Optimal design of engineered gas adsorbents: Pore-scale level

Chemical Engineering Science 69 (2012) 270–278

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Optimal design of engineered gas adsorbents: Pore-scale level Fateme Rezaei n, Paul A. Webley Department of Chemical Engineering, Monash University, VIC 3800, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 July 2011 Received in revised form 6 October 2011 Accepted 17 October 2011 Available online 28 October 2011

This study presents an optimization methodology and results for the structure of gas adsorbents at a pore level by evaluating the effect of pore geometry, size and overall adsorbent porosity on ultimate working capacity of adsorbents used in pressure swing adsorption (PSA) processes. Three model pore network topologies are studied: parallel, grid-like and branched structures. These are ‘‘near’’ optimal structures for an adsorbent particle and their relative performance is compared in a two-step PSA cycle. The macropore network of such structured adsorbents is optimized through maximization of an objective function i.e. working capacity WC, defined as the number of moles adsorbed per unit volume of slab. Molecular and Knudsen diffusion are considered as the sole transport mechanisms in the macropore channels. An unexpected finding of this optimization technique is that the branched structure with a porosity of less than 50% represents an optimum structure with highest working capacity for the system considered. Furthermore, for faster cycles the advantage of branched structures is more obvious, reflecting the advantages of the pore network in facilitating diffusion more efficiently than other structures. A macropore channel density (defined as the density of macropores per metre of external surface) of below 10 is suggested for optimum performance for both ‘‘fast’’ and ‘‘slow’’ PSA cycles. The theoretical results of this study will be used as a priori results for the design of adsorbents at the macro-scale (bed) level. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Optimal structure Working capacity Optimization Pore-scale level Nanoporous adsorbent Objective function

1. Introduction As an important prerequisite condition of techniques for optimizing adsorbents for gas separation, a detailed understanding of gas molecule diffusion into and out of nanoporous materials is necessary. Although the study of gas molecule transport in a porous matrix is a relatively mature topic of fundamental and engineering interest in different fields such as catalysis, separation, membranes, etc., relatively little attention has been paid to the optimization procedures of nanoporous materials for cyclic adsorptive separations. Recent work has mainly focussed on the synthesis and in some cases the optimization of nanoporous materials with controlled pore size and porosity; synthesis of hierarchical materials with ordered pore structure is among such attempts. There are different approaches for describing the complex structure of the cavities in porous media. The simplest model proposed in 1951 (Wheeler, 1951) was a bundle of straight cylindrical pores with smooth walls. Since then, more complexities were considered and concepts such as tortuosity and connectivity were introduced via more rigorous models such as Voronoi grid, Bethe–Lattice or

n

Corresponding author. Tel.: þ61 3 99053445; fax: þ61 3 99055686. E-mail address: [email protected] (F. Rezaei).

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.10.039

pore network models (Keil, 1999; Rieckmann and Keil, 1999; Sahimi et al., 1990, 1993, 1995). Petropoulos (1991) used a capillary network model consisting of a regular array of nodes joined by uniform capillaries to investigate the gas transport (Knudsen gasphase and surface diffusion) in bidisperse porous adsorbents. Their results showed that the value of surface diffusivity measured on the bidisperse adsorbent is correspondingly higher than that measured on a single constituent particle. In another study conducted by Satik and Yortsos (1996), the pore network model was utilized to investigate adsorption-desorption processes and hysteresis phenomena in porous adsorbents. Meyers and Liapis (1998, 1999) carried out pore network modelling of convective flow and diffusion of molecules in monoliths and in porous particles packed in a chromatographic column. They showed that the pore network model is a powerful tool to determine in an a priori manner the effective diffusivity and intraparticle interstitial velocity through the monoliths or particles for a given chromatographic column. They further investigated the dynamic behaviour of monoliths through a mathematical model for adsorption taking into account the effect of pore structure using pore network modelling analysis (Meyers and Liapis, 1998; Meyers et al., 2001). In catalysis, structural optimization has been extensively studied by both continuum and pore network models (Beeckman and Hegedus, 1991; Hegedus, 1980; Keil and Rieckmann, 1994; Morbidelli et al., 2001) and procedures using either fractal pore

