fig. 6. Step-limited (thin curves) and resulting (thick curves) hydrogen fluxes as function of the temperature. Pd-layer = 1 and 10 m. The non-reported steps provide very high rate-limited fluxes and are not plotted. See Table 2 for the other operating conditions.

Since the performances of a membrane are evaluated in terms of permeating flux, the two situations analyzed before (1 and 10 m) are compared in Fig. 6, showing the hydrogen flux as a function of temperature. Each thin curve is determined by considering the corresponding step as the unique rate-determining one and calculating all the others at the equilibrium. For instance, the H2 flux is much higher than the others if the ratedetermining step is the transport in the film (the highest-level curve). The resulting fluxes (thicker lines) are calculated by considering the influence of all the steps. The curve relative to the transport in the film on the feed side is almost flat, whereas the one relative to the transport in the support has an increasing profile. These steps are characterized by a nonlinear behavior in this type of plot, since they are non-activated processes. On the contrary, the diffusion in the Pd-alloy layer and of the surface phenomena are activated processes and their curves are straight, whose slopes are proportional to their respective activation energies. In particular, the curves of the diffusion limited for 1 and 10 m are shifted down by a factor exactly equal to the Pd-alloy thickness, since the dependence of the permeating flux on it is inversely proportional. In this type of plot, it is possible to visualize the effect of each permeation step in terms of hydrogen flux, because the overall fluxes curves tend to assume the slope of the controlling steps. For instance, at high temperature (> 350 ◦ C ca.) the overall curve at 1 m feels the effect of the transport in the support and of the diffusion in the Pd-layer. As seen before, the influence of the second step is greater than that of the first one. As a consequence, the slope of the overall curve is more similar to that of the corresponding diffusion-limited curve. When higher and higher temperature is considered, its slope tends to be similar to that of the support, since this step has increased its influence. Similarly, in the temperature range of 200.350 ◦ C the transition from the surface phenomena

A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

limited region to that limited by the diffusion is characterized by a significant change of slope. In this sense, this model can be very useful for maximizing the membrane performances when there is more than one rate-determining step, since it is able to recognize those steps whose influence can be minimized in dependence of the operating conditions, giving quantitative responses about the improvement in terms of permeating fluxes. 5. Comparison with the literature 5.1. Comparison with the simulation of Ward and Dao (1999) The comparison is made in the case of polarization, using the same operating conditions used by Ward and Dao (1999) (Re ≈ 10). However, not all the results can be compared, because they did not consider the effect of the support. For this reason, in order to be able to make a partial comparison, in this section the effect of the porous support was neglected, considering only the effects of the transport in the feed and permeate films (Fig. 7). Without the external mass transfer (absence of polarization), the difference between the two models is only in the mathematical description of the transport in the Pd-bulk, but it was demonstrated that for low atomic hydrogen concentrations, the two approaches are equivalent (see section “diffusion through the Pd-bulk”). The difference is very small in the whole temperature range considered for Pd-layer of 100 m, because the transport in the Pd-bulk limits the process. For thinner membranes, the effect of the external mass transfer is stronger, thus the behavior of the two models becomes increasingly different. The most important difference is that in Ward and Dao’s simulation there is a maximum value in the profiles at 1 and 10 m, while in present work the profile continuously increases with the temperature. This difference is due to the constant value of mass transfer coefficient considered by Ward and Dao (1999), a condition Temperature, °C 800

600

400

300

200

100

1 µm

1,000

JH2 = Kc(CHSurface − CHBulk )|Permeate 2 2

side

(35)

which, using the ideal gas law, becomes: JH2 =

Kc Surface − PHBulk )|Permeate (P 2 RT H2

side

.

