sepiolite catalyst for SCR of NOx

8:EWRONMEML ELSiiVItiR

Applied Catalysis B: Environmental 8 (1996) 299-314

Measurement of the effective diffusivity for a vanadia-tungsta-titania/sepiolite catalyst for SCR of NO, A. Santos a, A. Bahamonde b, P. Avila by*,F. Garcia-Ochoa a ’ Deparramenro de Ingenieria Q&mica, Facultad de CC. Quimicas, Universidad Complutense, 28040 Madrid, Spain b Instituto de Cata’lisisy Petroleoquimica, CSIC, Campus Universidad Authwma, Canroblanco, 28049 Madrid, Spain

Received 5 July 1995; revised 11 September 1995; accepted 12 September 1995

Abstract

Experimental measurement of the effective diffusion coefficient for a vanadia-tungstatitania/sepiolite catalyst, as employed for SCR of NO,, was carried out using a chromatographic technique in a packed column (under inert conditions) employing helium as carrier and nitrogen, argon and oxygen as tracers. Measurements were carried out at temperatures 298 and 473 K with particle sizes 0.4 and 0.5 mm. The values obtained for 0, are from 0.6. 10m6 to 0.94. 10m6 for the gas tracers and temperature range employed. Adsorption constant for the gas tracers above quoted have been also determined giving values of K, around 1.3, thus, a light adsorption strength existed. Because 0, depends on the gas temperature and composition, different structure models for the solid have been applied to calculate from 0, the corresponding structural parameters: T (in pore models) or Z (in percolation model), which only depends on the solid structure and can be employed for other gases and temperatures. The T values obtained from the bundle pore model (around 3) are smaller than calculated applying the cross linked pore model (7 between 5 and 6). The coordination number, Z, has a meaningless value when catalyst is analyzed as unimodal (Z < 3). On the other hand, if the experimental pore size distribution was taken into account Z yields appropriate values were between 5 and 6. Keywords: Diffusion coeffkient (effective); Selective catalytic reduction; Monolithic catalyst; V-W-Ti/Sep iolite; Pore model; Percolation model

’ Corresponding author. Fax. ( + 34-I) 5854760, e-mail [email protected]. 0926-3373/%/%15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0926-3373(95)00056-9

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1. Introduction Selective catalytic reduction (SCR) is a commonly used process for the removal of NO, from power plant stack gas, which is widely used all over the world due to its efficiency, selectivity and economy [l]. Commercial SCR honeycomb shaped catalyst currently used for the treatment of gaseous effluents from stationary power plants are basically prepared by supporting vanadium and tungsten oxides on high surface area titania [l-4]. However, in the commercial units of SCR of NO with NH,, the high activity of these materials must be balanced with the drawback of their low abrasion resistance. Attempts were made to minimize this problem by substituting the titania for other supports such as silica or alumina. The results were not considered satisfactory because large V,O, particles were formed when supported on silica [5] and the vanadia/alumina catalyst could not resist the presence of sulphur dioxide in the exhaust gases [6]. Another approach to this problem has been the utilization of binders such as strong organic or mineral acids [7] and different silicates [8] as mechanical promoters. Two decades ago, the use of monolithic catalysts for stationary source emission control was initiated [9]. The development of this type of catalysts was due to the fact that this configuration offered certain advantages. The main features of monolithic catalysts are: large external surface, low pressure drop, uniform flow, low axial dispersion and low radial heat flow (adiabatic) [9]. If a catalytic reaction is fundamentally considered as a surface phenomenon, the higher surface area per unit volume of the monolith in comparison to the pellet considerably reduces the catalyst volume. The low pressure drop in monoliths is also important in emission control applications where the reactions are usually carried out at high temperatures and very high space velocities. On the other hand, it is well known that the performance of commercial SCR catalysts is strongly limited by intraporous diffusion [ 10,111. For this reason, the introduction of macropores into the morphology of this type of catalysts [lo] is implemented in the latest generation of commercial SCR catalysts [ 121. Svachula et al. [12] found that the catalyst effectiveness factor could be increased by a factor 2 when the pore size distribution was bimodal, resulting from the inclusion of a fraction of macropores in the original structure of the unimodal catalyst. Thus, it is both important and necessary to measure and be able to predict the effective diffusion coefficient of the reactants in the SCR process in order to develop catalysts with improved NO, removal capacities. Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusion coefficient, 0,. If accurate prediction of the 0, value is not yet available, it must be determined experimentally. Among the methods developed to obtain the 0, value the chromatographic technique has been the most popular in the last two decades [13-171. Basically, the technique consists of injecting a pulse of diffusing component into the inlet

