differ in molecular composition, particularly with respect to olefins, from virgin gas oils. From the linearity of the data for the virgin gas oils, the following relationships can be derived :
From Figures 2 and 3, t’hevalues of a. and bo are 25 and -0.42, respect’ively, and the corresponding ones for a1 and bl are 20 and -0.46 for the range of data shown. These values were calculated by least-squares regression of the data for virgin gas oils. Relationships of K S to aromatic/iiapht.hene ratio did not correlate as well as the other rate constants. However, K z does not change drastically wit,h composition (Nace e t al., 1971), and it typically varies only from 1.5 to 2.5 at 900’F. The relationships between a , K O ,and K 1 vs. the ratio of aromat,icsjiiaphthenesas represented in Figures 1-3 appear to be independent of the level of the paraffin concentrations in the charge stocks. The data in these figures cover reasonably wide ranges of paraffin conceiit,rations. Since catalyst decay in catalytic cracking is considered to be intimately associated wit,h coke formation, it is interesting to examine the relationship between the catalyst decay constant, a . aiid cat.alyst coke. Figures 4-6 are plots of a, &, aiid K l vs. catalyst coke and show a increases and K o and K I decrease with increased catalyst coke. The relationship of the coke deposited to the decay constant is to be c.spectmedsince coke is the major contributor to catalyst decay. The decay constant, a , however, reflects all causes of decay such as the adsorption of aromatics and nitrogen compounds as well as t.hat coke which remairis after st.ripping. From Figure 5 and 6, it is clear that t’hosecharge stocks which produce more coke also have a decreased cracking and gasoline format,ion tendency. Iiit,erestingly, the slopes of KO and KI with coke are approsimately equal indicating the init,ial selectivity ( K l j K o ) is largely independent of coke make. ,lverage values of t,he nonstrippable catalyst coke in experiments wit8hcatalyst residence times of 5.0 minutes were used i n these plots (Kace e t al., 1971).
The Voorhies (1945) relationship between catalyst coke and catalyst residence time can be written as,
C
=
atcb
where C = catalyst coke, t, = catalyst residence time, aiid a and b are constants. The coefficient a is to a significant extent a measure of the coking tendency of a charge stock. I t is reasonable to expect some correlations e u t between a and the catalyst decay constants. Figures 7 and 8 are, respectively, plots of a vs. a and n vs. a. Both of the catalyst decay constants inciease n i t h incieasing a, which agrees with the trend shown 011 Figure 4. Summary and Conclusions
The kinetic rate constants for the catalytic cracking of gas oils and subsequent gasoline formation can be correlated with the aromatic/naphthene ratio for a wide range of virgin charge stocks. The catalyst decay constant also correlates directly Kith the aromatic/naphthene ratio. Finally, gas oils whose composition tends to produce more coke also give lower cracking and gasoline formation rates and higher catalyst decay rates. literature Cited
Akhmetshina, M. N., Levinter, ?*I.E., Tanatarou, 11.A., hlochalov, Y. D., A‘eftepererab.A’eftekhim. (hloscow), 1969 (8), 10. Crocoll, J . F., Jaquay, R. D., Petro/Chem Eng., C-24, November 1960. Fitzgerald, M. E., RIoirano, J. L., Morgan, H., Cirillo, V. A., App. Spectrosc., 24, 106 (1970). Nace. D. AI., Voltz. S. E.. Weekman. Jr..’ V. W.. Ind. Eno. Chem. Process Des. Develop., 16, 330 (1971). Reif, H. E., Kress, R. F., Smith, J. S., Petrol. Refiner, 40 ( 5 ) , 237 (1961). Service, W. J., PetrolChem Eng., C-32, November 1960. Shnaider, G. S.,llukhin, I. I., Chueva, R I . A , , Kogan, Y. S., Kham. Tekhnol. Topl. Masel, 1969 (I), 10. Voorhies, A., Ind Eng. Chem., 37,318 (1945). Weekman, V. W., Ind. Eng. Chem. Process Des. Dezelop., 7, 90 11968)
Weekman, 1’.W., ibid., 8 , 385 (1969).
