Ind. Eng. Chem. Fundam. 1981, 20, 54-59
54
-2
-1 0 MOVING COORDINATE
2
1
?
m
F
3 J
Y
Figure 6. Structure of the activity wave front.
used as a test, comparison wiith such "fitted" results is regarded as pointless. The effect of particle geometry on the position of the wave front is shown in Figure 5 for a Thiele modulus of 300. Note that 4 is based on the slab half-thickness or particle radius and not on a length given by the volume to surface area ratio (cf. Aris, 1957). Only after the activity wave has passed through the first 10% or so of the catalyst does the effect of geometry become noticeable. The detailed structure of the wave front is shown in Figure 6. The coordinates are moving with the front and the plot is valid for all (small) values of c and y. The notable feature is that to a first approximation the wave front possesses a self-preserving structure. Unlike the usual wave-type of problem, in which wave velocity is constant, here the wave velocity adjusts in precisely that way needed to preserve the structure of the wave invariant. In conclusion, through singular perturbation techniques, we have been able to present very simple asymptotic solutions to the catalyst self-poisoning problem for the case of large Thiele modulus. Our results agree with the numerical calculations of others, extend their conclusions to cylinderical geometry and, in addition, show that the detailed structure of the moving wave front is invariant with changes in the parameters and with time. Nomenclature a = catalyst activity A = nondimensional reactant concentration ( = C/Co)
C = reactant concentration Co = reactant concentration in the bulk fluid 2, = diffusivity of reactant I , = spatial subinterval behind the front [O,Xo) I , = spatial subinterval ahead of the front (Xo,l] k = reaction rate constant k , = deactivation rate constant L = characteristic particle dimension (half-thicknessfor slab, radius for cylinder and sphere) t = nondimensional time (= kt*) t* = dimensional time t = long time (= cy?) i: = long time (= e t ) u = intermediate variable (= aa /a{) u = intermediate variable (= lo FAo(s,E)ds) x = nondimensional distance scaled on L X o = position of activity wave 2, = velocity of activity wave Greek Letters a = integer (= 1, cylinder; 2, sphere) y = exponent in eq 27 t = k,jCo/k ( 6) (isomerization) R,. CzH4 + Rn-p
R,'.
01
R; + Rj. (i
+ R,.
-
(hydrogen transfer) (lob)
products
(11)
where R,. = CnH2n+l' (primary radical) and R,'. = C,H2"+!. (secondary radical). Kossiakoff and Rice (1943) assumed that radicals isomerized by a 1-5 intramolecular hydrogen transfer, that is
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
57
40 q
Table 111. Isomerization Equilibria at 500 "C
0
Kn
n 6 9 12 16
d
9.0
31.5 45.1 63.1
The R, equilibrium is corrected for the fact that the only possible secondary radicals that can be formed via a six-membered ring from primary radicals a r e w radicals. I
I
I
I
I
RICE-
I
I
I
I
I
I
I
L
3001
\
I
I 701,\,
I
2001
5 8 0
I
0.2
0.1
0.3
H$ PRESSURE (ATM)
i 1
'b
5
\
a reaction involving a six-membered ring transition state. This assumption was modified by Voge and Good (1949) to include internal hydrogen transfers involving larger than six-membered rings. These assumptions precluded any significant isomerization of radicals smaller than hexyl. A second assumption of Rice and Kossiakoff was that the isomerization reaction was rapid relative to decomposition or hydrogen transfer. Then the distribution of radicals of a given length would be characterized by an equilibrium constant K. The value of K was estimated by assuming that secondary radicals are 4 kcal/mol more stable than primary and by weighting the secondary vs. primary radicals by the relative number of abstractable hydrogen atoms. The ratio of secondary to primary radicals is given by
The equilibrium constants so calculated are given in Table I11 for various radicals at 500 "C. The absence of paraffins heavier than propane in the uncatalyzed cracking results suggests that at low hexadecane pressures, reaction 10 is slow relative to reaction 9 for n > 3. Thus only methyl, ethyl, and to some extent propyl radicals are participating in the hydrogen transfer reactions. If we make the same assumptions concerning isomerization as Voge and Good, and exclude any hydrogen transfer except for that by CH3. and C2H5.,we can predict a product distribution for hexadecane cracking at 500 "C. Figure 5 compares this prediction with the experimental results from Table 11. As with the prior work of Voge and Good (1949), the agreement is excellent.