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stream to the column packed with the catalyst and recording the effluent pulse. From the analysis of such a signal, the effective diffusitivity can be determined. Because of the 0, is affected by the composition, temperature and other properties of the fluid, it is more suitable to calculate a parameter from 0, which only depends on the porous solid structure (such as the tortuosity, T, from the pore model or the coordination number, and Z, from the percolation model). The aim of this work was to determine experimentally the effective diffusion coefficient for a catalyst of the same composition and morphology as that of a commercial monolith for the SCR of NO, [ 181 and by applying the pore models and percolation model to obtain the structural parameter, T or Z, respectively.

2. Effective diffusion coeffkient evaluation by chromatographic measurements The flow inside the packed column was assumed to be isobaric and isothermal. Under these conditions, the model of Kubin [19] and Kucera [20] can be applied to describe the mass transport inside the solid particle and through the packed bed. Since the tracer concentration in the gas-phase is small, the physical adsorption rate can be considered as first order. In this way, the Kubin-Kucera [19,20] model takes into account different phenomena in the pellet, such as adsorption and inter- and intra-phase diffusion, and axial dispersion in the bed, introducing the characteristic parameters: k,, O,, K, and DaX,respectively, in the differential equations describing the mass transport of the tracer gas in the bulk gas phase and in the pores, being the adsorption velocity of first order. By application of the Laplace transform to the differential equations of the Kubin 1191and Kucera [20] models, the moments of the tracer concentration-time curve can be calculated. Assuming that the adsorption-desorption equilibrium was reached, the first two moments of the tracer concentration-time curve were determined as in a previous work [21]. Then, the adsorption equilibrium constant, K,, was calculated by linear regression of p, vs. L/ui [14]:

2

CL1

1+

I-% [

(1- EL) 41 + 44)

I

EL

and the HEPT value can be calculated as: HEPT_

(W/-W

=- oZL

(2) 2 111 CL: with the relationship between HEFT and ui, similar to that found by Van Deemter [22]: HEFT=&

+B+Gu, ‘i

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where the coefficient G of Eq. (3) [14] is: 2

EL

15

R'P1

(l-EL)

G=

2 i

5 o+-e

(4) v,

1

I+$

i

0

I

The effective diffusivity, O,, was obtained from G values, which were achieved by Eqs. (2) and (31, from HEPT values at different values of Ui. For the 0, calculation, from G values using Eq. (41, it was necessary to estimate k, values by some of the equations given in the literature. In this work, the Eq. proposed by Wakao [23] was:

2kfRp -= 2 00 + 1 45Re’/2Sc’/3 D, * * p

forRe

P

< 100

(5)

Mass transport inside the particle has been usually described by applying the Fick equation where the effective diffusivity can be calculated as: D,=of

(6)

where 5 considering the transition regime, both molecular and Knudsen diffusion, can be calculated as: 1

1

qCrp>

DiK(rp)

-=

1 +

Tj

(7)

Parameter f in Eq. (6) is a correction factor to account for the complex internal structure of the solid. Different levels of complexity have been used to describe the orientation, size and interconnection of the pores. Among the models proposed in the literature two main groups can be distinguished: (a) Models based on capillary network range from a simple bundle of cylindrical capillaries of uniform cross section [24,25] to the more sophisticated random pore model or cross-linked pore model [26]. In the first model, the value of rp can be considered as an average of the values from different pore sizes evaluated as:

therefore 5 is obtained from Eq. (7) with the rp value from Eq. (8) to obtain the Knudsen diffusivity value.

