Ind. Eng. Chem. process Des. Dev, l g 8 3 , 22, 135-143
liquid-liquid dispersion. Since equal power per unit mass results in equal drop size, this implies that a bigger impeller can be employed to produce the same final drop size faster using the same amount of energy. Conclusion 1. Unsteady-state drop size, ita distribution, and the minimum transition time required to reach steady state during the initial period of liquid-liquid dispersion have been measured by using a microphotographic technique and a light transmittance method. 2. The average drop size was found to follow the exponential decay rule. 3. The drop size distribution changes from a very wide (multi-modal) distribution to a narrower (normal) distribution to, finally, a very narrow (skewed or log-normal) distribution as drop size becomes smaller and smaller. 4. The minimum time required to reach a steady state is very strongly dependent on impeller size and speed and on tank size. At the same power input per unit mass, a larger impeller is more efficient. Acknowledgment This material is based upon work supported by the National Science Foundation under Grant CPE-8006666. Nomenclature a = interfacial area per unit volume, m-l DI = impeller diameter, m DT = tank diameter, m d = particle or droplet diameter, m ds2= Sauter mean droplet diameter defined by eq 2, m dS2* = Sauter mean droplet diameter at steady state, m I = emergent light intensity Io = incident light intensity
135
ml, m2 = constants used in eq 1 n = number of drops N = impeller stirring speed, rpm T = fractional light transmittance = I/&,, dimensionless AT = fluctuation in T readings, dimensionless t = time, min t , = minimum transition time required to reach steady-state drop size, min Greek Letters a,6 = constants used in eq 4 y = constant defined in eq 5 t = rate of energy dissipation per unit mass of fluid, W/kg p = viscosity, N.s/m2 p = density, kg/m3 9 = Kolmogoroffs length scale, m u = standard deviation based on dS2,m ‘$d = volume fraction of dispersed phase, dimensionless Literature Cited Coulakglou, C. A.; Tavlarides, L. L. AIChE J . 1976, 2 2 , 289. Lee, J. M. Ph.D. Dlssertatlon, University of Kentucky, Lextngtm, KY, 1978. McCoy, B. J.; Madden, A. J. Chem. Eng. Sci. 1969, 2 4 , 416. Narslmhan, G.; Ramkrishna, D.; Gupta, J. P. A I C M J . 1980. 2 6 , 991. Roger. W. A.; Trlce. V. G., Jr.; Rushton, J. H. Chem. Eng. Prog. 1956, 52, 515. Ross, S. L.; V e h f f , F. H.; Curl, R. L. Ind. Eng. Chem. Fundam. 1878, 77, 101. Ramkrishna, D. Chem. Eng. Scl. 1974, 2 9 , 987. Skelland. A. H. P.; Lee, J. M. Ind. Eng. Chem. Process D e s . Dev. 1978, 77, 473. Skelland, A. H. P.; Lee, J. M. AIChE J . 1981, 27, 99. Sprow. F. B. A I C M J . 1967, 73, 995. Vermeulen, T.; Williams, G. M.; Langlols, 0. E. Chem. Eng. Prog. 1955, 57, 85-F.
Received for review March 23, 1982 Accepted August 16, 1982
Presented at the Fall 1981Annual Meeting of AIChE, New Orleans, LA,Nov 8-12,1981,
Model Building in Complex Catalytic Reaction Systems. A Case Study: p-Xylene Manufacture Nlgel H. Orr’ and Davld L. Cresswell’ Systems Engineedng Grow, E. T.H. Zentrum, CH-8092 Z m , Switzerlend
Davld E. Edwards I.C.I. Petrochemlcels Dlvlsiim H.O., P.O.B. 90, WHton, Cleveland TS6 &I Englend €,
A strategy of model building In complex catalytic reaction systems is described based on the deployment of different types of laboratory reactors and independent measurement of pore diffusion within a single-pellet diffusion cell. The approach is applied to the kinetic modeling of simultaneous isomerlzation and disproportionation of a mixture of xylenes over a commercial silica-alumina catalyst, which Is subject to continuous W i n g and pore diffusion limltatlons. A comparison between the predictions of product distributions In an integral reactor containing commerdal-size catalyst beads, employing a reactor model in which all effects were previously identified by independent experiment, and observed values shows that the model can pvdctp-xylene “lifts” to withi 16% error and integral selectivities to within 11% error over wide ranges of temperature, pressure, and space velocity.
1. Introduction
This paper is concerned with the quantitative &scription of the yield and selectivity of p-xylene formation by isomerization with simultaneous disproportionation reacBP Chemicals, Grangemouth, Scotland. 0198-4305/83/1122-0135$01.50/0
tions and catalyat deactivation by deposition of coke. Mass transfer through the diffusion film is shown to have no important influence on the complex chemistry. However, pore diffusion is of significance and is studied experimentally (e.g., with catalyst pellets of various sizes in both reactive and nonreactive conditions) as well as mathematically, using diffusion equations. 0 1982 Amerlcan Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
136
+
T SINGLE PELLET TRACER STUDIES
kinetics
I
\
I
I
r
f-
/
I
--
Table I. Catalyst Properties, Experimental Conditions, and Experimental Patterns Studied Catalyst ex-plant amorphous silica-alumina surface area (BET): 2.64 X lo5 m2/kg porosity: 0.5 m3 void/m3 bulk density: 1085 kg/m3 unimodal pore size distribution-mean pore radius of 4x m (40A ) Experimental Conditions temperature: 683-763 K total pressure: 101.3 kPa partial pressure of xylenes: 10.13-101.3kPa maximum contact time of catalyst batch: 40 h % conversion: highest, 9-10; lowest, 1-2;average, 3-4
I
Experimental Patterns
I
I
feed o-xylene+ N, m-xylene + N, p-xylene + N,
no. of pattern meas
no. of responses repli- fitted in cates kinet anala
1
147 12 r, 53 3 r*r rp 3 60 3 rm a Refers to rate of formation (subscripts m, 0,p refer t o m-, 0-,and p-xylene, respectively).