Figure 6. Effect of Ha preasure on the total molar yield of products from n-hexadecane cracking.
The effect of H2S on the pyrolysis can be explained by the addition of the following steps to the mechanism R,.
+ H2S
R,'.
+ H2S
HS.
--
+ HS.
(12a)
RnH + HS.
(12b)
R,H
---+
+ C16H34 HS- + Ri-
H2S + C16H33. products
(13) (14)
Reactions 12 and 13 in effect catalyze the hydrogen transfer reaction 10. If (10) is the rate-limiting step in the mechanism, the introduction of H2Sshould increase the observed cracking rate. It does so as Figure 4 indicates. The half-order dependence on H,S and first-order dependence on hexadecane agree with the steady-state rate expression obtained assuming first-order initiation and termination by reaction 14. The curvature in Figure 4 at low H2S concentrations is consistent with the view that the uncatalyzed and catalyzed mechanisms are not additive, but rather that the latter replaces the former as the H a concentration is increased and as thiyl radicals become the predominant chain carriers. As the H2S pressure is increased, the rate of hydrogen transfer (reaction 12) will become competitive with decomposition (reaction 9). Some higher alkyl radicals will be stabilized before they can decompose; the result is the appearance of C4+paraffms indicated in Figure 3. If fewer radicals decompose,the selectivity to light decomposition products should decrease; hence the reduction in methane, ethane and ethylene shown in Figure 2. A decrease in the amount of radical decomposition should also result in fewer total moles of product per mole of hexadecane cracked. Figure 6 indicates that as the H2S pressure is increased, the total molar yield of products does indeed decline. The uncatdyzed pyrolysis produces about 370 mol of product per 100 mol of hexadecane cracked. Thus a hexadecane molecule is cracked in an average of 2.7 places. At 0.2 atm of H2S, only 250 mol/100 mol cracked are produced indicating an average of 1.5 bond scissions per hexadecane converted. H2Sis clearly reducing the number of consecutive radical decomposition steps per molecule cracked. The relative rates of reactions 9 and 12 were estimated by a method similar to that used by Doue and Guiochon (1968) to compare reactions 9 and 10. They pyrolyzed hexadecane at a variety of pressures and then used the
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
58
yields of C5+paraffins and C3+olefins to determine relative rates for hydrogen transfer and decomposition. Their procedure has been modified slightly as follows. (a) Only hexyl and heavier alkyl radicals are assumed to isomerize; Doue and Guiochon allowed for isomerization of pentyl radicals as well. Isomerization to equilibrium according to the Rice-Kossiakoff mechanism is assumed. (b) Hydrogen transfer to both primary and secondary radicals is considered; Doue and Guiochon (1968) assumed that reaction 10b was unimportant. With these assumptions, the following results are obtained: rate of formation of C3+ olefins, assuming all secondary radicals crack at the same rate
I
*
o
I
I
1
0'1
/
,
-1
I
I
0.2
0.1
I 0.3
04
HzS PRESSURE IATMI
Neglecting initiation and termination, the rate of decomposition of hexadecane is given by UL
Figure 7. Determination of
k I 2 / k 9 b(see text).
