Pseudo-homogeneous

Most catalysts for solution processes are either completely soluble or pseudo-homogeneous all their catalyst components are introduced into the reactor as Hquids but produce soHd catalysts when combined. The early Du Pont process employed a three-component catalyst consisting of titanium tetrachloride, vanadium oxytrichloride, and triisobutjlalurninum (80,81), whereas Dow used a mixture of titanium tetrachloride and triisobutylalurninum modified with ammonia (86,87). Because processes are intrinsically suitable for the use of soluble catalysts, they were the first to accommodate highly active metallocene catalysts. Other suitable catalyst systems include heterogeneous catalysts (such as chromium-based catalysts) as well as supported and unsupported Ziegler catalysts (88—90). 

The second type of solution polymerization concept uses mixtures of supercritical ethylene and molten PE as the medium for ethylene polymerization. Some reactors previously used for free-radical ethylene polymerization in supercritical ethylene at high pressure (see Olefin POLYMERS,LOW DENSITY polyethylene) were converted for the catalytic synthesis of LLDPE. Both stirred and tubular autoclaves operating at 30—200 MPa (4,500—30,000 psig) and 170—350°C can also be used for this purpose. Residence times in these reactors are short, from 1 to 5 minutes. Three types of catalysts are used in these processes. The first type includes pseudo-homogeneous Ziegler catalysts. In this case, all catalyst components are introduced into a reactor as hquids or solutions but form soHd catalysts when combined in the reactor. Examples of such catalysts include titanium tetrachloride as well as its mixtures with vanadium oxytrichloride and a trialkyl aluminum compound (53,54). The second type of catalysts are soHd Ziegler catalysts (55). Both of these catalysts produce compositionaHy nonuniform LLDPE resins. Exxon Chemical Company uses a third type of catalysts, metallocene catalysts, in a similar solution process to produce uniformly branched ethylene copolymers with 1-butene and 1-hexene called Exact resins (56). 

The basic phenomenon was observed in modeling studies by Bjoreskov and Slinko (1965) that sudden increase in inlet temperature caused a transient drop of the peak temperature. The wrong-way response name was given by Mechta et al (1981) after they experienced the opposite a sudden of inlet temperature resulted in an increase of the peak temperature (which may eventually cause a runaway.) The work used a pseudo-homogeneous reaction model and explained the phenomenon by the different speeds of transient response in gas and solid. The example in the last part of Chapter 7.4 explained the speed difference by the large difference in heat capacity of gas and solid phases. For this a two-phase model is needed and spatial and time changes must be followed. 

Therefore, once again invokingthe pseudo-homogeneous approximation, the equations we mustsolveare ... 

Liquids of complex structure, such a polymer solutions and melts, and pseudo-homogeneous suspensions of fine particles, will generally exhibit non-Newtonian behaviour, with their apparent viscosities depending on the rate at which they are sheared, and the time for which they have been subjected to shear. They may also exhibit significant elastic... 

The internal structure of the catalyst particle is often of a complex labyrinth-like nature, with interconnected pores of a multiplicity of shapes and sizes, In some cases, the pore size may be less than the mean free path of the molecules, and both molecular and Knudsen diffusion may occur simultaneously. Furthermore, the average length of the diffusion path will be extended as a result of the tortuousity of the channels. In view of the difficulty of precisely defining the pore structure, the particle is assumed to be pseudo-homogeneous in composition, and the diffusion process is characterised by an effective diffusivity D, (equation 10.8). 

A highly concentrated suspension of flocculated kaolin in water behaves as a pseudo-homogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald-de Waele power law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced even though the flowrate of suspension is kept constant, Explain how this is possible in slug flow and estimate the possible reduction in pressure gradient for equal volumetric flowrates of suspension and air. 

Porous solid as pseudo-homogeneous medium 635 effective diffusivity 635 Portable mixers 306 Positive displacement pumps. 315... 

