Transport phenomena and chemical reactions in porous catalysts for multicomponent and multireaction systems

An Invited Paper-to

mark twenty years of publication of the journaI Zum 20. Jahrestag dieser Zeitschrift erbetenes Manuskript

229

Transport Phenomena and Chemical Reactions in Porous Catalysts for Multicomponent and Multireaction Systems Transportvorg5nge und chemische Reaktionen in poriisen Katalysatoren ftir Mehrkompohenten und Mehrfachreaktionssysteme ANDRZEJ

BURGHARDT

Polish Academy

of Sciences, Institute

of Chemical Engineering,

Gliwice (Poland)

Abstract The current studies concerning the interactions between chemical and physical phenomena in a porous catalyst pellet have been reviewed, with particular attention given to the following problems: (1) discussion of the basic laws and assumptions describing multicomponent mass transport in capillaries, and their use in building the models of mass transfer in porous media; (2) presentation of the physical principles of measurement techniques designed to determine the parameters of the models and to relate them both to the properties of the transferring species and to the porous structure of a peliet; (3) theoretical analysis of the mass and energy transport in a porous catalyst pellet for a multicomponent and multireaction system and the experimentelle iiberpriifung der Theorie. Kurzfassung Es wird iiber den Stand der Kenntnisse iiber die Zusammenhange zwischen chemischen und physikalischen Vorglngen in porijsen Katalysatorpartikeln berichtet, wobei folgende Punkte besonders behandelt werden: (1) die Diskussion der Grundgesetze und der Annahmen iiber den Mehrkomponententransport in Kapillaren und ihre Verwendung bei der Erarbeitung von Modellen iiber den Stofftransport in poriisen Medien; (2) die Darstellung der physikalischen Prinzipien dcr Messtechniken, die zur Bestimmung von Parametern fiir die Modelle entwickelt wurden und ihre Beziehung sowohl zu den Eigenschaften der transportierten Spezies als such zur Struktur der poriisen Partikel; (3) die theoretische Analyse von Stoff- und Energietransport in einer poriisen Katalysatorpartikel fiir Mehrkomponentenreaktionen und die experimentelle iiberpriifung der Theorie. 1. Introduction This review presents the current studies concerning the interactions between the physical and chemical phenomena occurring in a single porous catalyst pellet. Much attention has been given to this problem during the last two decades, and a number of papers have been published dealing both with theoretical analyses and with experiments. The major part of this work was summarized in an excellent, two-part monograph by Aris [ 1] : The Mathematical Theory p_+’Dijyiitsion and Reaction in Permeable Catalysts. The continuing avalanche of works and articles persuaded Carberry and Butt [2] to suggest a moratorium on the studies concerning the steady state effectiveness factor of a catalyst pellet. Nevertheless, despite the apparent abundance of information and data related to this problem, the mounting doubts and difficulties in explaining certain 0255s2701/86/%3.50

observations fostered, and did not hamper, related works, giving them quite a new direction. Thus, in the present review, I will try to prove that the actual state of knowledge concerning these phenomena is far from being as sufficient and complete as it is suggested in some papers. A combined process of chemical reaction and mass transfer of species taking part in this reaction is described by the following set of differential equations: Xi at

+ div Ni = v,(C.

T)

i= 1,2, . . ..n

(1)

There exists a number of solutions to the above equations [ 1, 3,4]. However, the majority of them is based on simplifying assumptions, leading to a mathematical description which is remote from the real picture of the process. In the above analysis, the mass flux of a species is usually defined as

Chem. Eng. Process., 21 (1986) 229-244

Q Elscvicr Sequoia/Printed

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230

Ni = -iIi

grad Ci

(2)

which is valid only for the binary equimolar diffusion or for the limiting case when the pore diameters are comparable with the mean free path lengths in the gas mixture (Knudsen diffusion). Moreover, the reaction rate is usually assumed to depend on the concentration of only one component, which further limits the problem to a simple isomerization reaction A f R. All these assumptions reduce eqns. (1) to only one differential equation, thus rendering the solution much easier to obtain. Tn industrial practice, the reaction mixture is very seldom binary, and the reaction rate depends on concentrations of all species taking part in the reaction. Consequently, the individual mass flux cannot be expressed using a relation as simple as eqn. (2). and has to incorporate both the interactions of all diffusing species and the possible convective flux resulting from the increase of the number of moles in a reaction (which, in turn, leads to the appearance of a pressure gradient in the system). What is more, in industrial catalysts a multicomponent mixture diffuses through a porous structure composed of irregular channels of various sizes, ranging from micropores, of dimensions much smaller than mean free path lenths of the gas mixture, to macropores. The relation defining the mass flux of a species in a catalyst pellet should remain valid over the whole range between the Knudsen diffusion in micropores and the molecular diffusion in large pores. Thus, the following problems of considerable practical importance arose in recent years: (1) to determine a relationship defining the mass flux for a multicomponent mixture in porous media; (2) to work out measurement techniques enabling one to determine the parameters of the models of mass transfer and to relate them to the properties of the diffusing species and to the porous structure of a pellet; (3) to develop methods of solving the set of differential equations describing the mass and energy balances of the processes occurring in porous catalysts for multicomponent and multireaction systems.

2. Models

of the mass transport

in porous

media

2.1. Mass flux in capillaries

It is commonly assumed that the full success of any model of a process or phenomenon lies in its ability to correctly predict the already known experimental and theoretical results. Because the conditions during the fluid flow in capillaries give no rise to turbulence, there exist three main mechanisms of isothermal transport of nonadsorbing gas in capillaries [S]. ( 1) Ej&sion now called free-molecule or Knudsen flow. In this case the pressure is so low or the capillary diameter so small that collisions between molecules can be neglected compared to collisions of molecules with the walls of the capillary. A species loses its momentum

transferring it directly to the wall as a result of moleculewall collisions. (2) Viscous (creeping)flow, also called pressure transpiration, in which the gas acts as a continuum fluid, the flow of which is caused by a pressure gradient. Here the pressure is high enough so that molecule-molecule collisions dominate over molecule-wall collisions. Thus the loss of momentum takes place by indirect transfer to the wall via a sequence of intermolecular collisions terminating in a molecule-wall collision. (3) Bulk diffusion, in which different species move under the influence of composition gradients. This is still a continuum phenomenon in the sense that molecule-molecule collisions dominate over molecule-wall collisions and momentum is lost as a consequence of collisions between pairs of unlike molecules. It should be emphasized here that all three mechanisms were described by Graham in his paper [6] (1863) in which he summarized and discussed his earlier experimental results. Furthermore, any model of mass transport in capillaries should correctly predict the known experimental results. The conclusion reached by Graham [7], and rediscovered in 1953 by Hoogschagen [S! 91, was that for steady state isobaric conditions the ratio of the fluxes of diffusing gases is given by the relation