where klo is again an "average" hydrogen transfer rate constant (klo = kloB+ (l/K)klob). Taking the ratio of (VII) to (VIII), we obtain kl2
7= 1.2
x 102
R 10
In the presence of H2S, the selectivity to C3+olefins in moles/mole cracked is then
For simplicity [k12b+ (l/Kn)k12a]is replaced by k12, an "average" hydrogen transfer rate constant which is assumed to be independent of n. Then 8=
selectivity to
c6+paraffins
selectivity to C,+ olefins - [Kl,/(l
+ Ki6)1
-
A plot of the left side of (VI) vs. H2S pressure is shown in Figure 7. The data lie on a straight line whose slope yields the result k 12
- = 4.5 atm-'
WII)
k9b
Doue and Guiochon (1969), using the slightly different assumptions outlined previously, calculated the ratio of hydrogen transfer to decomposition rates for the uncatalyzed reaction. Their result interpolated to 500 "C is kl" kRh
- = 0.037
atm-'
(VIII)
or that the average rate of hydrogen transfer to C6+alkyl radicals at 500 "C is approximately 120 times faster from H2Sthan from hexadecane. As the procedure outlined in this section indicates, these are average rate constants, derived by neglecting differences in decomposition and hydrogen transfer rates and isomerization equilibrium constants for alkyl radicals of different lengths. Nonetheless, this approach does allow a comparison of overall hydrogen transfer rates with and without H2S. Once the ratio of average hydrogen transfer to decomposition rates has been determined, it should be possible to insert relative probabilities for alkyl radical stabilization by hydrogen transfer and further decomposition into the Rice-Kossiakoff mechanism and predict the entire product distribution in the presence of H2S. Figure 8 compares such a prediction to the experimental results for one experiment with H2S. The agreement is excellent in that part of the product distribution that was used to determine k12/kgb,as might be expected, but is less satisfactory for lighter paraffins. Presumably an accurate prediction of individual product yields requires a more detailed picture of how the ratio of k12/kgbvaries with alkyl radical length. This analysis has assumed only homogeneous catalytic participation by H2S. For quartz reactors, this is probably sufficient, and indeed, it satisfactorily accounts for the results observed. However, for pyrolysis in metal reactors, the situation would be significantly more complicated. Dunkleman and Albright (1976) and Ghaley and Crynes (1976) have shown that surface reactions in metal reactors can have a significant effect on the kinetics and product distributions of pyrolysis reactions. H2Swill react with some metals to form sulfides, presumably altering the catalytic activity of the surface and, hence, the heterogeneous contribution to the pyrolysis. Thus, for pyrolysis in metal reactors, the effect of H2S on heterogeneous as well as homogeneous reactions needs to be taken into account. Conclusions This study has demonstrated that H2Saddition significantly alters both the rate and product selectivity of the pyrolysis of long-chain paraffins. The overall effects on hexadecane cracking are (1)an increase in cracking rate proportional to the square root of H2S partial pressure, (2) a decrease in the yield of light gases (methane, ethane, and
:I
Ind. Eng. Chem. Fundam. 1981, 20, 59-62
is also involved in the chain termination processes. These results indicate that the homogeneous catalysis of free radical cracking reactions is indeed possible and that by the selective catalysis of one of the steps in a complex chain mechanism, significant changes in product selectivity as well as in rate may result.
15
\.
i
59
I-ALKENES
Acknowledgment The author would like to thank Drs. W. H. Davis, Jr., and R. K. Lyon for helpful discussions. The excellent technical assistance of Mr. J. C. Dowling is also gratefully acknowledged.
o 1
2
3
4
5
6
7
8
9 1 0 1 1 1 2 1 3 1 4 1 5
WALKANES
1
2
3
4
5
6
7
8
9
i
1 0 1 1 1 2 1 3 1 4 1 5
NUMBER OF CARBON ATOMS IN PRODUCT
Figure 8. Comparison of distribution of products predicted b y a Rice-Kossiakoff mechanism modified to include reaction 12 and rates from Figure 6 with experimental data from the H,S-catalyzed pyrolysis of hexadecane.
