734
Energy & Fuels 1988,2,734-739
of free-radical stability and attack on the carbon-hydrogen bond is benzylic and allylic H > tertiary H > secondary H > primary H, it was expected that fuels with different oxidative tendencies would exhibit gross differences in their NMR spectra. To examine this possibility, the IH and 13Cnuclear magnetic resonance (NMR) spectra of fuels 1,2, and 4 were measured. The proton-type assignments were made according to the recommendations of Netzel and Hunter.18 Surprisingly, the relatively large difference in the reactivity with oxygen was not reflected in the NMR spectra of fuels 1,2, and 4. The spectra did not show any significant enhancement of the more reactive C-H bond types in fuel 4 compared to those in fuels 1 and 2. Apparently these differences in the strengths of C-H bonds in the test fuels are more subtle than expected and cannot be resolved by NMR spectroscopy. The absence of correlation is probably due in part to the complexity of fuel composition. However, it is also possible that the concentration of the active species (RH) in these fuels is relatively low. It was argued in the proposed mechanism that RH is in high concentration, based on the relative amount of peroxide formed. When it is realized that measured peroxide concentrations seldom exceed 3000 ppm, then an RH concentration of as little as 2-3% of the fuel could be considered to be relatively high. Yet, a species in the 2-3% concentration range is at the threshold of detection by NMR. These questions can only be answered by continued experimental work on the kinetics of autoxidation in jet fuels.
Conclusion The rates of peroxide formation in six model kerosenes were measured in the temperature range 43-120 OC with oxygen partial pressures ranging from 10 to 1140 kPa. To explain the rate of peroxide buildup in the fuels, a kinetic model of the autoxidation process was developed on the (17) Nixon, A. C. In Autoxidation and Antioxidants; Lundberg, W . O., Ed.; Interscience: New York, London, 1962; Chapter 17. (18) Netzel, D. A.; Hunter, P. M. DOE/LETC/RI-81-1, Laramie, WY,
May 1981.
premise that peroxide decomposition is the principal free-radical initiation step. In accordance with this model, it was found that the square root of the peroxide concentration was proportional to the stress duration. Global rate constants determined from the peroxide concentrationtime histories were independent of the partial pressure of oxygen but strongly dependent on the stress temperature. Arrhenius correlations of the global rate constants showed that the mechanism of peroxide formation remained unchanged in the temperature range 43-120 OC. The activation energies of the fuels ranged from 19 to 22 kcal/mol except that for fuel 3, which was 29 kcal/mol. Because there was a significant variation in the activation energies of peroxide formation, it is concluded that peroxide potential can not be predicted from single-point rate measurements at elevated temperatures. However, the results of this work encourage the development of a timely test method that predicts rates of peroxide formation at ambient conditions when two or more measurements are made at elevated temperatures.
Acknowledgment. This work was conducted at the Belvoir Fuels and Lubricants Research Facility, located at Southwest Research Institute (SwRI), for the Naval Research Laboratory (NRL) under Contract No. N00014-85-C-2520. For completeness, this paper includes the results of earlier work funded by the Naval Air Propulsion Center (NAPC). NAPC funding was provided by a Military Interdepartmental Purchase Requisition through the US.Army Belvoir Research, Development and Engineering Center under Contract DAAK70-85-C0007, with F. W. Schaekel as Contracting Officer's Representative. The project monitors for the Navy were Dr. D. R. Hardy of NRL and C. J. Nowack, G. E. Speck, and Lynda C. Turner of NAPC. We gratefully acknowledge the invaluable discussions and participation by colleagues G. H. Lee, 11, W. D. Weatherford, Jr., N. F. Swynnerton, Marilyn Voigt, K. B. Jones, J. J. Dozier, and Deborah Toles. Editorial assistance provided by J. W. Pryor, Marilyn Smith, Sherry Douvry, and LuAnn Pierce is also acknowledged with appreciation.
