Chemical Engineering Science, 1971, Vol. 26, pp. 1361-1366. Pergamon Press. Printed in Great Britain.
The role of axial dispersion in trickle-flow laboratory reactors DAVID E. MEARS Union Oil Company of California, Union Research Center, Brea, Calif. 92621, U.S.A. (First received 6 August
1970; accepted 25 November 1970)
dispersion or backmixing appears to be responsible for adverse mass velocity effects observed in trickle-flow laboratory reactors. At low Reynolds numbers typical of bench-scale units, the dispersion problem can be at least an order-of-magnitude more severe in trickle-flow than vaporphase operation. A simple perturbation criterion is derived for the minimum reactor length required for freedom from significant axial dispersion effects. It shows that the minimum length increases with both conversion and reaction order, and is inversely proportional to the Bodenstein number. Abstract-Axial
INTRODUCTION THE EVALUATION of kinetic data from fixed-bed catalytic reactors is usually b a s e d on the assumption of "plug" flow, that is, all reactants reside in the reactor a definite time determined by the flow rate and bed volume. H o w e v e r , m a r k e d deviations f r o m plug flow can o c c u r in shallowbed experimental reactors due to axial eddy dispersion. This superimposes on the overall flow an additional transfer which increases the effluent concentration of reactant (lowers the conversion). T h e resulting p h e n o m e n o n is c o m m o n l y called backmixing. Considerable effort has b e e n spent in evaluating the effect of backmixing on reactor efficiency [1-7]. F o r v a p o r - p h a s e reactions, the effect generally proves to be negligible except for cases of high conversion and short beds. T h u s while not important in commercial-scale reactors, it can be in laboratory units such as microreactors. Axial dispersion in concurrent trickle-flow reactors, in which a liquid reactant trickles down the bed while a gas reactant flows in the s a m e direction, has received little attention, In trickle-flow hydrotreating, we have o b s e r v e d decreases in reactor efficiency at low mass velocity which could not be attributed to transport limitations in the b o u n d a r y layer or "film" around individual particles. T h e present analysis examines the possibility that axial dispersion might be responsible and develops general criteria for predicting when such effects will be important.
ANALYSIS In single-phase flow through p a c k e d reactors, backmixing can usually be described by a onedimensional plug-flow model with superimposed longitudinal dispersion. According to this model, the net transport of reactant by bulk flow and longitudinal dispersion into a differential volume element must, at constant density, be equal to the amount c o n s u m e d by the reaction. Thus, the differential equation describing the steady-state concentration profile in an isothermal reactor is given by: D d2C
_ dC
a--~--- v--~-- r = O
(1)
where C is the concentration of reactant, x is the distance along the axis of the reactor, ~ is the superficial velocity of the fluid, Da is the longitudinal diffusivity (based on e m p t y bed volume), and r is the reaction rate per unit total bed volume. This equation can also be written in terms of the interstitial velocity and diffusivity, but the reaction rate must then be expressed, s o m e w h a t less conveniently, in terms of interstitial volume. T h e reaction rate per unit bed volume is a s s u m e d to be a power-law function of concentration and of catalyst dilution:
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r =k(1-~)C"
l+b
(2)
D. E. M E A R S
where k is the rate constant per unit catalyst particle volume, n is the order of the reaction, b is the ratio of diluent to catalyst volume, and e is the void fraction in the diluted bed. In dimensionless terms, Eq. (1) transforms to: 1 d2tk dtk dz
Pe dz2
R.qJ" = 0
(3)
where: C
x
pL