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

size distribution or bidisperse structure have been the subject of many theoretical studies. Some of their outcomes emphasized the remarkable performance of fractal catalysts. Coppens and coworkers (Coppens and Froment, 1997; Gheorghiu and Coppens, 2004; Johannessen et al., 2007; Wang and Coppens, 2008; Wang et al., 2007) have recently studied fluid transport optimization in hierarchical catalysts using the Thiele Modulus (O) method based on both continuum and pore network models. They showed that by introducing macropore networks into the pore structure, the diffusion limitations could be significantly reduced. Their findings revealed that by optimizing the macroporosity and size of the pores in a nanostructured catalyst, it is possible to significantly enhance the reaction yield with respect to commercial catalysts. They also found that catalysts with fractal-like structure operate very near optimality. The authors suggest that although in practice the finite structure of an adsorbent is not as simple and straightforward as the one considered in their studies, the performance of a true optimum structure is not significantly larger from this simplified pore network (Wang et al., 2007). Applying the Thiele Modulus (O) method for structural optimization in catalysts results in a simple and general approach to evaluate the optimal effectiveness factor (Z) and hence provides an insight to understanding what the optimal structure would be. In the case of adsorption separation, however, instantaneous adsorption equilibrium on the surface of active sites is assumed (local equilibrium) and the rate of adsorption is dictated by the overall mass transfer rate to the adsorption sites. Therefore, the conventional catalytic approach (O  Z) for structural optimization is not applicable to assessment of adsorption performance in cyclic systems. Instead, the parameter that best defines the process performance of an adsorbent should be based on its ability to load the undesired (or desired) component from the feed stream over the adsorption step and subsequently remove it on the desorption step. This parameter is frequently called the working capacity (WC). The working capacity defines the difference in the amount of material per unit volume adsorbed and desorbed at the end of the cycle, and can be calculated using the following: WC ¼ Dni ¼ ni,ads ni,des

271

nanoporous adsorbents requires simplified models of geometric shape for such irregular pores. Therefore, we represent the pore network as a regular geometry with porosity ea and channel density M as adjustable parameters which maps the pore structure from one type to another. These parameters were chosen as they adequately capture the expected trade-off between adsorbent properties, which govern the overall adsorbent performance, i.e. higher porosity or higher channel density result in faster diffusion rates however, there is less solid adsorbent in the structure. The geometry considered for nanoporous adsorbent particles is a 2D slab with given boundary conditions at its faces (a fixed adsorptive pressure pi0 at one face and a Neumann boundary condition at other faces, see Eq. (7)). It is possible to consider a wider range of configurations (including square, circle, etc.), however, as suggested by the study conducted by Gheorghiu and Coppens (2004) other geometries such as circular networks with radial and concentric channels display the same qualitative properties as slab ones. Therefore, in order to simplify the procedure, only 2D slabs with finite thickness are considered here. The pore network within the 2D slab particle in our study is represented by three different geometries i.e. parallel, grid-like and branched networks of macro-channels, as shown in Fig. 1. The branched network represents a hierarchical pore structure while the parallel and grid pores are regular pore networks with uniform channel sizes. The characteristics of the pore geometry are optimized numerically by taking into account the working capacity of the adsorbent as a function of pore network volume, size and shape. The goal is therefore to maximize the overall working capacity of all three adsorbent networks in a two-step PSA cycle by optimizing its pore network structure. It should also be noted that the optimal structure is not unique and depends on assumptions such as boundary conditions and details of the transport

2L

H

ð1Þ

ni is the loading of component i integrated over the adsorbent particle per unit slab volume. Strictly speaking, it is the surface excess but for moderate pressures such as those encountered in most PSA systems, it is very nearly equal to the absolute loading. For a two-step cycle (adsorption and desorption), this term defines the difference between the amount adsorbed at the beginning and end of desorption step. In this study, we propose a new structural optimization procedure for adsorbents used in adsorptive gas separation processes at the pore-scale level. In particular, the objective of this study is to provide a general criterion for the design of optimal adsorbent structures in terms of its optimal macropore channel size and shape, volume fraction occupied by channels and optimal channel density at the micro-scale (pore) level. It is clear that greater channel density and adsorbent porosity will improve mass transfer rates but only at the cost of decreased amount of adsorbent. Therefore, an optimum pore network geometry, channel density and overall porosity exist.