Analysis and Prediction of Product Distributions of the Fischer-Tropsc h Synthesis Timothy J. Donnelly, Ian C. Yates, and Charles N. Satterfield* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received March 14, 1988. Revised Manuscript Received June 18, 1988
A method is developed for calculating the three parameters needed to characterize the carbon number distribution of products of the Fischer-Tropsch synthesis. Experimental data are fit by a modified Schulz-Flory model that has two chain growth probabilities, using nonlinear regression. Excellent fit is shown for data from precipitated iron and fueed magnetite catalysts. The model is used to calculate selectivity information of interest in catalyst comparison and reactor design. Advantages of this method over asymptotic regression methods are discussed in detail. Introduction The r>roducts of the Fischer-Tror>sch synthesis are primariiy linear hydrocarbons distributed -over a wide range of carbon numbers. Herington' reported that a
model of stepwise addition of single-carbon units could predict the fraction of product at each carbon number. (1) Herington, E. F. G. Chem. Ind. (London) 1946, 65, 346.
0887-0624/88/2502-0734$01.50/00 1988 American Chemical Society
Fischer-Tropsch Synthesis
This and other early treatments discussed by Anderson et aL2are formulations of a condensation polymerization model developed independently and in another context by Schulz3and Flory.4 These matters are discwed in detail by Satterfield and Huffs5 In its simpler forms, the model suggests that a semilogarithmic plot of mole fraction of product as a function of carbon number should produce a straight line with a slope characteristic of the chain growth probability, a. Anderson et al.2developed a more detailed treatment that also considered isomer distributions. In extensive studies of six iron catalysts in a German pilot plant in 1943,the “Schwarzheide tests”, an increase or “break” in the slope of Schulz-Flory plots was observed a t a carbon number of about 10. This phenomenon received little further attention until recently, when several investigators reported the same effect under a variety of conditions, suggesting that it may be a rather general phenomenon. Two chain growth probabilities have been observed on iron catalysts by Konig and Gaube,B Huff and Satterfield,’ Schliebs and Gaube? Egiebor, et 81.: and Dictor and Bell,lo on cobalt by Schulz et a1.,l1 and on ruthenium by Inoue et a1.12 The causes of two probabilities for chain growth, frequently termed the “double-a”, are unclear, but an understanding of these causes might provide insights into methods for optimizing the Fischer-Tropsch product distribution. Konig and Gaubes suggested that two sites, one promoted by potassium and the other unpromoted, may cause such behavior. However, the work of Didor and Belllo and extensive data collected recently in this laboratory and soon to be published indicate that two chain growth probabilities exist even on catalysts not promoted by potassium. It is not easy to determine the experimental product distribution accurately over a wide range of carbon numbers. Mataumoto and Satterfield13discuss how calculated values of a may be affected by the selection of data used in calculations. Here, we develop and present a more general method for calculating the correct values of a1and a2 from data that include contributions by both chain growth probabilities. The method is not based on any assumptions about the cause of the double-a. Previous attempts to model Fischer-Tropsch product distributions with multiple values of a have several limitations. Rice and Wojciech~wskil~ have presented a comprehensive analysis of such product distributions. By assuming that the produds at each carbon number are of uniform molecular weight, they removed the difficulty of dealing with variations in alkene/&ane ratio with carbon number. We follow a similar assumption here. They developed an equation providing for contributions from two chain growth probabilities. By differentiating this equa(2)Anderaon, R. B.;Friedel, R. A.; Storch, H.H. J. Chem. Phys. 1961, 19,313. (3)Schulz, G . V. 2.Phys. Chem., Abt. B 1935,30,379. (4)Flory, P.J. J. Am. Chem. SOC.1936,58,1877. ( 5 ) Satterfield, C. N.; Huff, G. A. J. Catal. 1982,73,187. (6)Konig, L.;Gaube, J. Chem.-Zng.-Tech. 1983,55,14. (7)Huff,G.A.; Satterfield, C. N. J. Catal. 1984,85,370. (8)Schliebs, B.;Gaube, J. Ber. Bunsen-Ges.Phys. Chem. 1985,89,68. (9)Egiebor, N. 0.; Cooper, W. C.;Wojciechowski, B. W. Can. J. Chem. Eng. 1985,63, 826. (10)Dictor, R. A.;Bell, A. T. J. Catal. 1986,97,121. (11) Schulz, H.;Roach, S.; Gokcebay, H.Proceedings 64th CIC Coal Symposium; Al Taweel, A. M., Ed.; C.S.Ch.E.: Ottawa, Canada, 1982. (12)Inoue, M.; Miyake, T.; Inui, T. J. Catal. 1987,105,266. (13)Matsumoh, D. K.;Satterfield, C. N., submitted for publication. (14)Rice, N. M.; Wojciechowski,B. W. Can. J. Chem. Eng. 1987,65, 102.
