exptl criterion for absence of diff limit of heter cat reactions - madon-boudart

430 Ind. Eng. Chem. Fundam. 1902, 21, 438-447

Experimental Criterion for the Absence of Artifacts in the Measurement of Rates of Heterogeneous Catalytic Reactions

Rostam J. Madon' and Mlchel Boudart'

Department of Chemical Engineering, Stanford University, Stanford, California 94305

An experimental criterion has been developed to assess whether measured catalytic activity is independent of the influence of transport phenomena. A simple theoretical analysis and experimental verification of the criterion have been carried out. This test may be used when heterogeneous catalysis is carried out in batch, continuous stirred tank, and tubular plug-flow reactors, and is best suited to be used with supported metal catalysts. In this case, the test should be performed with two or more samples with different metal loading but of similar metal dispersion. Absence of physical artifacts is established by the constancy of turnover numbers. I f the reaction is exothermic, the test should be repeated at another temperature.

Introduction It has been stated (Boudart, 1972; Carberry, 1966;

Satterfield, 1970) that catalytic data should be free from all transport influences to obtain kinetic rate expressions and to yield the correct activity or selectivity of the catalyst. Though the dangers are known, it should be em- phasized that scaling down to laboratory studies is fraught with problems to eliminate concentration and temperature gradients both within catalyst particles (intraparticle gradients) and between the external surface of particles and surrounding fluid (interphase gradients).

In the past, a number of interesting and useful criteria have been developed to check for the influence of heat and mass transfer on observed catalytic rates. Most of these criteria have been explained in a detailed review by Mears (1971a). Several criteria have been assembled in Table I. The Weisz-Prater criterion (1954) for intraparticle diffusion, for which only directly observable quantities are re- quired, has been criticized for lack of generality by Pet- ersen (1965a). The Petersen criterion (Petersen, 1965a), however, does not contain only observables and is hence more difficult to use. The more generalized criterion of Bischoff (1967), where the integration of the rate expression is involved, or that of Hudgins (1968) and Stewart and Villadsen (1969), where the differentiation of the rate expression is necessary, all suffer from the fact that the rate expression must be known, be it the power law type or the Langmuir-Hinshelwood type law, and if it is the latter then numerical values for all equilibrium constants must be obtained before the criteria can be used. Hudgins (1972) derived a criterion for interphase diffusion, but it suffers from the same problem as stated above. The criteria to check for interphase and intraparticle temperature gradients are due to Mears (1971b) and Anderson (1963), respectively.

To use the several criteria, some of the following experimental properties are necessary: effective diffusivity in the pores of the catalyst, the heat and mass transfer coefficients a t the fluid-solid interface, and the thermal conductivity of the catalyst. A t the low particle Reynolds numbers used in laboratory packed bed reactors, the ac- curacy of the value of h obtained from known correlations is subject to question (Madon, 1975). Though numerous studies have been carried out for bulk diffusion and dif-

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fusivities of fluids, it is difficult to obtain the effective diffusivity for different catalysts. Satterfield and Saraf (1965) have shown that diffusional characteristics may change throughout a pellet due to compression of the catalyst powder in a pelletizing machine. Intraparticle criteria include surface concentrations and surface temperatures. These are usually not known and bulk values can be substituted only if there are no interphase gradients.

Though the heat transfer criteria can be used to check for temperature gradients in the presence of diffusion, the latter effect could cause the rate and activation energy to be low and the criteria to be obeyed. In the absence of diffusion with the true higher values of the rate and activation energy, the criteria could be disobeyed for the same reaction.

Finally, the particle size is another important parameter. Often powders with a wide size range are used in laboratory reaction studies, and Aris (1957) showed that the mean effectiveness factor is given by

B = CBia[ i

Aris has warned against using average values for particle size to calculate 7, and in this light the above criteria could not be used except by using the largest and hence the most conservative value for the radius.

Besides using the above correlations, experimental tests are often carried out to check for diffusion. In a flow system, if the space velocity is kept constant and then if the conversion remains constant as the flow rate is changed, the influence of external mass transfer is said to be negligible. A similar test is used in slurry reactors where the constancy of the reaction rate is checked as the agitation of the slurry is increased. Chambers and Boudart (1966) have indicated that at the low particle Reynolds numbers encountered in the laboratory the heat and mass transfer coefficients are quite insensitive to changes in flow rates, and hence the above diagnostic test may fail to indicate that the reaction rates are free from interphase concentration gradients. Moreover, this test cannot give any information regarding intraparticle diffusion.

If activity of a catalyst is affected by altering pellet size, then this activity is said to be influenced by pore diffusion. However, if there is no change in activity with pellet size one cannot say for sure that there are negligible intraparticle concentration gradients. The influence of internal diffusion in the small pores of a bimodal pore distribution may still be important, and the test may just indicate that there is no diffusional influence in the larger set of pores.

@ 1982 American Chemical Society

Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 439

Table I. Various Criteria Used t o Check for Transport Effects in Heterogeneous Catalysis For Intraparticle Mass Transfer

Weisz and Prater (1954) R2r

Cs De DaII = - G 1

Weisz (1957) Dan < 6 zero-order reaction

90.6 first-order reaction G0.3 second-order reaction

Petersen (1965a)

Thiele modulus < f i

where

ri(C1, C2, T)

ri(Cis, C2s3 e . . , TI ri(5) =

Hudgins (1 968) 0.75 1

Dan 4 -r(Cs) cs 1 dr(C)/dC I c=cs

cs I dr(C)/dC I c=cs

Stewart and Villadsen (1969) 0.77 1

Dan < -r(Cs)

Bischoff (1967)

For Interphase Mass Transfer Hudgins (1972)

R, 0.15 1 <-r(cb) =-

Cbk, Cb 1 dr(C)/dC 1 C=Cb

For Intraparticle Heat Transfer Anderson (1963)

[ - A H [ R2r RgTs - 0.75 Daw = < 0.75---

AT, E T For Interphase Heat Transfer

Mears (1971b) [ - A H 1 Rr RgTb - 0.15

X = < 0.15- -- hTb E yb

For Combined Intraparticle Heat and Mass Transfer Mears (1 971a)

1 (-AH) dr(C,T)

De dT -- (- )T=T, < 1

or for power law kinetics 1

DaII < - I n - 7 0 )

For Combined Interphase and Intraparticle Heat and Mass Transport"

Mears (1971a)

R lr 1 + 0 . 3 3 ~ 7 <

CbDe I n - 7bPb I(1 f 0.33nw) a Note. This overall criterion provides that the NET distortion of the observed rate will be within 5% of the rate under

isothermal, isoconcentration conditions. There is no guarantee of freedom from individual transport processes.

