1500 Shorter Communications

Acknowledgement-The authors wish to express their tangential surface velocity at time t, gratitude for the fellowship support provided by the cm/set National Research Council of Canada. We,,, modified Weber number

Greek symbols

NOTATION

dimensionless circulation number as defined in Eq. (7)

equivalent diameter as defined in Eq. (6) cm internal nozzle diameter, cm Halligan and Burkhart dimensionless Weber

c

De4 D,

F

number h drop height at time t, from nozzle exit to

drop apex k

m n

r, Re

Re, R.

V” ii,

experimental constant in Eq. (5), poises constant in Eq. (4) constant in Eq. (4) radius of curvature at drop apex, cm nozzle Reynolds number modified Reynolds number Internal nozzle radius, cm maximum nozzle velocity, cm/set average drop tangential surface velocity at

V* time t (equivalent to height h), cmlsec

= VJV” dimensionless average drop

VI

PI

[31

141

Dl

161

[71

PI

Popovich, A. T. and Hummel, R. L., Chem. Engng Sci. 1967 22 21. Halligan, J. E. and Burkhart, L. E., A.Z.Ch.E. J. 1%8 14 411. Garner, F. H. and Skelland, A. H. P., Znd. Engng Chem. 1954 46 1255. Rajan, S. M. and Heideger, W. J., A.Z.Ch.E. J. 1971 17 202. Coulson, J. M. and Skinner, S. J., Chem. Engng Sci. 1952 1 197. Sherwood, T. K., Evans, J. E. and Longcor, J. V. A., Am. Inst. Chem. Engrs 1939 597. Licht, W. and Conway, J. B., Znd. Engng Chem. 1950 32 1151. Rusin, G. Ph.D. Thesis, Michigan State University 1964.

interfacial tension, dynes/cm viscosity of continuous phase, poise viscosity of dispersed phase, poise equivalent viscosity as defined in Eq. (.5),

poise density of dispersed phase, grlcm’

REFERENCES

Chemical Engineering Science, 1974, Vol. 29, pp. 1500-1503. Pergamon Press. Printed in Great Britain

Theoretical analysis of poisoning in a single-pellet diffusion reactor for main reactions

not of first order

(First received 20 August 1973; in revised form 19 September 1973)

In some recent work, Hegedus and Petersen [ l-31 showed how a single-pellet diffusion reactor could be used as a diagnostic tool to study the mechanism of catalyst poison- ing. In their theoretical and experimental work, a first- order main reaction was studied. The orders of the poisoning reaction, however, were varied over a wide range of practical values. The purpose of this short com- munication is to investigate the differences in expected behavior if the main reaction were second- or half-order to ascertain if the semi-quantitative features of the analysis remain unchanged.

Details of the single-pellet reactor and the mathematical model representing its behavior are already available [l-4] and accordingly the description here will be reviewed only very briefly.

The single-pellet reactor consists of a single pellet of catalyst which separates the reaction chamber into two parts: a bulk phase chamber which contains the reactant

mixture and a center-plane chamber which is very small in volume compared to the bulk chamber. Interaction be- tween the above chambers is through the catalyst pellet wherein simultaneous transport and chemical reaction of reactants occurs. Simultaneous measurement of the initial reactant concentrations in the bulk and centerplane cham- bers permits the computation of the rate constant, effec- tive diffusivity, and the effectiveness factor directly, pro- vided the order of the main reaction is known.

When poisoning occurs, the center-plane concentration changes with time and the shape of a plot of the reaction rate vs. the center-plane concentration is characteristic of the poisoning process. From dimensionless plots of the above type, it is possible to discriminate among poisoning mechanisms.

THEORETICAL MODEL

The main reaction in the catalyst is represented by the

Shorter Communications 1501

stoichiometric equation

A *“B,

and the corresponding rate expression is, by assumption, of the powerlaw type:

rate., = - k,C.,“,

where n is the order of the reaction. Poisoning of the catalyst is idealized as occurring

through the following schemes: (i) Impurity poisoning: Poisoning occurs through a pre-

cursor contained in the feed (P). P is irreversibly ad- sorbed or reacted onto the active sites to produce a residual product W.