q' = -Coo'z = ~, ee = --~, R, = kbzCo"-' L(1 - E ) = L._.~u kb = k(1 --ec) z = ~(1 + b ) ( 1 - e c ) Co is the initial concentration, L is the bed length, Lu is the length undiluted catalyst would occupy, cc is the void fraction of undiluted catalyst, kb is the rate constant per unit catalyst bulk volume, and ~- is the space time (inverse space velocity). The latter two quantities are independent of both the dilution ratio and any changes in void fraction occurring as a result of dilution. The bed Peclet number is a measure of the ratio of convective to diffusive transport. Note that the first term of Eq. (3) diminishes in importance at high Peclet numbers, so that the equation approaches the plug-flow form. When the catalyst is diluted with inert solids, the bed length increases proportionately, assuming that the reactor diameter and fraction voids remain constant. Since kb and ~" are independent of dilution, the third term of Eq. (3) remains constant. However, dilution increases the Peclet number, thereby minimizing deviations from plug flow. Adding more catalyst, while keeping the space velocity constant, is at least equally effective in minimizing backmixing. In both cases lengthening the bed lowers the axial concentration gradient, and thus diminishes the dispersion. In trickle-flow operation, the reaction occurs in catalyst pores which are filled with liquid. The diffusional resistance of the liquid "film" around individual particles is generally negligible unless the catalyst is operating at a low effective-
ness factor internally[8]. The case to be considered here is that of cocurrent flow of gas and liquid. We shall assume that the vapor and liquid remain in physical equilibrium as they pass through the reactor. Reactants initially in the vapor phase are thus transferred to the liquid phase as the reaction proceeds, so it is only necessary to consider the dispersion in the slower moving liquid phase. Equation (3) is applicable provided that Peclet numbers obtained under trickle-flow conditions are used. A possible adverse influence to consider is "channeling", in which the fluid tends to flow through regions of the bed where the packing is more than usually open, e.g. near the wall. If the wall flow passed through unconverted, simple calculations show the effect on overall conversion could be significant. For example at 92 per cent conversion, a wall flow only 1.1 per cent of the total would be sufficient to increase the required reactor length by 5 per cent. However, radial mass transfer and exchange of liquid act to prevent high concentrations from being maintained near the wall. While the wall flow is greater in small-diameter reactors, the radial distance for the equalizing transport is also smaller. The net effect of channeling is probably to increase moderately the axial dispersion, which can be accounted for in the axial Peclet numbers assuming these are obtained at reasonably comparable radial aspect ratios (ds/dt). Perturbation solutions to Eq. (3) were obtained by Burghardt and Zaleski[6] for appropriate boundary conditions: 1 dqJ
1 = tk P e d z
at z = 0 + (4)
d__~_~= 0 dz
at z = 1-.
For small deviations from plug flow and first order reactions, their solution is given by:
C---£= e-n'[ l +-~e Ra2+ P-~-'~-- ( R 1 - 4 )
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" "]" (5)
The role of axial dispersion in trickle-flowlaboratoryreactors They show that the exact analytical solutions of Danckwerts [ 1] and Wehner and Wilhelm [2] can also be reduced to this form. It is often sufficiently accurate to terminate the series after the first two terms when RI/Pe is small; for plug flow all terms after the first are negligible. A longer space time is required to achieve the same conversion when deviations from plug flow occur. The ratio of space times, or reactor lengths, required can be compared by setting Eq. (5) equal to the plug flow solution, taking logarithms of both sides, and rearranging: L-L-a 1 [ 1+ ~ e R12] L = RR~l p = I --~-~in
(6)
where Lp and Rip are respectively the reactor length and reaction rate group which would apply if plug flow prevailed. If the deviation from the length required in plug flow is to be less than 5 per cent, the following relation must hold: 0-05 > ~1 In
[l+__l Pe Re].