2. Optimization procedure In our optimization procedure the problem consists in finding the optimal structure such that the working capacity WC is maximized. To find such a structure, an initial guess for the distribution of macropore channels within the nanoporous material is needed. However, the complex structure of cavities in

d

y

w/2

x 2L

H

d wy

d

y x

wx/2 2L

H

dx dy

y x

wy

wx

Fig. 1. 2D structures of pore network (a) parallel, (c) grid-like, (e) branched and (b, d, f) the their periodic unit structures, respectively. The channels are grey and the nanoporous material is white. (b), (d), (f) show the repeating unit for each structure.

272

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

mechanisms and diffusion limitations considered in the pores of the adsorbent material. Therefore, our results do not apply to kinetically based adsorptive separation (e.g. N2 purification on carbon molecular sieve) nor those in which a strong surface diffusion exists as the major component of mass transfer. A bidisperse pore structured adsorbent is assumed for all geometries, having micropores (smaller than 2 nm in size) within the nanoporous solid (the white regions in Fig. 1) and macropores (larger than 50 nm in size) between these solid islands (the grey regions in Fig. 1), adopting the IUPAC classification for pore sizes. For instance, the micropores may refer to the pore space within zeolite crystals or graphitic units in activated carbon adsorbents while the macropore channels resemble the pores between each crystal or the amorphous interspace between the graphitic units respectively. The physical properties of the solid (e.g. microporosity, micropore size, micropore diffusivity, etc.) are kept constant in the optimization procedure and the geometrical optimization problem is therefore to maximize adsorbent working capacity by adjusting the macropore network properties. The adsorbent macroporosity, ea, of the parallel channel structure with the repeating unit (in the x-direction) having a length of H and width of 2L (see Fig. 1b) is related to macropore channel size d and the thickness of nanoporous material w by the following expression: d d þw

unit volume of adsorbent and pi0 is the partial pressure of component i in the feed. Local equilibrium is assumed in the micropores within the adsorbent and hence, the adsorption derivative @ni/@t can be represented by the following: @ni @n @p ¼ i i @t @pi @t

@ni/@pi is the adsorption isotherm slope given by a linear isotherm assumed in the current analysis. Table 1 Adsorption parameters used in each two-step PSA simulation. K Dmicro H L Pi0 d0

yc T Solid density

0.9

The same expression holds for the grid-like structure Fig. 1c with equivalent repeating unit shown in Fig. 1d as follows: h d wx i 1þ ð3Þ ea ¼ ex þð1ex Þey ¼ ðd þ wx Þ H

0.8 0.7

Dimensionless CO2 Loading

where wx is the nanoporous adsorbent thickness in the x-direction and ex and ey given by ex ¼(d/(d þwx)) and ey ¼(d/H), are the porosities of macropore channels with equal diameters in x- and y-directions, respectively. For equivalent repeating units of a branched structure consisting of a main pore of diameter dx and N branches of diameter dy (as shown in Fig. 1f, where dy odx), the macroporosity can be estimated by a simple geometrical equation:   Ndy wx dx 1þ ð4Þ ea ¼ ex þð1ex Þey ¼ ðdx þ wx Þ dx H

¼ r

in the macropores ðnetwork channelÞ

subjected to the following boundary conditions:     @pi @pi ¼ 0, ¼0 pi0 9ðx,0Þ ¼ pi0 , @x ð 7 L,yÞ @y ðx,HÞ

0.3

0

grid branched parallel 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless Time

0.9

ð5Þ ð6Þ

0.7 Dimensionless CO2 Loadig

ei @pi

RT @t

0.4

0.8

The equations related to the diffusion and adsorption mechanisms in the adsorbent structures are presented as follows, assuming diffusion in the macropores and adsorption–diffusion in the micropores of the nanoporous material:

UN macro i

0.5

0.1

2.1. Diffusion and adsorption in the optimum structure

ea @pi

0.6

0.2

where ex and ey given by ex ¼(dx/(dx þwx)) and ey ¼(Ndx/H), are the porosities of main macropore and branches, respectively.