Energy & Fuels, Vol. 2, No. 6, 1988 735
1
tA a . I -
O E
.-c0
4 0.01
LL
-
H
I
0.001 0
I 5
I
I
I
IO
15
20
I
25
Corbon Number, Cn
Figure 1. Schulz-Flory diagram of data from a potassium-promoted precipitated iron cat.&at with asymptotic regressionlines.
tion, they established relationships between the observed product distribution and a1and az.Their model, however, does not provide a tractable method for determining al and a2 from experimental data, because it depends on numerical or graphical differentiation to estimate “local as‘ at each carbon number. Konig and Gaubes and Schliebs and Gaubes appear to use normalized weight fractions, rather than absolute mole fractions, to determine the values of the chain growth probabilities. This may lead to errors in the values of a and, in some cases, failure to describe the distribution of heavier products correctly. Stenger15extended Konig and Gaube’s theory that potassium causes multiple chain growth probabilities in a model that describes a as a function of local potassium loading, with a Gaussian-type distribution of potassium concentrations on the surface. Inoue et al.12 have shown that distributed-a and double-a models, each having three adjustable parameters, can ody be distinguished if reliable data at high carbon numbers are available. Accurate data at high carbon numbers may be difficult to obtain.13Js Novak et al.”J* developed several models to describe nonlinear Schulz-Flory distributions. In particular, they presented a model based on two active sites, one for hydrocarbon growth and the other for cracking. However, Schulz et al.l9and Pichler and Schulzmhave reported that cracking is not significant on iron or cobalt catalyata under the usual conditions of Fischer-Tropsch synthesis. Novak’s model requires four adjustable parameters as well as knowledge of surface rates and intermediates that is difficult to obtain experimentally. The most straightforward method for determining a1 and a2 from experimental data would seem to be linear regression of the asymptotes of a Schulz-Flory plot. Figure 1 shows a typical Schulz-Flory diagram from a potassium-promoted precipitated iron catalyst. The diagram shows regions with two distinct slopes. However, even several carbon numbers away from the intersection of the two lines, the values of a1 and a2 determined by linear Stenger, H.G. J. Catal. 1986,92,426. Huff, G. A. Sc.D. Thesis, MassachusettsInstitute of Technology, (17)Novak, S.;Madon, R. J.; Suhl, H. J. Chem. Phys. 1981,74,6083. (18)Novak, S.;Madon, R J. Ind. Eng. Chem. Fundam. 1984,23,274. (19)Schulz, H.;Rao, B. R.; Elstner, M. Erdol Kohle 1970,23,651. (20)Pichler, H.;Schulz, H. Chem.-hg.-Tech.1970,42, li62.
Donnelly et al.
736 Energy & Fuels, Vol. 2, No. 6, 1988
regression differ by 5-10% from those calculated by the rigorous statistical procedure outlined in this paper, demonstrating that simple linear regression over small carbon number ranges can only approximate ai and az. The model presented in this paper offers a method for determining ai and a2from rigorous statistical techniques. The error produced in analyzing data by linear regression is discussed. In calculating ai and az,this method uses the entire experimental product distribution including those points between the asymptotes. Excellent agreement between the regression and experimental data from iron catalysts is shown. For an unalkalized precipitated iron catalyst, experimental data from an overhead product stream are fit by the regression, and the value of a2 is shown to agree with that calculated from a slurry wax sample. In addition, product stream data from a fused magnetite catalyst are fit. For the purposes of this model, a is considered to be the probability that a C,,species will go on to become a Cn+l species. In the simplest case, with no products readsorbing and incorporating into growing chains and with no branching, this definition of a reduces to the traditional definition of r p / ( r p+ rt), where rp and r, are the rates of chain propagation and termination, respectively.