Table 11. The Koros-Nowak Criterion [rate of reaction]/[ concentration of active material] = constant

1. [rate of reaction (mol t 3 - I g-l of catalyst)]/[weight of active material per unit weight of catalyst] = r/f, (mol s'l g - l of active material) = constant

2. [rate of reaction (mol s - * g-l of catalyst)]/[surface area of active material per unit weight of catalyst] = r/fa (mol s-l cm-2 of active material) = constant

3. [rate of reaction (mol s- ' g-l of catalyst)]/[surface moles of active material per unit weight of catalyst] = turnover number = r/f, = constant

Furthermore, changing pellet sizes could cause distortion in the flow field wkich could affect contacting.

To circumvent the problems stated above, we wish to develop an experimental criterion which was first suggested by Koros and Nowak (1967) to check if reaction rates were free from mass transfer effects. It will be shown that this criterion can be used as a test for interphase and intraparticle mass and heat transfer either when they occur separately or when the physical events are coupled during the course of a reaction. I t will be shown that when this test is obeyed there is neither poisoning nor poor contacting of the catalyst due to bypassing and channelling. Also, deviations of plug flow behavior in tubular reactors due to transverse and longitudinal temperature and concentration gradients (interparticle gradients), backmixing, and velocity gradients, will cause the criterion to be disobeyed In all future discussion this test will be referred to as the Koros-Nowak (KN) criterion.

In the following sections, the test will be explained, analyzed, and finally demonstrated to show that the cri-

terion can be successfully applied when kinetic measurements are to be obtained. The Koros-Nowak Criterion

The criterion is based on making rate measurements on catalyst samples in which the concentration of the catalytically active material has been changed. The reaction rate, in the kinetic regime, is directly proportional to the concentration of the active material (see Theoretical Section). The concentration of catalytically active material may be defined as (a) the weight fraction of the active material in the catalyst, (b) the surface area of the active material per unit weight of the catalyst, or (c) the surface moles of active material per unit weight of the catalyst.

The Koros-Nowak criterion can then be explained by noting that in the kinetic regime (a) the ratio of measured reaction rates must equal the ratio of the weight fractions of the active material in the different catalyst samples, as long as the dispersion of the active material does not change from one sample to the next, or (b) the turnover number (defined as moles reacted per second per surface

440

mole of the active material) or the reaction rate referred to unit surface area of the active material must be invariant as the concentration of active material is changed. Table I1 summarizes the KN criterion. Another approach is to plot the logarithm of the reaction rate in mol s-l g-’ catalyst vs. the logarithm of the surface concentration of active material; a slope of 1 denotes that artifacts are not influencing rate measurements. Explanations regarding the deviation of this slope from unity are given in the Theo- retical Section.

Koros and Nowak (1967) have explained how one can change the concentration of active material without changing the dimensions and pore diffusion properties of the pellets. Their method consists of mixing catalyst particles with particles of an inert powder and then making pellets from this mixture. The initial particles in the mixture must be much smaller than the final pellets used in the reaction. The inert powder must have the same diffusional characteristics as the catalyst powder with which it is mixed. For both metallic and nonmetallic catalysts this test can be used conveniently as follows. If we dilute a given catalyst, by the method described above, by a certain factor a , then for the KN criterion to be obeyed the observed rate should change for the diluted catalyst over the initial undiluted catalyst by the same factor cy. It should be noted that no measurement of the number of surface sites is involved.

There are two cases when the K N test, carried out by diluting a given catalyst as described above, cannot be used. First, when the catalyst has a bimodal pore distribution, internal diffusion in the micropores may be important in both the diluted and undiluted catalyst samples, and if the KN criterion is obeyed it may just indicate that there is no diffusional influence in the larger set of pores. Secondly, if the particle size of catalysts used in the reaction is close to the size of the powder from which the “diluted” pellets are made, as is sometimes the case for catalysts used in liquid phase slurry reactors, then the KN test as described above cannot be used.

In the case of supported metal catalysts, concentrations of active material may he changed by changing the metal loading on the inert support but keeping the preparation and pretreatment procedure identical, and the dispersion of the metal (defined as the ratio of the number of moles of surface metal to total metal) close for all the catalyst samples. Techniques (Whyte, 1973; Benson and Boudart, 1965; Wanke and Dougharty, 1972; Benson et al., 1973) are now available to measure surface areas of the active metal in supported catalysts. The selective chemisorption of hydrogen or carbon monoxide has been used by Sinfelt (1969) on 11 different supported metal catalysts to obtain reaction rates per unit area of active material. The ti- tration of adsorbed oxygen by hydrogen first used to measure platinum surface areas by Benson and Boudart (1965) has been recently extended to obtain the surface areas of rhodium (Wanke and Dougharty, 1972) and pal- ladium (Benson et al., 1973). When surface concentration of the active material in different catalyst samples can be measured, the KN test may be carried out, with caution, for catalysts with bimodal pore distributions. Hence, if we have two catalysts, 1 and 2, with surface concentrations of active material f,, and f,,, respectively, where f,, = afmg, then if the concentration of active material in the micropores of catalyst 1 is different from that in catalyst 2 by the same factor cy, then the KN criterion cannot be used to check for the effect of intraparticle mass transfer in catalysts with bimodal pore distributions.

Let us now see how the KN criterion may be used in the

Ind. Eng. Chern. Fundarn., Vol. 21, No. 4, 1982

different kinds of reactors used to carry out chemical reactions. In all following discussions f, will be used to denote the concentration of active material. (The discussions, of course, will also be valid for f, or fa.)