@n þð1ei Þ i ¼ rUNmicro i RT @t @t in the solid matrix which includes micropores

30.0 mol/kg bar 0.6 1.66  10  4 cm2/s 1.0  10  3 m 1.0  10  6–1.0  10  3 m 0.1 bar 10  6, 10  7, 10  8 m 0.01, 0.1 300 K 785 kg/m3

ei

ð2Þ

ea ¼

ð8Þ

0.6 0.5 0.4 0.3 0.2 grid branched

0.1

parallel

ð7Þ

where both macropore and micropore mass transport occur simultaneously. In the above equations, ni is the amount of component i adsorbed on the solid walls in the micropores per

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Dimensionless Time

Fig. 2. Dimensionless CO2 loading as a function of dimensionless time at (a) yc ¼ 0.1 and (b) yc ¼0.01.

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

273

pore radius, therefore it may be argued that surface diffusion contributes to molecule transport in macropores along with pore diffusion as in carbon adsorbents however, in some adsorbents such as zeolites there is no evidence for surface diffusion therefore we neglect its contribution in our analysis. Therefore the total pressure drop @p/qy is assumed to be constant across the adsorbent material and the macropore flux is described by:

As a first approximation, in order to determine the molar flux of component i in macropores Nmacro and micropores Nmicro , a i i kinetic model is used based on rigorous kinetic theory represented by Maxwell and Stefan. In general, the total molar flux of adsorptive molecules through a nanoporous adsorbent can be given by the summation of contributions from pore diffusion (bulk and Knudsen diffusion), surface diffusion and viscous flow. In our model however, the contributions of viscous flow and surface diffusion to both Nmacro and Nmicro fluxes are neglected. i i The reason for neglecting viscous flow in the macropores of the adsorbent material is justified by taking into consideration that in real cyclic operations such as PSA systems with short cycle times, the pressure drop in macropores is not as significant as overall bed pressure drop. Moreover, the diffusion time constant for allowing viscous flow to occur is very short in comparison to process cycle time. Considering surface diffusion, as adsorption of gas molecules proceeds on the surface of adsorbent, the thickness of the adsorbed layer is increased resulting in the reduction of

Nmacro i Dka

þ

N macro ð1yi Þ 1 @pi i ¼ RT @y Dm

ð9Þ

or: ðNmacro =Dmacro Þ ¼ ð1=RTÞrðpi Þ, where Dmacro is the macropore i diffusion coefficient estimated by following expression: 1 1 1y ¼ ka þ m i D Dmacro D

ð10Þ

Dm and Dka are, respectively, molecular and Knudsen diffusivity of gas molecules in the macropore channels. Dm is calculated by the well-known Chapman–Enskog equation for a given pressure and

3000

2500

WC, mol/m3

2000

1500

1000 grid parallel

500

branched 1D effective model

0

0

100

200

300 M

400

500

600

Fig. 3. (a) Objective function WC as a function of macropore density M and (b) dimensionless CO2 loading as a function of dimensionless time at different macropore densities at yc ¼ 0.1.

274

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

temperature whereas Dka is estimated using Dm and mean free path. The Knudsen diffusivity is proportional to the pore diameter in both macro and micropore ranges and equal to Dm when the pore diameter is equal to the mean free path, d0: Dka ¼ Dm ðd=d0 Þ

ð11Þ

The adsorptive flux in the micropores of the nanoporous adsorbent is given by: Nmicro i Dmicro