(36)

As a consequence, their model establishes that in the case in which the external mass transfer is the rate-determining step, the permeating flux is inversely proportional to the temperature (decreasing profile). This behavior—flux decreasing with temperature—does not seem to have a physical justification for auto-supported membranes. On the contrary, the present model provides a monotonic behavior as a result, because the dependence of the mass transfer matrix on temperature and pressure was taken into account. A flux decreasing with temperature can be found only when a porous support is considered and at the same time the influence of the Knudsen diffusion in the porous support is much larger than the other diffusion mechanisms. 5.2. Comparison with some experimental data from literature In the following figure, the results of the model are compared with some experimental data from the literature relative to supported Pd-based membranes in different operating conditions. In particular, the comparison was made from literature data of: • Dittmeyer and co-workers (2001), who measured hydrogen flux as a function of the temperature for several feed pure hydrogen pressures. • Liang and Hughes (2005), for taking into account the behavior of the model at changing sweep-gas flow rate. The tortuosity and porosity (not specified in the two papers considered), are considered 1.2 and 0.4, respectively. Fig. 8 shows good agreement with experimental data taken from different sources. In particular, it can be noticed in the second plot (Liang and Hughes, 2005) that the permeating flux increases with the sweep rate only from 0 to ca. 0.5, whereas it remains at

10 µm

100

100 µm

10 1

This Work ----- Ward and Dao [1]

0.1 1

1.5

2

2.5

-1

1000/T, K

Fig. 7. Hydrogen permeating flux calculated considering the concentration Feed =101.3 kPa, PFeed =0 kPa, PPermeate ≈ 0 kPa, polarization phenomena. PH N H Permeate = 101.3 kPa. PN 2

removed in the present work. Their expression for external mass transfer in the permeate side is the following:

2

2

2

H2 Permeating Flux, mmol/m2 s

H2 Permeating Flux, mmol/m2 s

10,000

2157

100

400 Dittmeyer et al. (2001)

Liang and Hughes (2005)

Present Model

75

300 200

Pa

5k

50

35

Pa 55 k

2

100

25 Feed

P

0

350

= 155 kPa

450

550

650 0 0.5 1 1.5 2 N2 Sweep Flow Rate, mmol/s Temperature, °C

0

Fig. 8. Comparison with some literature experimental data. The values of kinetic parameters used for simulation are that reported in Table 4.

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A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

100 87%

Diffusio Resistances, %

n 68%

50 32%

upport

13%

S Porous

0 350

450 550 Temperature, °C

650

Fig. 9. Resistances of the permeating step calculated by means of the present model for some of the experimental data (Dittmeyer et al., 2001) considered in Fig. 8. PFeed = 255 kPa. The non-reported steps do not provide any appreciable contribution.

an almost constant value (55 mmol/s m2 ca.) for greater sweep rates. This happens because an increase of the sweep rate in the permeate side produces a decrease of the hydrogen partial pressure, reaching negligible values for very high sweep rate. In their experimental analysis, Dittmeyer and co-workers (2001) found that the membrane behavior did not follow the original Sieverts’ law. In order to investigate about the reasons for this discrepancy, the relative resistances of the permeation steps in their membrane were calculated by means of the present model, whose results are reported in Fig. 9. The unique relevant contributions are those of the diffusion in the Pd-based layer and the transport in the support. This means that, as in the present model, the explanation for the difference from the original Sieverts’ law is due to the presence of the support, whose resistance increases with the temperature. This type of evaluation is an example of the applicability of the present model to actual experimental cases. Since these evaluations cannot be made by the Ward and Dao’s model (1999) (where no support is considered), this model may represent a new tool to carry out the correct interpretation of the data analysis about the supported Pd-based membranes. However, it must be underlined that not all the Pd-alloy membranes have the same kinetic parameters, since the physical and chemical membrane properties strictly depend on the preparation method. Thus, an optimization procedure for calculating the model parameters is generally necessary in order to describe the permeation through a specific membrane.