A I I
I I
I model 7
Y
PLA T STUDIES
Figure 1. The overall strategy of model development.
The overall model-building strategy is summarized in Figure 1. Data gathered from the differential reactor permit the determination of isomerization and coking kinetics on crushed catalyst beads, operating in the chemical rate-controlling regime. These data, combined with product distributions measured over an integral reactor, also containing crushed catalyst beads, enable a lumped kinetic model of the slower disproportionation reactions to be determined. The reaction kinetic model is now combined with pore diffusivity data, obtained in a parallel study, to enable effectiveness factors to be computed and compared with measured values for particles of various sizes. Finally, the single-particle reaction-diffusion model is incorporated into a reactor model and the whole is integrated to give product distributions over an integral reactor bed packed with commercial-size catalyst beads. The predicted results are compared with measured product distributions, obtained from an experimental pilot plant, to provide a most stringent test of the whole modeling procedure. Several global iterations over this “flowsheet”prove to be necessary before satisfadory reaulta are obtained. In such a varied experimental program as this it is inevitable that a balance must be struck between the effort put into each stage and the need to keep the final engineering design objectives of the model in sight. Several opportunities therefore remain for model improvement, particularly in the description of coking and disproportionation kinetics if changes in the reactor feed composition were contemplated. 2. The Differential Reactor Studies In order to study the kinetics of xylene isomerization and to quantify the coking of the catalyst, a differential reactor was employed. This comprised a glass tube 0.025 m in diameter and 0.9 m in length, the central part of which was packed with crushed catalyst to a depth of 0.075
2
m; the remaining volume was filled with inert packing of similar size. The reactor and preheater were heated electrically by three separate coils, each one supplied with a temperature controller adjusted to maintain a uniform temperature through the catalyst bed. Vaporized xylenes plus Nzinert diluent passed downward through the bed, and the product was condensed, collected, and passed to a gas chromatograph for analysis. Preliminary experiments showed that pore diffusion and external gas film mass transfer resistances could be eliminated by crushing the catalyst beads to between 0.4 and 1 mm (which is 5-10 times smaller than original) and supplying a xylene flow rate >1.0 X g of liquid feed/s. The crushed packing was observed to be twice as reactive as the original catalyst beads. The ranges of experimental variables investigated, the catalyst micromeritics, and the experimental patterns adopted for determining the isomerization and coking kinetics are given in Table I. The choice of patterns involving a pure isomer plus inert N2diluent greatly facilitated the determination of reaction kinetics since, essentially, each pattern allowed the independent estimation of a small subset of kinetic parameters. Further, previous work on adsorption isotherms (Orr, 1981) had indicated low surface coverages and, therefore, only weak interactions between the xylene molecules on the catalyst surface under reaction conditions. 2.1. Catalyst Deactivation. Figure 2 shows all the data obtained at 723 K giving the measured rate of appearance of m-xylene starting from a pure o-xylene feed as a function of time from beginning a given run. Altogether, seven different batches of ex-plant catalyst were studied. Each batch had been regenerated at 773 K in an O2 flow for 24 h prior to being put in the reactor. In some of the runs, such as 30, 39, 51, 87, and 156, the catalyst had not previously been exposed to o-xylene in the reactor for different total times. In others, such as 32,34,38,148,154,155, and 158, the catalyst had already been exposed to o-xylene in the reactor for different total times. Before these runs were started, the catalyst was regenerated in situ at 773 K for 4-8 h. It is evident that the catalyst undergoes fairly rapid deactivation. Close inspection of Figure 2 shows that the deactivation is reversible-no permanent loss in activity was observed after a total of 40 h exposure. These results
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 197 31)-
0 mxylene
-Model prediction p-xylene
-1"-
Temperature Pressure, rim owline
t
0.xyt.m feed
particle diameter '7X10'4m.
"4
5 E
-Y 4> x
2.0-
I 0"
I
I
I
I
8
4
1.5-
4
b w
5K
'B
1.0-
-
pressure 100kPa. temperature723K. -2 m o l a r space velocity -5.5~10 +k particlediameter -7 X10-4,. g.ca . S ~ C
-o-xylene
feed
1
I
1
I 2
1 3
- 0
0
m
& . . A
0.25 0.50 0.75 Molar spacevelocity (xlOmol/kg.sec.)
,
1.0
Figure 3. Product distribution obtained from o-xylene feed.
0
n
x,
1
0.0
2 K 4
0"
723K 100kPa. 1 hr.