¼

1 rðpi Þ RT

ð12Þ

here the nanoporous adsorbent is treated as a continuum with a micropore diffusivity Dmicro corresponding to micropore channels. After substituting Nmacro and Nmicro , the dimensionless forms of i i Eqs. (5) and (6) are obtained by introducing p~ i ¼ ðpi =pi0 Þ, y ¼t/t, n~ i ¼ ðni =Kpi0 Þ, x~ ¼ ðx=LÞ and y~ ¼ ðy=HÞ:

ei H2 @p~ i tD

micro

@y

ea H2 @p~ i tD

macro

@y

þð1ei Þ

¼ M2

The equilibrium data for CO2 on NaX at 298.15 K were taken from Hyun and Danner (1982). Although the actual isotherm is nonlinear, the simulations we performed were based on the linear part of the isotherm taken at lower pressures (up to 10 kPa) with the equilibrium constant, K¼ 30 mol/(kg bar). A two-step PSA cycle of adsorption followed by counter-current desorption was considered for process simulation and obtaining working capacities. Isotherm and adsorbent parameters are shown in Table 1, along with individual simulation parameters.

RTKH2 @n~ i @2 p~ @2 p~ i ¼ M 2 2i þ micro @y tD @x~ @y~ 2

@2 p~ i @2 p~ i þ @x~ 2 @y~ 2

ð13Þ

ð14Þ

In the above equations M is the density of macropores per metre of external surface given by M ¼H/L, which can be calculated once H is specified. 2.2. Numerical solution of the optimization problem Simulation of the numerical models described in the previous section was performed using COMSOL Multiphysics 4.1a. The Chemical Engineering Module was employed to solve the set of partial differential equations using Lagrange Quadratic Finite Element Method. The number of mesh elements was increased until no further change was observed in the results. For all adsorbent simulations considered in this study, mass transfer resistance at the external surface of the particle was neglected. This is typical of PSA operation. The gas system used for calculations was 10% CO2 in N2 and the adsorbent was taken to be NaX.

2.3. General properties of optimal structures Johannessen et al. (2007) used the concept of macropore density, M, as one of the most important properties of the optimal network of catalyst particles indicating the relative optimal channel diameter and wall thickness and showed that the optimal networks consist of a very large number of narrow channels separated by thin walls. We use the same concept here with the same purpose in order to illustrate the role of macropore density on adsorbent properties. Given three different geometries are under investigation (all with the same overall voidage of 0.6), a two-step PSA was simulated and the evolution of CO2 loading from start up to cyclic steady state (CSS) for a dimensionless half cycle time (defined as dimensionless time for adsorption and desorption steps) yc ¼0.1 is shown in Fig. 2a. The model considered the dimensionless loading cycling between a maximum upper value of 1.0 and 0.0 for a particle size (H) of 1.0  10  3 m while a macropore size of 0.3  10  6 m and a wall thickness of 0.7  10  6 m were considered for all configurations. The mean free path of gas was calculated from kinetic theory of gases to be approximately 1.0  10  6 m at the average pressure of the system. The second dimensionless half cycle time was yc ¼0.01 operated over the same boundary conditions and the result is presented in Fig. 2b. As seen here, for both yc, the branched structure exhibited better performance (higher working capacity) under cyclic adsorption process compared to other structures characterized by higher loading at the end of the feed step and lower loading at the end of

Fig. 4. Dimensionless CO2 loading as a function of dimensionless time at different mean free path values.

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

the evacuation step. In fact, at faster cycles with shorter residence time in which mass transfer resistance becomes more important, the branched structure shows its capability in overcoming diffusion limitations (in the macropores) more effectively than in slower cycles. The number of branches was varied so that all structures have similar porosity (0.3) and hence the same amount of adsorbent. In order to evaluate the impact of macropore density on the working capacity of adsorbent configurations, the WC is plotted at different value of M as shown in Fig. 3a. To vary macropore density, H was kept fixed while L was ranged from 1.0  10  6 to 1.0  10  3 m. As can be seen from this figure, for all structures, WC increases by increasing the density of network channels as expected until it reaches a maximum value and then starts to decrease gradually for large values of M. This trend indicates that a large number of narrow macropores separated with thin walls contributes to higher working capacity to certain values, beyond which no gain is obtained by reducing the macropore diameters and wall thickness. It is expected as diffusion limitations decrease in thin walls and a large number of macropores facilitates the gas