When investigating heterogeneous catalytic reactions in a batch reactor, the system is assumed to be in a stationary state, and either reactant consumption or product for- mation is observed with respect to residence time. The KN test may be used in a batch reactor system as follows. Let us take w1 grams of a catalyst in which f m = f,, and note the conversion at a certain residence time t . In a second experiment, take w 2 grams of a catalyst in which f m = f m , SO that

~2 = wl(fm,/fm,)

and note the conversion at the same residence time t as the first experiment. If the conversions at the same residence time in the two experiments are identical, then the KN criterion is obeyed.

Rates are directly obtained (Boudart, 1968; Denbigh, 1966) in the case of continuous stirred tank reactors (CSTR) operating at the stationary state, and hence the KN criterion is easily used. The CSTR condition states that mixing is complete so that all the properties of the reaction mixture are the same in all parts of the reactor and in the exit stream. The concept of the CSTR has been used, by experimentalists in the laboratory, in the form of the recycle reactor pioneered by Temkin (1962) and the continuous stirred tank catalytic reactor used by Tajbl et al. (1966). The recycle reactor consists of a loop which includes a reactor containing a small amount of catalyst and a recirculation pump. At the stationary state, when the flow rate of chemical components in and out of the loop is small compared to the recirculation rate, the composition in the loop and exit line is practically identical, thus obeying the CSTR condition and allowing the rates to be obtained directly.

In plug flow reactors, rates cannot be measured directly unless the reactor is operated in a differential manner. Rather, the conversion of a reactant is obtained at a known space velocity which may be defined as the molar flow rate of the reactant divided by the weight of the catalyst. We must modify the way to use the KN criterion from the method described above to a method compatible with the observables in the tubular flow system. Let us define a quantity analogous to space velocity and denote it by p , the active space velocity

If two experiments are performed and f , is changed but p is kept constant in each experiment, then the KN criterion is obeyed if the observed conversion is invariant in both cases (see Theoretical Section). Another way is to obtain values of conversion at different flow rates and to plot conversion vs. l / p using a catalyst sample with f, = fm1. If this plot coincides with the plot when we repeat the experiment using a catalyst sample with f , = f,, where f,, # f m , then the KN criterion is obeyed.

Before concluding this section describing the KN criterion, it must be stressed that the test does not require a knowledge o f the number o f sites involved in the reaction. Theoretical Section

A. Intraparticle Mass and Heat Transfer. Three relations on which much of the following discussion will be based are

p = tiO/wf, (1)

r = kvdC,) (2) k 0: f m (3)


I 1 1 n@

Figure 1. Typical plots of In 7 vs. In $J for isothermal and nonisothermal exothermic and endothermic reactions (Weisz and Hicks, 1962).

Equation 2 indicates that the observed rate is dependent on the effectiveness factor which in turn is a function of the dimensionless Thiele modulus defined in eq 4. Equation 3 is characteristic for heterogeneous catalytic reactions taking place in the kinetic regime. The rate of a catalytic reaction, defined in moles per unit time, is an extensive property and is directly proportional to the amount (Le., weight, surface area, or surface moles) of active material; e.g., if the amount of active material is doubled then the rate should be doubled. Before pro- ceeding, we must clarify the above explanation by stating that the KN criterion is not the same as the well-known test in which if the weight of a catalyst is doubled, say in a slurry reactor, the reaction rate should be doubled if there is no influence of interphase mass transfer. In the KN criterion, the concentration of active material in the catalyst, rather than just the weight of the catalyst, is changed, and in the kinetic regime the reaction rate is directly proportional to the concentration of active material.

Let us consider the curves A, B, and C in Figure 1. The slope, of any curve, a t any point is

d l n q d l n v p=-=- d l n 4 d l n $ (-1 I p I a)

where

Using eq 2, 4, and 6

CY = 0 for A and B; a > 0 for C; q = v r = kd%!(c,)

s is the slope obtained on plotting In r against In f ; s will be referred to as the Koros-Nowak number or slope. When

p - 0 , s - 1 and when

p - -1, s - 1 / 2

The former limit corresponds to 7 - 1, i.e., no intraparticle diffusion, whereas the latter limit indicates severe

u 1 n@

Figure 2. A plot of In 7 vs. In $J for a nonisothermal exothermic reaction (Weisz and Hicks, 1962).

Figure 3. A plot of In 7 vs. In $J for a nonisothermal exothermic reaction (Weisz and Hicks, 1962).

intraparticle diffusion and low values of q. When diffusional influence exists but is small, the limits for curves A and B are in between those described above: -1 I p I 0 and ' I 2 I s I 1.

For the exothermic reaction a similar situation exists for part b to c (Figure 2). Between a and b, p is greater than zero and hence s is greater than 1. The largest (d In q)/(d In 4) will be at the inflexion point a' where s will also have its largest value. At point b, p is zero and hence s is unity, similar to the zero diffusional influence point a. The effectiveness factor at b is higher than unity, and the rate is hence higher than the true isothermal rate. An easy way to resolve this ambiguity is to perform the experimental criterion at another temperature which would move the point in question either to the left or right of point b. If s is unity at two different temperatures, then one can be assured that one is not concerned with the rather rare occurrence of being at point b. It is interesting to note that at point c' where q is unity and there is perfect compen- sation (the rise in the reaction rate due to the increase in the temperature within the pellet compensates the de- crease in the measured rate due to the influence of mass transfer), the value of s will tend to 1/2, indicating correctly that the exothermic nonisothermal reaction is being influenced by intraparticle diffusion. As the Arrhenius number

and the heat generation parameter

increase, graphs as shown in Figure 3 may arise (Weisz and Hicks, 1962). Here the values of p are greater than zero between points P and Q. At Q and R, (d In q)/(d In 4) - m and hence s - m. Between Q and R, p 0 and s < 1. It must be noted that if p is a large negative number s may become less than 0.5 and also zero or a negative number.

