Simulation of transient adsorption in MFI films

2007:099

M A S T E R ' S T H E S I S

Simulation of transient adsorption in MFI films

Manuel Rubio

Luleå University of Technology D Master thesis

Chemical Technology

Department of Chemical Engineering and Geosciences Division of Chemical Technology

ABSTRACT

The ability of zeolites to adsorb many gases is an important application for such a ceramic material. Specific channel size enables zeolites to act as molecular gas sieves and selectively adsorb such gases as Xylenes and carbon dioxide. Zeolites have the potential of providing precise and specific separation of gases where applied in conjunction with properly engineered systems.

Due to the heat effects on adsorption, nonisothermal kinetics may have to be used for the evaluation of diffusivities from rate of sorption measurements. A general analytical solution is derived, taking into account both the thermal conductivity within the adsorbent and the heat exchange with the surroundings.

In this work a simulation for the transient adsorption of CO₂ (and others molecules) in thin MFI films has been performed. Combined heat and mass transfer problem is modelled and simulated using the software packet MATLAB^® (MATrix LABoratory).

Mass transfer is described by micropore diffusion, while heat transfer is described in terms of conduction and convection. The temperature dependence of the diffusion coefficient, as well as the Henry law constant, is defined by an Arrhenius relationship.

It was found that for a high diffusion coefficient, the assumption of isothermal conditions was not a realistic model. Since the adsorption is extremely quick, heat generated inside the adsorbent cannot be easily exchanged with the surroundings in sufficient rate yielding an increase in the temperature. On the other hand, for a low diffusion coefficient, the time required to reach the maximum amount of the component is high and thus is relatively easy to exchange the heat formed during the adsorption with the surroundings. In that case, it is obvious that the temperature within the solid will remain almost constant and it can be consider as isothermal system.

It is also known that CO2 has a high diffusion coefficient. It will be appropriate to use a nonisothermal system. On the other hand, p-Xylene and isomers have lower diffusion coefficients. By comparing plots for each component, it will be made that topic clear.

Keywords: Adsorption kinetics; Simulation/MATLAB; Diffusion coefficient; Heat and mass transfer; Micropore diffusion; b-oriented MFI zeolites - CO₂, p-, o-, m-Xylene

ABSTRACT... 1

1. INTRODUCTION ... 3

1.1BACKGROUND... 3

1.2SCOPE OF THE PRESENT WORK... 3

2. LITERATURE SURVEY... 4

2.1ZEOLITES... 4

2.2ZEOLITE MEMBRANES... 5

2.3PHYSICAL ADSORPTION IN ZEOLITES... 6

2.4DIFFUSION IN ZEOLITES... 8

2.4.1 Transport diffusion ... 10

2.4.2 Self-diffusion ... 10

3. MATHEMATICAL MODEL ... 12

3.1BACKGROUND... 12

3.2MASS TRANSFER... 13

3.3HEAT TRANSFER... 14

3.4PARTIAL DIFFERENTIAL EQUATIONS (PDE) ... 15

3.4.1 Mass transfer ... 15

3.4.2 Heat transfer... 16

3.4.3 Boundary and initial conditions ... 16

3.4.3.1 Mass transfer ...16

3.4.3.2 Heat transfer ...17

4. RESULTS AND DISCUSSION ... 18

4.1ISOTHERMAL ADSORPTION... 18

4.2NON-ISOTHERMAL ADSORPTION... 20

5. CONCLUSION ... 28

ACKNOWLEDGEMENTS ... 29

6. REFERECES ... 30

APPENDIX A.2 ... 35

1. INTRODUCTION

1.1 Background

In the kinetic treatment of sorption processes in porous solids the effects of intermediate temperature changes due to evolution or consumption of heat on adsorption appears to have been excessively neglected. Already in 1939, Henry [1] as well as Wicke [2] had studied this phenomenon. However, only more recently the interest has been revived by conflicting results which were obtained for diffusivities in zeolites adsorbents [3-5].

The adsorption of carbon dioxide into a porous solid adsorbent, such as zeolite, results in the release of the heat of adsorption. Therefore, the temperature of the adsorbent rises, which reduces its CO2 uptake capacity. If the adsorption is quick, it is difficult to exchange this heat of adsorption with the ambient and thus the temperature of the adsorbent becomes higher.

Several mathematical treatments of a transient, nonisothermal adsorption process for various models have been reported with reference to zeolite adsorbents in recent years. So far, approximate solutions of the system of coupled differential equations for energy and mass transport were obtained for two concrete cases [6].

An analytical solution for transient nonisothermal adsorption, taking intraparticle heat conduction as well as heat transfer at the surface into account is reported in following sections. In order to solve the set of PDE (partial differential equations), some simplifications were taken into account. For instance, it was considered that the diffusion coefficient was not dependent on the temperature variations.

1.2 Scope of the present work

The scope of the present work was to model the transient adsorption of some gases in zeolites. A new technique [7] allows measure diffusitivity in zeolites. It is known that this technique works very well for measuring adsorption isotherms;

however, for diffusion measurements a suitable system must be identified. That is why it is useful model the process first.