275

transport through adsorbent particle. It also can be observed that over the range of macropore density studied here (1–1000), a branched structure exhibits better performance characterized by higher value of WC. In order to better demonstrate the effect of M on PSA performance, in Fig. 3b the dimensionless CO2 loading over a parallel structure adsorbent as a function of dimensionless time is presented at different macropore densities. The same trend was observed for other structures. For M¼1, molecular diffusion is the dominant transport mechanism in the large pore network, however, by increasing the macropore density, pore sizes are reduced so that d0 becomes comparable to pore diameter and hence pore transport is dominated by Knudsen diffusion. Therefore, for the design of an adsorbent being used in a particular system with known mean free path, macropore size should be large enough (larger than d0) to ensure low resistance against gas transport. The mean free path d0 was varied (by changing PT and T) to evaluate the effect of transport mechanisms on optimization procedure. For a parallel structure with a porosity of 0.3, CO2 loading as a function of cycle time (with a half-cycle time of yc ¼0.1) was obtained for different values of d0 (i.e., 1.0  10  6,

1800 parallel grid branched

1600

WC, mol/m3

1400 1200 1000 800 600 400 200 0 0.0

0.2

0.4

0.6

0.8

1.0

εa

Fig. 5. (a) Objective function WC as a function of macroporosity ea and (b) dimensionless CO2 loading as a function of dimensionless time at different macroporosites at yc ¼0.1.

276

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

1.0  10  7, 1.0  10  8 m) and the results are presented in Fig. 4. The obtained trends indicate that reducing d0 gives rise to higher loading and higher working capacity as a result of enhanced macropore diffusion. Fig. 5a shows the objective function of structured adsorbents WC, as a function of macroporosity, ea. In order to vary macroporosity, d was varied at constant values of H and L. As expected, increasing the macroporosity resulted in increasing WC due to larger surface area available for adsorption, until a maximum value is reached. Thereafter, WC decreases remarkably as a result of reduction in the amount of active material per unit volume of structured adsorbent. As with WC–M curves, to illustrate the effect of ea on adsorbent performance in a PSA unit, the dimensionless CO2 loading as a function of dimensionless time is shown in Fig. 5b for parallel structure adsorbent with different macroporosities.

2.4. Branching in hierarchical structures Fig. 6a shows the variation of objective function by number of branches N in branched structure when M¼ 1. As expected, by introducing the branches between solid islands, the working capacity of the adsorbent increases due to the decrease in diffusional limitations in smaller islands. In Fig. 6b, the working capacity as a function of macropore density for structures with and without

branches is plotted. Both structures exhibit the same behaviour when M-N, providing that introducing branches into a structure with an infinite number of macropores having very thin walls is not beneficial any more. The advantage of branched structures with N-N over a parallel structure with N-0 is obvious when the density of macropore channels is low. These results indicate that a fractal pore structure is beneficial over other configurations when diffusion time constants are comparable to cycle time (longer characteristic diffusion length). 2.5. 1D effective model When the nanoporous solid parts are very thin with a finite wall thickness, the transport limitation is negligible within the nanoporous adsorbent and therefore, there will be no concentration gradients in the x-direction. In this case, the 2D slab model can be simplified into a 1D effective model by the following correlation:

ea @pi RT @t

þ ð1ea Þrs

@ni @N total ¼ i @t @y

ð15Þ

The total molar flux term Ntotal , is adopted from the Maxwell– i Stefan equation with two mechanisms for intrapellet mass transfer i.e., molecular and Knudsen diffusions. Assuming the macropore

1600 1400

WC, mol/m3

1200 1000 800 600 400 200 0

0

5

10

15

20

N 3000 2500

WC, mol/m3

2000 1500 1000

N=0 500

N→∞ 0

1

10

100

1000

M Fig. 6. Objective function WC as a function of (a) branch number N and (b) macropore density M when N-N and N-0.