I 4
TIME( HRS)
I
1
I 6
I 7
for each isomer. This leads us to speculate that under reaction conditions (680-760 K)surface coverages of the different xylene isomers will be relatively low. For the reaction A F= B occurring on a single active site, the rate of formation of B would then be given by (1) rB = k d b A - PB/K) where k is the apparent rate constant for the forward reaction, 4 is the fraction of sites remaining active, p refers to partial pressure, and K is the equilibrium constant. For a differential reactor fed with a mixture of A plus inert diluent, the reverse reaction may be neglected with the result
Figure 2. Rate of m-xylene appearance vs. time "on-stream".
indicate that coking is the major, if not the only cause of deactivation-a not unexpected result on an ex-plant catalyst which had undergone many hours of contacting with xylenes and many hours of regeneration with oxygen. The data show considerable variation between runs, especially during the first few hours on stream. It is believed that the main cause of this variation resides in intrinsic differences in activity between the cataly& batches, which were taken from different sections of the plant reactor, and therefore had suffered different extents of sintering during frequent regenerations. Other contributory causes may be due to the following factors. (a) Total regeneration of the catalyst was not checked by, for example, COz measurement in the off-gas. Thus, the initial activity of the catalyst could have varied from run to run. (b) The temperature control on the apparatus was not sufficiently accurate (h3 K). 2.2. Product Distribution and Reaction Network. A convenient method for determining the proper network of isomerization reactions consists of feeding each xylene isomer in turn to the differential reactor and observing the product distribution a t different molar space velocities. One such experiment employing a pure o-xylene feed is shown in Figure 3. At high space velocities only m-xylene is obtained as product. The same result was found when p-xylene was used as feed. On the other hand, with mxylene feed, similar yields of o-xylene and p-xylene were observed at all space velocities. These results point toward the reaction scheme o-xylene m-xylene p-xylene with no direct interconversion between the ortho and para isomers, supporting earlier work by Hanson and Engel (1967) and Cortes and Corma (1978). 2.3. Isomerization and Coking Kinetics. Some prior measurements (Orr, 1981) on xylene adsorption at somewhat lower temperatures (548-613 K)revealed linear isotherms over the total pressure range considered (0-0.6atm)
where G is the mass flow rate of product B (molecular weight MB), AX is the difference in mole fraction of B between the reactor exit and feed streams, and W is the mass of catalyst in the reactor. Catalyst coking in xylenes isomerization has been studied in detail by Takaya et al. (1971, 1972a,b), who observed an abundance of trimethyl and tetramethyl diphenylmethanes in the coking produds. If we assume that two adjacent adsorbed xylene molecules are involved in the formation of a coke precursor P, ultimately leading to coke lay-down according to the scheme A.X + A.X slow P X coke + X
-+ -
then a balance on active sites leads to the relation (3) where k, is the apparent coking rate constant. For constant partial pressure of A through the reactor, eq 3 can be integrated to give
9(t) =
1
1
+ k$A2t
(4)
From (2) and (4) rB =
kpA
1
+ k$A2t
(5)
Equation 5 can be cast into the "linearn form 1 k$A2t rB k + - k
e=-
which is convenient for graphical testing of the proposed model and for providing preliminary estimates of the parameters k and k,. 2.3.1. Parameter Estimation. Joint estimation of the isomerization and coking kinetic parameters was carried
Ind. Eng. Chem. Process Des. Dev., Voi. 22, No. 1, 1983
138
Table 11. Sequential Reduction in Model Complexity no.of Smh X parameters
-
~
~
~~
~~
~
14 10
-7.11 -6.98
8
-7.02
6 5
-7.00 -7 .OO
~
~
comments
Equilibrium coefficients were introduced from thermodynamic data (Taylor et al., 1946). The rate of m-xylene coking is not significant; i.e., hcoke,M = 0. The confidence intervals of the 0 -and p-xylene coking rates at 723 K overlap. Putkcoke o - kcoke,p. The activation energies E, and E, are approximately equal. Set E, = E,.
out by minimizing the nonlinear multivariate objective function
s =2-INlIn Ml + N3In M 3+ NzIn (det M,)]
(7)
where N,is the number of measurements in pattern j and Mj is the moment matrix of residuals for pattern j . Now, for patterns 1 and 3 (Table I), each involving a single reaction, M is simply a scalar sum of squares of residual reaction rates. For pattern 2, on the other hand, involving 2 simultaneous reactions, M is a 2 X 2 matrix. If we represent the isomerization reactions in the following form Ortho-
ka, rM1 ' kl, r,, '
kl? ' p ,
vk,,rMa
Table III. Final Values of Parameters (*Indicate 95% Marginal Confidence Intervals) from Differential Reactor Analysis k , (723 K) = (1.01 r 0.07) X lo-' mol/kg of cat. s (kPa) k, (723 K) = (1.36 * 0.10) X lo-' mol/kg of cat. s (kPa) E, = E, = 100.5 t 10.5 kJ/mol k, (723 K) = (1.30 t 0.24) X (kPa)-'s-l E , = -72.4 f 32.2 kJ/mol n
. 18
I,
- a
:
r\
A
para-
-
763K 743)(
Molar space velocity Pressure Timeonline Particlediameter
= 5.5 X162mol/kgg.sec = 100kPa. = 7X10-4m n
= 1 hr.
723K 683K Model prediction
"
0
s
0
then 2
d y l e n e partial pressure (kPa)
whereas
-
Figure 4. Predicted and observed rates of m-xylene appearance.