(3) NZ

Ml

which remains valid even in conditions where bulk diffusion in capillaries is a dominant means of transport. A combination of the three basic mechanisms of mass transport produces interesting phenomena discovered experimentally. For instance, the combination of viscous flow and free-molecule (Knudsen) flow leads to the phenomenon known as ‘viscous slip’. This slipping of a gas over a solid surface was discovered experimentally by Kundt and Warburg [lo] Another combination is that between viscous flow and bulk diffusion, which gives rise to the phenomenon known as ‘diffusive slip’. first discussed in detail by Kramers and Kistemaker [ 1 I]. It is important to realize that the coupling between viscous flow and diffusion is much more common than is generally realized, for it occurs in almost all experimental arrangements for studying bulk diffusion where a simultaneous flow of the gas mixture is induced [I?]. Thus, to develop a model of mass transport in capillaries over a wide range of parameters we have at our disposal the relations defining the mass fluxes for the three basic transport mechanisms which, in a sense, may be regarded as limiting cases, For suitably low pressures or small capillary diameters we obtain Knudsen’s relations [13] J,=

_D_ i% !E RT do

where the Knudsen

(4) diffusion

coefficient

is given by

(5)

231

KO is a constant characteristic geometry. For long cylindrical

of the channel scale and capillaries of radius r

K. = 2r/3

(6)

For very small channel diameters which characterize capillaries and porous media, viscous flow of the gas mixture takes place, with the viscosity of gas influencing considerably the rate of flow. Thus, the total flux of the gas mixture is given by BOP _Q NV= - pRT dz where B. is a factor characteristic of the scaie and geometry of the pore structure, known as the permeability coefficient of the porous medium. For a cylindrical capillary of radius r we have the classical Poiseuille solution [ 14 ] Bo=r2/8 For a mixture of gases the viscous flux of species i is proportional to its mole fraction, since the flow is non.* separable,

:

Niv = xiNv

(9)

However, despite all the appearances, it is not &y to define precisely the bulk diffusion flux in ca$illal’i& The most logical approach would be to evaluate’ir’for isobaric conditions (grad p = 0), thus eliminati&‘$_tbe viscous flow. Unfortunately, the experiments carried out -by Graham [7], Kramers and Kistemaker [ 1 l] and McCarty and Mason [ 151 for such capillaries in which the’bulk diffusion should completely dominate all other types of transport showed that the countercurrent equimolar diffusion of two gases always causes the pressure gradient to develop. Hence, the most logically built relations defining the molecular diffusion fluxes are those of Maxwell [ 161 and Stefan [17], which are based on the differences between the diffusion velocities of individual species:

n xjzi - xizj -- Qi =RTZ d_z q)”

j=l

i=l,2,...,n

the relations between the diffusive fluxes and the gradients of the partial pressures of the species. As is well known, these researchers based their argument on momentum balances, or rather, strictly speaking, they determined the rate of the momentum transfer between pairs of unlike molecules (Maxwell-Stefan) or between the molecules and the wall (Knudsen). Evans et al., using the same line of argument, assumed that it is momentum transfers rather than fluxes that are additive. Hence, combining the rate of the momentum losses for the Knudsen flow and for the molecular diffusion we obtain

-- dPil& RT

n XjZi - XiZj

Zi

=-+I2 D&

(p

j=l

i=1,2,...,n

ij (11)

jfi

which is a set of equations defining the so-called ‘diffusive fluxes’ in a capillary. It is important to realize that these equations are linearly independent and may be applied to determine the fluxes Zi uniquely, although only (n - 1) of the Maxwell-Stefan relations areindependent. This is due to the fact that eqns. (11) contain the term Ii/D&i which characterizes momentum transfer to the walls as a boundary condition. It remains only to take into account the viscous flux described by eqns. (7) and (9). According to the assumptions made by Evans et al., diffusive and convective fluxes can be added, giving Ni=Zi

+Nfv

(12)

XiBoP dP Niv= _-_ /_LRT dz

(13)

where

The assumption of Evans and his collaborators follows from the rigorous Chapman-Enskog [22] kinetic theory of gases, and from the Curie theorem [23] which postulates that fluxes defined by quantities of different tensorial character do not couple. The above combining principles have a very simple electrical analogue (Fig. 1): diffusive fluxes combine like

(10)

ii

jfi We have a total of n Maxwell-Stefan equations, of which only (n - 1) are linearly independent. Physically this is not surprising, since they .describe only momentum exchange between pairs of species, and they lack the necessary boundary conditions defining the rate of momentum transfer to the capillary walls. The attempts to combine the three basic transport mechanisms described previously were undertaken by a number of investigators. Scott and Dullien [18] and Rothfeld [ 191 confmed attention to isothermal isobaric diffusion in binary mixtures and their method was later extended to multicomponent mixtures by Silveston

t

N,=Ji+Niv

INiv

KNUDSEN DIFFUSK)PI

WI.

The most complete method of combining various mechanisms was givkn by Evans and co-workers [5,2 11. In principle, they utilized and extended the approach employed by Maxwell, Stefan and Knudsen to derive

Fig. 1. Electrical analogue of the procedure of Evans and Mason.

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resistors in series (voltage drops add), whereas diffusive and viscous fluxes combine like resistors in parallel (currents add). It should be emphasized that the model of mass transport in capillaries described by eqns. (1 l)-(13) predicts correctly the phenomena found experimentally: the viscous slip, diffusive slip and Graham’s isobaric diffusion law. This last may be obtained easily by summing eqns. (11) for all n components and setting dp/dz = 0. Then n

ri

jF,x =O

(14)

which, on employing eqn. (5) describing the Knudsen diffusion coefficients, leads to Graham’s law generalized to multicomponent mixtures:

(15)

ig,l’dZ=”

However, the principle of additivity of momentum transfers, used by Evans and Mason to derive relations (1 l), does not have any theoretical justification. It is just a per anahgiam extension of Maxwell’s and Knudsen’s analyses to transport phenomena of different physical character. Furthermore, the model does not allow for the pressure diffusion, thermal diffusion or the diffusion under the influence of external forces. So far, in discussing the three basic mechanisms of mass transport, it has been assumed that molecules collide with each other or with the walls, then are reflected instantly in directions determined by the laws of collisions. llowever, if we take into account that all diffusing species are, to a greater or less extent, adsorbed at solid surfaces, then an additional motion of the adsorbed molecules might occur on the surface. If the mechanism of this surface migration is regarded as a ‘hopping’ from site to site when, of course, the adsorbed molecule has enough energy to surmount the energy of adsorption and desorption between adjacent sites, the surface flux might be defined by analogy to Fick’s law. Thus the surface diffusion flux is given by the expression [24,25]

(16) where the surface diffusion coefficient is usually dependent on the total surface concentration

made

i=l

or on the fraction of the surface covered molecules 126, 271 and defined by

by the adsorbed (17)

If the rates of adsorption and desorption are both large with respect to the surface diffusion rate, we may safely assume that the surface and bulk concentrations will be related by the equilibrium adsorption isotherm

c;

= fi(C,,

c2,

. . . , G)

i=l

-7 )_,...,

n

(18)

It follows from eqn. (18) that the surface flux of each substance is linearly related to all the concentration gradients in the bulk gas phase, and that the coefficients in this relation depend on the gas composition through the concentration dependence of Df. me problem may be simplified considerably when the bulk concentrations arc sufficiently small that the adsorption isotherm becomes a straight line. Then, for a cylindrical capillary of radius r, the surface flux in the axial direction, expressed in moles per unit crosssectional area of the capillary, is given by I;=

