m(R2509/82/030441~303.0310 Perslmon Press Lid.

ANALYSIS OF SO, OXIDATION IN NON-ISOTHERMAL CATALYST PELLETS

USING THE DUSTY-GAS MODEL

MARK E. DAVISt, GRAEME FAIRWEATHERS and JOHN YAMANIS* Department of Chemical Engineering, University of Kentucky, Lexington, KY 40506, U.S.A.

Abstract-Analysis of the experimental rate data[l3] for SO, oxidation on a commercial V,O, catalyst has yielded a model that describes the experimental data well while the rate constant follows the Arrhenius law. This model was subsequently used 10 adequately predict experimentally determined effectiveness factors using the dusty-gas model flux relations and a single value for the tortuosity factor. The latter analysis validates the excellent properties of the dusty-gas model to properly describe transport in complex systems.

INTRODUCTION

Since the pioneering publications by Thiele[l] and Zeldovich[2], on the mathematical modeling of diffusion and reaction in porous, solid catalysts, an enormous body of literature has emerged, and this body has been superbly brought together in the classical reference work by Aris 131. All but a few of these papers to date have been based on simple, uncoupled flux relations that are valid, strictly speaking, either for systems in which Knudsen diffusion is controlling or the reaction is a simple isomerization or a depolymerization one. Al- though these simple flux relations may describe a large number of real systems, these relations do not vigorously describe the overwhelming majority of real systems that involve multicomponent reactions and diffusion processes other than Knudsen flow. The modeling of the latter phenomena had to await the development of adequate flux relations that described the diffusion processes.

Following the development of the dusty-gas model for multicomponent mixturesIB, Jackson[S] and Jackson et al.&81 for example, as well as other investigators, Kehoe and Aris[9], Abed and Rinker [lo, 111, for exam- ple, have proposed and analyzed mathematical models that describe multi-component diffusion and reaction in isothermal and non-isothermal catalyst pellets. These models are very appealing from the point of view of solvability and of numerical analysis, and these desirable characteristics have been demonstrated in several papers[CS]. Because these models are comprehensive and solvable, it is important that they be subjected to further refinements and to testing in order to ascertain their adequacy in describing the behavior of real sys- tems. Except for the work by Wong and Denny[lZ], there is no other publication that deals with the testing or validation of these models which was one of the main objectives of the present study.

*Author to whom correspondence should be at: Allied Cor- poration, P.O. Box 1021-R. Morristown, NJ 07960, U.S.A.

tPresent address: Virginia Polytechnic Institute, Blacksburg, VA. 24061, U.S.A.

SDepartment of Mathematics.

In order to test the adequacy of the multicomponent diffusion and reaction models based on the dusty-gas flux relations, the experimental effectiveness factors for the oxidation of sulfur dioxide on an industrial V,O, catalyst as measured by Livbjerg and Villadsen[13] were used. Since this reaction was studied in the presence of nitrogen diluent, the above mathematical models were extended to include the presence of inert components. In addition, a new reaction rate model was developed that adequately described the experimental data[I3] which were obtained using thin pellets.

REACTION KIhWICS

Many investigators have studied the oxidation of sul- fur dioxide with the objective of establishing the kinetics of the reaction and a large number of models have been proposed. However, very few of these models show similarity of functional form and practically none agree in the values of the model parameters. Weychert and Urbanek[ 141 in a careful evaluation of rate models pro- posed up to 1968, pinpointed both the inadequacy of these models and the shortcomings of the experimental data used to establish them, and this inadequacy persists in more recent models [15].

Livbjerg and Villadsen[l3] reinvestigated the reaction with, presumably, the objective of establishing a rate model in order to subsequently study and model diffusion and reaction in large catalyst particles. In view of the wide disagreement of previously proposed models, undertaking the former task, that is, establishing a kinetics mode1 for their catalyst and their experimental conditions was indeed a very sound approach. However, Livbjerg and Villadsen did what most other investigators had done before them, i.e. they restricted their study to a very narrow range of experimental space by choosing to use a feed stream of constant composition. This restric- tion resulted in the correlation between the concen- trations of the reaction species, thus leading to sub- stantial loss of discriminating power in evaluating pre- viously proposed models.

Livbjerg and Villadsen tested twelve rate models, four of which fitted the data obtained in the range of 454-

447

448 MARK E. DAVIS et al.

484C relatively well, and the best of which was the following model initially proposed by Traina er a1.[16]

where ,kl = Ps,,/KpPmP g and 1 Is = 2. This model was considered best in the sense that it gave the minimum relative error for k based on a weighted sum of squares. However, the estimates of the rate constant, k, did not obey the Arrhenius law and its values from two different runs (Series 2 and 3[13]) at the same temperature differed by about 7%. These discrepancies prompted the present authors to take a closer look at the data, numerical values of which were generated by reading off the graphs presented by Livbjerg and Vifladsen after suitable magnification. It must be pointed out that esti- mates of k obtained from the generated data according to the articles procedure had an average absolute deviation of 1.3% from the reported values implying that the data were not significantly different from the original. The estimates of k at 453.6C are plotted against conversion in Fig 1, which clearly demonstrates not only the wide range of the data but also the very strong dependence of k on conversion. Since the rate constant of an adequate model must be independent of conversion, these data lead to the conclusion that the model of eqn (1) is inadequate.