An F test was made on the residuals for pattern 1, in which 12 replicates were carried out. The F value of 1.93 lies within the range 0 IF I2.45, which is the region over which the null hypothesis holds
with
52lack of fit
reject Hn:F =
and klpM u=l
)x
+ kcoke,@Mzt rpmeaad
-
In order to minimize the function S in eq 7, the computer program developed by Klaus and Rippin (1979) was utilized. Taking into account the temperature dependence of the isomerization and coking rate constants, 14 parameters are initially present in the overall model. However, by means of a sequence of analyses in which the complexity of the model is gradually reduced and the fit examined at each stage, nine of these parametem are in fact, redundant. The progress of t h i s sequential analysis is summarized in Table 11. Results from the final much-simplified 5-parameter model are given in Table 111.
pure error
=lforF=
S21,ck of fit S'pure error
>1
if F 2 F,(m,n) where a is the confidence level selected, m is the degrees of freedom of the lack of fit s u m of squares, and n is the degrees of freedom of the pure error (m= 131, n = 11, a = 0.95). Some of the fits of the model are given in Figures 3 and 4. It should be mentioned here that the differential reactor data do not permit discriminationbetween different coking models. A number of other forms of coking model were found to fit the data equally well. The particular form finally selected here was therefore based on independent work devoted specificallyto elucidating the main products of the coking reactions. There is clearly scope here for a more fundamental approach describing the deactivation as a function of the coke content of the catalyst. Introduction of this experimental variable, which is more directly linked to the cause of the deactivation, would probably allow a better prediction of the catalyst activity for different feedstocks. This alternative approach is described by Froment (1976). 3. Integral Reactor Studies Employing Crushed Catalyst The first set of experiments carried out in the integral reactor employed crushed catalyst material also, with the
Ind. Eng. Chem. Process Des. Dev., V d . 22, No. 1, 1983 198
i
tesr 723K ;
-model
5.:405 kPa
prediction
\
(1.0)
10
+
n inert packing
e3
catalyst
4
f
Figure 5. Integral reador assembly: (1) xylene feed; (2) pump; (3) vaporizer; (4) reactor; (5) thermocouples; (6) sample lines; (7) pressure gauge; (8) water condenser; (9) valves; (10) catch pot. Table IV. Experimental Conditions for the Integral Reactor temperature 673-773 K total pressure 132-405 kPa feed composition 60%m-xylene, 30%o-xylene, (mol %) 10%p-xylene run time 3h space-time ( W / F ) 0-180 kg of cat./(mol of mixed xylenes/s)
objective of portraying selectivity/conversionrelationships from which a simplified description of overall disproportionation kinetics could be evaluated. A schematic picture of the integral reactor assembly is shown in Figure 5. The reactor comprised a 1 X 0.038 m steel tube packed with alternate sections of inert packing and catalyst bed. Samples could be withdrawn from each of the five inert sections for GC analysis. A near-uniform temperature profile was established along the reactor by controlled electrical heating of the tube wall. The range of experimental conditions studied is given in Table IV. 3.1. Effect of Temperature and Pressure on Selectivity. Figure 6 displays integral Selectivity, defined as the ratio (p-xylene “lift”)/ (p-xylene toluene “lift”) vs. p-xylene “lift” over the reactor. The three sets of results presented indicate the relative effects of a 40 K increase in temperature and a threefold increase in total pressure. Temperature appears to have little, if any, effect on integral selectivity, whereas an increase of pressure lowers the selectivity noticeably. The numbers in the brackets refer to the time “on-line” in hours at which the sample was taken, starting with the fifth bed at 1h “on-line” and progressively working back to the first bed in Figure 5 at 2.7 h “on-line”. Over the total period of sampling the catalyst activity may have decreased by as much as 25% and this must be allowed for in the analysis of the data. The “pressure effect” would seem to indicate secondorder kinetics for xylene disproportionation at modest to low surface coverage. There are six possible bimolecular reactions of the form 2CJ&(CHs)z C&CH3 + C ~ H ~ ( C H S ) ~ that may occur between the three xylene isomers. Neg-
+
I
1
2
I
I
I
3 4 para-xylene *I
I
(mokj-
I
,
7
8
I
9
I
10
Figure 6. Integral selectivity vs. p-xylene “lift”relationships.
lecting the reverse reactions above at low conversions to toluene, that still leava 12 parameters for a comprehensive description of xylene disproportionation, which is far too many for engineering design purposes. 3.2. The Reactor Model. Only three of these reactions were chosen for the purposes of developing disproportionation kinetics ortho- + orthometa- + metapara- + para-
t
+ disproportionation
products
reducing the number of disproportionation kinetic parameters to 6 and the initial total number of parameters to 11. It should be stressed that the choice of these three reactions is entirely arbitrary. Since reactor feed composition, unlike temperature or throughput, is not a primary variable for optimization at plant level, being held within fairly narrow limits, this simplifying tactic is tolerable, just about. Nevertheless, it should be borne in mind @at the model now can only be applied to the given feed composition, indicative of that on the full-scale plant. The following molar balances on 0-,m-, and p-xylene and toluene may be written in terms of mole fractions xo, xm, x p , and XT over a differential mass of catalyst bed 6 W
-dx0- - +kl(Xo/Kmo dW
-hdW -m
- X M ) ~ J ~+T k6(xo6pT)2]/Fm
(8)
- {(h(Xo/Kmo- xm) + k2(xp/Kmp - xm))@T k7(xm@’PT)2j/Fm
-d X-P - -(kZ(xp/Kmp- Xm)@T + ka(xp@T)2)/Fm dW
-dXT - - { k g X , 2 + k,xm2 + kaX,2](6PT)2/2Fm dW
(9) (10) (11)
where Km, and Km are equilibrium constants, given by Taylor (1946); k6, I&, and ka are disproportionation rate constants for 0-,m-,and p-xylene, respectively; Fmis the feed molar flow rate. Within the increment of catalyst bed 6 W the activity of the catalyst is obtained from the equation (12)