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

dimension contains the limiting mass transport mechanism, the resulting form of the constitutive flux relationship is:

Nomenclature

Ntotal 1 @pi i ¼ RT @y De

d dx dy d0 De Dm Dka H K ‘ef f ‘ L M ni n~ i ni,ads ni,ads N Nmacro i Nmicro i Ntotal i pi p~ i pi0 PT R t T w wx WC x x~ y y~ yi y~ i

ð16Þ

In previous modelling studies published so far, the simplest and most commonly adopted approach for modelling diffusion and adsorption/desorption is to scale the diffusion coefficients by the equivalent voidage–tortuosity factor due to the lack of information regarding the ‘‘pathway’’ for gas transport within the adsorbent. In fact, the term tortuosity was employed to replace the curvature of diffusion path in studies where straight pathways were assumed (B ¼ ð‘ef f =‘Þ, where ‘ef f is effective mean free path in a porous medium and ‘ is the shortest distance in the same direction). In most studies, the tortuosity was not considered as a purely geometric factor, but dependent on pressure, temperature and diffusing species. In this study however, the principal approach is to relate the overall performance to pore structure by defining such pathways. Accordingly, the area averaged effective diffusivity is evaluated in following manner: De ¼ ea Dmacro þ ð1ea ÞDmicro and the boundary conditions are:   @pi ¼0 pi ðx,0Þ ¼ pi0 , @y y ¼ H

ð17Þ

ð18Þ

Johannessen et al. (2007) showed that 1D and 2D results are almost identical only when M-N and concluded that 1D effective model can be used to represent both 2D and 3D networks by relating the macroporosity to the channel diameter and the wall thickness in a proper way. As evident from Fig. 3a, the objective function (working capacity) evaluated by our 1D model is far less than values obtained by parallel structure but branched structures give a maximum WC much closer to 1D result. In practice, activated carbon adsorbents composed of graphitic layers with fast mass transfer kinetics may resemble the ideal structure.

3. Concluding remarks An optimization scheme which relates the geometrical properties to adsorptive performance of nanoporous adsorbents being used in cyclic adsorption processes was developed in this study. Based on this scheme, the pore space of the adsorbent particle was optimized so as to maximize its working capacity. Three different arrangements of macropore channels were considered and it was assumed that the optimal structure is a periodic assembly of a single, repeating unit. The gas transport in the channels was described by molecular and Knudsen diffusion mechanisms. Based on numerical results obtained from two-step PSA cycle simulations, the branched structure containing infinite branches represents near optimal structure. The optimal macroporosity of this structure is around 40–50%. The increase of channel density 10-fold leads to an increase in working capacity of three structures providing that both adsorbent amount and macropore size are at optimum condition. Simulating a two-step PSA cycle with different cycle times suggested that at ‘‘slow’’ cycles the difference between working capacities is not pronounced but at ‘‘fast’’ cycles the branched structures display higher working capacity. Introducing branches is only advantageous when macropore channel density is low and characteristic diffusion length is relatively long. In this study, the macropore channels with fixed cross section were considered, however, a more general case in which channel diameters are not position dependent, as with tree-shaped structures, should be considered in future optimization procedures.

277

macropore channel diameter, m diameter of main macropore, m diameter of macropore branches, m mean free path, m effective diffusivity, m2/s molecular diffusion coefficient, m2/s Knudsen diffusivity in macropores, m2/s height of repeating unit, m adsorption equilibrium constant, mol/kg bar effective mean free path in a porous medium, m shortest distance in the same direction of ‘ef f , m width of repeating unit, m macropore density moles of component i, mol/kg dimensionless moles of component i moles adsorbed during adsorption step, mol/kg moles desorbed during desorption step, mol/kg number of branches flux of component i in macropores, mol/m2 s flux of component i in micropores, mol/m2 s total flux of component i, mol/m2 s partial pressure of component i, bar dimensionless partial pressure of component i initial partial pressure of component i, bar total pressure, bar gas constant, 8.314 J/mol K time, s temperature, K wall thickness, m wall thickness, m working capacity, mol/m3 length in x-axis dimensionless length in x-axis length in y-axis dimensionless length in y-axis mole fraction of component i dimensionless mole fraction of component i

