A Complete Transport Model of Components through Zeolite
Membranes in a Wicke-Kallenbach Permeation Cell
翻唱红人
Carl Justin Kamp, Indra Perdana*, Derek Crear
Chemical Reaction Engineering, Chalmers University of Technology,
SE-412 96, Gothenburg, Sweden,
Fax: +46-31-7723035, *E-mail address: indra@chemeng.chalmers.
Abstract: This study discuss the development
of a complete mathematical model for multi-component mass transport through a zeolite membrane in a Wicke-Kallenbach permeation cell. A three dimensional permeation cell model was built in COMSOL Multiphysics. The general PDE module was ud to apply the Maxwell-Stefan equations for surface diffusion, the dominant transport mechanism in zeolite micropores, meanwhile the surface concentration
on the zeolite was predicted using the extended Langmuir isotherm. However, the simple matrix algebra required to define such a model has been found to be difficult for two reasons. First, the component-component interactions resulted in a lengthy Maxwell-Stefan equation. Second, the equation format in COMSOL did not allow the necessary matrix algebra operations for this ca.
In addition to diffusive transport through the membrane, the model also considered convective and diffusive transport of components in the permeate and retentate chambers. The work prents a systematic method for inputting and computing the necessary Maxwell-Stefan surface diffusion equations in COMSOL. Also, the prent work shows the u of a geometry and equation scaling approach to resolve numerical problems that can ari due to large differences
in domain geometries.
Keywords: Maxwell-Stefan surface diffusion, Wicke-Kallenbach cell, multicomponent diffusion
1.Discrete Maxwell-Stephan reprentation
ud for zeolite surface diffusion
1.1.Zeolite membranes
Zeolites are porous crystalline materials with pores of a consistent size (Falconer et al. 2006). The uniform sized pores are generally of molecular size and enable high lectivity and reduced energy requirements in industrial paration applications (Falconer et al. 2006). Furthermore, zeolites are thermally stable and have known surface properties (Vareltzis et al., 2003). Separation in zeolites is bad on dissimilarity of diffusivities and favored adsorptions between components (Falconer et al. 2006). Zeolites can be applied as powders, pellets, and in this ca, as thin films grown on inert support membranes with a larger pore size (Farooq et al., 2001). The support provides mechanical strength for the film.
1.2.Surface diffusion
For zeolite membrane paration, the dominant transport mechanism at low temperature and/or high pressure is referred to as surface diffusion. Surface diffusion can be simplified by breaking down the phenomenon into 3 steps; however some sources define 5 apparent steps (Dong et al., 2000). First, the molecules of a given chemical species adsorb to the surface of the zeolite membrane at the interface between the gas pha and the solid membrane (Falconer et al. 2006). Next, the chemical species ‘jump’ between sites on the surface and follow a path defined by the driving force, which is the difference in chemical potential (Falconer et al. 2006). Here, sites are defined as regions of low p
otential energy (Krishna et al., 1996). Finally, the chemical species desorbs from the surface of the zeolite at the interface between the solid zeolite and the gas pha permeate chamber (Falconer et al. 2006).
1.3.Existing COMSOL Maxwell-Stefan
convection and diffusion module and
the zeolite application
It is important to define the reasons for which the existing COMSOL Maxwell-Stefan Diffusion and Convection module cannot be ud in the context of zeolite membranes and surface diffusion in general. As pointed out by the COMSOL guide, the general mass balance for Maxwell-Stefan diffusion and convection is given by the equation:
()()i i i i R u j t
=+⋅∇+∂∂ρωρω (1) (COMSOL Chemical Engineering Guide 2005)
The Maxwell-Stefan convection and diffusion module then us an alternative form of equation (1) to
account for bulk diffusion and us the appropriate diffusion driving force such as pressure, and is given by:
())2(~1i i n j T i j j j ij i i R t
T T D p p x x D u =∂∂+⎥⎦⎤⎢⎣
⎡∇+⎟⎟⎠⎞⎜⎜⎝⎛∇−+∇−⋅∇∑=ρϖωρωρω(COMSOL Chemical Engineering Guide 2005)
This equation cannot be ud in the ca of surface diffusion becau the diffusivities of the species are a function of surface coverage, which cannot be expresd in this version.