where k, is the coking rate constant. The total molar flow
140
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
rate and the total pressure are effectively constant along the reactor. Because the catalyst deactivation is a “slow” transient process, it is safe to omit transient accumulation terms in the molar balances-a usual assumption made in problems of this type. The integrated form of eq 12, i.e. cp=
1 1+ kTT2(x,2
+ X:)t
(13)
may be used directly in the material balance equations as long as the term (x,2 + x:) is approximately constant throughout the reactor, as was the case in the integral reactor experiments. This approximation simplifies considerably the numerical integration of the reactor model. The set of eq 8-11 is integrated numerically from the initial conditions W = 0;X , = 0.3; X , = 0.6; xP = 0.1; XT = 0.0 (14) 3.3. Parameter Estimation. The 11parameters in the reactor model-the five isomerization and coking parameters introduced previously in section 2.3.1 plus the six disproportionation parameters (three preexponential factors pIus three activation energies) were jointly estimated by minimizing the multivariate objective function of Box and Draper (1965) S = N / 2 In (det M(k)) + i(k -k)TV-l(k - k) (15)
Altogether, six independent runs were made covering the range of variables in Table IV. Each run lasted 3 h. Within each run five samples were taken in total, from different points along the reactor bed and at different times on-line. Within each sample the mole fractions xo, x,, xp, and XT were measured by GC analysis. Thus, N = 30 and the moment matrix of residuals M(k) is the 4 X 4 symmetric matrix
where tJi = xJi(measd) - xJi(pred) and J = 0,m, p, and T, respectively. The second term in eq 15 accouqts for prior information on the mean parameter values k and their distributions, as embodied in the variance-covariance matrix V. These were obtained in section 2.3.1 for the isomerization and coking parameters. Terms corresponding to the disproportionation reactions were assigned zero values in the inverse V-l. A desired outcome of the analysis would foresee but marginal changes to the isomerization and coking parameters, reported in Table 111, plus well-determined disproportionation kinetic parameters. Minimization of the objective function S is now much more protracted than was the case previously through the necessity of repeated integration of the model eq 8-11. However, the procedure is rendered relatively straightforward, if not automatic, by the use of a specially developed computer program for treating problems of this complexity (Klaus and Rippin, 1979). Once again, some useful simplifications could be made to the disproportionation kinetics without any deterioration in the overall fit. Because of the low concentrations of p-xylene in the reactor relative to the other isomers, we
Table V. Final Values of Parameters from Integral Reactor Analysis (*Indicate 95% Marginal Confidence Intervals) Isomerization Parameters mol& of cat. s (kPa) h , (723 K) = (8.92 * 0.57) X mol/kg of cat. s (kPa) h , (723 K ) = (1.21 * 0.08) X E , = E , = 111.4 k 9.2 kJ/mol Coking Parameters h , (723 K) = (7.93 * 1.73) X lo-’ (kPa)‘2 s-l E, = -34.1 * 28.9 kJ/mol Disproportionation Parameters h , (723 K) = (1.29 0.23) X lO-’mol/kg of cat. s h, (723 K) = (2.37 * 0.41) X mol/kg of cat. s (kPa)2 E , = E, = 97.6 * 18.8 kJ/mol +_
Table VI. Standard Deviations of Model Residuals and Maximum Ranges of Variation in Measured Responses
response
max range of variation recorded, mol %
std dev, mol %
o-xylene m -xylene p-xylene toluene
22-30 46-60 10-20 0-7
0.85 1.6 1.2 0.6
may set k8 = 0, in spite of the fact that p-xylene is the most easily disproportionated isomer. Further, the activation energies of 0- and m-xylene disproportionation reactions can be set equal, Le., E6 = E,. This reduces the total number of parameters to 8. The final parameter values obtained by integral reactor analysis are given in Table V. 3.4. Discussion. The standard deviations of the model residuals for the various responses and the approximate maximum range of variation of the responses measured over an experimental run are shown in Table VI. Dividing the standard deviation by the range gives normalized standard errors of 10% for each response. Bearing in mind that the integral reactor experiments covered a 100 K range of temperature and involved extrapolation of the isomerization and coking kinetics from the atmospheric pressure differential reactor to -400 kPa (4atm) in the integral reactor, the errors are not unreasonably large. The predicted selectivities are compared with observed values in Figure 6 and give reasonable correspondence. The model fits to the atmospheric pressure runs were much better than indicated by the standard deviations shown in Table VI. These figures were inflated, in particular, by the experimental run made at 723 K and the largest pressure of 405 kPa, for which systematic errors in the model predictions were becoming noticeable. The activation energies (El, E2)in Table V lie within the practical range 70-120 kJ/mol, reported by earlier workers (Brown and Jungk, 1955; Hanson and Engel, 1967; Hansford and Ward, 1969; Norman et al., 1976). The activation energy for xylene disproportionation of 97.6 f 18.8 kJ/mol also appears reasonable in comparison with the spread of values between 60 and 120 kJ/mol reported for toluene disproportionation (Ai et al., 1965; Yashima et al., 1970; Iwamura et al. 1971). The activation energy for coking is poorly determined and the coking rate constants k, (723 K), as measured over differential and integral reactors, differ by 60%-a disappointingly large variation. These results emphasize the need for a more fundamental study of the coking mechanism than that followed here. 4. Intra-Particle Diffusion Measurements It was observed in section 2 that by crushing commercial catalyst beads (diameter 3.4-4.2 mm) to the size range 0.4-1 mm the rate of m-xylene formation from o-xylene
-
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
Table VII.