_2”r$5 r

dz

(19)

Measurements of surface diffusion coefficients were carried out by Wicke and Kallenbach [28], Barrer [29], Satterfield [3], Reed and Butt [30] and Gilliland et al. [31]. A review of the basic ideas, with refererice to more important papers, is given by Dacey [32] and Aris [ 11. The incorporation of surface diffusion into a general model of mass transport in capillaries does not present any problems, since the surface diffusion flux simply combines additively with the diffusion flux in the gaseous phase. However, some papers published recently [33.34] cast doubts on the validity of the mechanism and its mathematical description, and on the experimental techniques employed to measure the surface transport. 2.2. Mass flux in porous media A porous medium may be modelled in two different ways for the purpose of predicting effective mass fluxes. In the first the pore structure is viewed as an assembly of interconnected capillaries through which the mass transfer occurs. while in the second the analyses concern mainly the obstacles to gaseous motion presented by the solid matrix. The models belonging to the first group are called ‘capillary models’, those belonging to the second ‘globular models’. The model of mass transport in capillaries, presented earlier, contains two constants which depend on the scale and geometry of a pore: K0 and &,. Theoretically, they might be evaluated for a given geometry and pore size. However, the porous structure of a medium is an irregular network of channels of wide range of dimensions, randomly interconnected. Thus, for a given porous medium K0 and Bo may be evaluated only experimentally. It remains to define the effective bulk diffusion coefficient for a porous body, which would allow for the geometry of the medium. This definition is usually given in the form

(20) where the proportionality constant e/q is the ratio of the void fraction to the tortuosity factor. The above coefficients stem from a very simple model, in which a porous medium is visualized as a network of tortuous channels in a solid body. Free space for diffusion is thus represented by the void fraction F. while

233

each channel is, on the average, 4 times longer compared with the thickness of the porous medium measured in the direction of diffusion. Consequently, the simplest of Ihe capillary models contains three parameters which have to be determined experimentally: K,,, B,, and e/q. When it became possible to measure the distribution of the pore radii as a function of the void fraction of a pellet using mercury porosimetry and low-temperature adsorption, the attempts of the investigators turned towards predicting the geometrical parameters of a porous medium from this distribution. In an early work dealing with this problem, Wakao and Smith [35] presented a bidisperse model of random pores, in which they distinguished micropores of void fraction eP and mean radius rP, and macropores of void fraction fM and mean radius &. Such a model is a very realistic representation of the structure of pelleted materials, since a catalyst is often prepared by compressing powder particles which are themselves porous on a much smaller scale (Fig. 2). The model assumes that the mass transport takes place by three possible routes: through macropores, through micropores and through micro- and macropores arranged in series. According to the above model, the relation defining the effective diffusion coefficient is De =

eM2& + (1 - EM)~~cc +

4EMI(1

l/&

-

EM)

+ l/&

(21)

Fig. 2. Differential (A) and cumulative (B) curves of the porevolume distribution for two probes of porous alumina.

and D, are the mean where & macro- and micropores, respectively, equations 1 --ax1 -----+_ 012

~

1 -t +1

for the from the

-1

1 &a

coefficients calculated

(FM)

(22)

1

DKI

(23)

@/.I)

and (Y= 1 +NZ/N,

(24)

All these relations are valid in a relatively narrow range, as they concern only binary mixtures and isobaric conditions. The model was later extended by Cunningham and Geankopolis [36] who introduced two different sizes of macropores. However, their model suffers from the same limitations as its predecessor. The first model which allowed for the full distribution of the pore radii to evaluate the geometrical parameters was that of Foster and Butt [37], the ‘convergentdivergent pores model’. They divided the total volume of the pores into several parts, with a given pore size determined from the distribution curve E(T) ascribed to each of the parts. From these parts, they composed two sets of pores with varying radii, with the radius decreasing towards the centre of the pellet in one set and increasing in the other. The result is a complicated pore structure which requires numerical solution to determine the mass flux. The model is subject to the same limitations as the Wakao and Smith model for it uses the isobaric relations for the fluxes, and seems to be particularly ambiguous. The class of models formulated by Feng and Stewart [38,39] is conceptually much simpler and, at the same time, offers more possibilities of application than its predecessors. As the relation describing the mass flux was taken from the ‘dusty gas model’, it seems useful to discuss the basic assumptions of this very interesting globular model. Contrary to the equations determining the mass transport in the intermediate range in a capillary, the relations of the ‘dusty gas model’ have their full justitication in the kinetic theory of gases. The idea of this model was stated succinctly by Maxwell: “We may suppose the action of the porous material to be similar to that of a number of particles, fixed in space and obstructing the motion of the particles of the moving systems”. The idea was exploited by workers from Oak Ridge National Laboratory and the University of Maryland [40-44]. According to this model, a solid medium distributed along the surface of the pores may be represented as a set of very massive molecules, among which much smaller gaseous molecules have to diffuse. These large particles---the obstructions in a solid medium-are treated as large molecules which compose one of the species of the gaseous mixture. The Chapman-Enskog kinetic theory is then applied to the pseudo-gas composed of the n gaseous species and the (n + 1)th species, known as the ‘dust’.

234

From this viewpoint the gas-surface interactions are then taken into account by treating the dust particles as giant molecules in terms of the classical kinetic theory of gases. The starting point for developing the model is the set of diffusion equations defining the transport of a gaseous species in the presence of temperature, pressure and composition gradients, and under the influence of external forces. These equations are sometimes called the ‘generalized Maxwell-Stefan equations’: n+lXilj I: j=t

-XjIi

__~---

jfi

CQij

grad P = grad xi + (xi - Ui) ___ P

- “’

-

P

n~yjFj ;=1

i,

+ ng’xiXjaij

grad In T

(25)

i=l

ifi The basic assumptions of the model are the following. (1) The dust particles are immobile and uniformly dispersed in the system: &=O,

grad C, = 0

(26)

(2j The dust particles are constrained by external forces, preventing them from moving despite the existing pressure gradient in the gas. There are no external forces acting on the gaseous species. (3) An empirical modification is introduced to allow for the effects of porosity e and tortuosity 4 of a porous medium: Q3,“, = (f/S)cDij

parameters of the model, and the three parameters, E/q, K,, and &, have to be determined experimentally. The complete mass flux is obtained by adding the viscous flux predicted by the empirical relation (13). In the Feng and Stewart model the structure of the porous medium is represented by a statistically specified assembly of capillaries, closely interconnected, so that the smooth field assumption is valid. This assumption requires that for a given segment of a pore whose axial direction is defined by a unit vector I,, the derivative of the concentration of the gas along this axis should be given by the following relation: dCi ~.- = In-grad Ci (29) dl the function f(r, a) is introduced, Furthermore, where r is the radius of the pore, and fi denotes its orieptption in space. The function is defined in such a way. that the product f(r, a) dr dR gives the fractional void’space associated with pores in the radius interval dr and in the orientation interval d0. Thus, the total flux of species i in a given point of the pordus *medium is Ni = JJra A nr