Y = E/R,Ts (9)

P = (-WDeCs/XTs (10)

442

Table 111. Numerical Values for the Case of an Isothermal, Irreversible, First-Order Reaction on a Spherical Catalyst Pellet

Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982

rl d -p s = 1 i @/2)

0.059 0.14 0.27 0.37 0.48 0.67 0.81 0.94 0.98

50 20 10

7 5 3 2 1 0.5

0.98 0.95 0.90 0.83 0.75 0.55 0.35 0.12 0.032

0.51 0.53 0.55 0.58 0.62 0.73 0.82 0.94 0.98

q = l / A (13) where

Between R and S, p > 0 and s > 1. Beyond S the arguments are similar to the isothermal case.

For an irreversible, isothermal first-order reaction in a spherical catalyst pellet the above conclusions may be obtained analytically from results of others (Satterfield, 1970)

7 = #P = ”[ coth 4 - i] 4

d l n q d l n q - 4- d In 4 d4

--

2 - 4 coth 4 - 4’ csch2 4 4 coth 4 - 1 (12)

See Table I11 for numerical values. It is interesting to note that for the firsborder reaction considered, at effectiveness factors 10.8, the Koros-Nowak number is approximately equal to q.

Computations are not made so easily for other situations, e.g., nonisothermal reactions. Equations given by Weisz and Hicks (1962), Carberry (1961), and others have to be solved and numerical differentiation carried out to give (d In d / ( d In 4) = P.

It is essential to note that the above analysis for intraparticle diffusion holds not only for reactions obeying simple power rate laws but also the more complex Lang- muir-Hinshelwood rate expressions including strong product inhibition. Analysis, detailed by Satterfield (1970), and Hutchings and Carberry (1966), show that In 7 vs. In 4 plots similar in shape to the ones described above are obtained. Interphase heat or mass transfer influences can complicate the problem (i.e., when the Nusselt numbers are low) by being coupled with the intraparticle case. Hutchings and Carberry (1966) and Carberry (1961) have shown that interphase heat transfer influence raises q for exothermic cases whereas interphase mass transfer tends to reduce 7. Once again the plots In 7 vs. In 4 are similar to the ones described above. I n short, for all cases with intraparticle diffusional influence, with or without the complexity of strong product inhibition, intraparticle heat transfer, and interphase heat or mass transfer, p will deviate from zero and s will deviate from one.

It should be noted that the value of s equal to unity also means that the turnover numbers, obtained for different values off,, must be identical.

Large Intraparticle Concentration Gradients. Un- der extreme diffusion conditions only a thin layer of a catalyst particle is useful, and the local curvature and hence the shape of the catalyst particle is of minimal importance.

Aris (1965), Bischoff (1965), and Petersen (196513) have demonstrated that the effectiveness factor is asymptoti- cally h-l for a general case, where A is the normalized Thiele modulus

P’

and r (CJ = ?(CIS, C28, CSs, ..., n (15)

The apparent rate constant k may be taken out of the function r ( C ) . Hence

r(Cs) = kro(Cs) r0(CJ may be a power rate law or a Langmuir-Hinshelwood type expression.

7 oc k-lI2 (16)

From eq 2,3, and 16, it can be easily seen that under Severe intraparticle diffusion conditions for any rate expression

Before ending the section on intraparticle diffusion, a short corollary may be easily deduced from the above algebra.

In eq 7, if the concentration function g(CJ is described by Cisn, where n is the true reaction order, then

s =

where m is the observed reaction order. From eq 8 n = 1 + (m - l) /s (17)

Knowing s from the KN criterion and the observed concentration dependence m, one can obtain the true order of reaction n, for isothermal operation.

B. Interphase Mass Transfer. When the reaction rate is strongly dependent on the interphase transport of reactants to the active material on a catalyst

N = k,a(Cb - C,) (18) As k, is independent of the concentration of active material, changing the latter will not change the rate and the KN number s will be zero.

For porous catalysts, when there is a large interphase mass transfer influence the effectiveness factor is usually low, as only the active material at the mouth of the pores participates in the reaction.

(19) rate = qkg(Cs) = k,a[Cb - c,] It can be easily shown for a first-order reaction that

rate = k’Cb where

Remembering that for a severe internal diffusion problem 7 0: fm-lI2 and using eq 3 and 20, we have two bounds when the KN criterion is used 1. a k , << qk rate = constant Cb and s = 0

2. ak, >> qk rate = constant fm1/2Cb and s = y2 Though easily shown for a first-order reaction, the same

limits apply for more complex rate expressions. Values

of s between 0 and ‘ I2 would then indicate a coupled influence of the interphase and intraparticle mass transfer.

C. Interphase Heat Transfer. Mears (1973) has shown that interphase heat transfer effects become significant before the corresponding mass transfer effects unless /3 is very small in which case, of course, problems due to heat transfer will not arise. In the following discussion, it is assumed that there is no diffusional influence and C, is equal to c b . Though the arguments are for exothermic reactions, endothermic reactions may be treated similarly.

At steady state, the heat released by the reaction is equal to that transferred to the bulk fluid.

(21)

Let us change the concentration of active material on the catalyst from f,, to f,, where f,, > f,,, keeping all other conditions the same

(22)

(23)

(-mr = ha(T, - Tb)

( - W r i = ha(T,, - Tb)

( - m r 2 = ha(T,, - Tb)

As f,, > f,,, r2 > rl and hence T,, > Tal. T,, > T,, > T b indicates interphase heat transfer influence.

Let us use the KN criterion, N1 = N2, to check for interphase heat transport influence. Using the Arrhenius form for the rate constant and separating f, from the preexponential term, we get f m , A ’ exp(-E/R,T,,)g(C,,T,,)

f m ,

- -

f m , A ’ exp (-E/RgTa,)g(Ca, Ts,)

fm,

Divide by exp(-E/RTb)

This equality holds if either (case 1): T,, - Tb and T,, - Tb or (case 2): T,, - Tar.

Case 1 is easily explained, as when the surface temperatures tend to the bulk temperature there is a minimal gradient and hence negligible heat transfer influence.