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In the second case, w&en a more realistic pore size distribution, fir,>, considered, the value of D is obtained as:

/0

is

\

(

1

x

+

303

1 \-+D.., DiK(r,)

1

f(Qdrp

1~

where AT,> is a standard pore radius distribution which can be obtained from the gradient of the cumulative pore volume curve. Parameter f in Eq. (6) includes a tortuosity factor required to match the predictions to experimental data [26-281:

(b) The second main group are the stochastic models which consider that the catalysts are sufficiently disordered that a statistical approach can provide an accurate qualitative description of their connectivity and associated transport properties. In this category the percolation concepts has been widely applied [29] to describe the porous matrix and associated transport properties [30-321. In the percolation theory the accessible porosity, 6 A, is defined as the likelihood that any pore region is sufficiently well connected to the rest of those available for transport, and total porosity, E, is the sum of accessible porosity and porosity in isolated pores, &I. Evaluation of E A, 8’ and f can be made, employing the Bethe network as a model of pore space topology [33,34,3 I] (see Appendix). In the Bethe network a coordination number, Z, is defined as the number of bonds connected to a hypothetical site located at the centre of a branching pore, averaged over all sites.

3. Experimental 3.1. Catalyst preparation and characterization studies The catalyst used in this study was based on a mixture of metallic oxides (V205, WO,, TiO,) with an agglomerating agent based on a magnesium silicate (sepiolite), which has been described elsewhere [18]. The structure of this type of catalyst resembles that of a sepiolite, but with particles of TiO, impregnated with V,O, and WO,, scattered among the fibres of the sepiolite that build the skeleton of the catalyst [8]. The chemical composition of this catalyst by XRP is given in Table 1.

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Table 1 Morphological characterics and chemical composition of the catalyst Morphological characteristics Ss (BET), (rn’ g- ‘1 rm (A>

93

rM 01) VP, cm3 kg- ‘) VP,,,,(m3 kg-‘) VP,,
260

125 0.56.10-3

0.2856. lo- 3 0.2744. lo- 3 1100 0.616 43.21 4.22 0.16 27.2 9.3 2.6 1.9 21.41

The surface area values were measured by nitrogen adsorption using the BET technique in a Micromeritics 1310 ASAP and the pore size distribution by mercury-intrusion in a Micromeritics Poresizer 9320. The morphological properties of this catalyst are presented in Table 1 and the cumulative and incremental pore volumes of the studied catalyst are shown in Fig. 1. 3.2. Set-up for effective difision

coeficient measurements

The chromatographic technique in a packed column has been applied. A copper column 4.36 m in length and 0.0107 m internal diameter filled with the catalyst pellets was used inside an oven (the bed porosity was between 0.48 and 0.3

b nE 0.5: 0

'; OI

: :

g-0.4a : ; 0.3 y B t al 0.2.2 m -

g

:

s

:

ifi

IO.1

i E

ZO.l3

%

:0.2

P t

0.0. 0.001

0.01

0.1

1

-0

Pore Diameter, pm

Fig. 1. Cumulative and incremental pore volume of the SCR catalyst.

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Table 2 Conditions employed in the experimental measurements of 0, Tracer gas

Temperature (K)

d,. IO3 (m)

uI. 10’ (m s- ‘)

EL

Argon

298 473 298 473 298 473

5 4 5 4 5 4

4-32 4-32 4-32 4-32 4-32 4-32

0.45 0.48 0.45 0.48 0.45 0.48

Oxygen Nitrogen

0.451, with a Hewlett-Packard 5710-A chromatograph utilizing on line TC detector. A six way valve permitted an injection of 0.5 . low6 m3 of nitrogen, argon or oxygen as tracer into the carrier gas stream (helium). The experimental conditions of the chromatographic runs are listed in Table 2 and the scheme of the set-up is shown in Fig. 2. Experiments were carried out at 298 and 473 K and 0.93 atm. (absolute), with a carrier flow ranging from 1.7. 10e6 to 1.7. 10e5 m3 s-’ (at 298 K) under almost isobaric conditions. A portion of the column outlet stream was vented to carry to the detector a flow of between 0.85. 10e6 and 1.7. 10m6 m3 s-l. Because of the high linear velocities employed, a calming section consisting of a tube of 12 m in length, and 7 + 10m3 m internal diameter, was placed before the column to preheat the carrier gas to the oven temperature.

4. Experimental results For each run, carried out at a different gas carrier velocity, the signal obtained in the TC detector was stored in an on-line computer and values of the first and

l-0 Fl3

EXIT

Fig. 2. Experimental set-up for D, measurements.