XjZj

fVj(r, 52) =Ii(r,

Iz

j=l

ifi

=__

q)e

ii

1 KT

02) + Ni,(r,

n Dij(r)

=--ci=1 RT

(27)

(30)

a) + Iq(r, SZ)

ia *grad pj 2DgKi Ia*@ad ~ ~ r

xipr2

__ ln.gadp 8/*RT

XiZj

~

Q2).f(r, Qj dr da

To define this flux, Feng and Stewart assume the relations of the dusty gas model, with K0 = 2r/3, c/q = 1 and B0 = r2/8, that is, appropriate for a single capillary. Hence, the flux of species i in a capillary of radius r and orientation S1 is given by

Introducing the above assumptions into eqn. (25) we have, after some algebraic manipulations, R

N&,

Ci

(31) where the elements of the matrix of diffusion coefficients Dij(r) are calculated by solving eqn. (28) with respect

+j&

t0 Ii.

grad pi

‘g i=,

F C

(Y’ii grad In T

(28)

The Knudsen diffusion coefficient D& thus obtained predicts correctly the effect of pressure, temperature and molecular weight of the diffusing gas. Although it does not take into account the porous structure of the medium, it does indicate correctly the effect of the dust concentration. As this concentration increases, the Knudsen diffusion coefficient decreases and the Knudsen diffusion begins to control the transport. Although the dusty gas model is the only one which has a full theoretical justification, it is not free, however, of various shortcomings, the most important of which is the lack of clear correlation between the parameters of the model and the porous structure ofa medium. The level of kinetic theory approximation used makes it impossible to include the effect of porosity. Consequently, it is not possible to take into account the distribution of the pore radii in the calculated

After some transformations, the authors the most general formulation of the model: Ni = -

grad pj

2

. y-

j=l

-

finally give

I

/;“g

I&-grad p - DPKiy.grad

Ci

(32)

where u(r), B. and y denote a tortuosity tensor, a permeability tensor and a surface diffusion tensor, respectively. When the pore orientations are isotropically distributed, the tensors IC, B,, and y are also isotropic and therefore represent scalar multiples of the unit tensor. Thus, for the isotropic system we obtain Ni = -

2 IDij (r) i-1

- yR:

I ,

K(I) de(r)

IG

grad p - DFKfy grad Ci

(33)

235

The parameters which determine this most comprehensive model and which should be determined experimentally are the permeability Be, the geometrical surface diffusion factor 7, the tortuosity function I and the surface diffusion coefficients 0;. As the number of adjustable parameters is too large for practical purposes, Feng and Stewart propose simpler working forms of eqn. (32). The relation of most practical value assumes that the pores are isotropically oriented, and the pore size distribution is strictly bitnodal, with macropores of radius rr,, and micropores of radius rp. The tortuosity factor is a function of the pore radius. Hence

j=1

-

Xi PBO __ gradp-AigradCj MRT

The parameters which have to be determined experimentally are rM, r,,, ?VM, W,, B. and Ai. From the point of view of the porous structure of a medium, this model is identical with that of Wakao and Smith. However, it is not limited to binary mixtures and isobaric conditions. The common characteristic of all working forms of the Feng and Stewart model is the pore size distribution function e(r), which appears as a parameter and has to be measured by mercury porosimetry and low-temperature adsorption; it should be emphasized here that the method is not a simple one.

3. Experimental verification the methods of determining

of the models and their parameters

A number of investigators attempted to determine experimentally the range of validity and applicability of the dusty gas model, which is the most general and theoretically sound representation of the transport of gas in porous media. In one of the earliest works Gunn and King [45] found experimentally that this model is thoroughly applicable to binary mixtures. Omata and Brown [46] reach the same conclusion, using pellets with pores of large radii. They found, however, considerable discrepancies between the fluxes calculated and determined experimentally for the pores smaller than 5 nm. They suggest that the discrepancies should be attributed either to the surface diffusion, or to the molecule-wall interactions which are not taken into account by the model. Abed and Rinker [47] showed that there is a possibility of successfully applying the dusty gas model to pellets with bimodal pore size distribution, if the model is supplemented with an additional parameter T. This parameter may be measured directly and represents the ratio of the effective Knudsen radius measured for the pure gas in permeability experiments to the value of this radius estimated for the binary isobaric diffusion. An elegant verification of the dusty gas model for ternary mixtures was presented by Hesse and Kader [48]. Using the diffusion-reaction cell of Hugo and

Wicke [49], they found good consistency for pressures above 200 mmHg. At lower pressures, however, some discrepancies appeared, and the curve representing the permeation pressure drop as a function of pressure was found to pass through a minimum, as for Knudsen flow in long capillaries. Hesse and Kidder suggest that this may be due to the ‘end effects’ of many short pores. Remick and Geankopolis [50] employed the system of parallel capillaries of precisely defined diameter to verify the ‘intermediate region’ theory in multicomponent mixtures. Pate1 and Butt [51] conclude that the dusty gas model is the most useful to describe their experiments with ternary mixtures. Feng and Stewart [39] verified experimentally their working forms of the parallel pores model; the studies concerned mainly the porous media of a wide distribution of pore sizes. The comparison between the fluxes calculated and measured enabled them to determine the accuracy of the proposed special cases. Thus, it turned out that the most exact is the form with two integration constants WM and W,, which allows for the variation of the tortuosity factor with the pore radii (eqn. (34)). The estimated values of r,, and rM exhibited a clear tendency to assemble around certain points of the distribution curve V(r)[volume of pores/g]. These radii corresponded to the following fractions of the total volume of the pores: V(r,JV, V(r,)/V_

= 0.676 rt 0.034 = 0.977 * 0.018

Taking this into account, the number of adjustable parameters for the working form under discussion is decreased by two, i.e. r, and r,, , which may be assumed from the distribution curve e(r). Recently Rogut [52] carried out extensive studies concerning the gas transport in porous media of a highly non-uniform porous structure, for various gaseous mixtures and in a wide range of temperatures. The main objective of these studies was to verify the theoretical correlations between the effective mass transport coefficients and the porous structure of the medium. As the effective mass transport coefficients were measured using different methods, it was also possible to compare these methods basing on the experimental results obtained. In the measurements, six samples of porous alumina were employed, which varied considerably in pore size distribution-from monomodal through bimodal to a continuous distribution. The values of the specific surface areas of the samples covered the range 15-330 m2 g-l. The measurements were also carried out for five samples of the methanol synthesis catalyst, which were taken from the industrial reactor after various periods of operation. Three pairs of gases of different concentrations were used in the experiments (H2-N2, H2-Ar, N2Ar) with the temperature varying from 298 to 473 K. The following models were verified: (1) the model of parallel pores of Feng and Stewart, (2) the model of random pores of Wakao and Smith,