The above inadequacy of the best model,eqn (1) with l/s =2, led to testing other models, with particular attention given to the driving potential term of eqn (1). Since the partial pressure data were correlated, no attempt was made to alter the denominator term, and the

jaJ, 3.7 - x-

36 -

35-

. 3.4 - Y

2 x $* xx 3.3 -

s

x x

, x 32- xx _

8 0 3.1 - m

o-

Fig. 1. Dependence of the rate constant of eqn (1) on conversion. Data [13]: X, series 2; 0, series 3.

following model,

where Kss and Kso, are the functions of temperature reported by Traina ef ai.1161, was found (a) to bring the two estimates of k at 453.6C to within 1.7% of each other and (b) to yield rate constant estimates that fol- lowed the Arrhenius law to a very good degree as Fig. 2 clearly shows. Though a plot of k versus conversion showed some trend, the range of k was very much shorter than that of k, eqn (l), and this behavior is additional evidence in favor of the adequacy of eqn (2) in describing the available experimental data. The apparent activation energy for the sulfur dioxide oxidation ac- cording to eqn (2) is 14,185 cal/mol and the Arrhenius equation for k is given by:

In k = 2.965 - 7139 T

The model of eqn (2) is of a form that has most frequently appeared in the literature for the oxidation of sulfur dioxide and which corresponds to the stoi- chiometric reaction although it may not account for the elementary steps of the reaction which have not been elucidated as yet. So, in summary, eqn (2) is an empirical model which, however, satisfies the minimum require- ments that a kinetics model should fulfill, i.e. good des- cription of the experimental data and a rate constant that folIows the Arrhenius law, and in view of Urbanek and Trelas very recent review[lSl this is the only model proposed to date that possesses these properties.

The presently proposed rate eqn (2) was used to evaluate the equations describing diffusion and reaction in the large (6 X 6 mm) pellets, although eqn (I) was also used for comparison purposes.

MATEEMATICAL MODEL

Consider a mixture of n chemical species the first m of which participate in a single chemical reaction

2 viAi = 0

I5 I I I

0.51 I 1 I 1.31 133 I35 I 31

fx103 (K-l)

Fig. 2. Arrhenius plot of rate constant of eqn (2)

SO, oxidation using the dusty-gas model 449

in a porous catalytic particle, the remaining compounds being inert. In writing the material and energy balances in the particle it was assumed that the smooth field description for the fluxes, the concentrations, and the temperature is adequate and that the pellet takes one of three shapes, namely infinite slab with parallel plane faces (thickness Za), infinite cylinder of circular cross- section, and sphere. It was also assumed that external heat and mass transport resistances are negligible. Then the continuity equations are:

f&(zNJ=v,ri=l,....nt (5)

$&zNi)=O i=m+l,...,n (6)

(7)

where the mass fluxes are given by the dusty-gas model of Mason et nl.[4] in one dimension, i.e.

and where thermal diffusion and transpiration as well as heats of mixing have been neglected. Sensitivity analysis of the CO, methanation reaction[l7] showed that the viscous flow term does not have any appreciable effects on pressure variation or effectiveness factor, which is in agreement with the findings of other investigators[6,18] for other reaction systems. Based on these results the last term on the right hand side of eqn (8) has been dropped from the following developments.

For the inert components, eqn (6) with the appropriate boundary conditions yields

Ni=O i=mtl,...,n (9)

while for the simple geometrical shapes under con- sideration it has been shown[5] that

Equations (9) and (10) were used to express the flux relations aS follows in terms of component 1:

P dXi Xi dP _ N, ________ RT dz RT dz v,

where

where

j*i n

+v, z 3 i=m+j 0; (11)

= 1 , . . . , m, and

P dxi Xi do _ h, ----- mA RTdz RTdz --xxix j-1 0; (12)

=mtl,..., II. Equations (5), written for the key

component flux, (7), (11) and (12) along with the boun- dary conditions

dT N, =O,z=O,at.z=O (13)

completely specify the system, and their solution leads to the non-isothermal effectiveness factor, T, using the fol- lowing equation

t = (a + l)N, av,r

The above set of equations is more easily solved in its dimensionless form which is given by:

for i = 1,. , m - 1

fori=tntl,...,n

$ $@ = #?R/(l t D,e/D;z) (15)

rl = (a + l)N(l+ 0,=/D;,) 42

dt N=O,s=O,al=O

n=l.O,f=l.O,_ri=x,i=l ,..., m-l, WI+1 ,...,natJ=l

where the dimensionless quantities are defined in the notation, the Thiele modulus 4 is based on surface reaction, the Prater number fi is based on total surface concentration, and both 4 and B are based on a reference diffusivity defined by l/D = (l/D, t l/D;,).