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10

15

20

ui.102

25

30

m/s

Fig. 3. Representative results of HEPT vs. ui.

second central moments, p, and CT*, were obtained numerically from the effluent concentration curve. The HEFT values were calculated using Eq. (2). Some representative results are presented in Fig. 3. A relative influence of axial dispersion is found at the lower gas linear velocity range used, as can be seen from the shape of HEPT vs. ui values in this zone. When the velocity was increased, HEFT became more linear with ui, the internal diffusion being the main mass transport resistance to consider, at least in the range of velocities employed here. The K, values were calculated from ,u,. The values of the parameter G was obtained, according to Eq. (3), using non-linear regression [34], estimating k, X R, values by Eq. (5) and then calculating 0, according to Eq. (4). Experimental results obtained for K, and 0, are given in Table 3. As can be seen, a small value of the adsorption constant for all tracers employed was obtained that means a light adsorption strength. The almost negligible change of K, with the temperature in the temperature range studied indicates a low adsorption entalphy value. Experimental 0, values were interpreted according to Eq. (6) by means of a model of the porous solid structure to reach the corresponding structural parameters, T or 2. The models considered were the

Table 3 Experimental D, and KA values obtained T

(K)

Argon D; IO6 (m*

298

0.60

473

0.94

Oxygen

s- ‘)

K,

D; IO6 (m*

1.12 1.23

0.58 0.86

Nitrogen

s- ‘)

K,

D; 10” (m* s- I)

K,

1.22 1.38

0.75 0.94

1.33 1.23

A. Santos et al./Applied Table 4 Results applying

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the bundle pore model

Gas

T (K)

D; 1O-2 cm2 s-‘1

D,, lo4 cm2 s-‘)

DAB. IO4 (m2s- ‘1

5.106 (m2s- ‘)

f

7=&/f

Ar

298 473 298 473 298 473

0.60 0.94 0.58 0.86 0.75 0.94

3.2 4.0 3.6 4.5 3.8 4.8

0.78 1.56 0.80 1.60 0.75 1.49

3.1 4.0 3.4 4.4 3.6 4.6

0.193 0.235 0.171 0.195 0.208 0.204

3.18 2.62 3.61 3.15 2.96 3.01

0, N2

three described above: the bundle pore model, the cross-linked pore model and the percolation model, as representative of the two groups quoted above. 4.1. Bundle pore model According to this model, 5 is evaluated by means of Eq. (7) and rp the mean pore radius obtained by Eq. (8) using the experimental values of Ss and VP listed in Table 1. Values of molecular and Knudsen diffusion appearing in Eq. (7) were estimated for the He-tracers system at 298 and 473 k and 0.93 atm yielding the values given in Table 4. Estimating D according to the procedure described above, the parameter f was calculated from measured 0, values using Eq. (61, and the tortuosity factor, T, obtained applying Eq. (lo), by substituting E for the corresponding experimental porosity data listed in Table 1. The results obtained for 0, f and T are presented in Table 4. 4.2. Cross-linked

pore model

In this model, the mean coefficient 5 was evaluated from Eq. (91, integrating the D coefficient (which depends on the knudsen diffusivity, thus being a function of the pore radius, and also depending on the molecular diffusion) in the whole pore radius range, introducing the corresponding weight of the pore size distribution function - fl rp)- found experimentally by Hg porosimetry. For each pair of 0, and 5 data, f and T values were calculated, as in the bundle pore model, by means of Eq. (6) and (10) respectively. The results obtained are shown in Table 5. 4.3. Percolation

model

In the same way as in the cross-linked calculated from Eq. (91, after introduction

pore model, coefficient 5 of the corresponding pore

was size

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308 Table 5 Results applying Gas Ar 0, N2

Catalysis B: Environmental 8 (1996) 299-314

the cross linked pore model

T (K) 298 413 298 473 298 473

0; lo6 Cm* s- ‘) 0.60 0.94 0.58 0.86 0.75 0.94

E-106 5.490 7.158 6.084 7.955 6.416 8.424

f

r=.?/f

0.1092 0.1313 0.0950 0.1080 0.1170 0.1110

5.64 4.69 6.46 5.69 5.27 5.52

distribution, found experimentally. The f values calculated from Eq. (6) are listed in Table 6 as fexperim. In this model, the accessible porosity (which would correspond to that measured experimentally) and total porosity (which is not a directly measured variable) are distinguished, both porosities being connected by means of the percolation number value, 2. Fixing a coordination number value, 2, and with the experimental accessible porosity listed in Table 1 (E = 0.616) the total porosity (E) was obtained by means of Eqs. (A-2) and (A-3). Then the f parameter is calculated by Eqs. (A-5) or (A-6). Th’is procedure is repeated for some 2 values and the results