Greek letters

ea ex ey ei y yc

t B

macroporosity porosity of main macropore porosity of macropore branches microporosity dimensionless time dimensionless half cycle time cycle period, s tortuosity

Acknowledgement The authors would like to acknowledge the Monash Research Graduate School (MRGS) for providing scholarship support for Fateme Rezaei. References Beeckman, J.W., Hegedus, L.L., 1991. Design of monolith catalysts for power plant NOx emission control. Ind. Eng. Chem. Res. 30, 969–978. Coppens, M.O., Froment, G.F., 1997. The effectiveness of mass fractal catalysts. Fractals 5, 493–505. Gheorghiu, S., Coppens, M.O., 2004. Optimal bimodal pore networks for heterogeneous catalysis. AIChE J. 50, 812–820.

278

F. Rezaei, P.A. Webley / Chemical Engineering Science 69 (2012) 270–278

Hegedus, L.L., 1980. Catalyst pore structures by constrained nonlinear optimization. Ind. Eng. Chem. Prod. Res. Dev. 19, 533–537. Hyun, S.H., Danner, R.P., 1982. Equilibrium adsorption of ethane, ethylene, isobutane, CO2, and their binary mixtures on 13X molecular sieves. J. Chem. Eng. Data 27, 196–200. Johannessen, E., Wang, G., Coppens, M.O., 2007. Optimal distributor networks in porous catalyst pellets. I. Molecular diffusion. Ind. Eng. Chem. Res. 46, 4245–4256. Keil, F.J., 1999. Diffusion and reaction in porous networks. Catal. Today 53, 245–258. Keil, F.J., Rieckmann, C., 1994. Optimization of three-dimensional catalyst pore structures. Chem. Eng. Sci. 49, 4811–4822. Meyers, J.J., Liapis, A.I., 1998. Network modeling of the intraparticle convection and diffusion of molecules in porous particles packed in a chromatographic column. J. Chromatogr. A 827, 197–213. Meyers, J.J., Liapis, A.I., 1999. Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column. J. Chromatogr. A 852, 3–23. Meyers, J.J., Nahar, S., Ludlow, D.K., Liapis, A.I., 2001. Determination of the pore connectivity and pore size distribution and pore spatial distribution of porous chromatographic particles from nitrogen sorption measurements and pore network modelling theory. J. Chromatogr. A 907, 57–71. Morbidelli, M., Gavriilidis, A., Varma, A., 2001. Catalyst design: optimal distribution of catalyst in pellets, reactors, and membranes. Cambridge Ser. Chem. Eng. N.Y., 2001.

Petropoulos, J.H., 1991. Network model investigation of gas transport in bidisperse porous adsorbents. Ind. Eng. Chem. Res. 30, 1281–1289. Rieckmann, C., Keil, F.J., 1999. Simulation and experiment of multicomponent diffusion and reaction in three-dimensional networks. Chem. Eng. Sci. 54, 3485–3493. Sahimi, M., Gavalas, G.R., Tsotsis, T.T., 1990. Statistical and continuum models of fluid–solid reactions in porous media. Chem. Eng. Sci. 45, 1443–1502. Sahimi, M., 1993. Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65, 1393–1534. Sahimi, M., 1995. Flow and Transport in Porous Media and Fractured Rock. VCH, Weinheim, Germany. Satik, C., Yortsos, Y.C., 1996. A pore-network study of bubble growth in porous media driven by heat transfer. J. Heat Transfer 118, 455–462. Wang, G., Coppens, M.O., 2008. Calculation of the optimal macropore size in nanoporous catalysts and its application to DeNOx catalysis. Ind. Eng. Chem. Res. 47, 3847–3855. Wang, G., Johannessen, E., Kleijn, C.R., de Leeuw, S.W., Coppens, M.O., 2007. Optimizing transport in nanostructured catalysts: a computational study. Chem. Eng. Sci. 62, 5110–5116. Wheeler, A., 1951. Reaction rates and selectivity in catalyst pores. Adv. Catal. 3, 249–275.