2. Model
2.1. Model apparatus
This model was constructed in order to estimate the diffusion phenomenon through a very thin zeolite membrane (approximately 150 nm thick) in a Wicke-Kallenbach cell. This type of cell is frequently ud to experimentally determine diffusive transport properties of components through membranes. The cell is cylindrical in shape with a diameter of 19 mm and consists of a retentate gas
chamber, a permeate gas chamber, which are parated by a cylindrical, solid zeolite membrane usually held within a support system. The respective gas chambers are both 0.3 mm thick. This counter-current system feeds a concentrated gas flow into the retentate chamber and the chemical species reaches the zeolite surface. A portion of the chemical species diffus through the zeolite and is removed in the permeate chamber by feeding an inert sweep gas such as argon or helium. Depending on the flowrate of the sweep gas, a significantly reduced concentration can be assumed at the surface of the zeolite in the permeate side, which provides a driving force for diffusion. It should be noted that the sweep gas, in this ca, has been assumed not to experience counter diffusion. This apparatus has been modeled in 3D in COMSOL to investigate under what
conditions a lateral concentration gradient within the cell can develop and influence transport through the membrane.
2.2. Additional model information
As stated previously, the Maxwell-Stefan surface diffusion ud in this model was applied to a counter current Wicke-Kallenbach cell. The chemical species ud were hydrogen and carbon dioxide, while the non-diffusing sweep gas was argon. Relatively small concentrations of hydrogen a
nd carbon dioxide in an argon carrier are fed into the retentate gas chamber, while pure argon is fed into the permeate side. Due to the fact that the concentrations of H 2 and CO 2 are relatively small, the velocity profiles in the gas chambers, which were solved by the incompressible Navier-Stokes module in COMSOL, were assumed to be independent of the gas composition. Also, the sweep gas had a flow rate of roughly twice that of the H 2/CO 2/Ar mixture.
3. 3. Model development
3.1. Maxwell-Stephan surface diffusion
equation and Langmuir’s isotherm
Equation (2), shown in ction 1.3, describes the general mass balance from Maxwell-Stefan bulk diffusion, but us the Gibbs-Duhem relationship to replace the chemical potential driving force with a pressure driving force (Krishna et al., 1996). However, in the ca of zeolite membranes, bulk diffusion and the associated driving force (pressure) are not acceptable to describe the dominant transport mechanism, which has been introduced as surface diffusion. Surface diffusion is accounted for in the following equation (4) where a noticeable distinction can be made from equation (3) which displays the chemical potential driving force for general diffusion:
Ð
,t j i i
j i p T i
c N x N x RT x −=∇−
µ (3)
(4)
(Krishna et al., 1996)
()i i n
i
j j s i
s j
i s i j
RT
N
N
N µθρρθθ
∇−=+
−∑
≠=1s
ij sat b s ij
sat b Ðq Ð
q
Equation (4) introduces a variable, θi , which reprents the fractional surface coverage of sites by species i . The diffusion driving force is given by a surface chemical potential gradient and can be defined by the extended Langmuir isotherm, due to the assumption that the component partial pressures and respective surface occupancies of the adsorbed pha are in equilibrium (Piet et al. 1998). The extended Langmuir isotherm, in this ca, is given by:
tot i i i n
j j
j i
i sat
i i p b p b p b q q θθθ−=+==∑=1,11
* (5) (Krishna et al., 1996)
The extended Langmuir isotherm provides a relationship, in this specific model, to couple the convection and diffusion module to the general form PDE module.