PL’-w

(ii) Self-poisoning: The poison precursor is one or both of the reacting species A, B. The following three cases are considered:

A&B Parallel self-poisoning,

A&W

At’-B Series self-poisoning,

BL’-W

Triangular self-poisoning, A *‘B.

I@ /b W

The kinetics of the poisoning reactions is assumed to be first-order in the precursor concentration and first-order in the unpoisoned surface area. This assumption is justified because it limits the number of cases considered and pre- vious work [ l] showed the semi-quantitative behavior was not influenced by modest changes in the order of the poisoning reactions.

The reactant concentration is described by a conserva- tion equation written for an element of a porous flat slab of isothermal catalyst. A second equation is necessary to describe the variation in activity distribution with time. The two equations are coupled through reactant and pro- duct concentrations and local activity.

The mathematical form of these equations for the case of triangular self-poisoning mechanism, which of course includes parallel and series self-poisoning as limiting cases, is:

9 =-k(a --(I )C,” =0 Idx2 10 s (1)

k,(ao-ap)C, +k,(no-u,)Ce =& $ ( >

(2) v

where a0 and a, are, respectively, the initial and the poisoned areas per unit volume of catalyst and A, is the area rendered inactive by one mole of poison.

Implicit in Eq. (1) is the assumption that the time con- stant for the poisoning process is long compared to the time constant for the diffusion process. Otherwise, the time derivative of C, would not be zero. This is generally an excellent assumption. Furthermore, the diffusivity is assumed to be time independent. For most cases encoun- tered this is a good assumption, because the carbon con- content of a poisoned catalyst was 0.05 per cent by weight. However, this assumption may require some modification, for example, for heavily coked catalysts.

Equations (1) and (2) can be made dimensionless using the following definitions:

and

to give

Parallel poisoning results when k,/kz is small, and series poisoning results when k,/k, is large (effectively putting k, in the definition of 7).

Impurity poisoning can be described by adding another continuity equation in the impurity concentration (See Reference [ 11).

The boundary conditions on Eqs. (4) and (5) are:

O(O, n) = 1

#,%(r,O) = 1

$T, l)=O

where Z(h,, q) is the steady-state Thiele solution for the unpoisoned pore. This is consistent with the short time constant for the diffusion process.

RJMJLTS AND DISCUSSION Equations (4) and (5) with boundary conditions given by

Eq. (6) were solved numerically on a CDC 6400 computer by using the method of quasilinearization described by Lee [5]. Equation (4) was linearized around a trial solution matrix inversion and iteration.

Results are shown in Figs. l-4, where the relative rate is plotted vs a normalized center-plane concentration. These plots are convenient because time is eliminated as a vari- able, and they are equivalent to plots of an effectiveness factor vs a variable Thiele modulus.

The results are given for the same value of the initial Thiele parameter and for n = 2, 1, l/2.

Shorter Communications

PARALLEL SELF-POISONING (he ~2.5) TRIANGULAR SELF-POISONING (h, = 2.5)

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

,$

A I JI(T,II-JI(O,I)

I - +(o,l) Fig. 1. Relative rate vs normalized center-plane concen-

tration for parallel self-poisoning.

SERIES SELF-POISONING (h, = 2.5)

01 ’ ’ ’ ’ ’ ’ ’ ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

,# A

~ S(T,I)-eOtIl I -eo, I)

Fig. 2. Relative rate vs normalized center-plane concen- tration for series self-poisoning.

The results indicate that the relative rate is more affected by the order of the reaction for the case of parallel poisoning than for series poisoning. The effect of reaction order on triangular poisoning is intermediate be- tween the above cases. This effect follows directly from Eq. (5) where as k,,k,+ k,/k,, and parallel poisoning is approached, the poisoning depends mainly on the reactant concentration and of course is more affected by the reac-

Uniform Poisoning

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

,# A

_ er,l)-+(o,l) I - eo, 1)

Fig. 3. Relative rate vs normalized center-plane concen- tration for triangular self-poisoning. w = 1.