Weekman, V.W., Nace, D. hl., AIChE J., 16, 397 (1970). White. P. J.. PreDrint No. 24-68. API Division of Refining. 33rd 3lid-Year’hlee&ng,Philadelphia, Pa., May 1968. Y
RECEIVED for review October 29, 1970 ACCEPTEDMay 3, 1971
Tests for Transport limitations in Experimental Catalytic Reactors David E. Mears Union Oil Go. of California, Union Research Center, Brea, Calif. 9662i
I n esperiment,al studies of heterogeneously catalyzed reactions, one of the first objectives should be to determine whether the investigation is coriceriied with catalytic kinetics or with interactions between the kinetics and transport pheiiomena. T h e intrusion of temperature aiid concentration gradient,s can lead t o severe deviations in a catalyst’s performance, in many cases completely disguising the true kinetics of the reaction. To ensure that kinetic data obtained
in an experimental reactor reflect only chemical events, gradients must be virtually eliminated from three domains: Intraparticle within individual catalyst particle.3 Interphase between the external surface of the particles and fluid adjacent to them Interparticle between the local fluid regions or catalyst particles The latter doniaiii equally well could be called intrareactor Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
541
since it applies to gradients within the reactor as a whole. While the intraparticle domain has been extensively analyzed for several decades, the latter two domains only recently have received much attention. Criteria are n o x available to estimate whether transport effects in any domain are significantly altering the experimental results. I n deriving such criteria, perturbation analysis has proved particularly useful for detecting when deviations of the rate start to become appreciable. The method involves expanding the reaction rate expression in Taylor series in concentration, temperature, or both. Criteria involving various dimensionless Damkohler numbers (Damkohler, 1936) are then obtained which will hold deviations to a n acceptable tolerance. This paper compares perturbation criteria with those derived b y other approaches, and evaluates means of minimizing transport limitations in experimental reactors. Intraparticle Transport
Intraparticle transport has been analyzed for a wide variety of reaction kinetics, particle geometries, and thermal behavior, as well-reviewed elsewhere (Petersen, 196513; Satterfield and Sherwood, 1963). Generally the objective has been t'o calculate an effectiveness factor, 7 , defined as the ratio of the actual rate to that which would occur if the temperature and concentration were constant throughout the catalyst particle. The solutions show that TJ becomes inversely proportional to the characteristic dimension of the particle in the regime of strong diffusion influence. Cnfortunately, direct application of such solutions requires knowledge of the true kinetic behavior and the intrinsic rate constant. Jf7eisz and Prater (1954) presented a criterion for the absence of significant diffusion effects with irreversible reactions which is independent of the rate constant. I t is assumed that Fick's first law governs diffusion in the porous media, that the effective diffusivity, D e , remains independent of the nature of the reaction, and that the intrinsic catalytic activity is distributed uniformly throughout. T o ensure 7 2 0.95 in an isothermal spherical particle with a first-order reaction, the criterion requires:
where
is the observed reaction rate per unit particle volume,
C, is the reactant concentration a t the external surface of the particle, and r p is the radius of the particle. The dimensionless group, called Damkohler's number for diffusive transport, expresses the ratio of chemical reaction rate to diffusive flus. The merit of the criterion is that it is stated in terms of observables, and a parameter, D e , which is independently measurable. It also applies t o transport of a gas in liquid-filled pores if the effective diffusivity is correctly evaluated (Satterfield et al., 1968). Weisz (1957) later showed by means of a linear approximation of the concentration gradient a t the surface that the numerical value on the right is conservatively 0.3 for second-order reactions and 6 for zero order. The corresponding value for first-order reactions is 0.6 by this approsirnation. When strong inhibition by the reaction product occurs, Ailustin and 'CTTalker (1963) and Petersen (1965a) demonstrated that the Weisz-Prater criterion can fail. Hudgins (1968), using a perturbation method, derived a design criterion to ensure TJ > 0.95:
542
Ind. Eng. Chern. Process Des. Develop., Vol. 10, No. 4, 1971
where aisand Rots are, respectively, the rate expression and its first derivative with respect to reactant concentration evaluated a t the external surface. A numerical value of 0.75 appearing on the right side of the equation was rounded to 1 in this derivation. This criterion can be extended to negativeorder reactions by taking the absolute value of the derivative. For simple power-law kinetics (as= kc,"), the perturbation criterion becomes: (3) where n is the reaction order. A nearly identical result was recently obtained by Stewart and Villadsen (1969) as an illustration of their orthogonal collocation procedure, which is also applicable to other geometries and kinetics. Equations 2 and 3 fail for zero-order kinetics because the concentration deviations are beyond the scope of the perturbation approach. For this case, the value of 6 on the right obtained by Weisz is recommended. For reactions involving a change in the number of moles, the effectiveness factor relationships are shifted as a function of the transport mechanism. When the mode of transport is Knudsen diffusion, Otani et al. (1964) showed that a pressure gradient develops in the catalyst, but the shift in effectiveness factors is generally negligible. When molecular diffusion prevails, a decrease i n moles enhances the transport while an increase hinders it. Weekman and Gorring (1965) showed that the influence of a volume expansion or contraction can be expressed in terms of the volume-change modulus:
e
=
(.
-
i ) ~ ,
(4)
in which Y , is the mole fraction of reactant a t the catalyst surface and v is the stoichiometric coefficient in the reaction A + vB.Following Kubota e t al. (1969), the criterion for an isothermal particle can be modified to: (5) I n dilute cases ( Y s20-
20
= -111-
Pe,
C, Cf
in which r is the space time aiid C, and C/ are the initial and effluent concentrations. The axial Peclet number baqed on particle diameter:
is a measure of the ratio of convective to dispersive transport. The advantage of the second equality in Equatioii 30 is that it puts the criterion in terms of the observed conversion and Ind. Eng. Chem. Process Der. Develop., Vol. 10, No.
4, 1971
545
the independently obtainable Peclet number. Given in terms of reactor length, the new criterion is 20 times more conservative than the asymptotic criterion. For nonfirst-order reactions obeying simple power-law kinetics, the corresponding criterion is :
L 20 n ->-Ind, Peu
C, Cf
where n is t'he order of the reaction. This general expression gives Equation 30 for first-order reactions and zero for zeroorder reactions, which are not affected by axial dispersion. Kote that the axial dispersion problem becomes more severe wit8h increasing conversion or reaction order, or with decreasing Peclet numbers. Extensive studies have shown that the Peclet number is geiierally close to 2 for gases a t Reynolds numbers greater than 2 and to about 0.5 for liquids when Reynolds numbers are less t'han 10 (Levenspiel and Bischoff, 1963; Miller and King, 1966). I n trickie-flow, still smaller values are found in the liquid phase a t low Reynolds numbers (Hochman and Effron, 1969; van Swaaij et al., 1969). For example, at a Reynolds number of 4, typical of bench-scale react'or condit'ions, these correlations yield Peclet numbers of the order of 0.1. Coiisequently, the criterion shows that the minimum reactor length for freedom from backmixing effect,s may be a n order-of-magnitude greater in trickle-flow than in vaporphase operation. Empirical Tests for Transport Limitation