3.2. Derivation of discrete Maxwell-Stefan reprentation ud for zeolite surface diffusion
Next, it is important to discuss the Maxwell-Stefan equation form ud in this rigorous mathematical model. From equation (4), a new term can be introduced as Γ, which is referred to as a thermodynamic factor. This is en in the following equation:
∑=∂∂≡Γ∇Γ=∇n
j j i i
ij j ij i i
p RT 1
ln ,θθθµθ (6) (Krishna et al., 1996)
Next, the extended Langmuir isotherm, which has been briefly introduced earlier, can be ud to express the thermodynamic factor and is given by:
V
i ij ij θθ
δ+=Γ (7)
maradona(Krishna et al., 1996)
This thermodynamic factor is then ud with equation (4) to formulate a discrete matrix equation for the surface flux, N s . This flux equation is as follows:
()[][]()θρ∇Γ−=−1
s sat
b
s
B q N
(8)
(Krishna et al., 1996)
The elements of the matrix B s are defined by:
∑≠=−=+=i
j j i s
ij j s ii B B 1s ij
s ij s i Ð,ÐÐ1θθ (10)
(Krishna et al., 1996)
In this type of diffusion, the fractional surface coverage of sites strongly influences the Maxwell-Stefan micropore diffusion coefficients, as can be en in the Vignes equation:
()
[]
()
()[]()
j
哈尔滨外语学校
i j
j i i θθθ
θθθ++=/s
jV
/s
iV s ij 0Ð
0ÐÐ (11)惹怎么读
(Krishna et al., 1996, Falconer et al. 2006)
It should be noted that equation (11), which is esntially a logarithmic average between the two species diffusivities, makes the solution of this rigorous model to be quite stiff due to the fact that the surface coverages, θi and θj , are initially zero when solving.
4. COMSOL modeling development
湖南省四六级报名4.1. U of general PDE module to solve
Maxwell-Stefan surface diffusion
The COMSOL guide defines the general PDE form equation system as follows:
(12,13,14) (COMSOL Modeling Guide 2005) The first equation listed in this t,, reprents the partial derivatives in three dimensions for the flux through the zeolite membrane, which has been previously defined in equation (8) as N s . However, the source term, F l , is t to zero becau there are no chemical reactions in this ca.
In this model, the computational domain Ω refers to the zeolite membrane volume, and ∂Ω refers to t
he zeolite domain boundaries found between the permeate and retentate gas chambers and the zeolite surface respectively.
The Neumann boundaries, equation (13), in this model refer to the edges where no flux occurs, and are thus t to zero. The Dirichlet boundaries, equation (14), are ud at the interface between zeolite membrane surface and the respective permeate and retentate gas chambers. The Langmuir isotherm is ud here in order to calculate the surface coverage of sites at the interface between the gas chambers and the solid zeolite membrane. Thus, this relationship is defined in the Dirichlet boundary conditions where otherwi R m =0.
When solving the Maxwell-Stefan surface diffusion, scaling of the membrane thickness was required due to the large differences in the geometry dimensions in the model due to the geometry (Perdana and Crear, 2005). As stated before, the gas chamber was 0.3 mm thick whereas the membrane had a thickness of 150 nm. Scaling reduced the required mesh density in the membrane, ead the computations and their stability.
4.2. Velocity profile and mass balance in
retentate and permeate gas chambers
The incompressible Navier-Stokes module in COMSOL was ud to solve the velocity profile within the permeate and retentate gas chambers. The general equation that defines incompressible flow is given by:
(15)
(COMSOL Chemical Engineering Guide 2005)
Where, u reprents the velocity field. The velocity profile was then ud to solve the mass balance, via the convection and diffusion module in COMSOL, of the gas pha in the retentate and permeate chambers. The mass balance is solved within the respective gas chambers by using the following equation:
R cu c D t
c
ts
=+∇−⋅∇+∂∂)(δ (16)
Where, c reprents the concentration.
(COMSOL Chemical Engineering Guide 2005)
However, this model ud y instead of c, and multiplied by c t , where y is the mol fraction and c t is the total concentration.
