Fischer distribution

The distribution of a population s property can be introduced mathematically by the repartition function of a random variable. It is well known that the repartition function of a random variable X gives the probability of a property or event when it is smaller than or equal to the current value x. Indeed, the function that characterizes the density of probability of a random variable (X) gives current values between X and x -I- dx. This function is, in fact, the derivative of the repartition function (as indirectly shown here above by relation (5.16)). It is important to make sure that, for the characterization of a continuous random variable, the distribution function meets all the requirements. Among the numerous existing distribution functions, the normal distribution (N), the chi distribution (y ), the Student distribution (t) and the Fischer distribution are the most frequently used for statistical calculations. These different functions will be explained in the paragraphs below. 

The Fischer-Tropsch process can be considered as a one-carbon polymerization reaction of a monomer derived from CO. The polymerization affords a distribution of polymer molecular weights that foUows the Anderson-Shulz-Flory model. The distribution is described by a linear relationship between the logarithm of product yield vs carbon number. The objective of much of the development work on the FT synthesis has been to circumvent the theoretical distribution so as to increase the yields of gasoline range hydrocarbons. 

Liquid—Liquid Extraction. The tiquid—tiquid extraction process for the rare-earth separation was discovered by Fischer (14). Extraction of REE using an alcohol, ether, or ketone gives separation factors of up to 1.5. The selectivity of the distribution of two rare-earth elements, REI and RE2, between two nonmiscible tiquid phases is given by the ratio of the distribution coefficients DI and D2 ... 

SASOL. SASOL, South Africa, has constmcted a plant to recover 50,000 tons each of 1-pentene and 1-hexene by extractive distillation from Fischer-Tropsch hydrocarbons produced from coal-based synthesis gas. The company is marketing both products primarily as comonomers for LLDPE and HDPE (see Olefin polymers). Although there is still no developed market for 1-pentene in the mid-1990s, the 1-hexene market is well estabhshed. The Fischer-Tropsch technology produces a geometric carbon-number distribution of various odd and even, linear, branched, and alpha and internal olefins however, with additional investment, other odd and even carbon numbers can also be recovered. The Fischer-Tropsch plants were originally constmcted to produce gasoline and other hydrocarbon fuels to fill the lack of petroleum resources in South Africa. 

FIG. 23-24 Reactors with moving catalysts, a) Transport fluidized type for the Sasol Fischer-Tropsch process, nonregenerating, (h) Esso type of stable fluidized bed reactor/regeuerator for cracldug petroleum oils, (c) UOP reformer with moving bed of platinum catalyst and continuous regeneration of a controlled quantity of catalyst, (d) Flow distribution in a fluidized bed the catalyst rains through the bubbles. 

Therefore, the pH values of these solutions are between 11 and 12. The speciation model used by 8chwarzenbach and Fischer is certainly too simple but these authors have been the first to demonstrate the strong dependence of the polysulfide anion distribution on the alkalinity. According to Eqs. (26)-(28) higher pH values in dilute solutions will favor smaller anion sizes. 

Tytko KH, Mehmke J, Fischer S (1999) Bonding and Charge Distribution in Isopolyox-ometalate Ions and Relevant Oxides - A Bond Valence Approach. 93 125-317... 

Equation 3 is exact for fluids obeying Equations 1 and 2. However, in order to compute the density n(r) from the YBG equation one must know the relationship between density distribution and the pair correlation function of Inhomogeneous fluid. Such a relationship Is not available in general. However, an approximation introduced by Fischer and Methfessel (1.) has been shown to give fairly accurate predictions of the density... 

Heme and its immediate precursor, protoporphyrin IX (Figure 32-4), are both type III porphyrins (ie, the methyl groups are asymmetrically distributed, as in type III coproporphyrin). However, they are sometimes identified as belonging to series IX, because they were designated ninth in a series of isomers posmlated by Hans Fischer, the pioneer worker in the field of porphyrin chemistry. 

Figure 8.17. Hydrocarbon distribution of the products formed by Fischer-Tropsch synthesis over cobalt-based catalysts and by additional hydrocracking, illustrating how a two-stage concept enables optimization of diesel fuel yield. [Adapted from S.T. Sie,...

Fischer Christian-H, Hart Edwin J, Henglein Amim (1986) Ultrasonic irradiation of water in the presence of 181802 Isotope exchange and isotopic distribution of H202. J Phys Chem 90 1954-1956... 

Finally, it has been reported that carbon electrodes modified with thin polymeric films of polypyridyl metal complexes containing a dispersion of metal particles (Rh° or Pd°) can be used as electrocatalyst for reduction of C02 to hydrocarbons in MeCN. Apparently CH4 is the dominant reduction product (up to 18% of faradaic efficiency).123,124 It should be noted that the product distribution is reminiscent of a Fischer-Tropsch process since C2, C3, and C4 hydrocarbons are also formed. 

Huang, X. W., Elbashir N. O., and Roberts, C. B. 2004. Supercritical solvent effects on hydrocarbon product distributions from Fischer-Tropsch synthesis over an alumina-supported cobalt catalyst. Industrial Engineering Chemistry Research 43 6369-81. 

Tsubaki, N., Yoshii, K., and Fujimoto, K. 2002. Anti-ASF distribution of Fischer-Tropsch hydrocarbons in supercritical-phase reactions. J. Catal. 207 371-75. 

Temperature-programmed reduction combined with x-ray absorption fine-structure (XAFS) spectroscopy provided clear evidence that the doping of Fischer-Tropsch synthesis catalysts with Cu and alkali (e.g., K) promotes the carburization rate relative to the undoped catalyst. Since XAFS provides information about the local atomic environment, it can be a powerful tool to aid in catalyst characterization. While XAFS should probably not be used exclusively to characterize the types of iron carbide present in catalysts, it may be, as this example shows, a useful complement to verify results from Mossbauer spectroscopy and other temperature-programmed methods. The EXAFS results suggest that either the Hagg or s-carbides were formed during the reduction process over the cementite form. There appears to be a correlation between the a-value of the product distribution and the carburization rate. 

