Macroscopic and Microscopic Effects in Diffusion and

Defect and Diffusion Forum Vols. 283-286 (2009) pp 388-393 online at http://www.scientific.net © (2009) Trans Tech Publications, Switzerland Online available since 2009/Mar/02

Macroscopic and Microscopic Effects in Diffusion and Reaction in Catalyst Porous Particles António Martins1, a, Carlos H. Braga1,b, and Teresa M. Mata2,c 1

CEFT – Centre for Transport Phenomena Studies LEPAE – Laboratory of Process Engineering, Energy and the Environment Faculty of Engineering University of Porto, Rua Dr. Roberto Frias S/N 4200 465 Porto, Portugal 2

a

[email protected] , b [email protected], [email protected]

Keywords: Porous Media, Network Model, Diffusion, Reaction, Catalyst Particles, Boundary Conditions.

Abstract. This article presents and discusses a network model to describe and predict the behaviour and performance of catalyst particles. The differences and advantages of this approach when compared to the continuous models currently used in practice are highlighted and critically assessed. The local structure of the catalyst particle is modelled using a three dimensional network model made up of cylindrical pores and nodes of negligible volume. In the pores a homogenous first-order reaction takes place, coupled with the diffusion. For steady state conditions the concentration field can be obtained solving a sparse linear system of equations, obtained by solving the mass balance equations written for the network nodes and using the concentration profile in the network pores. The influence of the boundary conditions and the network sizes was investigated, showing the results in particular that the nature of the boundary conditions can have a profound impact in the predictions of the model. Introduction and significance of the problem Catalysis is currently one of the most important areas of fundamental and applied research. Most of the existing chemical processes depend or involve in some parts catalysts of different types and characteristics [1]. They are crucial to ensure a good conversion of the raw materials and a good selectivity to the desired products, minimizing losses and inefficiencies. Examples include processes like catalytic reforming and fluid catalytic cracking in the petrochemical industry, ammonia production, and automotive catalysts to reduce noxious emissions, among many others. A wide variety of chemical compounds and materials can act as a catalyst, including noble metals and inorganic molecules, depending on the operational and other specific conditions. Although in some processes homogenous catalysts are used, due to their high costs, potential adverse effects, and difficulties in separating them from the reaction products, in most cases the catalysis are supported in a solid phase. The materials used as support are normally highly porous materials, such as alumina or activated carbon. Also, the support plus catalyst particles can exist in many different shapes, making it possible to address specific operational needs, such as reduced pressure, improved heat transfer, etc. Notwithstanding its advantages and practical importance, heterogeneous catalysis it stills a very active area of research. From a qualitative point of view, it is a rather complex phenomenon that couples transport phenomena, in particular the diffusion of mass into and out of the catalyst particles, with chemical reaction. Concerning the mass transport, the particles porous structure can play a decisive role, for example limiting or enhancing the access of the reagents to the active sites. Therefore, this is a key factor that has to be taken into account when describing and modelling the performance of catalyst particles. Also, heat effects may be relevant in many situations, but in this work they are not considered. Most of the models used in practice assume that the catalyst particles are homogenous, and the diffusion and reaction can be described using effective coefficients spatially independent. Hence, it All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 193.136.33.211-02/03/09,13:17:24)

Defect and Diffusion Forum Vols. 283-286

389

is possible to use many mathematical tools and results available from other areas of physics. In particular, diffusion is characterized by and effective diffusion coefficient, De , that takes into account the irregular nature of the void space and how it impacts the transport of mass inside the catalyst particles. Assuming steady state and that only chemical reaction and diffusion are the relevant transport phenomena, the following mass balance equation can be written for steady state De ∇ 2 C − R (C ) = 0

(1)