Case 2. (a) When Tss > Tal > Tb, N1 # N2, there is interphase heat transfer influence. (b) T,, > Tal - Tb, Nl # N2. This occurs when by increasing the rate we may enter the heat transfer influence regime. (c) T,, - Tal > Tb. This is inconsistent with eq 22 and 23 because when their left-hand sides are different as r2 > rl, the right-hand sides cannot remain identical. Therefore, when T?, - Tal we must have T,, - Tal - Tb. Hence, the KN criterion, N1 = N2, can be used to establish the nonexistence of interphase temperature gradients.

I t is interesting to rationalize the value of s for interphase heat transfer without diffusion when rate expressions are of the type r = kg(C,). As explained above, when there is no transport influence and f, is increased from f,, to f,,, the rate will increase from rl to r2 so that N1 = N2 and s = 1. If there is external heat transfer influence then iff,, > f,,, TS2 > Tal and r i >> r;; it should be noted that due to the exponential nature of the temperature dependence the rate r i increases over r2 more than rl’ increases over rl. Hence N2 > N1 and s > 1 (see Figure 4).


-1 In f, (rurfoce mol g catalyst)

I Coupled Interphore Transfer I

Figure 4. Plots of In f, vs. In r , to indicate diagramatically the Koros-Nowak criterion for interphase heat transfer (exothermic reaction) and its coupling with interphase mass transfer.

As explained before, when there is only isothermal interphase mass transfer influence s = 0. These individual influences of external heat and mass transfer can be coupled as indicated diagramatically in Figure 4 in order that a value between s = 0 and s > 1 would be obtained. In the rare case when the coupling gives s = 1, experiments done at a new bulk temperature would shift the value of s from 1 if transport influences are important. For endothermic reactions as f,, > f,, and external temperature gradients are important, N2 < N1 and s < 1.

Mears (1973) has recently shown the relative importance of transport effects, and the order was interphase heat transport > intraparticle mass transport > interphase mass transport > intraparticle heat transport. It is useful to keep these generalities in mind when using the KN criterion. Figure 5 indicates usual values of N and s for the KN criterion.

D. Deviations from Plug Flow Behavior in Tubu- lar Reactors (Interparticle Gradients). Plug flow in tubular reactors may be defined as a state of flow when the flow rate and all fluid properties over any cross section normal to the fluid flow are identical and the residence time of all volume elements, entering the reactor, is the same. Deviations from plug flow are due to the influence of transverse and longitudinal temperature and concentration gradients (interparticle gradients), velocity gradients, and backmixing in the tubular reactor. Though longitudinal concentration gradients are always present in plug flow reactors, these gradients may influence the transport of reagents by causing them to reach the reactor outlet more rapidly than they would by bulk flow alone.

Denbigh (1966) has indicated that transverse temperature gradients most often affect and invalidate the plug flow assumption. If Tal and T, represent the axial temperatures when catalysts with the concentration of active material f, and f,, are used respectively, and iff,, > f,, and Tw is the temperature of the wall, we have

T,, > Tal > Tw

for a nonisothermal exothermic reaction, and Tap < T,, < Tw

444 Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982

in terp iare - rmc lnt ropar t ic la 0 Transport E'fect,

........... !: ...........I

. . I

...... . . . . 0 : 0 . 0 : ................

Figure 5. Interphase and intraparticle transport effects. The chart indicates the trend of the turnover numbers N1 and N 2 when f,, > f,,, and shows limits for the values of s when interphase and intraparticle effects exist separately or are coupled together.

for a nonisothermal endothermic reaction. As the dependence of the rate on temperature is expo-

nential, conversions in the plug flow reactor, where radial temperature gradients are significant, will change as f , is changed but p is kept constant. A similar analysis may be obtained for the problem of longitudinal temperature gradients.

The relation used for tubular reactors (Boudart, 1968; Denbigh, 1966) where the plug flow assumption is valid may be written as

where r here is expressed in mol sd g-' of catalyst. Mul- tiplying the above equation on both sides by f ,

If there is any significant deviation from plug flow behavior due to the influence of transverse and longitudinal temperature or concentration gradients, or due to backmixing and velocity gradients, the above equation will be invalid.

Hence, if in two experiments using different values of f, but keeping p constant the same conversion is obtained, then the KN criterion is said to hold and deviations from plug flow behavior will be negligible. In the rare event that

a coupling of various transport phenomena causes the test to be apparently obeyed, the KN criterion should be conducted at a second reaction temperature. If the test, at the second temperature, is not obeyed, then transport effects are influencing the reaction.

Of course a negative result for the test will also be obtained if besides nonideal flow behavior interphase or intraparticle concentration or temperature gradients are present.

E. Poisoning and Poor Contacting of the Catalyst. Poor contacting of the catalyst may be due to the fluid channelling and bypassing the catalyst. This case and the case of catalyst poisoning would result in a certain number 6 moles of active material not being used for the reaction. I t will be shown that when this occurs the measured turnover numbers, using catalyst samples with different values off,, will not be the same and the KN criterion will therefore be disobeyed. In two separate experiments let the measured rates in mol d 8 - l of catalyst be r< and r2/ and the concentration of active material in surface mol g-l of catalyst be f , and f,,, and the weights of the catalysts taken be w1 and w2, respectively.