2. LITERATURE SURVEY

2.1 Zeolites

Zeolites are porous alumino-silicates. The three-dimensional network of the zeolite is built of [SiO₄]^4- and [AlO₄]^5- tetrahedral sharing all the corners with each other. Zeolites may be found both in nature as a mineral or can be synthesized in the laboratory. The chemical composition of zeolites may be expressed as:

O yH xSiO AlO

M₁ⁿ_/⁺_n ⋅ ₂⁻⋅ ₂ ⋅ ₂ [8]

where: M – counterion. It may be a metal, an ammonium or an alkylammonium cation

n – counterion valence x – silicon/aluminum ratio y – content of hydrate water

Zeolites have different channel geometry [9]. For that reason, it was useful to divide zeolites into different structure type groups. The geometry of the channel may be elliptical, circular and tubular or it can contain periodic cavities, straight or zigzag.

Depending on the structure, the size of the pore is in the range 3 to 13 Å [9]. Thus, the International Union of Pure Applied Chemistry (IUPAC) considers zeolites as microporous materials.

In terms of research and industry, it is clear that the most common structures are MFI, A and FAU type. The channels in the MFI structure are formed by five-member ring building units linked together (Figure 2.1) [10].

Examples of MFI structures are Silicalite-1 and ZSM-5. The difference between them is the amount of aluminum that they contain. The ratio for ZSM-5 is in the range 10-200 Si/Al [11]. The material is called Silicalite-1 when this ratio is larger than 200.

The fact that silicalite-1 does not content aluminum, or only a small amount, makes that this material is considered as a molecular sieve and not as a zeolite, although both ZSM- 5 and Silicalite-1 have an analogue structure.

Zeolites are of great interest due to their large potential in a variety of applications, such as [12]:

- Membranes - Sensors - Catalysts - Adsorbents

The use of zeolites as membranes may be useful to carry out several difficult separations. The reason of using zeolites in sensors is to improve both the sensibility and selectivity. Synthetic zeolites are widely used as catalysts in the petrochemical industry, for instance in fluid catalytic cracking and hydro cracking.

Figure 2.1: The MFI channels system viewed along [010].

In order to identify and have a proper control of each zeolite, The Structure Commission of the International Zeolite Association has assigned framework type codes (consisting in three capital letters) to all zeolites of known structure [10].

2.2 Zeolite membranes

Zeolite membranes are compounds that are able to separate mixtures by adsorption, diffusion or molecular sieving. Due to the small thickness of the zeolite film, they cannot be self-supported. For that reason, the zeolite layers are frequently deposited on different supports, such as flat disc or tubular shaped [13].

The good quality of a zeolite membrane depends on:

- Zeolite thickness - Amount of defects

- Resistance to permeance by the support

The general preparation procedure of zeolite films membranes involves three main steps [14]:

1. Pre-treatment [15]

2. Synthesis procedure

3. Adjust of pore structure [16]

Zeolite membranes are generally used for processes in which one or both sides of the membrane are in gas phase (i.e. pervaporation or gas separation). At the present, there is a large research activity, which is reflected in the large number of published papers and filed patents, on this subject.

Zeolite membranes have also a great potential as a component in membrane reactors [17]. A membrane reactor can be used to simultaneously carry out reaction and separation in a continuous process [18].

The majority of the applications involving zeolite membranes are of relatively large-scale. For instance, for separation process, excellent results were obtained for isomer separation such as xylenes [19]. The main inconvenient for using zeolite membrane in industrialization of large-scale process is the high cost for the membranes.

2.3 Physical adsorption in zeolites

The solid surface is a singular region, which is responsible or at least determines most of their properties. The atoms located in the surface do not have the bond strength balanced, as the atoms located inside the solid have, see Figure 2.2.

At high distances, the interaction between the molecule and the surface is almost zero. Thus, the trend is for the system energy to go to zero. As the molecule is moving

closer to the surface, the system energy becomes lower due to the equilibrium of the bond strength for the atoms located on the surface.

Figure 2.2: Bond strength in a solid and in a solid with gas adsorbed [20].

When the distance between the surfaces and the molecule becomes closer, repulsion strengths (due to the proximity of the electrons on the surface with the electrons in the free molecule) are important. Thus, the energy system is minimum, see Figure 2.3 in a concrete distance. This distance is known as physical adsorption distance. The high adsorption efficiency of zeolites is related to the high internal surface that they have [20].

Figure 2.3: Potential energy evolution of a molecule moving closer to a plane surface [20].

The amount of adsorbed gas for a constant temperature and for different partial gas pressures is known as an adsorption isotherm. Adsorption isotherms are useful for the characterization of porous solids. The IUPAC admits six adsorption isotherm types [21]. Figure 2.4 shows a scheme of each one.

Type I isotherms are typical of microporous solids and their main characteristic is that adsorption is produced at low partial pressures. Type II isotherms are typical of macroporous and not porous solids. Type III isotherm occurs when the interaction sorbate-adsorbent is low. Type IV isotherm is typical of mesoporous solids. In this type of isotherms, the amount adsorbed rises at relatively middle pressures. Type V isotherm, as well as type III, occurs when the interaction sorbate-adsorbent is low, but the

difference between III and IV is that the last one does not present an asymptotic trend in the final section.

Figure 2.4: Scheme of each adsorption isotherm type [20].

The majority of zeolites present adsorption isotherm type I. At low pressures, the amount adsorbed versus the partial pressure will present a linear relationship [22]. This relationship is often referred to as Henry’s law:

p K C = _H·

where: C (mmol/g) is the adsorbate loading KH (mmol/g Pa) is Henry’s law constant

p (Pa) is the partial pressure of the adsorbate in the gas.

2.4 Diffusion in Zeolites

Diffusion is the phenomenon of random motion causing a system to decay towards uniform conditions. For example, diffusion of particles causes a net movement of particles from areas of higher concentration to areas of lower concentration until equilibrium is reached. Diffusion is caused by the thermal motion and subsequent collisions of the molecules. Two types of diffusion can be distinguished: transport diffusion, resulting from a concentration gradient, and self–diffusion, which takes place in a system that is at equilibrium.