Many solutions of the general mass balance equation exist for various geometries, in particular the slab, sphere and infinite cylinder, for different boundary conditions, for example with or without interfacial mass transfer resistances, and simple kinetic laws [1]. A dimensionless number, the Thiele Module, φ , can be defined as the ratio between the transport of mass by diffusion and chemical reaction. Its value is a measure of the relative importance of both processes. For low values of this parameter, reaction is the controlling process and the catalyst is said to operate in the chemical regimen. For high values of φ , diffusion is controlling and a regimen is said o be diffusional. In most situations the results of this model are expressed in terms of a effectiveness coefficient, that quantifies the deviation between the ideal situation and the cases with reaction only and with other effects, in particular diffusion. The previous model has some drawbacks when it is applied to describe the behaviour of catalyst particles. One of the most important concerns is De , which value is strongly dependent on the structural characteristics of the particle. The models proposed in literature can give wildly different estimates for this parameter, and large errors between experimental and predicted values are common. Also, the hypothesis of uniform reactivity inside the particle and the assumption of ideal geometries are not adequate in many cases. For example for oddly shaped particles, small sized pellets where diffusion in all spatial dimensions have to be taken into account, or catalysts where the local structure or active sites spatial distribution, were engineered [2,3]. A way to circumvent these problems is to consider explicitly the local characteristics of the porous media and how they influence the reaction and the mass diffusion. Since an exact and rigorous model of the void structure will be to difficult to use, either because there are no experimental methods to characterize it, or the system of mass balance equations will be impossible to solve, a simpler model is needed. A network model made from the combination of different elements is a good option, because it can capture the essential features of the porous media, and yet be simple enough to be mathematically treated. In other systems where porous media are a key part, such as fluid flow, porosimetry, and mass transport, among others, this approach has been quite useful [4,5]. For catalysis network models were also proposed in the literature, but most of them assume that the particle size and geometrical effects are negligible, or there are no spatial variations of the kinetic constant [6-9]. Also studied in the literature is the influence of the catalyst deactivation on the diffusion of mass inside a catalyst pellet [10,11], where the variation of the porous structure has to be taken into account. Although some of the key aspects of the influence of the network characteristics in the particle catalysts behaviour are tackled in those works, there is still the need to address some key aspects, as for example size effects, relevant with the advent of nanocatalysts, particles with a special engineered spatial distribution of active sites, or the influence of different values of concentration in the fluid-solid boundaries.

etwork model The local structure of a catalyst particle was approximated by a tridimensional network of cylindrical pores with null volume mixing nodes and possessing a regular cubic structure, as shown in Figure 1. The pores are characterized by its diameter d j , that follows a given size distribution and length l j . The average value of the pore length is determined assuming a porosity value for the network, ensuring that the ratio between total and void volume is maintained between the network

390

Diffusion in Solids and Liquids IV

and a real catalyst pellet. The network size to reach the spatial dimensions can be varied by changing the number of nodes lines in each spatial direction. The local structure can be varied by using different size distributions of the pore diameter and length, or by randomly removing pores, allowing different distribution of pore connectivities to be studied. As only diffusion and reaction are considered, there is no need to model the fluid flow inside the network. In this work, it is assumed that diffusion in the pores can be described by the Fick law, using the diffusivity coefficient, and chemical reaction is first order, isothermal and irreversible. For these assumptions, and assuming unidirectional diffusion in the pores, for each pore the following mass balance equation can be written

Figure 1 - Three dimensional network model

D

d 2C − kC = 0 dx 2

(2)

where D is the diffusivity coefficient, k is kinetic constant for the pore j , and x is the axial dimension in the pore. The network model implemented in this work has the possibility of define kinetic constants for different pores. Although a similar procedure can be considered for the diffusion coefficient, in this work that possibility is not considered. The previous equation should be solved for each pore for the solution boundary conditions:

C = C1 , x = 0 ; C = C 2 , x = l j

(3)

where C1 and C1 represent the concentrations in the ends of the pore, which may be the concentration at the nodes or at the boundaries of the network, depending on the spatial position of pore in the network. The previous equation can be solved analytically to give the concentration profile inside the pore in the form

 sinh(φ j x l j ) sinh(φ j x l j ) k C = C1 cosh(φ j ) − ;φ j = l j  + C2 tanh φ j  sinh φ j D 

(4)

being φ j the Thiele modulus for pore j . Writing the mass balance equations for the network nodes it is possible to determine the concentration values in the network nodes. For a giving node, the flux of mass arriving should be equal to the flux exiting. As the flux in a pore can be calculated from the concentration profile using Fick’s law, the following expressions can be obtained for the mass flux, j , at the nodes and respective mass balance equations

j =

π 4

dj

2

C2 kD  C1 −  d j  tanh φ j sinh φ j

π

2 ∑j =0⇔ ∑ dj

4

   

C2 kD  C1 −  d j  tanh φ j sinh φ j

(5)  =0  

(6)

Note that in the previous equation the sum refers to all channels connected to a giving node, value that may change depending on the pore connectivity. Equation 6 is a linear system of equations that can be solved to obtain the values of D for the mixing nodes. The system is sparse, symmetric and positive definite, and a conjugate gradient method was used to solve it.