Rx 2
> Bo(exp0-05 R1-- 1)
(8)
where the Bodenstein number, Bo, is a Peclet number based on the particle diameter (vd,/Da). For small values of R1 (less than 90 per cent conversion) the criterion reduces to: L
> 20
B~
20kr 20 In Co = Bo -- Bo Cf
C Co
- - =
. . •
]
(10)
Pn = l-- (l--n)Rn. As before, we can set Eq. (10) equal to its corresponding plug-flow solution and solve for the minimum reactor length to keep the deviation within 5 per cent. Making use of the approximation: ( 1 - - 0 " 0 5 ) 1/1-n~ 1
0"05 l--n
(11)
there results upon simplification the general criterion: L 20n lnp, = ~20n, m Co • ds > B o ( n - - 1 )
(12)
Note that this expression correctly gives Eq. (9) for first-order reactions and a minimum length of zero for zero-order reactions, which are not affected by axial dispersion. Petersen[5] applied the asymptotic solution method to obtain an approximate criterion for axial dispersion. For first-order reactions, the analysis indicated that the ratio Lp/L is close to unity for values of: ct =
(9)
which is also conservative at higher conversions. The last equality puts the criterion in terms of observables. Note that the minimum length for negligible deviation increases with conversion or decreasing Bodenstein number. Thus no simple rule, such as the frequently quoted L / d s > 30, is adequate for all cases. Burghardt and Zaleski[6] also derived per-
[ 1 ninp,~_ (O")m-n 1-~ Pe n--1 O.
where
(7)
Solving for the Peclet number yields a criterion for the minimum L/d, ratio: L
turbation solutions to Eq. (3) for non-first-order reactions. At large bed Peclet numbers, the solution for a reaction of order n is as follows:
V ~ a
9
< 1
(13)
When put into the form of Eq. (9), this criterion becomes: L kz as-;-> B--o"
(14)
Comparison shows that the new criterion, allowing only a 5 per cent deviation, is more conserva-
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D. E. MEARS
tive by a factor of 20. In terms of t~, it would be more conservative by the ~ As noted by Petersen in a similar case[16], an asymptotic criterion is not to be interpreted as a design criterion, but as a guide to determining which of the regions the catalyst operates: the kineticcontrolled region, a ~ 1; the diffusion-influenced region, a >> 1; or the transition region, a ~ 1. In single-phase flow, the Bodenstein number is usually close to 2-0 for gases (at Re > 2) and to about 0.5 for liquids in laminar flow (Re < I0) [9,10]. In trickle flow [l l-14], Bodenstein numbers for the liquid phase are only slightly influenced by the gas in both cocurrent and countercurrent flow until the flooding point is approached. The values also approach 0-5 at high ratios of dynamic to static holdup, but decrease in proportion to the holdup ratio at low Reynolds numbers[13]. This effect can be estimated through use of the Otake-Okada correlation for dynamic holdup [ 15], or the latter can be included in the Bodenstein number correlation [11, 14]. Due to the low Bo valves at a Reynolds number of 10, the minimum bed length for freedom from significant axial dispersion effects is about ten times longer in trickle flow than in vapor phase operation.
the catalyst volume was 35 cc, but in the second run the catalyst was uniformly diluted with an equal volume of quartz. In the last two duplicate runs the catalyst volume was increased by a factor of four and no diluent was added. The pretreatment procedure consisted of heating in flowing nitrogen to 600°F, followed by presulfiding with a 10 per cent H2S-90 per cent H2 mixture at 371°C, 1 atm, and 500 v/v hr for 2 hr. The unit was then pressurized with hydrogen before introducing the West Coast straight-run gas oil feed, whose properties are briefly summarized in Table 1. For each run process conditions were maintained as follows: Temperature, °C Pressure, atm H2/oil, (STP)/cc L H SV , v/v hr
Under these conditions, the oil was about 37 per cent vaporized. The liquid viscosity and density were estimated to be about 0-2 cP and 0.91g/cc, respectively. The catalyst reached steady activity after an induction period of about two days. Table 1. Properties of West Coast gas oil feed
EXPERIMENTAL
A study of axial dispersion effects in trickleflow hydrotreating was performed in a conventional bench-scale unit. The 1.1 cm i.d. x 152 cm reactor tube was enclosed in a large fluidized sand bath to maintain isothermal operation. The commercial nickel-molybdena/A12Oa catalyst used was in the form of a 0-059 in. dia. extrudate with an average length of about 0-15 in. Its equivalent spherical diameter is given by the formula: d, = ~v/dc 1c + d~2/2 = 0-26 cm.