236

(3) the model employing the mean pore radius of Weisz and Schwartz. The analysis of the experimental results led to the conclusion that both for alumina and for the methanol synthesis catalyst the best correlation was that of Feng and Stewart. The optimum values of the tortuosity factor estimated using Feng and Stewart’s model fell within the range q = 2.7-3.1 thus confirming the value of q = 3 estimated by Stewart for the isotropic media. No change of the tortuosity factor with temperature was found. The second in quality was the correlation of Wakao and Smith, particularly when applied to bidispersed structures. Rudimentary correlations, such as the WeiszSchwartz relationship, appear to be useful only for monodispersed structures. Passing on to a concise discussion of the measurement methods concerning the effective mass transport coefficients in porous media, we shall concentrate on the description of their physical basis, trying not to dwell upon the measurement techniques and the parameter estimation. To determine the permeability coefficient R, and the Knudsen diffusion coefficient D&, it is most convenient to measure the flux of pure gas passing through a porous membrane under the influence of an imposed total pressure gradient. The principle of the method is shown in Fig. 3. The method is based on measuring the permeation flux of the pure gas, passing through a flat porous partition at a constant pressure difference. By carrying out the permeation experiments in a wide range of pressures it is possible to determine both the values of the effective Knudsen diffusion coefficients and the permeability coefficient Be. The total flux of the pure gas is given by

Hence, the overall permeability

coefficient

K=D&+B&/_i

lag’ method [56-601. The principle of the method is illustrated in Fig. 4. The method is based on a step change of pressure in one of the chambers (VI); the pressure-time dependence is then measured in the second chamber. This dependence makes it possible to determine the effective mass transport coefficient (the Knudsen diffusion coefficient + the permeation coefficient) in the unsteady region and in the so-called ‘pseudo-stationary’ region. The mass transport coefficient in the unsteady region is calculated based on the measured value of the relaxation time 0, while in the pseudo-stationary region-from the slope of the curve, dp,/dt, in its linear range. It is often necessary to employ several pellets in series, in order to increase the accuracy of the measurements of Q and dpz/dt by increasing the thickness of the porous septum. To measure the effective mass transport coefficient when the bulk diffusion is present, it is necessary to carry out experiments using the mixtures of gases. The schematic diagram of the measurement system for the Wicke Kallenbach stationary method [61] is shown in Fig. 5. The fluxes of species diffusing through a porous plug are calculated fr-om the mass balances of these species in streams Fr and Fa, which flow around the front surfaces of the pellet analysed. It is thus esscntial to determine very accurately the compositions of the streams at the inlet and outlet of both chambers VI and Vs. Smith and co-workers [62-651 and Gibilaro et al. [66] modified the Wickc-Kallenbach stationary method, adapting it to dynamic experiments. Two streams of car-rier gas flow past each face of a porous pellet at a constant flow rate. A pulse of tracer gas is injected into one of these streams, Fr, and the response curves are monitored by the detectors situated at the outlets of chambers VI and Va. Interpretation of the moments of the response curves enables the effective diffusion coefficients to be deter-

K, (36)

should depend linearly on pressure p, with the slope equal to Be/p and the intercept Dg. A detailed description of the method is given by Schneider [53 551. The permeation experiments for pure gases were also carried out under unsteady conditions using the ‘time

Fig. 4. Ideological scheme of the ‘time lag’ method.

,“owrr

.?=o

Z=t_

Fig. 3. Ideological scheme of the permeation method.

Chamber

,‘. gt?t?@&oJ

Fig. 5. Scheme of the measuring device for the impulse method of Smith and Burghardt and the stationary method of Wicke and Kallenbach.

231

mined. A thorough theoretical analysis of such a measurement system was given by Burghardt and Smith [65], together with some modifications of the system and with the analysis of their parametric sensitivity. The authors reach an interesting conclusion, pointing to the possibility of determining the effective diffusion coefficient from the zeroth moments of the response curves. The advantage of this procedure compared with the use of the first moment is due to the fact that the zeroth moment can usually be determined with much higher accuracy. This method of estimation of the effective diffusion coefficients was employed by Rogut [52] and Baiker er al. [67]. Much better measurement opportunities are provided by the chromatographic method (Fig. 6). Pellets of the catalyst under investigation, of mean radius R, are placed in a column of length I and diameter dk. Carrier gas flows through the column with constant linear velocity. At the starting point of the experiment, t = 0, a pulse of tracer gas is introduced to the stream of flowing gas at point z = 0. Detectors located at the inlet and outlet of the column record the composition of the gas stream, that is, the curves of concentration as a function of time. The moments of these curves are then used to determine the effective diffusion coefficients. A detailed description of the measurement system and experimental techniques was given by Rogut and Schneider [68], while the mathematical model of the process was presented by Kubin [69], Kucera [70], Haynes [71] and others [72-781. The interpretation of the results obtained using the chromatographic method is much more complicated than for the membrane methods (Wicke-Kallenbach and Smith-Burghardt). This is due to the fact that the tracer concentration in the outlet gas has to be determined by solving two separate mass balances: one for the interior of a pellet, and the other for the interstices between the pellets. Thus, it is necessary to take into account the axial dispersion in the bed and the convective mass transport from the gas to the pellet. Consequently, the experiments have to be carried out in the regions where these phenomena are negligible compared with the diffusion inside the pellet. The comparison of the methods described shows that it is the chromatographic method which most correctly reproduces the conditions of the real process in a catalytic reactor. This is due to the fact that in the chromatographic method the column is packed with

Cil

pellets of an industrial catalyst in amounts sufticient to form a representative sample of the whole portion of the catalyst studied. The measurements may be carried out over a wide range of temperatures and pressures. Because of the dynamic character of the method the measurements are fast, as it is not necessary to wait for the system to reach equilibrium. A serious drawback of the method is the difficulty in interpreting the measurement results; the effective diffusion coefficient can be evaluated only from the second moments of the response curve. In the membrane methods the preparation of the measurement system, and in particular the sealing of the lateral walls of a pellet forming a porous septum, may present problems. Moreover, the membrane has to have a suitable shape (most often cylindrical of finite length). The advantages of these methods consist in a much simpler interpretation of the experimental results which, as in, for example, the Smith and Burghardt method, may be based only on the zeroth moments of the concentration distribution curve. The advantage of all permeation methods (both steady state and dynamic) is the possibility of employing pure gases and hence the elimination of the necessity to determine the gas composition, The only measuring instruments required are manometers and flowmeters. The interpretation of the results is also straightforward. The studies carried out using the various methods show that the effective diffusion coefficients obtained in both the steady state and unsteady state experiments and in the measurements with a chemical reaction sometimes differ considerably from each other. As yet, it has not been possible to explain fully these discrepancies. However, the effect of the pore structure on the measurement procedure was often emphasized, with particular regard to the presence of the dead-end pores and their position in relation to the through pores.

4. Theoretical analysis of mass transport chemical reaction in a catalyst pellet

and

The review of the theoretical works dealing with the processes of mass transport and chemical reaction in a single catalyst pellet has been limited to the studies employing the mass flux relations in their most general form. Particular attention has been given to multicomponent and multireaction systems. We shall start from the system in which a single reaction takes place, described by the stoichiometric equation ~UjAi=O

(37)

i=l

If the process is assumed to be stationary, described by the following set of differential diV

Fig. 6. Ideological scheme of the chromatographic method.

Ni = ~ir(C, T)

i=1,2,...,n+l

Using one of the relations discussed defining the mass fluxes Ni (e.g. the dusty we find that it is impossible to decouple

it may be equations: (38) previously, gas model), eqns. (38).