Table 6 Results applying Unimodal Gas Ar 0, N2

Bimodal Gas Ar

02

N2

the percolation

model

E* = 0.62 T (K) 298 473 298 473 298 473

folperim 0.1092 0.1313 0.0950 0.1080 0.1170 0.1110

2 3 4 5 6 7 8

fcalcularcd

2 5 6 5 6 5 6 5 6 5 6 5 6

D . lo6 m’s_ o.e49 0.66 0.64 0.86 0.55 0.73 0.71 0.95 0.58 0.77 0.76 1.01

0.24 0.35 0.43 0.47 0.50 0.54

.s&,, = 0.31 T (K) 298

D. lo6 m’s_ OJXI

473

0.94

298

0.58

473

0.86

298

0.75

473

0.94

’exper.

’calculated

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1

0.1

Fig. 4. Values of f for E A = 0.62 and 0.3 1.

obtained for f vs. Z (at E~ = 0.62) are shown in Fig. 4 and in Table 6. Comparing the experimental f values with those predicted by changing Z a meaningless coordination number value Z < 3 was obtained, as seen in Table 6 and Fig. 4. Thus, the percolation model could not be applied in this manner for this catalyst. A different approach can be made considering a bimodal pore distribution, that was experimentally found for the catalyst under study (see Fig. 1). In this case, two effective diffusivities can be considered, one for each pore Tegion: D,, for the micropore zone (with an average micropore radius of 125 A) and DeM for the meso-macro pore region (with an average radius about 260 A>. Applying the approach of Whitaker [35] the global effective diffusivity was calculated as a function of De,,, and De,, according to:

(11) with the local diffusion coefficient for each zone calculated as follows:

fy(++-p

Y =m,M

(12)

where f,,
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The global 0, value was then calculated using Eq. (1 l), where local diffusivities were estimated by means of Eq. (12) and the same value for the coordination number 2 of micro and macropore regions was assumed. These predicted 0, values are compared with the experimental 0, data and the coordination number Z matching both sets of data. As can be seen from Table 6, a value of Z between 5 and 6 suits quite adequately the predicted and experimental results.

5. Discussion From the 0, values obtained with the vanadia-tungsta-titania/sepiolite catalyst by means of the chromatographic technique (unsteady-state measurements) some considerations can be outlined. When different pore models are applied to analyze the 0, value, the tortuosity factor is calculated, which is the match parameter of the considered models. Lower values of r were obtained from bundle pore model (around 3) than from the cross linked pore model (7 between 5 and 6). This could be explained by the unrealistic mean pore size assumed for the bundle pore I_aodel,which was calculated according to Eq. (8), yielding a value of r of 120 A. This value is very close to the micropore mean radius experimentally found by Hg porosimetry, thus, yielding an inadequate evaluation of transport in the macropore zone. Because of the low r value, a small 5 was achieved and a smaller tortuosity value was obtained. This unrealistic description of bimodal catalysts by the bundle pore model in the evaluation of transport parameters was also detected by Wang and Smith [37] and Garcia-Ochoa and Santos [21]. The tortuosity values obtained in this work, employing both pore models, are larger than those calculated by Bee&man 1381for a Titania supported catalyst. Beeckman [38] found 7 close to 2, employing a steady state measurement technique. Some authors [ 141 have found that lower values of T result when steady state measurements are used to calculate 0,. In those conditions, this could be due to the smaller contribution of micropores under steady state conditions than under unsteady state conditions. Indeed, the different values obtained for r can be also due to the different properties of the catalyst under study. Other authors [39] have recently calculated r under reaction conditions for a Pt-Rh/y-Al,O, mon$ithic catalyst. In this case the catalyst had a smaller mean pore radius (r = 35 A) and a value of 7 = 1.5. A meaningless value of the percolation number (Z < 3) was obtained when the catalyst was taken as unimodal. However, Z values of between 5 and 6 were achieved if the bimodality was included. This effect was also detected by Santos [4O] analyzing other bimodal catalysts. Using the tortuosity factors calculated from experimental 0, values obtained in this work and taking into account the kinetic model previously determined

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[41] for the SCR of NO, with the catalyst under study, the simulation of the monolithic reactor was accomplished [42], considering external and internal mass transfer limitations (at 623 K, linear velocity of 0.12 m s- ‘; NOJNH 3 ratio = 1, [NO,linlet = 1000 ppm; Ozinlet= 3 vol.-%). It was concluded that the NO concentration reached an almost zero value at wall thicknesses greater than 0.2 mm with an effectiveness factor lower than 0.1, so there were no differences in NO conversion when the wall thickness was larger than this value.