141
Helium Effective Diffusivities
D e e x IO', mz/s Single Pellet Diffusion Cell regenerated catalyst a 3.48, 4.01, 4.27, 3.34, 3.95
w=o
Single Pellet String regenerated catalyst 3.62 * 0 . 1 6 coked catalyst (22%wt gain) 2.92 * 0.95 Five different pellets.
doubled. Further reduction of particle size was found to have little or no effect on reaction rate. These observations were interpreted in terms of pore-diffusion intrusions. In order to scale up particle size, therefore, it is desirable to predict the effective diffusivity of the catalyst to xylene transport. The usual method employed consists of measuring the effective diffusivity of some inert nonadsorbing tracer, such as helium, from which a tortuosity factor ( 7 ) for the pellet can be determined. Once the tortuosity factor is known, the effective pore diffusivities of other adsorbinglreacting molecules can be estimated. This much simplified approach to characterizing intraparticle diffusion is considered adequate for preliminary engineering design purposes. Helium diffusivity measurements were made on single particles sealed into a miniature diffusion cell using a transient technique similar to that described by Dogu and Smith (1975). Two important improvements were made to the basic design. First, two micro-katharometer detectors were used, one at the outlet of each chamber. This has the effect of making the difference between mean times of appearance of tracer at the outlet of the two chambers independent of the form of the injected tracer pulse. Secondly, through judicious cell design, the dynamic response of a small commercial-size pellet (3.2 X 3.2 mm cylinder) can be accurately determined, thereby removing the need for usinq large pressed pellets, which are unrepresentative of the particles of interest. For purposes of comparison, diffusivity measurements were also made over both regenerated and coked catalyst pellets held in a single pellet string column (Scott et al., 1974). A comparison of results is given in Table VII. The good agreement obtained between the two independent techniques lends confidence to the models and woIking procedures employed. Pellet to pellet variations of effective diffusivity are significant and, when using the diffusion cell approach, it is expedient, though extremely tedious, to carry out measurements on a number of particles. The packed-bed method, on the other hand, cannot detect pellet to pellet variations of effective diffusivity, but rather provides an estimate of DeHaveraged over several hundred particles. This limitation is offset by greater flexibility and ease of use. Catalyst coking leads to a relatively small reduction in the effective diffusivity and the porosity of the particle. A tortuosity factor ( T ) for the pellet may be determined from the equation 7
= PDsp/Deff
value of 7 = 3.8 was estimated. Equations 16 and 17 were then used to calculate the effective diffusivities of the xylenes, toluene, and trimethylbenzenes under reaction conditions (723 K, 150-450 kPa pressure) giving Deff(xylenes) = 1.33 X Deff(toluene) = 1.43 X
m2/s
lo-'
m2/s
Deff(trimethylbenzenes) = 1.25 X
(18) m2/s
These values of the effective diffusivities are incorporated into a mathematical model of a single catalyst pellet, which is used to explore the relationship between catalyst effectiveness factor and particle diameter for regenerated catalyst during the first few hours on stream, in which time the catalyst weight gain is only 1-2%. 5. Differential and Integral Reactor Studies Employing Commercial Catalyst Beads At this stage we have available the necessary data for prediction of integral reactor performance using commercial catalyst beads-isomerization, disproportionation, deactivation, and pore diffusion parameters. In principle there are no adjustable parameters remaining. When the reactions are influenced by pore diffusion, the reactor model must include the particle radial coordinate as an additional dimension. The numerical solution then becomes considerably more complicated. Material balances must be written for an individual catalyst bead contained within a differential element (6 W) of catalyst bed. The pellet equations are coupled to the interparticle gas phase balances through boundary conditions at the pellet surface. 5.1. Integral Reactor Model. A molar balance for component i (i = 0-,m-or p-xylene) in the interparticle gas phase over the differential element of catalyst bed 6W (Figure 7) gives
The term on the right hand side of eq 19 expresses the molar transport rate of component i from the interparticle gas to the catalyst pellet. For a spherical catalyst pellet surrounded by interparticle gas of uniform composition the combination of pore diffusion and chemical reaction leads to the equation
(16)
where Dsp is the Knudsen diffusion coefficient of helium in a straight cylindrical pore of radius rp, given by Dsp = 9.70r,(T/M)1/2 m2/s
AW
Figure 7. Differential elements forming basis for reactor model.
(17)
The bulk diffusion coefficient of He in N2is approximately twenty times greater than Dsp and it is therefore safe to assume that Knudsen diffusion prevails in the fine pore structure ( r p= 4 X 10" m (40 A)). From eq 16 and 17 and the measured values of Dd on regenerated catalyst, a mean
where y is the normalized radial coordinate y = rJR. Superscripts f and s refer to interparticle and intraparticle mole fractions of component i . Appropriate boundary conditions on eq 20 are ax;
-= 0
(21)
y = 1 (pellet surface) xis = X {
(22)
y = 0 (pellet center)
aY
142
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
Table VIII. Data Employed in Effectiveness Factor Comparisons
PT
= 100 kPa; T = 733 K ; p s = 1085 kg/m3 x o f = 1; x m f = x 0 K,, = 0.781 exp[-3k3/T) Kmp = 0.45 Do = D, = D, = 1.33 x 10.’ m*/s h , , h , , h,, k,, h , (see Table V) time “on-stream” = 3600 s f -
The rate equations to be employed in eq 20 are for i = 0-, m-, and p-xylene, respectively r o = (k,(x,B/&o - xmB) + ks(X,”)2dJpT)dJpT
4 I
r~ = ( k l ( x m a - x,”/Kmo) + k2(xma - xpB/Kmp) + k,(xma)2dJpTWT
rp = k,(x,”/Kmp - xma)dJpT (23) The local intraparticle activity function dJ is defined by
I
I
I
I
I I I I
a? 1 Dp (mm.)