Diffusion in zeolites is different from diffusion in fluids or large porous solids.

Due to the molecules have to move through channels of molecular dimensions, there is a constant interaction between the zeolite framework and the diffusing molecules. For that reason, the molecular motion is highly influenced by both the size and the shape of these channels instead of temperature and concentration only.

Depending on the pore diameter, the diffusion of molecules in pores can be classified in a number of different regimes (see Figure 2.5). For large pore diameters, of the order of one µm or larger, usually called macropores, molecular diffusion is the dominant mechanism. That is because the collisions between the molecules occur much more frequently than collisions with the wall. As the size of the pores decreases, the number of collisions with the wall increases until it finally becomes smaller than the mean free path of the gas molecules. At this point, Knudsen diffusion takes over and the mobility starts to depend on the dimensions of the pore [23]. At smaller pore sizes, in the range of 20 Å and smaller, the interaction between the molecules and the walls is constant. Diffusion in the micropores of a zeolite usually takes place in this regime, and is called configurational diffusion [24].

Figure 2.5: Effect of pore size on the diffusivity [25].

The diffusivity in this regime will depend strongly on the pore diameter, the structure of the pore wall, the interactions between the surface atoms and the diffusing molecules, the shape of the diffusing molecules and the way the channels are connected. The values of the diffusion coefficients vary in an enormous range from 10⁻⁸ to as low as 10⁻²⁰ m²·s⁻¹ [26]. The diffusivity of the molecules inside the zeolite channels is greatly reduced compared to the ones in the gas phase (typically around 10⁻⁵ m²·s⁻¹), and the temperature dependence is often stronger.

2.4.1 Transport diffusion

In the 19th century, Fick laid the foundations of the theory of diffusion. In one dimension, the flow of a certain species can be related to the gradient of the concentration according to Fick’s first law [27]

(2.1)

where: J (mol/m²·s) is the molar flow C (mol/m³) is the concentration x (m) is the spatial coordinate

Dt (m²/s) is the (transport) diffusion coefficient

The diffusion coefficient is thus defined as the proportionality constant between the flux and the concentration gradient. Although the above equation is a convenient starting point, it does not reflect the true driving force of diffusion. As diffusion is nothing more than the macroscopic manifestation of the tendency of a system to approach equilibrium, the driving force should be the gradient of the chemical potential µ [41].

Assuming a concentration-independent diffusion constant, Eq. 2.1 can be transformed into a diffusion equation known as Fick’s second law:



 





∂

− ∂

∂ =

∂

2 2

x D C t C

t

This equation gives the change of concentration in a finite volume element with time.

2.4.2 Self-diffusion

Self-diffusion is an equilibrium process. The difference between that and transport diffusion is that for transport diffusion a gradient in the chemical potential was necessary. This type of diffusion can be monitored by labelling some of the molecules



 





∂

− ∂

= x

D C

J _t

inside the zeolite pores and following how the labelled and unlabeled molecules are mixed. Eq. 2.1 can again be used to describe the flow of the labelled components [41]:

where the asterisk refers to the labelled component and D_s in this case is the self- diffusion constant.

The relation between the self-diffusion and the transport coefficient is shown in the Equation 2.2. This equation implies that the self- and transport-diffusivity coincide at low concentrations [29].





 



= 

q d

p D d

q

D_{t s}

ln ) ln 0 ( )

( (2.2)

In this relation, q is the concentration of the species adsorbed in the pores. This equation implies that the self- and transport-diffusivity coincide at low concentrations.

t cons i C s

i x

D C J

tan

*

*

∂ =

− ∂

=

3. MATHEMATICAL MODEL

In this section, the set of equations describing the model of the combined heat and mass transfer in a consolidated zeolite layer will be illustrated. Figure 3.1 shows a scheme of the studied system. The following basic assumptions were made:

1. The heat and mass transfer problem is considered one-dimensional.

2. The diffusion coefficient is considered non-temperature dependent.

3. The gas temperature is considered constant during the adsorption process.

Figure 3.1: Scheme of the studied system (M.T.: mass transfer, H.T.: heat transfer)

3.1 Background

For an isothermal, as well as one-dimensional diffusion process, the combination of Fick’s law and the continuity equation for the diffusion component gives the following equation:

₂

2

z D C t C

∂

= ∂

∂

∂ (3.1)

where C (mol/kg) is the concentration of the penetrant D (m²/s) is the diffusion coefficient

t (s) is the time coordinate z (m) is the space coordinate

Substrate z = 0

z = L

Zeolite layer Gas phase M.T.

H.T.

An appropriate set of initial and boundary conditions describing the process is the following:

C=0 at t=0 and 0 < z < L =0

∂

∂ z

C at z=0 and t ≥ 0

p K C

C = _eq = _H ⋅ at z=L and t ≥ 0

where K_H is the Henry law constant (mol/kg·Pa), p is the partial pressure of the component in the gas phase (Pa) and L is the thickness of the film (nm).