While the analytical criteria are useful, estimation of parameters involved can be difficult a t times. For such cases a well-established empirical criterion is to run the reaction 011 progressively smaller particle sizes, usually produced by crushing. If the reaction rate per unit particle volume changes with diameter, strong transport influence is indicated. Graphical (Weisz and Prater, 1954) and analyt'ical (Gupta aiid Douglas, 1967) techniques are available for estimating the effectiveness factors in this case. When the reaction rate remains iiidependent of the particle size over an appreciable range, the reaction is considered free from intraparticle gradients on the macroscopic scale. An alternate empirical approach, suggested by Koros and Nowak (1907), is based on the fact that the rat>eof reaction in the kinetic regime is proportional to the number of sites per unit volume (S), but proportional to S to the 0.5 power in the internal diffusion regime. The test involves pelletizing mixtures of finely divided catalyst with a n inert powder of comparable diffusional characteristics. If the fraction of catalyst in the mixed pellet is f , the ratio of rates must also be equal to f in the kinetic regime. This method avoids distortions in the flow field which changing pellet' sizes may cause. A common empirical test for the possible iiifluence of ext,ernal heat aiid mass transfer limitations consists of checking the effect of flow rate on conversion a t consbarit space velocity. If no such effect is found, it is concluded t'hat transfer of heat and mass to the particles does not influence the reaction rate. Chambers and Boudart (1968) warned that this diagnostic test may fail because of its lack of sensitivit,y a t Reynolds numbers of the order of 10. This observaof the interphase heat transfer tion \\'as based on a reanal data of D e Xcetis and Thodos (1960). More recent data (Gliddon and Cranfield, 1970; Littman et al., 1969; Petrovic and Thodos, 1968) iiidicate that both h and k , remain proportional to between the 0.95 and 0.64 powers of G to Reynolds 546 Ind. Eng.
Chem. Process Des. Develop., Vol. 10, No. 4 , 1971
numbers as low as one. However, the interparticle heat, transfer coefficient (Yagi and Kunii, 1959) becomes quite weakly dependent on G a t low Reynolds numbers. Thus, as Chambers and Boudart noted, this diagnostic test requires very precise conversion data or an extended range of flow rates. Design Criteria for Fixed-Bed Reactors
I n designing a laboratory reactor, transport limitations can be minimized through proper choice of the parameters. Because of the large Arrhenius effect of temperature on the reaction rate, maintaining isothermal operation is by far the most critical considerat'ion. The criteria reviewed provide a means for assessing the best ways to achieire this. Reducing the reactor radius, a n obvious choice, is particularly important since the interparticle criterion depends on it both explicitly and implicitly. The latter dependence occurs because the interparticle heat transfer coefficient becomes effectively proportional to the mass velocity, G, a t Re > 100, and G is inversely proportional to RO2 (at constant mass flow). The int'erparticle heat transfer criterion then becomes effectively proportional to the fourth power of R, and the interphase criterion to the 1.2 power a t higher Reynolds numbers. Hence decreasing the reactor radius is a vital step in minimizing both transport' resistances. The question then arises as to how far the radial aspect ratio (RO/rP)may be reduced before wall effects become serious. Because the packing is more open near the wall, a larger fluid velocity and lower catalyst bulk density occur there. As a result the conversion is lower near the wall, causing an apparent lower activity for the bed. Kondelik and Boyarinov (1969) calculated the effect of radial void fraction distribution on reactor efficiency as a function of Ro/rp. Their model divides the reactor into annular channels based on statist'ical distribution data and utilizes the Kozeny equation (Carman, 1956) to compute the volumetric flow rates in individual channels. Radial mass transfer is accounted for by a lumped radial mass transfer coefficient D. I n the Roirp range of 5-10, the calculations show a minimum in the effective bed activity whose depth increases with conversion. But the minimum is reduced with improved radial transfer, and the maximum value assumed for D still appears to underestimate the contribution from eddy diffusion. Consequently, the gain in isothermality through reducing the