6. Summary Effective diffusivity coefficient has been calculated for a vanadia-tungstatitania/sepiolite catalyst by means of an unsteady-state technique employing He-Ar, He-O, and He-N, systems. Tortuosity factors (in pore models) and coordination number (in percolation model) have been subsequently calculated from these 0, values. They were obtained smaller tortuosity factors with the bundle pore model (T = ca. 3) than for the cross-linked pore model (T = ca. 5.5) because of the non realistic mean pore radius was assumed by the fist model. It has been concluded from the analysis employing with the percolation model, that when the catalyst was bidisperse, as here, the bimodality should be included in the model to obtain appropriate coordination numbers. Under this assumption the Z values obtained are between 5 and 6. In any case, the structural parameter could be contemplated as a merely matching factor, hence, it is suitable for calculating the 0, value when it is used with the solid model from r or Z was obtained.

7. Nomenclature A: B:

0: D(r,): De: Dij: Di,: Di,: f:

fcr,):

Parameter in Eq. (3) Parameter in Eq. (3) Average composite local diffusivity value cm2 s- ‘1. Composite diffusivity value cm2 s- ‘1. Effective diffusivity (m2 s- ‘1. Molecular diffusivity cm2 s- ’>. Knudsen diffusivity (m* s- ‘1. Local diffusion coefficient (m2 s- ‘1. Structural correction factor in Eq. (61, defined by Eqs. (101, (A-5 or A-6). Normalized pore radius distribution function (m-l).

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312

G: HEPT: K,: k,:

L: P:

R,: Re,: 2: ss: T: Ui: VP: 2:

Catalysis B: Environmental 8 (1996) 299-314

Parameter in Eq. (3), defined by Eq. (41 Equivalent height of theoretical plate (m) Adsorption constant. Interphase mass transfer coefficient (m s-i). Chromatographic packed column length cm>. Pressure (atm). Particle Radius (m). Particle Reynolds number. Pore radius (m, A>. Schmidt Number Specific surface (m2 g-i). Temperature (K). Interstitial velocity (m s- ‘> Pore volume (m3 g-i) Coordination number

8. Greek symbols 8:

&*: & .fi

;k; I4.2:

Pp: u2: 7:

Total porosity. Accessible porosity. Percolation threshold. Isolated porosity. Bed porosity. Root of polynomium in Eq. (A-3). First and second moments of tracer responses, (s, s*, respectively). Pellet density (g mw3). Second central moment or variance of tracer response (s*). Tortuosity factor.

9. Subindexes a: f: *. 1. j b: M:

m: PI

Related to adsorption in catalyst surface. Relative to interphase. for compound i. for compound j. Relative to the bed. i Relative to macropores. Relative to micropores. For pellet or pore.

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Appendix

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313

A

Percolation threshold, accessible and isolated porosity: cc, .sA and E’, respectively, can be calculated in the percolation model according to the following equations:

(A-1) (A-2) with CR (1 - ER)(Z-2) _ C( 1 _ E)(z-2) = 0

(A-3) &I =&-&A (A-4) If the porosity value is close to cc, then f can be obtained as proposed by Stimchombe [36]: f= l.522(zl):(C- EC)2 (A-5) (Z- 2) while when porosity is in the far-percolation region parameter f can be calculated as indicated by Stimchombe [36]:

(Z- 1) f=-

(z_

2)

(A-6)

c;Gs

where: G,=

-(c-

&)

G, =0 8 - EC)(l

- &)

c 3

‘1;

G,- - -(&-

EC)*(l

-

&)[&(2&_ 1) +2(&-

EC)(l - E)] (A-7)

c4

G4 = --+ (1; +(&-

+

-

Ec)*(l - 4[3(1-

C)‘(.s-

EC) - 241 - e)

&“)(l - 3E+ 3E2) + 5(&_ E’)2(1 - &)

$(e-Ey3(l - e)‘]

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