238

Thus, it is necessary to solve the full set which, together with the boundary conditions, forms a boundary value problem defining the process analysed. It should also be borne in mind that the relationships describing the mass fluxes in multicomponent systems have an implicit form and do not deiine the fluxes explicitly. The solution of such a boundary value problem, which in a general case is a set of partial differ-ential equations, is very difficult to find. Thus, to solve this type of problem it is essential to know the algebraic relations between the individual mass fluxes, defined by the stoichiometry of a reaction: Ni /vi = Nj /Vi

(39)

Equation (39) enables one not only to obtain the explicit expression defining an individual mass flux, but also, in many cases, to define (n - 1) concentrations in terms of the concentration of a chosen reference component. Thus, relations (39) simplify considerably the solution of the boundary value problem discussed. The proof that eqns. (39) are valid for pellets of high degree of symmetry (infinite cylinder, sphere, infinite slab) is relatively simple and was presented by Jackson [791. Similarly, Hite and Jackson [80] proved the validity of the stoichiometric relations for a pellet of any shape. However, they analysed only binary mixtures and the pellet of very high permeability, that is, the region of bulk diffusion at a negligibly small absolute pressure gradient. The proof defining unequivocally the conditions under which the stoichiometric relations remain valid for pellets of any shape and for a general form of the mass flux equation was carried out by Burghardt [81]. It should be emphasized once again that these relations should hold at any point of the pellet, as the proof that they remain true for the pellet as a whole is quite straightforward. The discussion based on vector analysis led to the following theorem. The stoichiometric relations are valid in a steady state at any point of a simply connected region V,, bounded by a smooth surface Fp, if the following conditions are satisfied: (1) variables of state Y = (p1,p2,. . ., pn, 2’) are constant on the bounding surface Fp, i.e.

Yi(Q) = 5s

for

Q E 4,

i=1,2

) . . ..?I + 1

(2) gradients of the variables of state are collinear vectors at every point of the region V,. The second condition may be substituted by a stronger one, namely, (2a) coefficients Fit(Y) in the relation determining the individual mass flux Ni=-

yj(r)

=&[yi(Q>,

Y,(r). i=

q(Q)] I,2 ,...)

(40) n+l,

j#l

QEF, The above functions do not depend either on the shape of the pellet or on the expression defining the reaction rate, and, employed in the boundary value problem (37), reduce the number of its differential equations from (n + 1) to one. As particular cases we have, for example, invariant relations between the concentrations of species in the region of Knudsen diffusion : (41)

Q E Fp or Prater’s equation 1821 defining the relationship between the temperature and concentration of the reference component in a pellet in the region of Knudsen diffusion :

D$[C{(r) - Cj(Q)] =& ,

F’-(r)- T(Q)1

(42)

Further studies were directed towards the generalization of the existing analyses to a set of chemical reactions occurring in a catalyst pellet and towards the formulation of the mathematical and physical limitations sufficient for the stoichiometric relations to hold in the pellet of arbitrary shape. Let us assume that R linearly independent chemical reactions take place in the pellet; they are defined by the following stoichiometric relations:

i: VirAi = 0

r= 1,2,

. . ..R

(43)

i=l

n+1 xF;:j(Y)ggradq

Thus, the differential the form

j=l

are independent of the variables of state. Now let us discuss the possibility of satisfying above conditions under real physical situations,

during mass transport accompanied by a chemical reaction in a porous pellet. The constancy of the variables Yi on the surface Fp bounding the region VP is assured only when the pellet is surrounded by the fluid of unifonn constant composition and of constant temperatur-e, and the resistances to mass and heat transfer to the surface of the pellet are negligibly small. This situation is often encountered in catalytic reactors, The second condition is always satisfied for isotropic pseudo-homogeneous porous media, for which we may assume the smoothness of the fields of concentration and temperature. If the stoichiomctric relations in a catalyst pellet are valid, it is possible to determine the invariants of the combined process of chemical reaction and mass transport, that is, the relations between the variables of state in the form

the i.e.

div Ni = 5 ~i,.r,. r=1

mass and energy

i=1,2,...,n+l

balances

take

(44)

239

The set of differential equations (44) also contains the energy balance equation with flux N,,,, and coefficients V, + i, r = (-M,). The variables of state take constant values at the pellet surface Y=Y,

on

Fp

On introducing metric coefficients VT,

Vll

vzl

_’

.

VZR :

1VlR

(45) the augmented

. . .

VRl

. . .

VRR !

matrix

vR+l,l

a.*

VR+l,R j

of the stoichio“nl

1

(--LwlI

. . . VnR

i-A&Z) :

(46) we may write form

the set of eqns. (44) in a concise

matrix

div N = or

(47)

which makes the subsequent analysis clearer. (Incidentally, the analysis is very similar to that for a single reaction and leads to identical conditions, that is, the constancy of the variables of state at the surface of the pellet and the collinearity of their gradients inside the pellet.) The stoichiometric relations for the system analysed have the form K=

R XvYtrAr

i= 1,2,

. . ..n

(48)

r=1

or, in the matrix N=uAR

notation, i=l,2,...,n+l

(49)

From these equations it follows that all (n + 1) fluxes in the system studied may be expressed in terms of R hypothetical fluxes, where R is the number of linearly independent reactions. Thus, for a single reaction we obtain the simple relation iV* =

i = 1,2,

VilAl

. . ..N+

1

div Ar = r,.(Y)

r=1,2

, -0-7R

$,.

r=1,2 R equations

div grad J/, = V2$, = -r,(Y)

Yi= Yi,

(54)

l,&...,n+l

l=R

for

r=l,2,...,R The symmetry of the pellet boundary conditions on A,: A,=0

for

imposes

r=1,2

I=0

(55) the following

, .-., R

(56)

Consequently, the process analysed is described by (iV + 1 + R) differential equations instead of (2N + 2). The solution of these equation yields (N + 1) variables of state and R fluxes A,. From the previous analysis it follows that A, = -d&/d1

(57)

Using this relation, the set of equations sion can be written as

under discus-

(58) Y = Y*,

for

i=1,2,...,n+l

IL1=0

r=1,2

(51)

thus reducing the number of equations of the boundary value problem (44) from (n + 1) to R. Furthermore, Burghardt and Patzek [83] showed that each of the hypothetical fluxes A,. can be expressed in terms of a suitable scalar potential G,.(r)

and, consequently,

i=

(50)

Substitution of (49) into (47) and use of the condition of linear independence of the R reactions give the set of differential equations

A, = -grad

Unfortunately, for multireaction systems the situation becomes much more complicated, for the R scalar potentials J/, are not independent (as was proved by Stewart [84] and Burghardt and Patzek [83]). Based on a very ingenious argument Stewart [84] shows that from the relation between the scalar potentials $,. it follows that the contours of these potentials should lie on parallel surfaces, that is, on parallel planes, co-axial cylinders or concentric spheres. Hence, the following theorem may be formulated. The stoichiometric relations are valid in multireaction systems (R > 1) only for pellets of simple geometrical symmetry, like infinite slabs, infinite cylinders or spheres. Taking this into account, we may write the set of differential equations defining mass and energy transport in a symmetrical catalyst pellet, ln which R linearly independent chemical reactions take place, as follows:

, .--3 R

which tions

are to be solved

subject

to the boundary for

$i=&=...=J/R=o

3.-., R

(59) condi-

l=R

(52)

(51) may be written r = 1,2,

. . ..R

as (53)

This might suggest that the boundary value problem (44) reduced to R linearly independent equations defining R independent scalar potentials ti, (for a single reaction there was just one scalar potential).