1
Y
0 /
3
/ I 4 5
Figure 8. Observed and predicted effectiveness factors. Table IX. Comparison between Observed and Predicted Values for Integral Reactor Containing Commercial-Size Catalyst Beads
W/Fm = 177.8 kg of cat./(mol of feed/s); Time “On-Stream” = 3600 s
Finally, the model is completed by specifying the feed and initial conditions W = 0; x,f = 0.3; xmf= 0.6; xpf = 0.1; t L 0
t = 0; ’$ = 1; w10;0 5 y 5 1 (25) The mole fraction of toluene, which is also measured experimentally, is found by overall material balance = -(1 1 - X,f - X,f - x p 3 2 5.2. The Computational Procedure. A detailed description of and a comparison between various numerical methods of integrating the model equations is given by Orr (1981). A three-dimensional grid is placed over the ( W,t,y) space and each node within the grid is labeled. The computational method involves “marching out” the solution in the time domain, starting from t = 0, and integrating iteratively one time step At at a time. At each time step, the two-dimensional array of node values within the (W J ) space is computed iteratively. The most satisfactory way of doing this is also to “march” along the reactor, starting from W = 0, and integrating iteratively by Euler’s modified predictor-corrector method one A W step a t a time. At each step in W the pellet equations are solved iteratively by orthogonal collocation (Villadsen and Michelsen, 1978) using the N zeros of the Nth degree Jacobi polynomial P N ( 1 ~ o Y yas2 )internal collocation points. In practice, converged solutions were obtained well within the accuracy of experimental measurements with A W = WT/15 N 0.01 kg, At = 900 s, and N = 3. Computation times ranged between 10 and 20 s on a CDC 6400/6500. 5.3. Comparison of Model Predictions with Experimental Data. 5.3.1. Catalyst Effectiveness Factors Measured over the Differential Reactor. The rate of formation of m-xylene from o-xylene was measured for particles of various sizes within the range 0.35-4 mm in the differential reactor. A t a particle diameter of 0.7 mm the reaction rate showed no increase on further size reduction. This asymptotic value waa taken as the intrinsic rate of reaction, unhindered by pore diffusion. The effectiveness factor for a pellet of given size is defined by the ratio 71 = (observed rate of reaction for given pellet size)/(asymptotic rate of reaction for catalyst powder) XTf
The effectiveness factor can be predicted by integrating eq 20 to 24 for a single catalyst pellet. Necessary data for
p-xylene “lift”, mol % reactor pressure, kPa obsd pred temp, K 673 723 773 723 723 723
136 136 136 205 308 411
3.4 6.1 6.4 6.6 6.1 6.3
2.0 5.1 7.7 6.3 7.0 7.5
integral selectivity, % obsd
pred
73.9 70.8 56.4 62.6 56.2 45.5
74.1 70.1 60.2 62.4 54.9 52.3
W / F , = 132.8 kg of cat./(mol of feed/s); Time “On-Stream’’ = 5040 s 673 136 2.5 1.4 76.9 16.5 723 136 5.3 3.9 73.9 72.2 773 136 5.7 6.7 60.3 63.9 723 205 6.3 5.0 68.1 66.7 723 308 6.3 5.7 64.6 62.6 723 411 6.9 6.1 54.1 61.4
W / F , = 56.2 kg of cat./(mol of feedls); Time “On-Stream” = 8280 s 673 136 0.9 0.5 78.0 75.7 723 136 2.5 1.9 76.9 74.0 773 136 3.7 3.9 64.0 69.6 723 205 3.2 2.4 72.7 72.7 723 308 3.6 2.7 63.2 68.9 723 411 3.8 2.9 67.3 70.4
the comparison of observed and predicted effectiveness factors are summarized in Table VIII. Observed and predicted effectiveness factors are shown as a function of particle diameter (D,) in Figure 8. It is apparent that beads of commercial size (3.4-4.2 mm) display relatively low effectiveness factors (-0.5), indicative of pore diffusion limited reaction rates. The predicted trend of q vs. D, is in reasonable agreement with that observed. 5.3.2. Product Distributions Measured on the Integral Reactor. Altogether, six runs were undertaken on the commercial-size beads (3.4-4.2 mm diameter) contained within the same integral reactor as used previously and covering the same range of conditions as in Table IV. A comparison of the observed and predicted p-xylene “lifts” and integral selectivities at various values of the space-time W/Fmand time “on-stream”t is given in Table IX. The standard deviations of the residuals between observed and predicted values are 0.96 and 3.5 in the case of the p-xylene “lift” and the integral selectivity, respectively. The ranges of these variables covered in the experiments were 6 mol % and 32.5%, respectively. Dividing
-
predicted
R.... Regenerating
1 ‘
’--o-s-o-c I
2Days%line4-
.---
toluene I
6
7
Figure 9. Comparison of predicted and observed product distributions for a commercial reactor.