Hence, the calculated turnover numbers will be

In the case when the reaction is carried out under no poisoning conditions and good catalyst contacting

The "true" rate rl is then related to rl' by

61 r,' = rl - N- W1

Dividing (28) by f,, and using (26) and (27)

N(fml - 6 1 / ~ 1 ) N,' =

fm1

N f m , - W W 2 )

fmz

Similarly

N,' =

N,' and N,' are then equal only when

This is not possible except in the case

61 = Cyf,,W1

and 6, = afm2W2

(29)

(30)

To check for this case the weights of the catalysts should be changed by an unequal amount (say, w1/2 and w2/3) and the measurement of the rates repeated. If again turnover numbers are found to be equal, then there is no catalyst poisoning or poor catalyst contacting. Experimental Results and Discussion

The liquid phase hydrogenation of cyclohexene was studied over supported Pt/Si02 catalysts. The details of the apparatus, materials, procedure, and results are given elsewhere (Madon et al., 1978). The reaction was tested via the above criterion for any transport effects.

in fm (rvrface mol Pt g-1 catalyst)

-I 1 IO I

0 1 .50% Pt/SiO2 B l O O % 0 2.30% Pt/SiO2 W 62% 0 0.38% Pt/SiO2 BlOO% V 0.53% Pt/Si02 E= 56%

Figure 6. The Koros-Nowak criterion applied in the study of the liquid phase hydrogenation of cyclohexene on Pt/Si02 catalysts. Conditions: r corrected for 101.3 kPa H2, catalyst particle size > 200 mesh, solvent-cyclohexane = 20 cm3, at 275 K cyclohexene added = 2 cm3, at 307 K cyclohexene added = 0.5 cm3.

The reaction was carried out with three different size fractions of Pt/SiO, catalysts: (a) greater than 200 mesh, (b) 100 to 200 mesh, and (c) 60 to 100 mesh. Observed reaction rates on the three separate catalyst size fractions have been plotted in Figures 6 and 7, which are diagram- matic representations of the KN criterion written as r a f," and indicate clearly whether mass transport effects play a dominant role. From Figure 6, we ascertain that with the catalyst size fraction (a) there is negligible transport influence as s is equal to 1, whereas with the two larger size fractions (Figure 7) the observed reaction rates are found to be severely affected by intraparticle concentration gradients, the value of s being 1/2. Calculations to sub- stantiate these observations are given elsewhere (Madon, 1975).

I t is also interesting to note the values of the turnover numbers when catalysts with size fraction (a) were used. The 1.5% and 0.38% Pt/SiOz samples had the identical dispersion of 100% and N was found to be 2.67 s-' and 2.51 s-' at 275 K and 9.16 s-l and 9.02 s-' at 307 K. The 2.3% and 0.53% Pt/SiO, samples had dispersions of 62% and 56%, respectively, and N was found to be 2.40 s-' and 2.33 s-l at 275 K and 8.67 s-' and 8.51 s-l a t 307 K. It should be noted that only catalysts with dispersions that are identical or very close must be compared when the KN criterion is used. The identity of the turnover numbers a t two different temperatures is another way of looking at the test and indicates, as does Figure 6, that the reaction rates are not influenced by any physical events.

The activation energy of the reaction, using the small size particles, i.e., when s = 1, was found to be 24.4 kJ mol-l whereas using the larger size (100 to 200 mesh) particles when s = 1/2, a value of 18.9 kJ mol-' was obtained.

The rate expression for the liquid phase hydrogenation of cyclohexene may be written in the form r = kvg(C,). For


In i, (ruriace mol pt g-l catalyst)

-9 - 1 0 -1 I I I

0 1 .50% Pt/SiOp D=lOO% 02.30% Pt/SiO2 E= 62% v0.53% Pt/BO2 W 56%

I

Figure 7. The Koros-Nowak criterion applied in the study of the liquid phase hydrogenation of cyclohexene on Pt/SiOz catalysts. Conditions: r corrected for 101.3 kPa Hz, solvent-cyclohexane = 20 cm3, cyclohexene added = 0.5 cm3,

severe internal diffusion, 7 = 1/4, and using eq 4, r cc (kDe)'/2. One may express k and De by simple Arrhenius relationships (Satterfield, 1970; Satterfield et al., 1968)

De = A, exP(-ED,/R,T)

k = A exp(-E/R,T)

Hence the observed activation energy for the severely internal diffusion influenced reaction is given by '/z(EDe + E). Using an activation energy of 12.6 kJ mol-' for the diffusion of H2 in cyclohexane (Satterfield, 1970; Satter- field et al., 1968), the activation energy of the reaction in the 100 to 200 mesh Pt/SiO, particles is '/&4.4 + 12.6) = 18.5 kJ mol-'. Though this is close to the observed value of 18.9 kJ mol-', it must be stressed, however, that using values of activation energy alone to explain diffusional influence could lead to errors.

Mears and Boudart (1966) studied the liquid phase dehydrogenation of isopropyl alcohol on nickel catalysts reduced with sodium borohydride and on Raney nickel in a slurry reactor. The slurry was agitated either by stirring it with a propeller or, to study the effect of mass transfer under different hydrodynamic conditions, by shaking it. As the slurry agitation rate was increased the reaction rate increased and then leveled off to a plateau. It is usually inferred (Satterfield, 1970) that there is no influence of external mass transfer on the reaction rates at the plateau value. Mears and Boudart argued that, a t the plateau value, though the possibility of external mass transfer influence was eliminated, the rates could still be affected by pore diffusion. To check the latter point, they prepared nickel catalysts of different surface areas and noted that chromium acted as a structural promoter and increased the specific surface area of the active nickel component. Surface areas of the nickel catalysts were determined by the fatty acid adsorption method of Smith and Fuzek (1946) and by the nitrogen BET technique. At high agitation intensities, the rate constant per unit surface area remained essentially constant over a sevenfold variation in the specific area of the catalyst. Mears and Boudart concluded that this observation provided strong evidence

446

against any influence of internal or external diffusion. They argued that at low agitation intensities the reaction rate was reduced as a result of internal (and to a certain extent external) mass transfer due to catalyst agglomeration and clumping, and with increased agitation the reaction rate increased due to better suspension of the catalyst in the slurry with breakup of catalyst aggregates. The test, performed with several catalysts in which the surface area of the active component was varied and used to conclude that reaction rates at the plateau value were free from both internal and external mass transfer, was in effect the diagnostic criterion later proposed by Koros and Nowak (1967).

The Koros-Nowak criterion has also been used successfully on vapor phase reactions: hydrogenation of ethylene (Schlatter and Boudart, 1972), cyclohexene (Segal et al., 1978), cyclopropane (Boudart et al., 1966), and oxygen (Hanson and Boudart, 1978), decomposition of NO (Amirnazmi et al., 1973) on supported Pt catalysts, and ammonia synthesis on supported Fe catalysts (Dumesic et al., 1975).