For the sulfur dioxide reaction studied by Livbjerg and Villadsen[l3], the system comprises four components, three of which are reactive. All the experiments were performed at one atmosphere total pressure using an internal recycle reactor under operating conditions that eliminated interphase gradients. Under the latter con- ditions the values of mole fractions at the external sur- face of the particles are easily calculated from the material balances, the conversion, and the feed com- position which was set at 7 vol.% SO*, 11 vol.% O,, and

450 MARK E. DAVIS et al.

82 vol.% NT for all the runs. Intraphase diffusion-limited experiments using 6 x 6 mm cylindrical pellets were car- ried out at 453.6 and 484.W. The set of eqns (15) were solved using rate models (1) and (2), for comparison purposes, and effective diffusivities calculated from

D; =; [97OOr,( ?7Mi)]

for the Knudsen diffusivity [ 191, and

10~3T-75(1/M, + l/Mj) I (17) P[(Z &)il f u&.);3]2

for the binary diffusivity[20]. For calculating the effective diffusivities, physical property data reported in [13] for fresh catalyst were used, namely, pc = 1.33 &m, e=O.44, r,,, =3190x Wcm, and k= 7.0 X 1OW cal/cm/seclC, while 7, the tortuosity factor, was treated and an unknown parameter which would be estimated from the data on experimental effectiveness factors given in the same paper.

NUMERICAL SOLUTION METHOD

Solution of the boundary value problem defined by the set of eqns (15) was achieved by the development of a generalized software package, named EFFECT, in which the user can specify reaction rate model, chemical spe- cies, and parameters to suit his/her reaction system. Code EFFECT is available upon request from the authors.

The program EFFECT achieves solution of eqns (15) by calling COLSYS[Zl-231 which is a code designed to handle boundary value problems for mixed order sys- tems of ordinary differential equations. The latter code is based on spline collocation at Gaussian nodes, the pie- cewise polynomial approximation being given in terms of B-splines. Collocations sound theoretical footing has resulted in the derivation of effective algorithms for error estimation and adaptive mesh refinements, which are implemented in COLSYS. The problem is solved on a sequence of discrete meshes until user-specified error tolerances are satisfied. For the case of this paper, the error tolerances were taken to be 10m4. This choice was based on the results of preliminary experiments [ 171. All computations reported in this paper were performed in double precision on an IBM-370-165 computer.

RFSULTS AND DISCUSSION

The mean-free path of SO, at the experimental con- ditions explored by Livbjerg and Villadsen (454-48OC) is of the order of magnitude of the average pore radius of the catalyst used by the above investigators, and, there- fore, transport into the large pellets was almost equally affected by molecule to wall collisions as well as mole- cule to molecule collisions, and these processes are correctly reflected by the model of eqns (15). These equations account for Knudsen streaming and bulk diffusion as well as for pressure, temperature and com- position variation in the pellets, and the effects of the

latter variables on transport and reaction. It should, therefore, be expected that a single value of the tortu- osity factor, 7, the only unknown parameter, would result in adequate description of all the experimental data (Series 2, 3, and 5) for the large particles, provided the catalyst pellets had not undergone some drastic change in chemical or physical properties.

The experimentally determined effectiveness factors for Series 2 and 3 were determined at the same bulk temperature and overlapping, for most part, conversion range and found to have significantly different values. In particular, the factors of Series 3 were significantly lower than the factors of Series 2, and of the same order of magnitude as those of Series 5, the latter having been determined at temperature 30C higher. Since the effectiveness factor at lower temperature must be higher than those at higher temperature, all else remaining the same, the data for the large particles of Series 3 indicate that they had been affected by unknown factors and thus were not used in evaluating the adequacy of eqns (15).

Based on the above arguments, it was decided to estimate the tortuosity factor, 7, from one experimentally determined effectiveness factor, namely, that of Series 2 at 50% conversion. The estimate of 2.70 was obtained by requiring the effectiveness factor predicted by eqns (IS) to match the experimental value. Though this estimate of 7 is not the best in a least-squares sense, it should still be expected to describe the experimental data well, pro- vided that the model is adequate and the data have been obtained from catalyst of the same physical and chemical quality. Figure 3, shows experimentally determined and predicted, using eqns (2) and (IS), effectiveness factors for 453.6 and 484C, with the tortuosity factor kept constant at 2.70. The close agreement between the experimental points and the predicted curves is rather striking, especially striking if it is kept in mind that the average pore size, r,,,, the porosity, @, of the particles, and the tortuosity factor have been kept constant, although these properties should, on physical grounds, be expec- ted to vary for different catalyst batches and they do [13].