the standard deviation by the range gives normalized standard errors of 16% for p-xylene “lift” and 11%for integral selectivity. Bearing in mind that the model contains no adjustable parameters, every effect having been identified from previous experiments, these levels of predictive accuracy over the wide ranges of experimental conditions studied are very encouraging. The destructive effect of pore diffusion on selectivity is considerable. By comparing the entries in Table M with the values in Figure 6 for corresponding operating conditions on crushed catalyst, selectivity losses of between 10 and 25% are apparent. 6. Comparison of t h e Model Predictions with P l a n t Data The last and most important part of the model development is concerned with the prediction of product distributions from actual plant reactors. Commercial production of p-xylene is carried out in “dustbin” type large diameter adiabatic reactors operating on a “swing” system-two reactors working in parallel while a third is being regenerated. The commercial reactors differ from the laboratory integral reactor in the following respects. (a) The space-time WIF, is about twice that of the laboratory reactor. (b) The reactor feed composition is slightly different from that employed in the laboratory, since a few percent ethylbenzene is present. However, this is essentially inert with respect to the catalyst used. (c) The surface area of the catalyst in the plant reactors was less than that of the catalyst studied. (d) The time “online” in the laboratory was a few hours compared with 2 days on the plant. However, the main variables of temperature, pressure, and particle size were within the range studied. The plant catalyst was also silica-alumina and its mean pore size had been increased by “steaming” to reduce the effeds of pore diffusion. It is assumed that the chemistry occurring on this catalyst is identical with that studied in the laboratory: reaction rate constants are merely adjusted downward in relation to the relative surface areas. Effective diffusivities are increased in relation to the relative mean pore radii and pellet porosities. From Figure 9, which shows the product distribution of one of the reactors of the “swing” system vs. days “on-line”, it can be seen that the predictions are good when the reactor first comes “on-stream” but they deterioate progressively as the catalyst deactivates. In view of the vastly different times “on-stream” between the laboratory and the plant reactor, the uncertainties surrounding the coking kinetics and pore size differences between laboratory and plant catalysts, such discrepancies are only to be expected.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 143
Models capable of predicting the time “on-stream” reactor performance, which are needed for optimizing the scheduling of a “swing”-reactor system, necessitate a much deeper study of coking mechanics than that presented here. Acknowledgment Our thanks are expressed to the Northern Ireland Department of Education for their financial support during this project. We are indebted to I.C.I. Petrochemical Division for financial support and for the use of laboratories and equipment. In particular, thanks are due to J. Gloyn for helping to set up the project; to D. Williams for continuing that interest; to D. Deans for his guidance with the diffusion work; to R. Goddard and D. Clarke for their interest and encouragement; and to the staff at I.C.I. whose help, patience, and interest made this work possible. Thanks are also due to Professor D. W. T. Rippin for enabling this work to continue at E.T.H. Zurich. Nomenclature D,K = effective diffusivity of helium in catalyst pellet, mz/s Di = effective diffusivityof component i in catalyst pellet, mz/s DsP = effective straight pore diffusivity, mz/s E = activation energy, kJ/mol F, = total molar flow rate of hydrocarbons, mol/s kl, kz = isomerization rate constants, mol/ (kg s kPa) k6,k7,k8 = disproportionation rate constants, mol/(kg s ma2) k, = coking rate constant, kPa-2 s-l K,,, K, = equilibrium constants M = moiecular weight p = partial pressure, kPa PT = total pressure, kPa ri = rate of ith reaction, mol/(kg s) = mean pore radius, m = gas constant R = particle radius, m T = temperature, K t = time, s W = catalyst mass, kg X,,,X,,X~,XT = mole fractions of 0-,m-, p-xylene, and toluene y = normalized pellet radius Greek Symbols B = pellet porosity 7 = effectiveness factor p s = pellet density, kg/m3 4 = fraction of sites remaining active 7 = pellet tortuosity factor Registry No. o-Xylene, 95-47-6; m-xylene, 108-38-3;p-xylene, 106-42-3; xylene, 1330-20-7. Literature Cited
k
AI, M.; Echlgova, E.; Ozakl, A. Bull. Jpn. Pet. Inst. 1985, 7 , 46-51. Box, 0. E. P.; Draper, N. R. BIOmetrika 1985, 52, 355-365. Brown, H. C.; Jungk. H. J . Am. Chem. Soc. 1955, 7 7 , 5579-5564. Codes, A.; Corma, A. J . &tal. 1978, 57. 338-344. DOgU, G.; Smkh, J. M. A I M J . 1975, 27, 56-61. Froment, G. F. Roc. 6th Int. Congr. &tal. (London) 1978, 7 , 10-31. Hansford, R. C.; Ward, J. W. J . Catal. 1969, 73,316-320. Hanson, K. L.; Engel, A. J. A I C M J . 1987, 73. 260-266. Iwamawa, T.; Otanl, H. Bull. Jpn. Pet. Inst. 1971. 73, 116-122. Klaus. R.; Rippin, D. W. T. Comp. Chem. fng.1979, 3 , 105-115. Norman, G. H.; Shlgemura, D. S.; Hopper, J. R. Ind. fng. Chem. process Des. OW. 1978, 75, 41-45. Orr, N. H. Doctoral Thsls, E.T.H. ZLirlch. 1961. Scott, D. S.; Wey Lee; Papa, J. Chem. Eng. Sci. 1974, 29, 2155-2167. Takaya, H.; Todo, N.; Hosoya. T.; Mlneglshl, T.; Yoneoka. M.; Oshlo, H. Bull. Chem. Soc. Jpn. 1971, 44,2296-2301. Takaya, H.; Todo, N.: Hosoya, T.; Mlneglshl. T.; Yoneoka, M.; Oshlo. H. Bull. Chem. Soc. Jpn. 19720, 45, 2337-2343; 1972b, 45, 3262-3267. Taylor, W. J.; Wagmann, D. D.; WIIkms, M. S.; Pitzer, K. S.; Rosslnl. F. D. Nan. BW. stand. J . Res. 1948, 37, 95-121. Vllladsen. J.; Mlchelsen, M. L. “Solution of Dlfferentlal Equation Models by Polynomlal Approxlmation”; Prentlce-Hall; Engelwood Cliffs. NJ. 1978. Yashlma, T.; Moslehl, H.; Hare, N. Bull. Jpn. Pet. Inst. 1970, 12. 106-111.
Received for review September 17, 1981 Revised manuscript received July 5, 1982 Accepted July 15, 1982