Development of the Model As discussed above, the hydrocarbon products of the Fischer-Tropsch synthesis are generally taken to follow the Schulz-Flory distribution. For carbon number n , the mole fraction of product M,, as determined by a single chain growth probability, is given by
M,,= ( 1 - CY)&-')
(1)
Normally, product mole fractions are plotted on a semilogarithmic scale since the chain growth probability CY can be calculated from the slope of such a plot. Again, for a single-a model In M,,= n In a In [ ( l- a)/aI (2)
+
If two chain growth probabilities contribute to the total product distribution, the appropriate equation is Mn = Aal(fl-l)+ Ba 2 (n-1) (3) and the appropriate logarithmic expression is In Mn = In [Aal("-l)+ Baz("-l)]
(4)
I t can be seen from eq 4 that the product distribution can not be treated strictly as the sum of two logarithmic terms. That is, two straight lines on a Schulz-Flory diagram can not be added to generate an experimental product distribution, as such. This suggests that asymptotic linear regression is an inappropriate technique for calculating ai and a2.Instead, we note that at the break point on a Schulz-Flory diagram, the contributions of each term in eq 3 are equal. Aal(n-l) = B a 2 b-1) (5) This allows us to calculate the ratio of A to B, which must be known to characterize the product distribution. If [ is defined as the break point, then B = A(al/a2)(~')
(6)
[ is not necessarily an integral carbon number. This is a somewhat different approach than has been taken previously by Schliebs and Gaube6 and by Huff.16 In their models, A and B were assumed to correspond directly to the fractions of products produced from ai and a2 re-
1
a , / a 2 = 0.5
0 . 7 0 r
0.50 o.60
~
0
1
2
3
4
5
6
7
Separation from Carbon Number at Breakpoint
Figure 2. Effect of break point separation, I(n - [)I, on aloc/q or % c / ~ 2 .
spectively. Instead it is noted that, since the sum of the mole fractions over all carbon numbers is unity m
0)
n=l
,,=I
E M n = C [Aal("-l)+ Ba2(n-i)]= 1
(7)
Evaluating the geometric series, this equation can be expressed as A [ 1 / ( 1 - ai)]
+ B [ 1 / ( 1 - 041 = 1
(8) Equations 5 and 8 are two linearly independently equations that are used to solve explicitly for the values of A and B: A=
1 1 / ( 1 - a1) + (a1/az)(E-l'[1/(1- %)I
(9)
B is then calculated from eq 6. Substituting these values of A and B into eq 3 gives
This method places no artificial constraints on the system. The equations presented above allow computation of theoretical product distributions based on chosen values of ai, az, v d 6. Differentiatingthe logarithm of equation 10 with respect to carbon number, it is possible to obtain a "local a",ab, which is the slope of the Schulz-Flory curve and is a function of carbon number. This slope accounts for contributions by both chain growth probabilities. al&) = exp[d[ln M,,]/dn] = exp[ In a1 + (a2/al)("-t) In a2 (11) 1
+ (az/a1)("f)
]
At low carbon numbers, the local a approaches ai, and at high carbon numbers, it approaches az. The differences between local as and the true values of a1and a2in the region near the break point cannot be eliminated experimentally. Figure 2 shows how the ratios a,,Jal and a,Ja2 vary with separation between the carbon number n corresponding to aloe and the break point, I(n - [)I, for values of a1/a2= 0.5 and a1/a2= 0.7. The second case, a l / a 2 = 0.7 corresponds to the commonly reported values of al near 0.6 and a2near 0.9.