391

The model results are the concentration values for the network nodes. The network efficiency, η , is determined from the total flux that enters the network with diffusional effects, that by mass balance is equal to the total amount of reagent consumed in the particle, and the quantity of reagent consumed without diffusional effects, given by the expression

η=

Total Flux of mass entering network Re action rate without Diffusion

(7)

where Ej represents the flux that enters the network, calculable from the system results, and VV the void volume of the network. Because we are using a pore size distribution, different realizations of the network will lead to different results. To ensure that the obtained results are meaningful, the implemented simulator in each simulation runs several realizations of the network and samples the results to obtain the average values of the relevant parameters. Also, because a procedure to randomly remove pores of the network was implemented to change the pore connectivity, an algorithm was also implemented in the simulator to ensure that at least there is path for the reagent molecules o cross the network.

Results The model was implemented in Fortran 95 in both Windows XP and Linux environments. Special care was taken to ensure that the code comply by the Fortran standard and can be compiled in a wide variety of operating systems and compilers. In the near future and after improving and documenting it the code will be publicly released as Open Source as a GPL license. Only preliminary results are shown in this article. Most of the results presented in this work used D = 10 −8 m 2 s , a mean that follows a Gaussian distribution with pore radius of 25 nm with a standard deviation of 10 nm, a porosity value of 0.7, and a mean coordination number of 5.0. The pore length distribution was sampled from a Gaussian distribution with a standard deviation of 10 nm, scaled in such way that the network porosity is equal to 0.7, as stated before, to ensure that the results are statistical significant. Properties were sampled over a set of 5 independent networks with each one having 20 independent pore size distributions. No periodic boundaries are imposed in the network, meaning that all six values of concentration have t be defined to the simulator. To assess the influence of the boundary conditions, in particular differences in the concentration values defined in each of the network sides, simulations were performed varying those values. Four networks are considered: 20×20×20, 10×50×50, 10×30×30 and 10×15×15; for two cases: one assumes that the same values of concentration occur at all sides of the networks and in this case equal to 15 mol/m3 (this situation is equivalent to a uniform boundary condition), and in the second case a concentration value of 150 mol/m3 is imposed in one of faces of the network with a constant value of x . The value of the kinetic constant was varied to change the Thiele modulus, and be able to study the behavior of the model in the reactional and diffusional regimens. The results for the 20×20×20 network are presented in Figure 2 in form of a network effectiveness factor as a function of the average channels Thiele module. Qualitatively, the results are in agreement with the predictions of the homogenous models and experimental values [1,2]. However, as the network structure is far from homogenous, no comparison was attempted to compare the network predictions with the predictions of other models.

392


When the values of η are compared for 1.00 uniform and non uniform boundary predictions, Uniform although qualitatively similar, large variations η Non Uniform can be observed. In particular, the values obtained for non uniform networks are smaller than those obtained for uniform networks. This 0.10 can be attributed to differences in the flux of mass that cross the boundary for each case. In the case of non uniform boundaries, the largest value of concentration leads to higher reaction 0.01 rates and consequent fluxes in the faces with φ 10.00 0.01 0.10 1.00 higher concentration values. The other faces have much lower fluxes, and as η is calculated Figure 2 - η as a function of φ for a 20×20×20 for the largest boundary concentration, η will with uniform and nonuniform boundary necessarily lower. For uniform boundary conditions. conditions the flux into the network are similar in each face, and the reaction will be much more uniform within the network, leading to higher values of η . To confirm the conclusions made before, in Figures 3 and 4 the xy concentration profiles are presented in Figures 3 and 4, for a case in the intermediate regimen (not chemical neither diffusional). Note that the concentration profiles are averaged in the z direction. Comparing the profiles it can be clearly that the boundary conditions have a profound impact on the concentration profiles inside the network. In particular, in the non uniform case it can be observed that the concentration profile varies significantly close to the boundary, and for the rest of network the reagent concentration is low, as concluded before.

