Chain termination probabilities

The mass fraction of polymer with a chain length j within a certain interval dj is given by yjdj. The chain-termination probability, q, can be estimated from Mayo s equation ... 

An alternate approach that we use here describes FT products in terms of individual chain termination probabilities for each chain size, j3 . 

Carbon number distributions are qualitatively similar on all Ru catalysts used in our study. Chain termination probabilities (/8 ) for small chains... 

Chain termination probabilities initially decrease with increasing chain size (Fig. 2b) product distributions are non-Flory on all catalysts. This reflects an increase in readsorption rate as larger a-olefins become increasingly difficult to remove from liquid-filled catalyst pellets (4,5,14,40,41,44). Large olefins readsorb extensively and leave catalyst pellets predominantly after they form n-paraffins in sequential chain initiation and termination steps. As larger olefins (n > 30) disappear from the products, the chain termination probability reaches a constant value and product distributions become predominantly paraffinic and obey Flory kinetics (Fig. 2b). The asymptotic termination probability (/3=o) reflects the intrinsic probability of... 

The previous discussions have shown that chain initiation by readsorbing a-olefins readily reverses hydrogen abstraction termination steps and leads to a net decrease in the overall chain termination probability (j8 ). The re-... 

Chain termination probabilities are readily separated into the individual contributions from each type of chain termination [Eq. (4)]. The net termination to olefins becomes the difference between jSo, the probability of forming an a-olefin, and j8r, the probability that it will readsorb before leaving the catalyst bed. On both Co and Ru catalysts, the net olefin chain termination probability (j8 - j8r) decreases rapidly with increasing carbon number (Figs. 9a and 10a). This termination probability becomes zero for hydrocarbon chains larger than about C25 because olefins disappear from the reactor effluent, even at very short bed residence times. Bed residence times shorter than 2 s (and CO conversions less than about 10%) do not affect chain termination probabilities even for short hydrocarbon chains, because fast convective transport removes a-olefins from the catalyst bed before readsorption occurs. Yet, readsorption within pellets occurs even at short bed residence times, because intrapellet residence times do not depend on space velocity consequently, chain termination to olefins and the olefin content in products still decrease strongly with increasing hydrocarbon size. 

Fig. 9. Carbon number effects on olefin and paraffin chain termination probability (catalyst Ru/Ti02, 1.2 wt% Ru, 48% Ru dispersion 476 K, 560 kPa, H2/CO = 2.1). (a) < 2-s bed residence time, < 5% CO conversion (b) 12-s bed residence time, 60% CO conversion.

Intrapellet transport restrictions can limit the rate of removal of products, lead to concentration gradients within pellets, and prevent equilibrium between the intrapellet liquid and the interpellet gas phase. Transport restrictions increase the intrapellet fugacity of hydrocarbon products and provide a greater chemical potential driving force for secondary reactions. The rate of secondary reactions cannot be enhanced by a liquid phase that merely increases the solubility and the local concentration of a reacting molecule. Olefin fugacities are identical in any phases present in thermodynamic equilibrium thus, a liquid phase can only increase the rate of a secondary reaction if it imposes a transport restriction on the removal of reacting species involved in such a reaction (4,5,44). Intrapellet transport rates and residence times depend on molecular size, just as convective transport and bed residence time depend on space velocity. As a result, bed residence time and molecular size affect chain termination probability and paraffin content in a similar manner. 

Ethylene and propylene intrinsic readsorption rates exceed those of larger olefins (4). This leads to the low apparent chain termination probabilities for C2 and C3 molecules shown in Fig. 8. These lower /8t values reflect a more effective reversal of chain termination to olefins (/3o) for these smaller but more reactive molecules (Figs. 9 and 10). The high rate of ethylene and propylene readsorption accounts for their appearance below the rest of the distribution in carbon number distribution plots on both Co and Ru catalysts (Fig. 5). 

Fro. 13. Site density effects on selectivity and chain termination probability Co/TiOa (catalyst A 11.7 wt% Co, 1.5% Co dispersion, 79 h site-time yield, site density I.O /ig-atom surface Co m catalyst B 12.1 wt% Co, 5.8% Co dispersion, 82 h site-time yield, site density 3.3 /cg-atom surface Co m" 473 K, 2000 kPa, H2/CO = 2.1, <10% CO conversion). (a) C5+ selectivity (b) olefin chain termination probability (c) paraffin chain termination probability. 

The initial increase in C5+ selectivity as x increases arises from diffusion-enhanced readsorption of a-olefins. At higher values of CO transport restrictions lead to a decrease in C5+ selectivity. Because CO diffuses much faster than C3+ a-olefins through liquid hydrocarbons, the onset of reactant transport limitations occurs at larger and more reactive pellets (higher Ro, 0m) than for a-olefin readsorption reactions. CO transport limitations lead to low local CO concentrations and to high H2/CO ratios at catalytic sites. These conditions favor an increase in the chain termination probability (jSr, /Sh) and in the rate of secondary hydrogenation of a-olefins (j8s) and lead to lighter and more paraffinic products. 