Conclusion An experimental criterion to check for true reaction rates

in heterogeneous catalysis has been developed. If the KN test is obeyed, at two different temperatures, one can be sure that the observed kinetic data are free of artifacts such as mass or heat transport, catalyst poisoning, and contacting. If the test is not obeyed, it may be difficult to say exactly which physical event is affecting the rate data, though good characterization of the catalyst used and some understanding of the reaction being studied may help one to understand which artifact influences the reaction rate.

The KN criterion cannot be used for catalysts with a bimodal pore distribution if one uses the dilution method suggested by Koros and Nowak (1967).

The test may be used with batch, continuous stirred, or plug flow reactors, and during differential or integral rate measurements. A simple analysis of the criterion has been given, and its use in the laboratory demonstrated. Though in general the test may be used for any catalyst if an ap- propriate inert diluent can be obtained, it is most readily and easily used for supported metal catalysts where the support is inert and for heterogenized homogeneous or- ganometallic catalysts.

Nomenclature

a = external surface area of catalyst per unit volume of the catalyst, cm-l

A' = preexponential factor of the rate constant


A = A'fm, A'fa, or A'fw A, = preexponential factor in the Arrhenius relationship for

D. c ==concentration, mol cm-3 DaII = Damkohler number of the second kind DaIv = Damkohler number of the fourth kind De = effective diffusivity, cm2 s-l E = activation energy, kJ mol-' f, = concentration of active material, surface moles of active

f, = concentration of active material, cm2 of active material

f, = concentration of active material, g of active material g-'

g(CJ = g(Cls,C2s,CBs...) = concentration function in the rate

h = heat transfer coefficient, J cm-2 s-l K-l A H = heat of reaction, J mol-' k , = mass transfer coefficient, cm s-l k = rate constant

material g-' of catalyst

g-' of catalyst

of catalyst

expression

k' = defined by eq 20 m = observed reaction order n = true reaction order AO = molar flow rate, mol s-' N = turnover number, s-' N = molar flux, mol s-l cm-3 p = defined by eq 5 r = reaction rate, mol s-l 8-l (or ~ m - ~ ) of catalyst r ( C ) = rate expression, defined by eq 15 R = radius, cm R, = gas constant s = Koros-Nowak criterion number or slope S, = external surface of the catalyst particle, cm2 t = time, s T = temperature, K V, = volume of the catalyst particle, cm3 w = weight of the catalyst, g x = conversion Greek Letters a = any integer or fraction a' = volume fraction p = heat generation function, defined by eq 10 y = Arrhenius number, defined by eq 9 6 = unused moles of active material, see eq 28 rate 7 = effectiveness factor, defiied as the ratio of the actual rate

for a catalyst particle to the rate evaluated at external surface conditions

X = thermal conductivity of the catalyst, J cm-' s-l K-' h = normalized Thiele modulus, defined by eq 14 v = stoichiometric coefficient

p = active space velocity, defined by eq 1 I$ = Thiele modulus, defined by eq 4 x = defined in Table I w = defined in Table I Subscripts a = property at the axis of a tubular reactor b = bulk property i = species 1, 2, 3, ... s = surface property eq = equilibrium Literature Cited Amirnazmi, A.; Benson, J. E.; Boudart, M. J. Catal. 1973, 30, 55. Anderson, J. B. Chem. Eng. Sci. 1963, 78, 147. Ark, R. Chem. Eng. Scl. 1957, 6 , 262. Aris, R. Ind. Eng. Chem. Fundam. 1965, 4, 227. Benson, J. E.: Boudart, M. J. Catal. 1965, 4 , 704. Benson, J. E.; Hwang, H. S.; Boudart, M. J. Catal. 1973, 30, 146. Bischoff, K. B. AIChE J. 1965, 1 1 , 351. Bischoff, K. B. Chem. Eng. Sci. 1967, 22, 525. Boudart, M.; Aidag, A.; Benson, J. E.; Dougharty, N. A,; Harkins, C. G. J.

Boudart, M. "Kinetics of Chemical Processes"; Prentice-Hall: Englewood

Boudart, M. AIChE J. 1972, 18, 465. Carberry, J. J. AIChE J. 1961, 7, 350. Carberry, J. J. Ind. Eng. Chem. 1966, 58(10), 40. Chambers, R.; Boudart, M. J. Catal. 1966, 6, 141. Denbigh, K. 0. "Chemical Reactor Theory"; Cambridge University Press:

Dumesic, J. A.; Topsm, H.; Khammouma, S.; Boudart, M. J. Cafal. 1975,

Hanson. F. V.; Boudart, M. J. Catal. 1978, 53, 56. Hudglns, R. R. Chem. Eng. Scl. 1968, 23, 93. Hudgins, R. R . Can. Chem. Eng. 1972, 50, 427. Hutchlngs, J.,; Carberry, J. J. AIChE J. 1966, 12, 20. Koros, R. M.; Nowak, E. J. Chem. Eng. Scl. 1967, 22, 470. Madon, R. J. Ph.D. Dissertation, Stanford Unlverslly, 1975. Madon, R. J.; O'Connell, J. P.; Boudart, M. AIChE J. 1978, 24, 904. Mears. D. E.; Boudart, M. AIChE J. 1966, 72, 313. Mears, D. E. Ind. Eng. Chem. Process Des. Dev. 1971a, 10, 541. Mears, D. E. J. Catal. 1971b, 20. 127. Mears, D. E. J. Catal. 1973, 30, 283. Petersen, E. E. Chem. Eng. Scl. 19658, 20, 587. Petersen, E. E. "Chemlcal Reaction Analysis"; Prentlce-Hall: Englewood

SatterfleM, C. N.; Saraf, S. K. Ind. Eng. Chem., Fundam. 1985, 4 , 451. Satterfleld, C. N.; Ma, Y. H.; Sherwood, T. K. Inst. Chem. Eng. (London).

Satterfield, C. N. "Mass Transfer In Heterogeneous Catalysis": MIT Press:

Schlatter, J. C.; Boudart, M. J. Cafal. 1972, 24, 482.

t = Ci/Ci,

Catal. 1966, 6 , 92.