reactor radius appears to more than compensate for the slight channeling loss to R J R , ratios as low as 4. However, wall effects may be more serious in trickleflow, so larger ratios should be used for this operation. Finally, in an existing reactor of larger diameter, dilution of the catalyst bed with inert particles can be utilized to reduce heat generation per unit volume. Linearly decreasing dilution ratios are st,rongly recommended for integral reactors. But in diluting a bed of fixed length, it should be remembered that the mass velocity is reduced proportionately a t constant space velocity. The interparticle criterion shows that dilution will be advantageous in minimizing radial gradients only if the reactor is operating a t Reynolds numbers sufficiently low that the effective thermal conductivity is relatively insensitive to the mass velocity (Le.) Re < 100). Conclusion
The elimination of transport influeiices in studies of catalytic kinetics is a complex problem, The criteria presented provide tools for evaluating and minimizing the various transport resiatances. Like any diagnostic test, each must be applied with due caution and understanding. If carefully used,
the criteria can help the experimenter to achieve a more rigorously defined experimental system . Nomenclature
a = superficial (outside) surface area of catalyst particle per unit particle volume, cm2/cm3 dilution ratio, cm3inert solid/cm3 catalyst b Bi, s thermal Biot number for catalyst particle, = hd,/h Bi, = thermal Biot number a t the reactor wall, = hwdt/ke C* = equilibrium concentration of hydrogen in liquid, g-mol/cm3 C h = concentration of reactant in bulk fluid, g-mol/cma cf = effluent concentration of reactant, g-mol/cm3 CO =-c initial concentration of reactant, g-mol/cm3 C , = concentration of reactant at outer surface of particle, g-mol/cm3 d, = diameter of catalyst particle, cm D, = axial eddy diffusivity, cmZ/sec De = effective diffusivity in a porous catalyst, cmZ/sec E = intrinsic activation energy for chemical reaction, g-cal/g-mol f = fraction of active catalyst in mixed pellet G = mass velocity, g/sec.cm2 of total or superficial bed cross section h = heat’ transfer coefficient between gas and particle, g-caljsrc’ cm*. “C h, = heat t i , > i i i f i f w coefficient at the reactor wall, g-cal/ sec. cni2 ”(’ AH = heat of cLheinica1 reaction, g-cal/g-mol IC = intrinsic rate constant per unit particle volume, sec-1 for first-order reaction kb = apparent rate c~onstantper unit bulk catalyst volume, = k ( l - ~ ) qsec-l , k c = mass transfer coefficient between gas and particle, cm/sec = effective thermal conductivity across reactor bed, g-cal/sec. cm. “ C Iz LS = overall mass transfer coefficient through liquid film, cm/sec L = reactor bed length, cm L o = reactor bed length rvith undiluted catalyst, cm n = integer exponent in power law rate expression Pea = axial Peclet number, = d,fi/D, T P = particle radius, cm R = gas constant, g-cal/g-mol. O K R e = Reynolds number, = cpd,/p R O = radius of tubular reactor, cm R = observed reaction rate per unit particle volume, g-niol/sec. em3 = reaction rate per unit bed volume, g-mol/sec. cm3 = derivative of reaction rate expression with respect t,o concentration, see-‘ = derivative of reaction rate expression with respect to temperature, g-mol/sec. cm3.“K = active sites per unit particle volume, S.V. = space velocity, = o/L,, sec-1 t = time, sec T = absolute temperature, “K D = superficial velocity, cm/sec Y , = mole fraction of reactant a t catalyst surface z = distance in axial direction, cm I
s
GRI:EK LETTERS cy
P
= =
dimensionless axial dispersion number, = d h X / o dimensionless maximum temperature rise, = ( - A H )
D,C,/XT,
dimensionless activation energy, = E / R T , relative experimental error in conversion E = bed void fraction 7 = effectiveness factor e = volume change modulus = (1 - v ) Y 8
7 = 6 =
X = thermal conductivity of partick, g-cal/sec. cm . “C
viscosity, g/sec cm stoichiometric coefficient density,g/cma = space time = inverse space velocity, see = Damkohler number for interphase heat transport,
p = Y = p = 7
x
=
(- A H ) &r,/hTb (cr
=
Damkohler number for interphase mass transport, @r,/k,Cb
=
literature Cited
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