4.3. Mathematical organization of matrix
flux equation terms
The diffusing species in Maxwell-Stefan surface diffusion have an incread interaction with one another. This incread interaction creates long and complex matrix elements ud in the flux equation defined in equation (8). Thus, special attention was given to the organization and methodology of the flux calculations. MATHCAD was first ud to symbolically solve the various matrix variables from equation (8), and then to complete the necessary matrix algebra.
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5. Results
Maxwell-Stefan surface diffusion has been modeled in COMSOL and applied to a Wicke-Kallenbach cell. From this model, veral important characteristics were visualized.
0.05
0.1
0.15
0.2
0.25
0.3
Figure 1. Velocity and CO 2 concentration profiles in the gas pha from the plane in the center of the gas chamber for the retentate (right) and the permeate (left) chambers from COMSOL.
()F
accountforp u u u t
u =∇+∇⋅+∇−∂∂ρηρ2
It has been stated before that the chemical species modeled in this simulation are hydrogen and carbon dioxide. The following plots will display the concentration and surface coverage gradients for carbon dioxide due to the fact that this species is preferred over hydrogen for surface diffusion and gives a larger gradient.
The Incompressible Navier-Stokes module was ud in COMSOL to solve the momentum balance. Figure 1 shows that laminar flow is prent within the gas chambers. Since the velocity arrows in this plot are proportional by length, it is obrved that the highest velocities occur in a direct path between the inlet and outlet gas streams. This means that the gas flow along the edges experiences a longer residence time and could have incread transport with the preferred species, which is CO 2 in this ca.
The retentate chamber, en on the right of figure 1, displays CO 2 entering in the H 2/CO 2/Ar feed, and is shown as the red species. Converly, the blue region found in the left of figure 2 shows the lack of CO 2 where the sweep gas enters the permeate gas chamber.
It is apparent, from this model, that there is no homogeneous concentration in gas chambers, with the given conditions.
It should be noted that the concentration profiles show that the membrane might not be ud to its maximum potential. This is due to the prence ‘dead zones’ in the gas chambers where either H 2 or CO 2 are not located. This would suggest that the value of the zeolite surface area would need to be corrected in such a Wicke-Kallenbach cell when using the operating conditions.
6.797e-30.050.10.150.20.239
Feed inlet
Feed outlet
Sweep gas inlet
Sweep gas outlet
Figure 2. CO 2 surface coverage in zeolite membrane (colors) and gas pha velocity profile (streamlines ) from COMSOL.
Figure 2 reprents one of the fundamental reasons for modeling in 3D; a lateral concentration gradient can be obrved. This plot shows surface coverage variation in the membrane not only in th
e vertical direction, but in the lateral direction as well. The reason lateral variations exist in the membrane is becau concentration variations exist in the gas chambers. In addition, this plot shows that the general form PDE module for Maxwell-Stefan surface diffusion has been successfully coupled with the convection and diffusion module in COMSOL.
From this work, advancements can be made in the field of zeolite membrane modeling. Also, this study suggests that precedence should be given to determining the optimal experimental conditions.
6. Notation
b i Langmuir isotherm parameter, 1/atm B ji s inverted Maxwell-Stefan diffusivity,
s/mm 2
c t total concentration, mmol/mm 3 Đij s Maxwell-Stefan micropore
countersorption diffusivity, mm 2/s
ĐiV s (0) Maxwell-Stefan diffusivity at zero
coverage mm 2/s
D T i generalized thermal diffusion
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evercoefficient g/(mms)
ij D ~
multicomponent Fick diffusivity for ij
component, mm 2/s
F volume force field ud in
incompressible Navier-Stokes module (F =0 in this model)
F l source term for general form PDE
module (F l =0 in this model)
G l coefficient in general form PDE
module
j i mass flux of species i , g/(mm 2s) n normal vector in general form PDE
module
N i molar flux of species i , mmol/(m 2s) p pressure, atm
p i partial pressure of species i , atm q* equilibrium concentration of
adsorbed species, mmol/g
q sat total saturation concentration of
adsorbed species, mmol/g
R gas constant, latm/(molK)
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