Cliffs, NJ, 1968.

Cambridge, 1966.

37, 503.

Cliffs, NJ. 1965b.

Symp. Ser. 1966, 28, 22.

Cambridge, MA, 1970.

Ind. Eng. Chem. Fundam. 1982, 21, 447-451 447

Segal, E.; Madon, R. J.; Boudart, M. J . Catai. 1976, 52, 45. Sinfelt, J. H. C9fel. Rev. 1989, 3 , 175. Smith, H. A.; Fuzek, J. F. J . Am. & e m . Soc. 1946, 68, 229. Stewart, W. E.; Vllladsen, J. V. AIChEJ. 1969, 75, 28. TaJbl, D. (3.: Simons, J. B.; Carbeny, J. J. Ind. Eng. Chem. Fundam. 1966,

Weisz, P. B. Z . Phys. Chem. (Frankfurt am Main) 1957, 1 7 , 1. Weisz, P. B.; Prater, C. D. Ab.. Cafal. Relat. Sub/. 1954, 6, 143. Welsz, P. B.: Hicks, J. S. Chem. Eng. Sci. 1962, 77, 265. Whyte, T. E. Catal. Rev. 1973, 8 , 117.

c .-. 3. I 1 I.

Temkin, M. I . Klnet. Kafal. 1962, 3 , 509. Wanke, S. E.: Dougharty, N. A. J . Cafai. 1972, 24. 367.

Receiued for reuiew September 28, 1981 Accepted May 7, 1982

Mechanism of Stickiness in Hygroscopic, Amorphous Powders

Galen E. Downton,' Jos6 L. Flores-Luna,2 and C. Judson King'

Department of Chemical Engineering, University of Callfornia, Berkeiey, California 94720

Sticking, caking, and agglomeration tendencies of hygroscopic, amorphous powders are interpreted in terms of a mechanism of viscous flow driven by surface energy during particle contact. The mechanism predicts that stickiness should occur for combinations of temperature and moisture content corresponding to viscosities of the amorphous material below some relatively constant value, which for short-time contact should be within the range loe to 10' Pas. The viscosity is affected by both moisture content and temperature, thereby explaining the inverse relationship between sticky-point temperature and moisture content for such powders. Both bulk-solution and powdered specimens of a 7:l w/w mixture of sucrose and fructose were prepared, with moisture contents in the range of 2 to 7% w/w. Viscosity measurements of the bulk-solution specimens were made with both a falirg-baii method and a rheogoniometer. Viscosities corresponding to experimentally measured sticky-point temperatures fell consistently within the range 0.3 X lo7 to 4.0 X lo7 Paos, thereby lending support to the proposed mechanism and quantitative model.

Introduction Dehydration of liquid materials is widely used to pro-

duce stable, easily handled and stored products in an economical fashion. Several techniques, such as spray drying, freeze drying, and drum drying, are available for producing dried powders from a wide range of raw materials including liquid foods, plastics, detergents, fertilizers, and pharmaceuticals. One phenomenon frequently encountered during production and storage of these dried powders is stickiness.

A familiar example of stickiness is caking during storage. Hygroscopic dried foods such as instant coffee or drink mixes tend to form hard cakes when subjected to high temperatures and/or humidities. The powder is no longer free-flowing and, because the cakes are less porous, they reconstitute less readily. Another example of stickiness, which can be turned into an attribute, is agglomeration, a process commonly used with dehydrated powders to obtain larger, porous particles with better rehydration and handling characteristics and/or more pleasing and con- sistent color, shape, and appearance. This operation is redly one of controlled stickiness; particles are deliberately subjected to steam, moist air, or a fine mist of water, as well as repeated contact, so that surfaces become sticky and clusters form via particle collisions and adherence. In spray drying, stickiness can be a major problem, which occurs when particles which are insufficiently dry collide with one another or with the walls of the drying apparatus and become stuck. This can lead to lower product yields,

Procter and Gamble Company, Coffee Division, The Winton Hill Technical Center, 6210 Center Hill Road, Cincinnati, OH 45224.

'Mezquital de Oro, Apdo. Postal 138, Hermosillo, Sonora, Mexico.

operating problems, and powder-handling difficulties. For heat-sensitive products this can also lead to overheating, resulting in unpleasant sensory characteristics and/or degradation. These problems are preventing some materials-notably fruit juices which contain large amounts of hygroscopic, amorphous sugars-from being successfully dehydrated.

Coping with stickiness in spray dryers and in powdered products and achievement of successful agglomeration have largely been matters of trial-and-error experimentation to find conditions which avoid or control the sticky characteristics of a given material. Cooling the dryer walls and flushing them with cool air have been employed to avoid stickiness in spray dryers (Lazar et al., 1956; Gupta, 1978a,b). Various additives, often of high molecular weight, have also been used to combat stickiness in spray drying (Brennan et al., 1971; Gupta, 1978a,b) and in powders (Peleg and Mannheim, 1969,1973,1977).

There is as yet no quantitative model which describes the mechanism of stickiness in dried powders and explains the success of failure of past approaches to the problem. Previous mechanistic speculations are nearly all qualitative and incomplete. White and Cakebread (1966) recognized dried liquid foods as amorphous glasses, i.e., metastable, supercooled liquids below their glass-transition temperature, Tg. At temperatures below Tg, the viscosity is ex- tremely high, of the order of 10l2 Pa.s (Jones, 1956). A t higher temperatures and/or humidities, the glass assumes more of a liquid nature, with a lower viscosity. Stickiness defects, such as lumpiness and caking, were attributed to this incipient liquid state.

Peleg and Mannheim (1977) attributed caking of onion powder to moisture absorption and proposed a humidi- ty-based caking mechanism. In their model, water absorbs on particle surfaces, forming a saturated solution and thereby making the particles sticky and capable of forming

0196-4313/82/1021-0447$01.25/0 0 1982 American Chemical Society

exptl criterion for absence of diff limit of heter cat reactions - madon-boudart

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