Energy & Fuels, Vol. 2, No. 6,1988 737
Fischer-Tropsch Synthesis
Regression Method By iteratively calculating distributions based on different combinations of al,a2,and 5 and comparing them to experimental data, one can determine the best fit theoretical distribution. The method used here is minimization of the sum of square errors as recommended for this type of regression by Churchil121
0.50 ~
n=m
9 = C [In M , - In mnI2 n=3
(12)
Here, M, is the theoretical mole fraction of product at Cn and m, is the experimental value. The summation is taken from C3 to C,, where C, is the highest carbon number at which reliable data are available. The sum begins at C3, since experimentally obtained C1 and C2 products frequently do not obey a Schulz-Flory m e c h a n i ~ m . ~ J ~ J ~ l ~ At higher carbon numbers, experimental uncertainties may be introduced in continuous-flow slurry and fixed-bed 0.55 0.65 0.75 0.85 reactor systems. In fixed beds, less volatile products ac=I cumulate throughout the bed and, specifically, in catalyst Figure 3. Calculated effect of a1on product selectivity for a2 pores. Huff and Satterfield22give theoretical analyses = 0.90, € = 10. regarding the time on stream necessary for the exit stream to be truly representative of the products being synthescarbon numbers to determine the value of az.As discussed ized. above, such data may be difficult to obtain. In addition, In continuous-flowslurry reactors, nonvolatile products their method involves graphical or numerical differentiaare retained in the slurry liquid. Volatile products are tion, which makes calculations very sensitive to any local flashed overhead, but a significant fraction, particularly scatter in the experimental data. in the range ClS-C&, is distributed between the two phases A second advantage of our method is that it allows inin a manner that changes with time on ~ t r e a m .Experi~ clusion of all reliable product distribution data, including mental data are usually based on volatile overhead prodthat near f , which cannot be used in asymptotic fitting ucts, occasionally supplemented by analysis of slurry liqmethods.6J2 As mentioned, when volatile products of the uid. Higher molecular weight products require longer time synthesis are used to determine product distributions, on stream to reach steady-state vapor-liquid equilibrium. vapor-liquid equilibrium effects cause the Schulz-Flory Thus, C, is the carbon number above which product model to deviate from data at carbon numbers greater than distributions are observed to change with time on stream. roughly C15 (Huff1% Therefore, inclusion of, for example, This carbon number can be affected by such variables as c7-cl4 mole fractions effectively doubles the quantity of catalyst activity, catalyst loading, and reaction temperadata that can be used to calculate chain growth probature. HufP6 indicates that for representative experimental bilities. Three parameters are needed to characterize the conditions, apparent deviations from the Schulz-Flory s y ~ t e m . ~ , * JAdditional ~J~ points in the analysis greatly diagram may be expected for products above roughly C15. improve the statistical significance of the regression. Removing C1 and C2 products to fit theoretical distriThe regression routine allows selective exclusion of inbutions to data leads to the following modifications to eq dividual data points within a given carbon number range. 7 and 9: There are at least two instances in which this feature is m m useful. First, if hydrocarbons of a given carbon number Mn = [ A c ~ l ( ~ - l )Ba2(n-')]- A ( l + al) - B(l + a2) are deliberately added to the synthesis gas feed, the data n=3 n=l from that carbon number may be excluded from the fitting (74 routine. Second, if an impurity of some sort is known to occur at a given carbon number, the data from that carbon or number need not be regressed. m E M n = 1 - ml - m2 (7b) n=3 Results Parametric Study on Theoretical Product Distriand butions. Representative values of al, a2,and 5 can be used A = (1 - ml - m 2 ) / [ l / ( l - al)- (1 + al)+ to generate theoretical product distributions, which can be used to determine selectivity to different product cuts. ( a l / a 2 ) V 1 / ( 1 - a2)- (1 + 4 1 1 ( 9 4 Generally, the product cuts of interest are methane (CJ, B is still calculated from eq 6. light gases (C2