One solution to Equation 3.1 with these initial and boundary conditions is [28]



 +



 



− +

+ ×

− −

− =

−

∑

= L

z n L

t n

D n

C C

C z t C

n

n

eq 2

) 1 2 cos ( 4

) 1 2 exp (

1 2

) 1 ( 1 4

) , (

2 2 2

0 0

0 π π

π (3.2)

3.2 Mass transfer

In the previous section, the most general equation for an isothermal adsorption was presented. It is relatively easy to obtain expressions for the uptake curve for some more complex cases involving more than one diffusional resistance, Figure 3.2. For the combination of micropore diffusion within the particle and the external fluid film resistance, the solution for the uptake curve is [29]

^∑ [ ]

∞

= + −

− −

=

1

2 2

2 2

2

) 1 (

) /

· exp(

1 6 ) (

eq n A A

L t D A

m t m

β β

β (3.3)

where

KD

A = kL and β represent the roots of the equation

0 1 )

cot(β + A− = β

where k (m/s) is the external mass transfer coefficient.

Figure 3.2: Scheme of the studied system (1. External film control, 2. Intraparticle diffusion control)

For A→∞ (large k, small D)β →nπ and Equation 3.3 reverts to the general Equation 3.2, the solution for intraparticle diffusion control. (3.4) In the other limit forA→0,β is small and Equation 3.3 reverts to the solution for external film control:



−

−

= KL

kt m

t m

eq

exp 3 ) 1

(

In order to calculate k, the Sherwood number will be used. Assuming the worse value for this number (Sh=2) it will be easy to find the mass transfer coefficient:

D K

D A Sh

D L

Sh k ^{A B}

B

A ⋅

= ⋅

⇒

⋅ =

= ⁻

−

2

where k (m/s) is the external mass transfer coefficient

DA-B (m²/s) is the diffusion coefficient of the studied compound within the gas phase.

K is the dimensionless Henry law constant

D (m²/s) is the diffusion coefficient between the gas and the solid.

3.3 Heat transfer

Although heat conduction through an individual crystal is generally fast enough to maintain a uniform temperature within the particle, the possibility of a temperature

z = 0 z = L

Zeolite layer Gas phase 1

Substrate 2

difference between the particle and surrounding fluid resulting from the shortage of time to evacuate all the heat produced must also be considered.

If the absorption process is slow enough, the heat produced due to the adsorption is exchanged with the surroundings. Thus, there is no variation in the temperature inside the zeolite.

On the other hand, if the adsorption process is very fast, the heat produced due to the adsorption could not be evacuated from the zeolite and then it is obvious that the zeolite will rise its temperature, see Figure 3.3.

Figure 3.3: Scheme of the studied system (1. Slow diffusion process, 2. Fast diffusion process)

3.4 Partial Differential Equations (PDE)

3.4.1 Mass transfer

Assuming intraparticle diffusion control with a constant diffusivity, the response of a microporous particle or zeolite crystal to a small differential step change in sorbate concentration at the external surface is described by the following equation:

₂

2

z D C t C

∂

= ∂

∂

∂ (3.1)

where C is the concentration of the gas inside the solid (mol/kg), D is the diffusion coefficient (mn²/s), t is the time coordinate (s) and z is the space coordinate (nm).

z = 0 z = L

Zeolite layer Gas phase

1

Substrate

2 Heat Prod

3.4.2 Heat transfer

The non-isothermal adsorption process with the heat production is presented by the following equation [31]

t C C

H z

T t

T

p ads

∂

∂

∆ + −

∂

= ∂

∂

∂ ( )

2

α 2 (3.5)

where:

Cp

= ⋅ ρ

α λ (3.6)

λ(J/s·m·K) is the thermal conductivity of the zeolite layer.

ρ(kg/m³) is the density of the zeolite.

C (J/kg·K) is the heat capacity of the zeolite. p

Hads

∆ (J/mol) is the heat of adsorption.

It is the result of combining the energy and mass balance equations and the definitions given by Dawoud [30] for the specific internal energy of the zeolite.

3.4.3 Boundary and initial conditions

3.4.3.1 Mass transfer

Regarding the boundary conditions, it is assumed that there is no mass transfer from the bottom of the zeolite layer, resulting on Equation 3.7. At the gas-zeolite interface, the concentration in the zeolite is assumed to follow Henry’s law, resulting in equation 3.8. Accordingly, the following two boundary conditions can be formulated for the mass transfer

(0, )=0

∂

∂ t

z

C (3.7)

C(L,t)=K_H ⋅ p (3.8)

where exp( ) RT K H

K_{H o} −∆ ^ads

⋅

=

Regarding the initial condition, it is assumed that at the beginning of the process the amount adsorbed is equal to zero. Accordingly, the initial condition can be formulated for the mass transfer

C (z,0) = 0

3.4.3.2 Heat transfer

Regarding the boundary conditions for heat transfer, at the substrate-zeolite interface the temperature will remain constant, and equal to the reference temperature (T_ref), since it can be assumed that heat conduction coefficient for the substrate is higher than the one for the zeolite. Thus, the heat produced at the bottom of the zeolite can easily be exchanged with the substrate, Equation 3.9.

The energy exchange between the uppermost zeolite sublayer and the gas phase is assumed to take place with an overall heat transfer coefficient (h). The value of the overall heat transfer coefficient was estimated by using dimensionless numbers such as the Nusselt and the Prandtl numbers, Equation 3.10.

The following equations are in agreement with the previous explanation

T (0, t) = Tref = constant (3.9)

(3.10)

Regarding the initial condition, it is assumed that at the beginning of the adsorption process the zeolite temperature is set equal to the room temperature.