371 100 89 2-0.
(16)
In one run the catalyst was uniformly diluted with an equal volume of 12-20 mesh quartz which had an average diameter of about 0-1 cm. The catalyst and diluent volumes were chosen to give varying bed lengths. In the first two runs
Density, g/cc Sulfur, X-ray, ppm Total nitrogen, ppm ASTM distillation, °C IBP 50% EP
0.914 11,300 1920 256 350 449
RESULTS
The product concentrations of unconverted organonitrogen compounds increased significantly with decreasing bed length, as shown in Table 2. At the shortest bed length, the reactor efficiency, expressed as a ratio of space times (plug flow to actual) required to achieve 86 per cent conversion, amounted to only 80 per cent. Note that adding quartz diluent to increase the bed length, while holding mass velocity constant,
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The role of axial dispersion in trickle-flow laboratory reactors Table 2. Effect of bed length and mass velocity on conversion of organonitrogen compounds Unconverted nitrogen, ppm
Bed length (cm)
Mass velocity (g/cm 2 sec)
Rez
Pe
Exper.
Calc.
39 67(a) 152
68 68 274
2 2 8
8.2 22 84
257 175 155, 148
246 196 162
(a) Diluted with quartz.
improved the conversion markedly. No change should have occurred in this case if transport limitations across the liquid film surrounding individual particles were responsible for the poor efficiency. In addition, if either interphase or interparticle heat transport limitations were involved, the exothermic reaction should have been most rapid in the low-mass-velocity, undiluted run. The data can be interpreted with the axial dispersion model utilizing trickle-flow Bodenstein numbers from the correlation of Hochman and Effron [12]. After correction for operating holdup variations, the values check within 10 per cent those obtained from the correlation of Sater and Levenspiel[ll]. On the basis of previously determined[17] pseudo-first-order kinetics in unconverted nitrogen, Eq. (5) was used to compute product nitrogen values for the various bed lengths. Reasonably good agreement between the calculated and experimental values is seen in Table 2, supporting the conclusion that axial dispersion is the most likely cause for the poor performance of the shorter beds. The minimum length required for freedom from significant backmixing effects is readily calculated from Eq. (9). At ReL = 8, the minimum L/d8 ratio is 350, which corresponds to 90cm. Obviously, there is a fair amount of uncertainty in this estimate, which is based on Peclet numbers for different packings. But used conservatively, the new criterion offers a useful tool for evaluating the possibility of axial dispersion effects. Acknowledgements-The author wishes to acknowledge helpful discussions with his colleagues at Union Research Center and Professor Thomas K. Sherwood.