partial pressure profiles in the Pd-alloy thickness, in the porous support and in the adjacent gaseous films were calculated as a function of the temperature and of the membrane thickness, evaluating the relative importance of each permeation elementary step and the hydrogen flux. The analysis shows that, for a membrane thickness of 1 m, the surface phenomena (adsorption and desorption) control the hydrogen permeation at very low temperatures (< 230 ◦ C ca.), whereas at a temperature included in the range of interest (350.500 ◦ C), the diffusion in the Pd-alloy bulk and the transport in the porous support are the rate-determining steps. For a thicker membrane (10 m), the diffusion in the Pd-alloy bulk represents the rate-determining step in the whole temperature range considered (relative resistance of ca. 85%). The simulation based on the present model was also compared with that of Ward and Dao (1999) (in terms of flux vs. temperature) for self-supported membranes. The results of the present model showed that the hydrogen permeating flux is an increasing function of the temperature, whereas the model of Ward and Dao (1999) establish the flux profiles to be non-monotone with temperature (owing to the use of constant mass transfer coefficients, removed in the present model). Since the model developed in the present work proved able to quantify the influence of each permeation step, especially of the multilayered porous support, it represents a useful tool to improve the flux prediction and optimize the performance of actual separation systems which the supported thin Pd-based membranes are involved in. Notation B0 C dequivalent DH Dij Dij d Pore E EAdsorption ESurface to bulk EYoung JH k K Kc Kw H

6. Conclusions The hydrogen permeation through supported Pd-alloy membranes was described by means of a model, considering several elementary steps of the process, accounting for in a multilayered porous support and the transport in the films adjacent the membrane (concentration polarization). The model prediction shows a good agreement with the experimental literature data in quite a large range of operating conditions. The hydrogen

Mw N Nb NS P R

geometrical factor, m2 molar concentration, mol/m3 equivalent diameter, m diffusion coefficient of atomic hydrogen in Pd-bulk, m2 /s binary diffusivity, m2 /s multicomponent diffusivity, m2 /s support mean pore diameter, m activation energy, J/mol heat of adsorption on the Pd-surface, J/mol heat of absorption in the Pd-bulk, J/mol Young’s modulus of palladium, Pa flux of atomic hydrogen, molH /m2 s Boltzmann constant, J/mol K kinetic constant, m2 /molH s mass transfer coefficient, m/s probability constant, dimensionless non-ideality correction factor, dimensionless molecular weight, kg/mol molar flux, mol/m2 s octahedral sites concentration in Pd-bulk, molPd /m3 superficial sites concentration on Pd-surface, molPd /m2 Pressure, Pa ideal gas constant = 8.31451 J/mol K

A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

vD

Re = eq S() S0 Sc = D12 T Kc Sh = Film

Reynolds number, dimensionless sticking coefficient, dimensionless clean sticking coefficient, dimensionless Schmidt number, dimensionless temperature, K Sherwood number, dimensionless

VH

hydrogen partial molar volume in Pd-bulk, m3 /mol pairwise interaction energy, J/mol interaction energy of two adjacent hydrogen atoms in the Pd-bulk, J/mol molar fraction, dimensionless stress parameter, Pa high transport correction matrix, dimensionless



D12

w WHH x Y= Z

EYoung 1−P

Greek letters

H  j 0  Support i , i , ij , ij  HH OO H  = P 

  

hydrogen activity coefficient, dimensionless molecular bombardment rate, mol/m2 s jump frequency, s−1 thicknesses, m porosity of the support, dimensionless ˚ Lennard–Jones parameters: , K; , A Viscosity, Pa s surface coverage, dimensionless probability of twoadjacent filled sites, dimensionless probability of two adjacent empty sites, dimensionless hydrogen chemical potential in palladium, J/mol cinematic viscosity, m2 /s2 Poisson’s ratio, dimensionless atomic hydrogen concentration, molH /molPd permeance, mol/m2 s Pan density, kg/m3 hydrostatic pressure in the metal lattice, Pa tortuosity, dimensionless collisional integrals, dimensionless