Figure 3 - xy concentration profile for uniform boundary conditions.

Figure 4 - xy concentration profile for non uniform boundary conditions

In Figure 5 the values of η as a function of φ are presented for the 10×50×50, 10×30×30, and 10×15×15 networks. Qualitatively all the networks show the same behavior in all regimens. However, comparing the results obtained for non uniform and non uniform boundaries differences in the behavior predicted by the model can be seen. For uniform boundaries, it can be seen that for chemical regimen (low values of φ ) there is no significant size effects. For diffusional regimen it can be seen that the smaller the network sizes, the larger the value of η . For large values of φ , diffusion is the limiting step, thus, the smaller the network the further the reagent can enter inside the network, resulting in larger values of η . Also, for smaller networks the influence of the corners is more pronounced, and mass can diffuse more inside the network before being consumed.


393

For non uniform boundaries a different behavior can be observed. In particular, the larger the network the larger the value of η . This is the result of smaller concentration variations for larger networks, leading to higher reaction rates throughout the network, in particular close to the boundaries with higher concentration value. Thus, under those conditions, it is expected that η will be higher, as confirmed in Figure 5.

Conclusions In this work a three dimensional network 1.00 model was presented and discussed that is aimed to describe diffusion and reaction in η porous catalyst particles. Contrary to current used homogenous models, the network model can easily take into account not 0.10 uniform boundary conditions, and variations in the characteristics of the particle porous 10 × 50 × 50 10 ×15 × 15 10 × 30 × 30 Non Uniform Non Uniform Non Uniform structure. Qualitatively the results are in Uniform Uniform Uniform agreement with the behaviour predicted by 0.01 other models and observed experimentally. 0.001 0.010 0.100 1.000 φ 10.000 The results show that the effects of irregular boundaries can have a profound impact in Figure 5 - η as a function of φ for fixed length of the behaviour predicted by the model, and network in the x direction with uniform and should be accounted for whenever necessary. nonuniform boundary conditions. In future studies the model will be extended to include more complex kinetics, in particular competitive and consecutive reactions where selectivity can be an issue and using more realistic kinetic laws, and study the influence of the structural network characteristics. Also, heat effects due to the chemical reaction will be added in the future.

References [1] L.K. Doraiswamy and M.M. Sharma: Heterogeneous Reactors: Analysis, Examples, and Reactor Design. Volume 1: Gas-Solid and Solid-Solid Reactors (John Wiley & Sons, 1984). [2] E. Johannesen, G. Wang and M.O. Coppens: Ind. Eng. Chem. Res Vol. 46 (2007), p. 4245 [3] G. Wang, E. Johannesen, C.R. Kleijn, S.W. deLeeuw and M.O. Coppens: Chem. Eng. Sci. Vol. 62 (2007), p. 5110 [4] V. Mata., J.C.B. Lopes and M.M. Dias: Ind. Eng. Chem. Res Vol 40 (2001), p. 3511 [5] A.A. Martins, P.E.L Laranjeira, M.M. Dias, J.C.B. Lopes: AIChE J. Vol 53 (2007), p. 91 [6] P.N. Sharrat and R. Mann: Chem. Eng. Sci. Vol. 42 (1987), p. 1565 [7] R. Mann and P.N. Sharrat: Chem. Eng. Sci. Vol. 43 (1988), p. 1875 [8] S. Arbabi and M. Sahimi: Chem. Eng. Sci. Vol. 46 (1991), p. 1739 [9] M. Davdar and M. Sahimi: Chem. Eng. Sci. Vol. 57 (2002), p. 939 [10] C. Rieckman and F.J. Keil: Chem. Eng. Sci. Vol. 54 (1999), p. 3485 [11] S. Arbabi and M. Sahimi: Chem. Eng. Sci. Vol. 46 (1991), p. 1739 [12] F.J. Keil and C. Rieckmann: Chem. Eng. Sci. Vol. 49 (1994), p. 4811 [13] U.A. El-Nafaty and R. Mann: Chem. Eng. Sci. Vol. 54 (1999), p. 3475

394


Diffusion in Solids and Liquids IV doi:10.4028/3-908454-50-6 Macroscopic and Microscopic Effects in Diffusion and Reaction in Catalyst Porous Particles doi:10.4028/3-908454-50-6.388