Accordingly, the initial condition can be formulated as

T (z,0) = T_ref

) ) ( ( ) ,

(L t h T L T_ref z

T = −

∂

−λ∂

4. RESULTS AND DISCUSSION

4.1 Isothermal adsorption

In this section, the amount adsorbed versus time for an isothermal process is plotted. By using Equation 3.2, two plots were obtained. The aim of this section was to compare the time to reach equilibrium for different diffusion coefficients. As stated previously, the higher the diffusion coefficient the shorter the time to reach equilibrium.

Regarding the adsorption of CO₂ in MFI films, Plot 4.1 is presented. The adsorption process for 2000 nm thickness is extremely fast and just 0.1 seconds are needed to reach the equilibrium. On the other hand, the adsorption of p-Xylene in Silicalite (presented in Plot 4.2) needs more than 100 seconds to reach equilibrium.

Table 4.1 shows the data for each adsorption process at 298 K.

Carbon dioxide p-Xylene

Sorbent MFI Silicalite

L (nm) 2000 2000

z (nm) 1000 1000

D298 K (nm²/s) 2·10⁺⁸ [33] 8·10⁺⁴ [32]

Table 4.1: Data for Carbon dioxide and p-Xylene at 298K.

As we can see in the table above, the diffusion coefficient for Carbon dioxide is thousand times higher than the one for p-Xylene. That is why the time to reach equilibrium is extremely different for each adsorption process.

It is obvious that, for the adsorption of Carbon dioxide, thermal effects will take place. Accordingly, it is not convenient to speak about isothermal adsorption. In this case, due to the high diffusion coefficient, the process is extremely fast.

0 10 20 30 40 50 60 70 80 90 100

0,0000 0,0200 0,0400 0,0600 0,0800 0,1000 t (s)

C/Ceq(%)

Plot 4.1: Percentage adsorbed versus time for carbon dioxide in MFI film (298K).

0 10 20 30 40 50 60 70 80 90 100

0 50 100 150 200

t (s) C/Ceq(%)

Plot 4.2: Percentage adsorbed versus time p- Xylene in Silicalite-1 film (298K).

Regarding p-Xylene adsorption, more than two minutes are needed. As said previously, the main reason for such a different situation is the diffusion coefficient. In Table 4.1, there are data for both gases at the same temperature.

Table 4.2: Data for a mixture of Carbon Dioxide-Helium that is adsorbed in MFI at 298K

As it was said in the Section 3.2, some more complex cases involving more than one diffusional resistance may be possible. So far, all the work has been done supposing micropore diffusion within the particle and neglecting the external fluid film resistance.

In order to confirm that assumption, Relation 3.4 will be used. First, parameter A will be estimated. The needed data for this calculation can be found in Table 4.2

∞

→

⋅

⋅ =

⋅

⋅

= ⋅

⋅

= ⋅

⇒

⋅ =

= ⁻

−

6 8

13

10 10 2

2 31 , 0

10 5 , 5 2

2

A

D K

D A Sh

D L

Sh k ^{A B}

B A

Following Relation 3.4, for A→∞ Equation 3.3 reverts to the general Equation 3.2, the solution for intraparticle diffusion control. Accordingly, the assumption proposed is correct.

In the next section, non-isothermal adsorption will be studied. As explained before, external mass transfer will be neglected, as intraparticle diffusion is the rate- limiting step.

4.2 Non-isothermal adsorption

The aim of this section is study the temperature variation inside the zeolite film.

The equations, which fit to this process, were presented on Section 3.4. In order to solve that set of equations, software packet MATLAB was used. MATLAB has a powerful function called PDEPE, which allows solving these kinds of PDE’s.

PDEPE solves initial-boundary value problems for systems of parabolic and elliptic PDE’s in the one space variable and time. The ordinary differential equations

Carbon dioxide-Helium

Sherwood number 2

DA-B (nm²/s) 5.5· 10⁺¹³ [34]

Dimensionless Henry law constant (K) 0,31

D (nm²/s) 2·10⁺⁸ [33]

(ODE’s) resulting from discretization in space are integrated to obtain approximate solutions at times specified in a variable. The PDEPE function returns values of the solution on a mesh provided in another variable. PDEPE solves PDE’s of the form





 





∂ + ∂



 



 



 





∂

∂

∂

= ∂

∂

∂



 





∂

∂ ₋

z u u t x z s

u u t x f z x t x

u z u u t x

c , , , ^{m m} , , , , , , (4.2)

where c, m, f and s are parameters that must be introduced in sub-functions, which belong to the main function PDEPE [35].

It is obvious that the equations to be solved must be of the same form of Equation 4.2. Equation 3.1 is in perfect agreement with that equation but Equation 3.5 nevertheless has different structure than Equation 4.2.

Mass transfer equation:

2 2

z D C t C

∂

= ∂

∂

∂ (3.1)

Heat transfer equation:

t C C

H z

T t

T

p ads

∂

∆ ∂ + −

∂

= ∂

∂

∂ ( )

2

α 2 (3.5)

In order to solve that problem, an alternative method was proposed. Since the main problem is the term

t C

∂

∂ in Equation 4.2, it was suggested to solve this derivative by using the analytical solution of Equation 3.1



 +



 



− +

+ ×

− −

− =

−

∑

= L

z n L

t n

D n

C C

C z t C

n

n

eq 2

) 1 2 cos ( 4

) 1 2 exp (

1 2

) 1 ( 1 4

) , (

2 2 2

0 0

0 π π

π (3.2)

The derivative of Equation 3.2 looks as follow





 



 +

⋅

 

 +

⋅



 



− +

⋅

−

∂ =

∂

∑

= 2 2

2 2

0

) 1 2 ( 2

) 1 2 cos ( 4

) 1 2 exp (

) 1 ) (

, (

L n D L

z n L

t n

C D t

z t C

n

n eq

π π

π (4.3)

In order to have an idea about the shape of Equation 4.3, Plot 4.3 is presented. In this plot, the variation of

t C

∂

∂ versus time is performed. The need data for that calculation is shown in Table 4.3.