Experimental number distributions (Fig. 16) and chain termination probabilities (/3 and jSn) (Fig. 17) on Co catalysts at low values of bed residence time (<2 s, <10% CO conversion) are accurately described by the model. We reported previously a similar agreement on Ru catalysts (4). The model quantitatively describes the observed curvature of carbon number distribution plots (Fig. 16) and also the constant values of j3/y and the decreasing values of )3o observed as hydrocarbon size increases (Fig. 17). Such effects arise from the higher intrapellet fugacity and the higher residence time of larger a-olefins within transport-limited pellets. 

Fig. 17. Comparison of model and experimental results. Carbon number effects on olefin and paraffin chain termination probability (experimental/model same as in Fig. 16).

On catalysts with large values of x> he chain termination probability reflects the kinetics of the primary hydrogen addition termination step, because olefin termination steps are completely reversed by extensive readsorption within a catalyst pellet. In this case, bed residence time does not affect the product distribution because olefins seldom reach interpellet voids before converting to paraffins. 

The olefin readsorption model also accounts for the observed decrease in CH4 selectivity as x values initially increase (Fig. 21). C2+ product distributions are simulated using the olefin readsorption model and methane selec-tivities calculated using a chain termination probability to CH4 (jScH ) of 1.1, a value experimentally observed ( 0.05) on all Co catalysts, irrespective of bed residence time or of the presence of olefins in H2/CO reactants. Such )3ch4 values were also constant but much lower (0.60) on Ru catalysts (4,5). The model predicts a minimum CH4 selectivity at x values greater than lO m, consistent with our experimental findings (Fig. 21). 

Our readsorption model shows that carbon number distributions can be accurately described using Flory kinetics as long as olefin readsorption does not occur (/3r = 0), because primary chain termination rate constants are independent of chain size (Fig. 24). The resulting constant value of the chain termination probability equals the sum of the intrinsic rates of chain termination to olefins and paraffins (j8o + Ph)- As a result, FT synthesis products become much lighter than those formed on Co catalysts at our reaction conditions (Fig. 24, jSr = 1.2), where chain termination probabilities are much lower than jS -I- Ph for most hydrocarbon chains. The product distribution for /3r = 12 corresponds to the intermediate olefin readsorption rates experimentally observed on Co/Ti02 catalysts, where intrapellet transport restrictions limit the rate of removal of larger olefins, enhance their secondary chain initiation reactions, and increase the average chain size of FT synthesis products. 

Carbon number distribution plots also become linear when olefins readsorb very rapidly (large /3r) or when severe intrapellet transport restrictions (large ) prevent their removal from catalysts pellets before they convert to paraffins during chain termination (Fig. 24, jSr = 100). In this case, chain termination to olefins is totally reversed by fast readsorption, even for light olefins. Chain termination occurs only by hydrogen addition to form paraffins, a step that is not affected by secondary reactions and for which intrinsic kinetics depend only on the nature of the catalytic surface. The product distribution again obeys Flory kinetics, but the constant chain termination probability is given by )8h, instead of po + pH- Clearly, bed and pellet residence times above those required to convert all olefins cannot affect the extent of readsorption or the net chain termination rates and lead to Flory distributions that become independent of bed residence time. 

Chain termination probabilities for a-olefins (j8o), n-paraffins ()8h), and alcohols (j8a) are plotted as a function of carbon number in Fig. 29b. The net rate of chain termination to olefins decreases with increasing carbon number without a proportional increase in the probability of chain termination to paraffins, except perhaps for very light paraffins. This suggests that secondary hydrogenation reactions cannot account for the selective disappearance of olefins as the carbon number increases on Fe catalysts. Chain termination to paraffins decreases slightly for short chains but then remains essentially independent of chain size the slight decrease in /3h with increasing chain size disappears if we include isoparaffin products in our /3h term. Therefore, except for an initial decrease in /3h values for short chains, the observed trends are similar to those observed on Co and Ru catalysts. 

It is proposed in [92], that the summary of polymer MWD can be nsed for the Tikhonov regularisation method [93, 94] to derive AC distribntion by chain termination probability. No assumption has been made about the type of this distribution. 

Where P s the inverse value of M and the measnre of chain termination probability. P = VM Tfjtp at the tp polymer growth rate and the total chain termination rate rg, (M is the MW of a polymer). 

In the case of lumped processes, hybrid model design focuses on reducing a complex set of static relationships with mat r fit parameters by overall fiizzy sub-models. These sub-models can be derived from measurements. A polymer reactor has been used as an example. In the liquid phase reactor, which is ideally stirred, monomer reacts to polymer by means of a single site metallocene catalyst. The inputs of the reactor are listed in Table 30.2 the load Fin, the hydrogen fraction of the feed xh2, the activated catalyst fraction Xc,a, the non-activated catalyst fraction Xc,na, the monomer fraction and the jacket temperature 7 adfe,. The most important outputs are the molecular weight distribution MWD (or q, the chain termination probability which is inversely proportional to MWD) and the conversion or (feed mass fraction conversion). The model is described in detail by Roffel and Betlem (2003). The requirements for the model of this reactor are ... 

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