The effectiveness each reaction as

factor

of a pellet

is defined

for

(60)

240 and for the geometries m + 1 A,(R) “Ir = __ R

r, (Y,)

studied

takes the form

(61)

The form of the matrix H depends on the model of the porous structure of a pellet. Values of the elements of this matrix (H,,) for the dusty gas model were evaluated by Burghardt and Patzek [83 ] . Without discussing in detail the solution of the set of differential equations presented earlier we should emphasize that in dealing with a multireaction system it is more convenient to employ eqns. (54)--(M), which contain the hypothetical fluxes A,.. However, to solve the multicomponent system with a single reaction, eqns. (58)-(59) are most suitable, as in that case we have just one scalar potential &, the others being equal to zero (l&=$3=...= gR = 0). The solution of the problem is then accomplished in two stages: (1) solution of the initial value problem, eqn. (58): (2) solution of the boundary value problem, eqn. (59). It should be pointed out that the initial value problem is solved only once for the given conditions at the boundary of a pellet. Hence vector Y = Y(Y,, G1) is obtained, which is subsequently used in eqn. (59) to calculate r(Y(9,)). Before analysing rhe existing solutions of the boundary value problem studied, it seems worth while to discuss the effect of the total pressure gradient in a pellet, as this effect is surrounded with much controversy and confusion. It is easy to show that every reaction must be accompanied by pressure gradients in a pellet; only in very special cases is this gradient equal to zero. The total pressure gradient in the pellet provides an additional degree of freedom, thus releasing the fluxes from the constraints imposed by the Graham law. Without this degree of freedom the fluxes cannot be adjusted to the stoichiometry of the reaction. The estimates of pressure variations in catalyst pellets carried out by Hite and Jackson [85] show that the highest values may be expected for the region of Knudsen diffusion. For example, if Av = 1, the absolute pressure at the centre of the pellet may be about 40% greater than that at the surface. For the region of bulk diffusion, when the permeability coefficient B,, is very large, the pressure variations become negligible. This result, howeve,;, should not be understood as implying that for B. + -, dpld2 --z 0 and pressure gradients have no physical significance in a pellet of high permeability. Indeed, the product B,, X dp/dZ defines the finite value of the viscous flux necessary to alter the individual fluxes to the values required stoichiometrically. In a very interesting paper Hite and Jackson [85], discussing two earlier articles [86, 871, show how the incorrect neglect of pressure gradients in the flux equations may lead to substantial errors in calculating the effectiveness factor of a pellet (Fig. 7). In their three publications Krajewski and Burghardt [88] and Krajewski [89,90] presented an instructive analysis concerning the expected magnitudes of errors resulting from the use of an inappropriate mass flux

Fig. 7. Effectiveness factors of a catalyst pellet at the limit of bulk diffusion control~&./DKI = 0.01. relation to calculate the effectiveness factor of a pellet. To provide a basis for the comparison of their calculations with the results already existing in the literature, Burghardt and Krajewski introduced certain simplifying assumptions to the dusty gas model. These assumptions made it possible to characterize the interactions of- the diffusing species using just one additional dimensionless parameter 0 = EAXAO =x*,

(-Av)

(62)

VA

which allows for the non-equimolarity of the stoichiometric equation. Strictly speaking, the analyses presented in refs. 88, 89 and 90 are valid for mixtures with components only slightly differing in molecular weights. The calculations were carried out for an isothermal process (Fig. S), for an isothermal catalyst pellet with the resistance to heat transfer taken into account (Fig. 9), and for a non-isothermal process (Fig. 10). All these Figures show a considerable effect of the nonequimolarity of a reaction on the effectiveness factor of a pellet. In the region of high diffusive resistance or high activity of the catalyst (large Thiele moduli), neglect of the parameter Q may lead to errors as high as 100%. To determine the effectiveness factor of a catalyst pellet for the ammonia synthesis reaction, Burghardt and Patzek 1831 employed the complete dusty gas model and the balance equations (SR) and (59). The calculations were performed for a three-stage adiabatic ammonia reactor and for a five-component mixture (argon and methane as inert gases). The comparison of the effectiveness factors calculated and detemlined experimentally showed only minor discrepancies (Fig. 11).

Fig. 8. Effectiveness factor of a spherical catalyst pellet for an irreversible reaction of the third order.

241

7 ioo 50 20 i0 50

2.0 1.0 ^_ “3 0.2 0.1 a2

2.0

5.0

IO

20

Fig. 9. Effectiveness factor of an isothermal, spherical catalyst pellet for an irreversible reaction of the third order and various nonequirnolarity parameters 0 (7 = 10, a/Bi = 0.05, BiM= -).

Fig. 10. Effectiveness factor of a cylindrical catalyst pellet for a non-isothermal irreversible reaction of the third order (0 = 0.5, 7=20,Bi=-,BiM=-).

Fig. 11. Comparison of the dusty gas (curve 1) and Dyson’s (curve 2) correlation for the effectiveness factor of a spherical catalyst pellet of radius I = 0.003 m.

It should be pointed out that the diffusive interactions of the components changed the molar ratio HaNa from 3: 1 on the surface of the pellet to 6:l at its centre of symmetry. Klusacek and Schneider [91] investigated experimentally the influence of multicomponent diffusion on the dehydration of methanol in a specially prepared model pellet of known porosity and of known mean pore size and tortuosity factor. The results fully con-

firmed the possibility of describing the mass transport in such a multicomponent system using the MaxwellStefan equations (large pore sizes). The values of the tortuosity factors obtained experimentally were in satisfactory agreement with those calculated for the given geometry of the pellet. A very interesting verification of the dusty gas model was presented by Davis and Fairweather [92]. Based on the experiments carried out in a reactor with recycle, they evaluated the effectiveness factors for the oxygenation of SOa on the industrial catalyst VsOs. The pellets employed were cylindrical (6 mm X 6 mm), of void fraction e = 0.44 and of mean pore size r,= 319 nm. In the calculations the constant value of the tortuosity factor, ~7= 2.7, was assumed. An excellent agreement between the effectiveness factors calculated and determined experimentally (Fig. 12) led the authors to the conclusion that the properties of the dusty gas model make it particularly suitable to describe the mass transport in very complicated systems. The effectiveness factors were computed using the set of balance equations (58)-(59). In two papers Jackson et al. [93,94] presented a procedure to solve the differential mass and energy balances (eqns. (54) and (55)) describing the mass and energy transport in porous catalyst pellets for a multireaction, multicomponent system. As may be seen from the boundary conditions for this set of equations, it is a two-point boundary value problem which, generally, requires an iterative procedure. Under isothermal conditions, when the concentration variations in a pellet and, consequently, the changes of the reaction rate are not too drastic, these authors recommend the use of the global orthogonal collocation method based on the zeros of the Legendre polynomials. The original set of differential equations is thus transformed into an equivalent set of non-linear algebraic equations, which is subsequently solved using the Newton-Raphson method. Jackson et al. also described two approximate semianalytical procedures which, in principle, are based on an erroneous assumption that the scalar potentials $, are independent. The authors, treating these quantities as

0.6

a4

a2

--

0.0 0

0.2

0.4

a6

aa

x

Fig. 12. Comparison of calculated and experimentally determined effectiveness factors for the reaction of SO* oxidation.