0 100 200 300 400 500 600 700 800 900 1000

0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035

t (s) dC/dt

(mol/kg·s)

Plot 4.3: Variation of t C

∂

∂ versus time for Carbon dioxide in MFI.

As we can see in the plot above, for short times the variation of the concentration with time is very high. The maximum of the variation is achieved about at 0,001 s. From this time, the variation decreases until a time value in which the variation becomes zero. By comparing this plot with Plot 4.1, it can be stated that this way for calculating the derivative is correct.

Carbon dioxide

max

Ceq ^(mol/kg) 4.8 [34]

L (nm) 2000

z (nm) 1000

D (nm²/s) 2·10⁺⁸ [33]

Table 4.3: Data for Carbon dioxide in MFI at 298 K applied in the derivative calculation.

Regarding the p-Xylene adsorption, the shape of the plot the same. The main difference is both the time and concentration variation. In this case, the maximum of the variation is almost 0,001

s kg

mol

⋅ . This little variation is the responsible of the isothermal performance.

So far, theoretical methods as well as data have been provided. This is the previous step before starting with the simulation process. Now we are in situation to introduce the equations into the software and receive results. The data input is provided in Table 4.4.

Carbon dioxide o-Xylene m-Xylene p-Xylene

z (nm) 2000 2000 2000 2000

teq (s) 0.02 - - 100

λzeolite (W/m·K) [36] 0.2 0.2 0.2 0.2

Cp,zeolite (J/K·kg) [37] 1400 1400 1400 1400

ρzeolite (kg/m³) 1760 1760 1760 1760

α (nm²/s) 10⁺¹⁰ 10⁺¹⁰ 10⁺¹⁰ 10⁺¹⁰

D (nm²/s) 2·10⁺⁸ [33] 4·10⁺² [32] 8·10⁺¹ [32] 8·10⁺⁴ [32]

L (nm) 2000 2000 2000 2000

Hads (J/mol) -25000 [34] -80000 [39] -100000 [40] -80000 [29]

max

Ceq ^(mol/kg) 4.8 [34] 2.5 [39] 3.5 [40] 0.7 [38]

h (W/nm²·K) 2.6·10^-14 2.6·10^-14 2.6·10^-14 2.6·10^-14 KH (mol/kg·Pa) 4·10^-5 [34] 0.01 [39] 0.1 [40] 0.4 [38]

Table 4.4: Data for Carbon dioxide, o-Xylene, p-Xylene and p-Xylene (298K).

The time to equilibrium was chosen by analyzing Plots 4.1 and 4.2. Hardly adsorption is taking place after this time. The thermal diffusivity (α ) was calculated using Equation 3.6. The heat transfer coefficient (h) was calculated taking in account setup measures. More details for that calculation are provided in Appendix A.1.

Taking in account all the previously said, Plots 4.4, 4.4.a and 4.4.b are showed.

These plots are represented in 3D format. Plot 4.4 shows the temperature variation in function of the distance from the substrate and time for Carbon dioxide while Plot 4.4.a shows the temperature variation in function of the time. Plot 4.4.b shows how the concentration varies with time and distance.

Plot 4.4.: Temperature variation versus distance and time for Carbon dioxide in MFI (P=0.5 atm).

Plot 4.4.a: Temperature variation versus time for Carbon dioxide in MFI (P=0.5 atm).

The simulation showed above was carried out taking in account a partial pressure of zero point five atmospheres. That pressure is low due to it is considered that we are using Henry’s law. For higher pressures, more gas is adsorbed and consequently the temperature inside the zeolite increases. Since more gas is adsorbed, more heat of adsorption is produced and consequently the increase of the temperature.

Plot 4.4.b: Concentration (%) variation versus time and distance for Carbon dioxide in MFI.

So far, carbon dioxide simulation was presented. In order to explain the important role of the diffusion coefficient in the adsorption process, o-, m- and p- Xylene simulations are provided. As it was set out in previous sections, diffusion coefficient of p-Xylene is thousands times lower than the one for carbon dioxide. The other isomers have even lower diffusion coefficient than p-Xylene. That fact will be reflected on the almost isothermal behaviour of the simulation, Plot 4.5.

As we can see in plot 4.5, the temperature variation within the zeolite film is almost zero. As it was explained previously, the reason for this isothermal behaviour is due to the low diffusion coefficient as well as the amount adsorbed within the zeolite.

Since the rest of isomers have lower diffusion coefficient than p-Xylene, the temperature variation is even lower. For that reason, plots for m- and o-Xylene are not

presented in this work. Data for these isomers, as well as for Carbon dioxide and p- Xylene is showed in Table 4.4.

Plot 4.5: Temperature variation versus time and distance for p-Xylene in MFI (P=10 Pa).

Plot 4.6: Concentration (%) variation versus time and distance for o-Xylene in MFI

As it was done for carbon dioxide, concentration variation versus time and distance is also presented for o-, m- and p- Xylene. Since there is no variation in temperature during adsorption, equations for isothermal adsorption will be used. Plot 4.6 to 4.8 shows the variation of the concentration (in percentage) versus time and distance for o-, m- and p-Xylene respectively.