Acknowledgments The “Repubblica Italiana, Ministero degli Affari Esteri (MAE), Direzione Generale per la Promozione e la Cooperazione Culturale” is gratefully acknowledged for co-funding this research. This work is part of the FIRB-CAMERE (RBNE03JCR5) co-funded by the “Ministero dell’Universitá e della Ricerca (MIUR)” that is gratefully acknowledged. A. Caravella is grateful to the “Repubblica Italiana, Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR)” in the framework of the Center of Excellence CEMIF.CAL (CLAB01TYEF) for the financial support. The authors thank

2159

Prof. F.P. Di Maio of The University of Calabria for the very useful discussions. Appendix A. Correlations used for the calculation of the physical properties A.1. Viscosity of a single gas The viscosity of a single gaseous component is calculated by the kinetic theory of gas developed independently by Chapmann and Enskog. For relatively low densities, the following relation is valid: √ Mi T . (37) i = 1 2 i  The collisional integral  is a function of the group kT /i and its values are provided in form of tables (Hirschfelder et al., 1949). However, in the calculations all these tabled values were transformed into a continuous function, interpolating them by the following functional expression:  =

a1 3 + b1 2 + c1  , 3 + d1 2 + e1  + f1

≡

kT . i

(38)

The calculated values of the parameters (a1 , . . . , f1 ) reported in Table 5 provide a correlation coefficient practically equal to unity in the range kT /i = {0.30, . . . , 10}. A.2. Viscosity of a mixture For calculating the viscosity of a multicomponent gas mixture, the Wilke semi-empirical correlation (Wilke, 1950) was used: mix =

Species N

x i i , j =1 xj ij

n

i=1

(39)

where the coefficients ij are given by ⎡ ⎤2

0.5     Mj 0.25 i 1 Mi −0.5 ⎣ ⎦ . (40) 1+ ij = √ 1 + Mj j Mi 2 2 A.3. Binary mixture diffusivities The binary mixture diffusivities were calculated by this relation:   1 1 0.5 + T 1.5 Mi Mj Dij = 2 . (41) 2 PTotal ij d Table 5 Correlation parameters for the calculation of u and i

1

a1

b1

c1

d1

e1

f1

2.67E − 05

0.427

58.13

55.89

73.30

−4.765

2.754

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A. Caravella et al. / Chemical Engineering Science 63 (2008) 2149 – 2160

at a T Ref . Thus:

Table 6 Parameters of the correlation for d and Dij calculation

2

a2

1.86E − 03

0.404

b2 36.30

c2

d2

35.79

49.29

R = Normalized overall resistance e2 −0.974

f2 1.107

=

Permeance at T Ref Overall resistance at T . = Permeance at T Overall resistance at T Ref

In this case a T Ref of 350 ◦ C was chosen. By analogy with the case of the viscosity calculation, a continuous expression for d was adopted (Table 6): a2 3 + b2 2 + c2  , d = d2 3 + e2 2 + f2  + g2

kT ≡ . ij

(42)

The values for ij and ij can be obtained from those of pure compounds in this way: √ ij = i j , ij = 21 (i + j ). (43) A.4. Knudsen diffusivity The following expressions were used to calculate the Knudsen diffusivity: T Knudsen = 4850d pore , (44) Di Mi Knudsen Di,effective = DiKnudsen

Support . 

(45)

A.5. Density of a pure compound and gas mixture For the calculation of the densities, all components were considered ideal gases and thus the ideal gas law was used. Appendix B. Calculation of the resistances B.1. Relative resistances of the elementary steps The relative resistance of each step was calculated by the following relation: (j )

Relative resistance(j th step) =

PH2

PHTotal 2

,

(j )

where PH2 is the hydrogen partial pressure drop in the j th step and PHTotal is the hydrogen partial pressure drop between 2 the feed and permeate fluid bulks. B.2. Normalized overall resistance The overall resistance in the membrane at the generic temperature T is the inverse of the permeance: Flux|T = |T P =

1 P . R|T

In order to compare the values at different temperatures, the overall resistance has been normalized with respect to its value

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