Diffusional restrictions

In several papers (51, 84, 96, 104) the decrease of the polymerization rate with time was assumed to be caused by the decrease of C as a result of diffusional restrictions due to the formation of a polymer film on the catalyst surface. However, as a matter of experience in work with heterogeneous catalysts for ethylene polymerization, it is known that even for polymerization with no solvent, the formation of a solid polymer is possible at high rates (thousands of grams of polymer per gram of catalyst per hour) that are constant until large yields are reached (tens of kilograms of polymer per gram of catalyst). 

The calculation of C according to (6) shows (95) that if the catalyst splitting results in the formation of catalyst pellets about 1000 A in size, then even under the most unfavorable conditions (the concentration of the active centers is equal to the total chromium content in the catalyst, 2r2 = ) the diffusional restriction on the primary particle level is negligible. 

Due to the small dimensions of the micrograin [<1 X Iff-2 cm according to the porosity measurements (55)], diffusional restrictions at this level may occur only at rather high (tens of kilograms per gram of catalyst per hour) polymerization rates. 

Thus, two factors may be pointed out that determine the possibility of obtaining high yields of crystalline polyethylene on a solid catalyst with no diffusional restriction (1) the splitting up of the catalyst into small particles (< 1000 A), possible when using supports with low resistance to breaking (2) the formation of polymer grains with polydispersed porosity. 

If the dissolving of a portion of the polymer takes place, diffusional restriction may occur as a result. Such a case was observed in (98) where a decrease of the polymerization rate (slurry process in cyclohexane) with temperature rise from 75° to 90°C was found despite the increase in the number of propagation centers. At a further increase of the polymerization temperature (>115°C) polymerization becomes a solution process that may also proceed with no diffusional restrictions (94). 

Modeling of pore diffusion phenomena can be a helpful tool mainly in terms of catalyst design considerations but also in terms of understanding the effects caused by diffusional restrictions. For example, a modeling study by Wang et al.7 demonstrated a negative impact on selectivity by particle diffusion limitations. 

FIGURE 19.12 Considerations for the interpretation of SSITKA data. Case 1 Three formates can exist, including (a) rapid reaction zone (RRZ)—those reacting rapidly at the metal-oxide interface (b) intermediate surface diffusion zone (SDZ)—those at path lengths sufficient to eventually diffuse to the metal and contribute to overall activity, and (c) stranded intermediate zone (SIZ)—intermediates are essentially locked onto surface due to excessive diffusional path lengths to the metal-oxide interface. Case 2 Metal particle population sufficient to overcome excessive surface diffusional restrictions. Case 3 All rapid reaction zone. Case 4 For Pt/zirconia, unlike Pt/ceria, the activated oxide is confined to the vicinity of the metal particle, and the surface diffusional zones are sensitive to metal loading. 

In addition to forming close contacts with the PM, the SR network also comes into close contact with the mitochondria (Nixon et al 1994, Rizzuto et al 1998), forming yet another diffusionally restricted space (Fig. 4, panel E-G). This space, approximately 60—80 nm wide and sandwiched between the SR and mitochondrial membranes, also appears to be functionally important. As the SR network penetrates deeper into the cell, it inserts into the nuclear membrane such that the lumen of the perinuclear SR network is continuous with the lumen of the nuclear envelope (Somlyo 1985). 

This may be partly the result of increased steric crowding in the transition state of transalkylation. Another contributory factor to the increased selectivity in ZSM-5 is the higher diffusion rate of ethylbenzene vs m-/o-xylene in ZSM-5 and hence a higher steady state concentration ratio [EB]/[xyl] in the zeolite interior than in the outside phase. Diffusional restriction for xylenes vs ethylbenzene may also be indicated by the better selectivity of synthetic mordenite vs ZSM-4, since the former had a larger crystal size. 

If the rate of reaction obtained with no diffusional restrictions is denoted by the symbol l then an effectiveness factor may be defined as ... 