Plot 4.7: Concentration (%) variation versus time and distance for m-Xylene in MFI

Plot 4.8: Concentration (%) variation versus time and distance for p-Xylene in MFI

5. CONCLUSION

A simulation for the transient adsorption of CO2 (and others molecules) in thin MFI films was performed. A general analytical solution was derived, taking into account both the thermal conductivity within the adsorbent and the heat exchange with the ambient.

The time to reach the maximum amount of adsorbed gas for carbon dioxide was extremely different that the one for Xylenes. The high diffusion coefficient for carbon dioxide determines its non-isothermal behaviour. On the other hand, Xylenes have a lower diffusion coefficient and thus it can be assumed their isothermal behaviour.

On Section 4, the simulation for both compounds was carried out. Those simulations were in agreement with the fact that the adsorption velocity determines the reaction time and thus the time to evacuate the heat of adsorption produced during the process.

Regarding carbon dioxide, its low time to keep equilibrium causes the impossibility to evacuate all the heat produced. Although carbon dioxide presents a non- isothermal behaviour, it can be considered for low pressures as isothermal. Just four degrees of variations is insignificant. In terms of p-Xylene, its slow adsorption makes possible the exchange of the heat with the surroundings. The temperature remains constant.

The time to achieve the equilibrium highly determines the possibility to measure the diffusivity. If there is a significant rise in temperature inside the zeolite, this will affect the determination of the diffusivity as well. For these reasons, it could be easy to measure the diffusivity by using that technique [7] for carbon dioxide.

ACKNOWLEDGEMENTS

I would like to be grateful to:

• My supervisor, PhD Mattias Grahn for helping me with all the difficulties during this work.

• My co-supervisor, Professor Jonas Hedlund for allowing me to do this Master’s Thesis in this Department and for your advice and guidance during this work.

• All the friends at the Department of Chemical Engineering and Geosciences with special thanks to Iván Carabante for your support and good jokes.

• My friends from Björskatan for the amazing moments that we shared during these months and for helping me to get over difficult times.

• My friends from Barcelona for your support, although you were far away.

• My parents and brother for your love and encouragement.

Finally, I would like to thank to my girlfriend Beatriz for having such patience with me and for your perpetual smile.

6. REFERECES

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[5] (a) Doelle, H.J., and Riekert, L., in “Molecular Sieves II” (J.R. Katzer, Ed.), Vol. 40, p. 401. ACS Symp. Ser., Washington, D. C., 1997. (b) Doelle, H.J., and Riekert, L., Angew. Chem. 91, 309 (1979).

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348. No. 2, February 1984.

[7] Grahn, M., Wang, Z., Lidström-Larsson, M., Holmgren, A., Hedlund, J., Sterte, J., Silicalite-1 coated ATR elements as sensitive chemical sensor probes, Microporous and Mesoporous Materials 81 (2005) 357–363.

[8] Bonhommr, F., Welk, M.E., Nenoff, T.M., CO2 selectivity and lifetimes of high silica ZSM-5 membranes, Micropores and mesopores materials, 66, pp 181-188 (2003).

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Doctoral thesis. 1998:33 ISSN: 1402-1544.

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URL: http://topaz.ethz.ch/IZA-SC/StdAtlas.htm

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[16] Zheng Wang. Molecular Sieve Films and Zoned Materials. Luleå University of Technology. Licentiate thesis. 2003:37 ISSN: 1402-1757.

[17] McLeary, E.E., Jansen, J.C. and Kapteijn, F., Micropor. Mesopor. Master., 90 (2006) 198.

[18] Seracco, G., Neomagus, H.W.J.P., Versteeg, G.F. and van Swaaij, W.P.M., Chem.

Eng. Sci., 54 (1999) 1997.

[19] Hedlund, J., Jareman, F., Bons, A.J., Anthonis, M., Memb. Sci., 222, (2003) 163.

[20] Garcia, M.J., Materiales zeolíticos: síntesis, propiedades y aplicaciones, Informe Interno, Dep. de Química Inorgánica, Universidad de Alicante, España (2002).

[21] Sing, K.S.W., Everett, D.H., Haul, R.A.W., Moscou, L., Pierotti, R.A., Rouquerol, J. and Siemieniewska, T., Pure Appl. Chem. 57, (1985) 603.

[22] Ruthven, D.M., Principles of Adsorption and Adsorption Processes. 1st ed., John Wiley & Sons: New York, 1984.

[23] Wakao, N., Kaguei, S., Heat and mass transfer in packed beds, Gordon and Breach Science, London (1982).

[24] Weisz, P. B., Chem. Tech. 3, 498–505 (1973).

[25] Post, M. F. M., Diffusion in zeolite molecular sieves, in: Introduction to zeolite science and practice, van Bekkum, H., Flanigen, E. M., Jansen, J. C., eds., vol. 58 of Studies in surface science and catalysis, pp. 391–443, Elsevier, Amsterdam (1991).

[26] Chen, N. Y., Degnan Jr., T. F., Smith, C. M., Molecular transport and reaction in zeolites - design and application of shape selective catalysis, VCH Publishers, New York (1994).

[27] Fick, A., Ann. Phys. 94, 59–86 (1855).

[28] Crank, J., Park, J. S., In Diffusion in Polymers; Academic Press: New York, 1968.

[29] Kärger, J., Ruthven, D.M., Diffusion in Zeolites and other microporous solids, John Wiley & Sons; 1st edition (April 1992)

[30] Dawoud, B., Thermische und Kalorische Stoffdaten des Stoffsystems Zeolith MgNaA-Wasser, Ph.D. Dissertation, RWTH-Aachen, Germany, 1999.