The TPR-XAFS technique confirmed that doping Fischer-Tropsch synthesis catalysts with Cu and alkali (e.g., K) remarkably promotes the carburization rate relative to the undoped catalyst. The EXAFS results suggest that either the Hagg or e-carbides were formed during the reduction process over the cementite form. A correlation is observed between the a-value of the product distribution and the carburization rate. 

In this chapter a two a selectivity model is proposed that is based on the premise that the total product distribution from an Fe-low-temperature Fischer-Tropsch (LIFT) process is a combination of two separate product spectrums that are produced on two different surfaces of the catalyst. A carbide surface is proposed for the production of hydrocarbons (including n- and iso-paraffins and internal olefins), and an oxide surface is proposed for the production of light hydrocarbons (including n-paraffins, 1-olefins, and oxygenates) and the water-gas shift (WGS) reaction. This model was tested against a number of Fe-catalyzed FT runs with full selectivity data available and with catalyst age up to 1,000 h. In all cases the experimental observations could be justified in terms of the model proposed. 

The readsorption and incorporation of reaction products such as 1-alkenes, alcohols, and aldehydes followed by subsequent chain growth is a remarkable property of Fischer-Tropsch (FT) synthesis. Therefore, a large number of co-feeding experiments are discussed in detail in order to contribute to the elucidation of the reaction mechanism. Great interest was focused on co-feeding CH2N2, which on the catalyst surface dissociates to CH2 and dinitrogen. Furthermore, interest was focused on the selectivity of branched hydrocarbons and on the promoter effect of alkali on product distribution. All these effects are discussed in detail on the basis... 

The carbon number distribution of Fischer-Tropsch products on both cobalt and iron catalysts can be clearly represented by superposition of two Anderson-Schulz-Flory (ASF) distributions characterized by two chain growth probabilities and the mass or molar fraction of products assigned to one of these distributions.7 10 In particular, this bimodal-type distribution is pronounced for iron catalysts promoted with alkali (e.g., K2C03). Comparing product distributions obtained on alkali-promoted and -unpromoted iron catalysts has shown that the distribution characterized by the lower growth probability a, is not affected by the promoter, while the growth probability a2 and the mass fraction f2 are considerably increased by addition of alkali.9 This is... 

With the exception of methane and the C2 fraction, the carbon number distribution of Fischer-Tropsch products can be well represented by superposition of two ASF distributions ... 

In Fischer-Tropsch synthesis the readsorption and incorporation of 1-alkenes, alcohols, and aldehydes and their subsequent chain growth play an important role on product distribution. Therefore, it is very useful to study these reactions in the presence of co-fed 13C- or 14 C-labeled compounds in an effort to obtain data helpful to elucidate the reaction mechanism. It has been shown that co-feeding of CF12N2, which dissociates toward CF12 and N2 on the catalyst surface, has led to the sound interpretation that the bimodal carbon number distribution is caused by superposition of two incompatible mechanisms. The distribution characterized by the lower growth probability is assigned to the CH2 insertion mechanism. 

Zhan, X., Davis, B. H. 2002. Assessment of internal diffusion limitation on Fischer-Tropsch product distribution. Applied Catalysis A General 236 149-61. 

Donnelly, T.J., Yates, I.C., Satterfield, C.N. 1988. Analysis and prediction of product distributions of the Fischer-Tropsch synthesis. Energy Fuels 2 734. 

Schulz, H., Claeys, M. 1999. Kinetic modelling of Fischer-Tropsch product distributions. Appl. Catal. A 186 91. 

Kuipers, E.W., Scheper, C., Wilson, J.H., Vinkenburg, I.H., and Oosterbeek, H. 1996. Non-ASF product distributions due to secondary reactions during Fischer-Tropsch synthesis. J. Catal. 158 288-300. 

The primary product from Fischer-Tropsch synthesis is a complex multiphase mixture of hydrocarbons, oxygenates, and water. The composition of this mixture is dependent on the Fischer-Tropsch technology and considerable variation in carbon number distribution, as well as the relative abundance of different compound classes is possible. The primary Fischer-Tropsch product has to be refined to produce final products, and in this respect, it is comparable to crude oil. The primary product from Fischer-Tropsch synthesis can therefore be seen as a synthetic crude oil (syncrude). There are nevertheless significant differences between crude oil and Fischer-Tropsch syncrude, thus requiring a different refining approach.1... 

The Arge Fe-LTFT syncrude (Table 18.8)29 was much heavier than the syncrude of the two German Co-LTFT processes (Table 18.2). The Arge Fe-LTFT syncrude exemplified a high a-value Fischer-Tropsch product with a significant linear paraffinic wax fraction. The syncrude (Table 18.8) from the Kellogg Fe-HTFT synthesis was very similar in carbon number distribution to that of Hydrocol Fe-HTFT synthesis (Table 18.5). 

The Fischer-Tropsch synthesis, which may be broadly defined as the reductive polymerization of carbon monoxide, can be schematically represented as shown in Eq. (1). The CHO products in Eq. (1) are any organic molecules containing carbon, hydrogen, and oxygen which are stable under the reaction conditions employed in the synthesis. With most heterogeneous catalysts the primary products of the reaction are straight-chain alkanes, while the secondary products include branched-chain hydrocarbons, alkenes, alcohols, aldehydes, and carboxylic acids. The distribution of the various products depends on both the type of catalyst and the reaction conditions employed (4). 

---