ethylene), and (3) an increase in the yield C4+paraffin products as H2S pressure is increased. These effects are consistent with a mechanism in which H2S catalyzes hydrogen transfer reactions between alkyl radicals and the reactant. The overall kinetics observed suggest that H2S
Literature Cited Doue, F.; Guiochon, G. J . Chim. Phys. 1088, 64, 395. Doue, F.; Guiochon, 0.Can. J . Chem. 1989, 47, 3477. Dunkleman, J. J.; Aibright, L. F. ACS Symp. Ser. 1078, 32, 241-273. Lah, R. I.; Borsanyi, A. S.; Satterfield, C. N. Id. Fabuss, B. M.; Smith, J. 0.; Eng. Chem. PrOcessDes. Dev. 1982, 1 , 293. Ghaly, M. A.; Crynes, B. L. ACS Symp. Ser. 1978, 32, 218-240. Groenendyk, H.; Levy, E. J.; Sarner, S. F. J. Chromatug. Sci. 1970, 8 , 115. Hutchings, D. A.; Frech, K. J.; Hoppstock, F. H. ACS Symp. Ser. 1976, 32. 178-196. Imai, N.; Toyama. 0. Bull. Chem. SOC. Jpn. 1981, 34, 328. Kossiakoff, A.; Rice, F. 0. J. Am. Chem. SOC. 1043, 65, 590. Large, J. F.; Martin, R.; Nickuse, M. C.R. Acad. Sci. Paris 1972, 2746, 322. McLean, P. R.; McKenney, D. J. Can. J. Chem. 1070, 48, 1782. Nlciause, M.; Martin, R.; Baronnet, F.; Scacchi, G. Rev. Inst. Fr. Pet. 1088, 27, 1924. Niclause, M.; Baronnet, F.; Scacchi, G.; Muller, J.; Jezequei, J. Y. ACS Symp. Ser. 1078, 32, 17-36. Porchey, D. V.; Royer, D. J. US. Patent 3803 260, Apr 9, 1974. Scacchi, G.; Baronnet, F.; Martin, R.; Nickuse, M. J. Chim. Phys. 1988, 65, 1671. Scacchi, G.; Dzierzynski, M.; Martin, R.; Niciause, M. Int. J . Chem. Kinet. 1070, 2 , 115. Tischier, L. G.; Wing, M. S. U.S. Patent 3773850, Nov 20, 1973. Voge, H. H.; Good, G. M. J . Am. Chem. SOC. 1949, 77, 593.
Received for review December 5, 1979 Accepted October 24, 1980
Effects of Number and Size of Milling Balls on the Mechanochemical Activation of Fine Crystalline Solids Masaru Shlnozakl and Mamoru Senna' Faculty of Engineering, Keio University, Hiyoshi, Yokohama 223, Japan
y F e 2 0 3was vibro-milled in cyclohexane using a varying number, N, of steel balls of diameter d. The amount of stored energy, AHs,varied with Nand d as AHs = 24.2Nd,2.57,where d, is the ratio of d to the diameter of the milling pot. The exponents of Nand d, were very similar to those in the equation expressing the work required for the specific surface increase in a simple comminution process. For the enhancement of the rate of transformation into a-Fe203 on subsequent heating, the effects of Nand d, were considerably different from those in the comminution process, because of the threshold energy of the transformation.
Introduction Comminution of fine crystalline solids in a rotational or vibrational ball mill is one of the most widely used operations in ceramic, pharmaceutical, and many other industries. Aside from a simple size reduction of particles, crystalline solids are subject to deformation, giving rise to lattice imperfection and amorphization during milling. These in turn result in an increase in the catalytic activity and the reactivity of the solids themselves (Kubo, 1978; Schrader and Hoffmann, 1973; Thiessen et al., 1973). Such mechanochemical effects are important, not only by
themselves with regard to the activation of materials, but also in evaluating the efficiency of milling equipment, since the energy dissipation for such a mechanochemical process can often be one of the significant factors in the energy required in a grinding process (Rumpf, 1973). Nevertheless, the activation of solids through mechanical treatment has been previously discussed mostly on a qualitative basis, owing to the difficulty of quantitative measurement of the effective mechanical energy absorbed by the solid particles. The present authors have studied polymorphic transformation processes on PbO (Senna and Kuno, 19711, 0 1981 American Chemical Society