[31] Vedder, U., Nichtisotherme Adsorptionskinetik von Wasserdampf an einer kompakten Zeolithschicht, Dipl.-Ing. Thesis, Lehrstuhl für Technische Thermodynamik, RWTH Aachen University, Aachen, Germany, 2003.

[32] Hedlund, J., Öhrman, O., Msimang, V., van Steen, E., Böhringer, W., Sibya, S., Möller, K., The synthesis and testing of thin films ZSM-5 catalyst, Chemical Engineering Science 59, p. 2647-2657 (2004).

[33] Bakker, W. J. W., van den Broeke, L. J. P., Kapteijn, F., Moulijn, J. A., Temperature Dependence of One-component Permeation through a Silicalite-1 Membrane, AlChe J., 43, 2203-2213 (1997).

[34] Delgado, J.A., Uguina, M.A., Sotelo, J.L., Ruiz, B., Gómez, J.M., Fixed-bed adsorption of carbon dioxide/methane mixtures on silicalite pellets, p. 11, Adsorption (2006) 12:5-18.

[35] Help of MATLAB^© The Language of Technical Computing, Version 7.

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[37] Heinke, L., Chmelik, C., Kortunov, P., Shah, D.B., Brandani, S., Ruthven, D.M., Kärger, J., Analysis of thermal effects in infrared and interference microscopy: n- Butane-5A and methanol-ferrite systems, Micropor. Mesopor. Mater, (2006), doi:10.1016/j.micromeso.2006.11.017.

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[39] Huang, Q., Vinh-Thang, H., Malekian, A., Eic´, M., Trong-On, D., Kaliaguine, S., Adsorption of n-heptane, toluene and o-Xylene on mesoporous UL-ZSM5 materials, Microporous and Mesoporous Materials 87 (2006) 224–234.

[40] Lachet, V., Boutin, A., Tavitian, B., Fuchs, A.H., Computational Study of p- Xylene/m-Xylene Mixtures Adsorbed in NaY Zeolite, J. Phys. Chem. B 1998, 102, 9224- 9233.

[41] Schuring, D., Diffusion in zeolites: towards a microscopic understanding, Doctoral thesis, Technische Universiteit Eindhoven, 2002. ISBN 90-386-2624-X

APPENDIX A.1

In order to calculate the heat transfer coefficient (h), several dimensionless numbers were used. It was assumed for this calculation that the gas phase was formed by a mixture of helium (the majority) and the desired compound. The set of equations used in this calculation is shown bellow

L h Nu Nu

L C v

S v Q

p gas

gas

λ λ µ µ

ρ

·Pr ·

·Re 6 , 0

· 2

Pr ·

· Re ·

3 / 1 2 /

1 → =

=

=

=

=

•

where v is the gas velocity (m/s),

•

Q is the volumetric flow rate (m³/s), S is the cross sectional area (m2), ρ is the density of the gas (kg/m3), µ is the gas viscosity (Pa·s), λ is the thermal conductivity of the gas (W/K·m), L is the thickness of the zeolite (m), C is the heat capacity of the gas (J/kg·K), Re is the Reynolds number, Pr is the Prandtl p

number and Nu is the Nusselt number.

Figure A.1: Schematic representation of the device in which the gas flows.

The needed data, as well as the results, are shown in Table A.1. A schematic representation of the device, just before the air gets in contact with the atmosphere, is

4 mm

13 mm

Q = 0,01-1 l/min

provided in Figure A.1. The value of the volumetric flow rate is the range 0,01-1 L/min.

In this calculation, it is assumed an average value for that parameter.

Helium

S (m²) 5,2·10^-5

•

Q (m³/s) 1,67·10^-5

v (m/s) 0,321

ρ (kg/m³) 126

L (m) 2·10^-6

µ (Pa·s) 1,9·10^-5

Re 4,25

λ(W/m·K) 1,6·10^-2

C_p (J/K·kg) 850

Pr 1

Nu 3,24

h (W/m²·K) 2,6·10⁺⁴

Table A.1: Data and result for the heat transfer coefficient estimation.

APPENDIX A.2

m = 0;

x = linspace (0,2000,20);

t = linspace (0,0.02,100);

sol = pdepe (m,@pdex7pde,@pdex7ic,@pdex7bc,x,t);

% Extract the first solution component as u.

u = sol (:,:,1);

surf (x,t,u)

xlabel ('Distance x') ylabel ('Time t')

function [c,f,s] = pdex7pde(x,t,u,DuDx)

alfa = 10^10;

H = -25000;

Cp = 1400;

D = 2*10^8;

c = 1;

f = alfa*DuDx;

L = 2000;

K = 4*10^-5;

P = 5*10^4;

Ceq = K*P;

sum = 0;

for i = 0:100

sum = sum + (Ceq *(-1)^i * cos ((2*i+1) * pi * x/(2*L)) * (D*(2*i+1)*

* pi/(L^2)) * exp((-(2*i+1)^2 * pi^2*D*t)/(4*L^2)));

sum2 = sum;

end

s = -H*sum2/Cp;

end

function u0 = pdex7ic(x);

Tref = 298;

u0 = Tref;

end

function [pl,ql,pr,qr] = pdex7bc(xl,ul,xr,ur,t)

Tref = 298;

H = 2.6*10^-14;

K = 2*10^-10;

pl = ul-Tref;

ql = 0;

pr = (ur-Tref)*h;

qr = k;