NOTATION
b dilution ratio of inerts to catalyst Bo Bodenstein number, = dsPlDa C concentration of reactant, moles/cm3 dc diameter of cylindrical catalyst particle, cm d8 equivalent spherical diameter of catalyst particle, cm dt diameter of reactor tube, cm Da axial eddy diffusivity, cm2/hr k apparent rate constant per unit particle volume of catalyst, hr -1 for first-order reaction kb rate constant per bulk catalyst volume, = k ( 1 - Ec), hr -1 for first-order reaction 1~ length of cylindrical catalyst particle L length of reactor bed, cm Lp length of reactor when plug flow prevails, cm LHSV liquid hourly space velocity=P/Lu, hr-I n order of reaction Pe bed Peclet number = L e/Da r local reaction rate per unit bed volume, moles/hr cm 3 bed R e Reynolds number = ~, ds p/lz R n reaction rate group = kb • Co ~-1 superficial velocity, cm/hr x distance in axial direction, cm z dimensionless distance in axial direction -- x/L Greek symbols ot dimensionless axial dispersion number = DalP bed void fraction ~c void fraction of undiluted catalyst
365
D. E. MEARS /x p p. r q,
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [I 1] [12] [13] [14] [15] [16] [17]
viscosity offluid, g/cmhr d e n s i t y o f fluid, g / c m a parameter for non-first-order reactions = 1 (l--n) R. space time = inverse space velocity, hr d i m e n s i o n l e s s c o n c e n t r a t i o n = CICo
Subscripts f final v a l u e L liquid phase o initial v a l u e u undiluted
REFERENCES D A N C K W E R T S P. V., Chem. EngngSci 1953 2 I. W E H N E R J. E. and WI LHELM R. H., Chem. Engng Sci. i 956 6 89. CARBERRY J. J., Can. J. chem. Engng 1958 36 207. L E V E N S P I E L O. and B I S C H O F F K. B., Ind. Engng Chem. 1959 51 1431 ; 1961 53 313. PETERSEN E. E., Chemical Reaction Analysis. Prentice-H all, Engelwood Cliffs, N.J. 1965. B U R G H A R D T A. and ZALESKI T., Chem. Engng Sci. 1968 23 575. M E C K L E N B U R G J. C. and H A R T L A N D S., Chem. Engng Sci. 1968 23 57. S A T T E R F I E L D C. N., PELOSSOF A. A. and S H E R W O O D T. K.,A .I.Ch.E. JI 1969 15 226. L E V E N S P I E L O. ChemicalReaction Engineering, p. 275. Wiley, New York 1962. MILLER S. and KING C. J.,A .1.Ch.E.J11966 12 767. SATER V. E. and L E V E N S P I E L O., Ind. Engng Chem. Fundls 1966 5 86. H O C H M A N J. M. and E F F R O N E., Ind. Engng Chem. Fundls 1969 863. VAN SWAAIJ W., C H A R P E N T I E R J. C. and V I L L E R M A U X J., Chem. Engng Sci. 1969 24 1083. F U R Z E R I. A. and M I C H E L L R. W.,A .1.Ch.E. J11970 16 380. O T A K E T. and O K A D A K., Chem. Engng (Japan) 1958 22 144. PETERSEN E. E., Chem. Engng Sci. 1968 23 94. MEARS, D. E., Unpublished results. R~sun~- I1 semble que la dispersion axiale ou le contre-m61ange soit responsable des effets adverses de v61ocit6 de masse observ6s dans les r6acteurs de laboratoire goutte-~-goutte. A des nombres de Reynold faibles, typiques des unit6s de table, le probl~me de la dispersion peut ~tre d'un ordre de grandeur au moins plus s6rieux dans l'op6ration goutte-~-goutte que dans l'op6ration en phase de vapeur. Un simple crit~re de perturbation est d6riv6 pour la longueur minimale de r6acteur requise en vue de se lib6rer compl~tement des effets significatifs de dispersion axiale. Ce crit~re montre que la longueur minimale augmente avec rordre de r6action et de conversion et est inversement proportionnelle au nombre de Bodenstein. Zusammenfassung-Axialdispersion oder Gegenmischung scheint fiir die in Rieselreaktoren im Laboratorium beobachteten nachtefligen Stoffgeschwindigkeitseffekte verantwortlich zu sein. Bei niedrigen Reynoldsschen Zahlen, typisch fiir laboratoriumsmiissige Einheiten, kann das Dispersionsproblem in Rieselanlagen um mindestens eine Gr6ssenordnung schwerer sein als bei Arbeiten in der Dampfphase. Es wird ein einfaches Sttirungskriterium abgeleitet zur Feststellung der minimalen Reaktori~inge, die erforderlich ist um wesentliche Axialdispersionseffekte zu vermeiden. Es ergibt sich, dass die MinimallSnge mit der Ordnung der Umwandlung sowie der Reaktion zunimmt, und dass sie der Bodensteinschen Zahl umgekehrt proportional ist.
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