IMPURITY POISONING (h, = 2.5)

0.6

g 0.5 IL

0.4

0. I

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

$

A ~ Hr,l)-+Io,l)

I - S(0, I)

Fig. 4. Relative rate vs normalized center-plane concen- tration for impurity poisoning.

(o- $A) and the effect of reactant concentration is smaller.

At corresponding Thiele parameters, second order kine- tics gives the largest relative rate for a given value of the normalized center-plane concentration. This appears to result from the denominator of &J being smaller for the second order reaction, and as a consequence 4 tends to be greater for a given value of %/Se,.

tion order. On the other hand series poisoning depends on Impurity poisoning is only slightly affected by the reac-

Shorter Communications 1503

tion order of the main reaction, which appears reasonable EDUARDO WOLF because the poisoning process depends only upon the EUGENE E. PETERSEN poison concentration. A similar effect was shown for pore Dept. of Chemical Engineering, mouth poisoning [6]. Unioersity of California,

These results indicate that the semi-quantitative be- Berkeley, California, 94720 havior of poisoning on 3/R, vs 6 plots is similar for 2, 1 U.S.A. and l/2 order main reactions, which indicates that the diagnostic value of these plots does not depend upon a first-order main reaction.

There is an interesting consequence of a non-first-order main reaction in diagnosing the mechanism of poisoning. Hegedus and Petersen[2,3] found that a minimum of three experiments were required to discriminate among the poisoning mechanisms. The reason three experiments were required was that in order to vary h,, the reaction temperature had to be varied. This gave a number of possibilities depending upon the activation energies of the various processes.

It is possible, in principle, to diagnose the mechanism in two experiments if the main reaction is not of first order because h, can be varied by changing the bulk concentra- tion. The apparent order can be established from the ini- tial values of the two runs[7], whereupon the values of h, can be established. Since the ratio k,/k, does not change with changes in the bulk concentration, it is possible to fit the data of one run to the triangular model (or either limit- ing case) and use the same parameters to fit the second case.

It should be emphasized here, however, that a mechan- ism based upon two experiments would be most uncon- vincing and many runs would be made to establish a mechanism of poisoning. The importance of the above discussion is to emphasize the large amount of informa- tion contained in a single experimental run using a single- pellet diffusion reactor.

Acknowledgement-This work was supported in part by the National Science Foundation, under grant GK-34228, and in the form of a fellowship from the Convenio be- tween the University of Chile and the University of California.

X

I) 8

VI

[21

[31 r41

VI

[61

[71

NOTATION

initial catalytically active area, cmz/cm3 pellet poisoned surface area, cm2/cm3 pellet surface area rendered inactive by one mole of

poison precursor W, cm’ concentration of species i, mole/cm’ effective diffusivity of species i, cm’lsec Thiele modulus defined in Eq. (6) rate constant, cm3”-*/mole”-’ set thickness of catalyst pellet, cm rate of the reaction, as a function of time time, set distance coordinate, cm dimensionless distance x/L fraction of the catalytic surface area remaining un-

poisoned dimensionless time dimensionless center-plane concentration dimensionless concentration of species i defined by Eq. (3)

REFERENCES Hegedus L. L. and Petersen E. E., Chem. Engng Sci. 1973 28 69. Hegedus L. L. and Petersen E. E., Chem. Engng Sci. 1973 28 345. Hegedus L. L. and Petersen E. E., J. Catal. 1973 28 150. Hegedus L. L., and Petersen E. E., Ind. Engng Chem. Fundls. 1972 11 579. Lee E. S., Quasilineatization and Invariant Imbed- ling. Academic Press, New York, 1968. Petersen E. E., Chemical Reaction Analysis. Prentice- Hall, New York, 1965. Balder J. R. and Petersen E. E., Chem. Engng Sci. 1%8 !3 1287.