A typical effect observed in the synthesis of linear polymers by a free-radical mechanism is the auto-acceleration process. At a particular conversion, when sufficient polymer has accumulated in the system for the viscosity to reach a certain level, the rate of the bimolecular termination reaction begins to fall because of diffusional restrictions to the encounter of two chain ends. However, the initiation and growth rates are hardly affected. 

The propagation step can be characterized by a single specific rate constant that decreases continuously with conversion due to diffusional restrictions. 

For T = Tg, the value of kc increases with the selected cure temperature while the value of kd remains constant. This means that diffusional restrictions are observed earlier at high temperatures. But in this particular system, they become significant only when Tg > T. 

Figure 5.9 shows the evolution of Tg as a function of In (time) at different cure temperatures. Diffusional restrictions show their influence by a decrease in the slope of the experimental curves at Tg values below the corresponding cure temperatures. 

Figure 5.9 Glass transition temperature (Tg) vs In (time) for the catalyzed cyclotrimerization of a dicyanate ester at different cure temperatures. Full lines represent the fitting with a kinetic model that includes diffusional restrictions. (Simon and Gillham, 1993 - Copyright 2001 - Reprinted by permission of John Wiley Sons, Inc.)...

Actually this is a simplification. As a result of random fluctuations of pore radii (21, 22) larger pores in the interior are not available without severe diffusional restrictions. 

The effects of diffusional restrictions on the activity and selectivity of FT synthesis processes have been widely studied (32,52,56-60). Intrapellet diffusion limitations are common in packed-bed reactors because heat transfer and pressure-drop considerations require the use of relatively large particles. Bubble columns typically use much smaller pellets, and FT synthesis rates and selectivity are more likely to be influenced by the rate of mass transfer across the gas-liquid interface as a gas bubble traverses the reactor (59,61,62). 

Three broad regimes are thus possible (i) reaction takes place on the surface of the liquid, (ii) reaction takes place homogeneously in the bulk of the liquid catalyst without diffusional restrictions, and (iii) reaction takes place homogeneously in the liquid catalyst, but is limited to a layer near the liquid surface because of diffusional effects. 

A reduced reaction rate may result from external diffusional restrictions on the surface of carrier materials. In stirred tanks external diffusion plays a minor role as long as the reaction liquid is stirred sufficiently. Further, partition effects can lead to different solubilities inside and outside the carriers. Partition has to be taken into account when ionic or adsorptive forces of low concentrated solutes interact with carrier materials [81 - 83]. The most crucial effects are observed in porous particles due to internal or porous diffusion as outlined below. 

On the other hand, approximate calculations of ti can also be used [85].In this way, cumbersome solutions of the differential equations are not required if one merely wishes to obtain a general idea of substrate-related diffusional restrictions. A profound insight into effectiveness factors is given by Kasche [86] where a good correlation between calculated and experimental data is demonstrated. 

In practice it is desirable to be able to detect substrate-related porous diffusion reliably by simple means. A popular method is to assay the enzyme activity at varying substrate concentrations. At low substrate concentrations diffusional restrictions are likely to be predominant. They can be detected by graphical evaluation. Instead of straight lines somewhat curved Hues are obtained in the case of v/S vs. v-plots according to Eadie-Hofstee [89]. However, the extent of the curvature is not necessarily as great as would be required to detect diffusional control beyond all doubt. Thus one caimot exclude diffusional effects if nonlinearity is not observed. 

Ruckenstein [96] has calculated the resulting diffusion-restricted enzyme activities at high substrate concentrations. He teaches us in simple terms that pH-shifts and resulting reductions in activity occur even at substrate concentrations high enough to exclude any substrate-related diffusional restrictions,... 

Unfortunately, most enzymes do not obey simple Michaelis-Menten kinetics. Substrate and product inhibition, presence of more than one substrate and product, or coupled enzyme reactions in multi-enzyme systems require much more complicated rate equations. Gaseous or solid substrates or enzymes bound in immobilized cells need additional transport barriers to be taken into consideration. Instead of porous spherical particles, other geometries of catalyst particles can be apphed in stirred tanks, plug-flow reactors and others which need some modified treatment of diffusional restrictions and reaction technology. 

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