242

independent variables, transform eqn. (58) in such a way as to obtain a total differential in terms of the &.. They illustrate their methods by two examples of irreversible reactions C* Ha + l-l* = C,H4 (1) C&+

Hz=CzHb

and 2H1 + 4CO = CZ H‘, + 2C02 (2) 3Hz f 6C0 = C,He + 3C02 of power kinetics. In the first example, there occurs an interesting increase of the effectiveness factor of the second reaction above the value of one, despite the fact that the process is isothermal and the reaction kinetics is very simple. In the second example, pronounced reverse diffusion took place, that is, the transport of hydrogen against its own concentration gradient. The method of global orthogonal collocation becomes impractical (the necessity to use a large number of collocation points) when the reaction rate varies considerably within the pellet, which may be expected for non-isothermal processes. In that case, Jackson et al. propose a transformation of the boundary value problem into the initial value problem by guessing the missing values of Y&l = 0) = Yi, and multiple integration of eqns. (54) and (55) from I = 0 to I= R, subject to the initial conditions Y;(O) =

Yj* for

i=1,2,...,n+l

until the conditions at the matched. The method was illustrated reversible reactions: CO + 3Hz = CH,+

surface

of the

pellet

using the following

are

set of

show the possibility of multiple steady states occurring even under industria1 conditions. Moreover, the concentration ratio H2:C0 inside the pellet is 500 times higher than that at the surface of the pellet. Recently Sdrensen and Stewart [95 ] proposed another method of solving the boundary value problem, employing hyperbolic functions rather than polynomials as trial functions. As has been pointed out previously, with the global collocation method based on the polynomials it is necessary to use a large number of internal collocation points for very fast reactions and drastic variations of temperature and concentration. Although the use of ‘splines’ enables this difficulty to be bypassed, the choice of the connecting points of the polyr~omials requires an additional analysis of the problem, as these points cannot be guessed in a simple way. Thus, S&ensen and Stewart assumed that the natural basis of trial functions for the problem discussed is provided by the eigcnfunctians of the linearized set of differential equations for a suitably selected vector of the variables of state called the ‘state of reference’. It should be emphasized that the eigenvaiues and the corresponding eigenfunctions for a linearized set of differential equations have to be determined separately for each case and cannot he tabulated, unlike those for the global polynomial collocation. The most crucial point of the method under discussion is the appropriate choice of the state of reference for which the set of differential equations is linearized, as this choice determines the eigenvalues, and consequently the collocation points of the procedure. The results of computations carried out for the catalytic reforming of n-heptane. where six reactions take place and six reactants are present in the system, indicate that the procedure proposed converges faster than the polynomial collocation method.

l-I20

CO+H20=COz+H2

5. Suggestions for further research

with the kinetics of the Langmuir-Hinshelwood type. The plots of effectiveness factor versus Thiele modulus are shown in Figs. 13(a) and (b). The Figures clearly

The experimental results presented in this paper show clearly that all the models of mass transport in porous media worked out until now, including the most general dusty gas mndcl, fail when the pore sizes ar-e less than 5 nm. The discrepancies become particularly large in zeolites, which are more and more frequently used as catalysts owing to the possibility of controlhng their selectivity in a desired direction. Because of the very small pore sizes in zeolites (OS--l nm), the movement of matter in such pores cannot be completely chaotic and, consequently, the diffusion mechanism based on the kinetic theory of gases is useless in this case. As has been pointed out, the molecular configuration of a species plays a very important part in the transport processes in zeolites. Therefore, there is an urgent need to develop a model which would explain the mass transport phenomena in zeolites. Also, the mechanism of surface diffusion has not been fully explained. Particular controversy surrounds

(a) @I Fig. 13. Effectiveness factor as a function of Thiele modulus for (a) reaction I and (b) reaction 2.

243

the procedure of estimation of the surface diffusion coefficients. Thus, it is essential to work out an independent experimental technique to measure these parameters. As to the measurement procedures concer~ng the mass transport in porous media, two problems remain unsolved. The first refers to the explanation of disthe mass transport parameters crepancies between measured using steady state methods on the one hand and dynamic methods or methods employing chemical reactions on the other. The second problem is of a more general nature, as it concerns the basic principles of the measurement procedure from the point of view of its practical usefulness. Thus the question arises which of the two would be more helpful: the development of a fast routine method for measuring the effective mass transport coefficients for any catalyst, or the attempt to correlate these quantities with the porous structure of a medium and physical properties of the diffusing species using appropriate correlating equations. The method presented here, concerning the solution of the set of differential equations describing the mass transport in a multicomponent, multireaction system, refers in principle only to the steady state processes. As yet, only one paper [96] has been published, dealing with the dynamics of this process in a very simplified form. The problem of multiple steady states and their stability remains completely untouched.

rate of rth reaction, mol m-’ s-l column matrix of reaction rates, mol m-3s-1 = 8.314 J mol-’ K-l, gas constant pellet radius, m time, s temperature, K mass fraction of component i volume, m3 mole fraction of component i degree of conversion column matrix of state variables coordinate along pore axis, m coefficient of thermodiffusional transport _ (-~)D;“C? , Prater number heTo density, kg me3 = E/RT, density of component i, kg m-’ porosity of pellet effectiveness factor of catalyst pellet for rth reaction tortuosity function effective heat conducti~ty of pellet, W m-l K--l hypothetical mass flux, mol m” s-’ viscosity, Pa s stoic~omet~c coefficient of component i in rth reaction matrix of stoichiometric coefficients Thiele modulus scalar potential of mass transport in pellet

Nomenclature permeability factor = aR/Ae, Biot number = &R/D&, Biot number for mass transfer total concentration, kmol m-’ column matrix of concentrations, km01 me3 concentration of component i, kmol mu3 diffusion coefficient of binary pair i-j, m2 s-l Knudsen diffusion coefEcient, m2 sm.’ surface diffusion coefficient, mz s-l element of the matrix of multicomponent diffusion coefficients, m2 s-l activation energy, J mol-’ surface area, mz force acting on particle i, N g-’ heat effect of tih reaction, J mol-’ diffusion flux of component i, mol rn-‘sF1 geometrical factor of Knudsen diffusion, m equilibrium constant of adsorption unit vector in direction 52 coordinate, m = 0, 1, 2, number characterizing pellet geometry molar mass, g mol-’ total flux of component i, mot mm2 s-l column matrix of fluxes, mol mF2 s-l total pressure, Pa partial pressure, Pa tortuosity factor pore radius, nm

Superscripts e S

denotes effective value refers to surface of catalyst pellet

Subscripts M p P V

refers to macro-values refers to micro-values refers to catalyst pellet denotes viscous flux

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