Figure 4 shows typical component profiles in the pellet. In all the simulations, the total pressure in the

~r-----l OB

c t

o-- I / 0 02 0.4 0.6 08 ID

Conversion I a

Fig. 3. Plot of predicted (solid curves) effectiveness factor. Data [131: A, series 2: 0, series 5.

SO, oxidation using the dusty-gas model 451

particles was almost constant, being in the range of 0.97 to 1 atm, and the temperature rise was less than 3C. Figure 5 shows a plot of effectiveness factor vs Thiele modulus with conversion as parameter at bulk tem- perature of 453.6C. It should be noted that the low effectiveness factor at low Thiele modulus is due to the fact that the modulus has been defined in terms of reaction rate at the external surface. Approximately 60 sec. of CPU time was spent in generating the data of Fig. 5, which shows how modest the computer time requirements in solving the set of eqns (15) is. However, this is not always true. At high conversions and high Thiele modulus, the system of equations becomes stiff and the CPU demand becomes high, and this is the reason why the predicted effectiveness factor curves in Fig. 3 based on rate equation (2) were not extended beyond 90% conversion. Other computer runs using rate equation (1) predicted the same behavior as in Figs. 3-5, on the scales of these figures.

In summary, the close agreement between the experimentally determined and the predicted effective- ness factor is strong evidence that the mathematical model given by eqns (15) is indeed very adequate in describing the oxidation of sulfur dioxide in an industrial

20, I I

0 05 IO

Dlmenslonless Radius. z

Fig. 4. Plots of predicted component profiles in a catalyst pellet at 70% conversion, 726.6 K and Thiele modulus equal to 0.4.

catalyst, and this evidence, in turn, validates the excellent properties of the dusty-gas flux relations to properly describe transport in complex systems.

Acknowledgement-Ackwledgement is made to Anne Leigh of the University of Kentucky Computing Center for valuable assistance in the debugging of EFFECT.

NOTATION

reaction component radius of the cylindrical pellet, spherical pellet,

half the thickness of the plate shaped pellet, cm permeability of the pellet, cm2 effective Knudsen diffusivity of i, cmlsec effective binary diffusivity of i in j, cm/sec (l/D, + l/D;*) heat of reaction, cal/gmole rate constant, eqn (l), gmolelseclg catlatm rate constant, eqn (2). gmolelseclg cat/atm2 effective thermal conductivity, cal/cm/sec/Y (D,)lDi molecular weight of i, g/mole number of reacting components total number of components rt 2 m molar flux of i, gmol/cm/sec aN,RT/v,DIP dimensionless key flux total pressure, partial pressure of ;, atm universal gas constant mean pore radius, cm reaction rate function = pcrso2, gmole/cm3/sec reaction rate function of SO1, gmolelgcatlsec rjr dimensionless reaction rate absolute temperature, K T/T dimensionless temperature atomic diffusion volume of i mole fraction of component i space coordinate, cm

Greek symbols geometric factor: for slab = 0; cylinder = 1;

sphere = 2 AH,DP/R (TI)k, Prater Number DW;,) effectiveness factor, calculated, experimental

porosity of pellet viscosity stoichiometric coefficient of i P/P dimensionless pressure density of the catalyst, g/cm tortuosity factor a[rR7j PD)], Thiele Modulus

Superscripts 0 at reactor feed conditions s at pellet external surface conditions

O,lU 0.1 Thiele Modulus, .+

Fig. 5. Plots of predicted (solid curves) effectiveness factors at different conversions: 1 at 70%; 2 at 50% 3 at 30%; 0, data from

1131.

REFERENCES

[I] Thiele E. W., Ind. Engng Chem. 1939 31 916. (21 Zeldovich Ia. B., Zhur. Fiz. Khim. 1931 13 163. [31 Aris R., The Mathematical Theory of Diffusion and

Reaction in Permeable Catalysts, Voi. 1. Clareidon Press, Oxford 1975.

452 MAFX E. DAVIS et al.

[4] Mason E. A., Malinauskas A. P. and Evans R. B. III, I. I161 Traina F., Cucchetto hf., Cappelli A., Collins A. and Dente C/rem. Phys. 1967 46 3199 and Refs. therefrom. A., Chim. & Ind. 1970 52 329.

[S] Jackson R., Transport in Porous Catalysrs. Elsevier, Am- I171 Davis hf. E., Ph.D. Dissertation, University of Kentucky, sterdam 1977. Lexington 1981.

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13.