Strictly gas-phase CSTRs are rare. Two-phase, gas-liquid CSTRs are common and are treated in Chapter 11. Two-phase, gas-solid CSTRs are fairly common. When the solid is a catalyst, the use of pseudohomogeneous kinetics allows these two-phase systems to be treated as though only the fluid phase were present. All concentration measurements are made in the gas phase, and the rate expression is fitted to the gas-phase concentrations. This section outlines the method for fitting pseudo-homogeneous kinetics using measurements made in a CSTR. A more general treatment is given in Chapter 10. 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

In order to exemplify the potential of micro-channel reactors for thermal control, consider the oxidation of citraconic anhydride, which, for a specific catalyst material, has a pseudo-homogeneous reaction rate of 1.62 s at a temperature of 300 °C, corresponding to a reaction time-scale of 0.61 s. In a micro channel of 300 pm diameter filled with a mixture composed of N2/02/anhydride (79.9 20 0.1), the characteristic time-scale for heat exchange is 1.4 lO" s. In spite of an adiabatic temperature rise of 60 K related to such a reaction, the temperature increases by less than 0.5 K in the micro channel. Examples such as this show that micro reactors allow one to define temperature conditions very precisely due to fast removal and, in the case of endothermic reactions, addition of heat. On the one hand, this results in an increase in process safety, as discussed above. On the other hand, it allows a better definition of reaction conditions than with macroscopic equipment, thus allowing for a higher selectivity in chemical processes. 

Temperature and concentration differences between gas and catalyst can be neglected to give a pseudo-homogeneous model,... 

The units of rv are moles converted/(volume-time), and rv is identical with the rates employed in homogeneous reactor design. Consequently, the design equations developed earlier for homogeneous reactors can be employed in these terms to obtain estimates of fixed bed reactor performance. Two-dimensional, pseudo homogeneous models can also be developed to allow for radial dispersion of mass and energy. 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

Illustration 12.7 indicates how to estimate an effective thermal conductivity for use with two-dimensional, pseudo homogeneous packed bed models. 

Pseudo homogeneous models of fixed bed reactors are widely employed in reactor design calculations. Such models assume that the fluid within the volume element associated with a single catalyst pellet or group of pellets can be characterized by a given bulk temperature, pressure, and composition and that these quantities vary continuously with position in the reactor. In most industrial scale equipment, the reactor volume is so large compared to the volume of an individual pellet and the fraction of the void volume associated therewith that the assumption of continuity is reasonable. 

Pseudo homogeneous models require global rate expressions of the type discussed in Section... 

The term pseudo homogeneous model has been popularized by Froment (94, 95). 

The amount of A that leaves the volume element during the time increment At consists of three similar terms. The amount of A that accumulates within the volume element during time At is (ACa)2tiR(AR)(Az), and the amount of A that is generated by chemical reaction within the volume element is vArv2nR(AR)(Az)(At), where rv is the reaction rate expressed in pseudo homogeneous form [i.e., the number of moles transformed per unit time per unit of total reactor volume (voids plus solid)]. 

At steady state, the right side vanishes, and one obtains an equation that describes in pseudo homogeneous fashion the concentration profile in the reactor. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

The One-Dimensional Pseudo Homogeneous Model of Fixed Bed Reactors. The design of tubular fixed bed catalytic reactors has generally been based on a one-dimensional model that assumes that species concentrations and fluid temperature vary only in the axial direction. Heat transfer between the reacting fluid and the reactor walls is considered by presuming that all of the resistance is contained within a very thin boundary layer next to the wall and by using a heat transfer coefficient based on the temperature difference between the fluid and the wall. Per unit area of the tube... 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

The units on the rate constants reported by DeMaria et al. indicate that they are based on pseudo homogeneous rate expressions (i.e., the product of a catalyst bulk density and a reaction rate per unit mass of catalyst). It may be assumed that these relations pertain to the intrinsic reaction kinetics in the absence of any heat or mass transfer limitations. 

The first type of model considers the heat transfer surface to be contacted alternately by gas bubbles and packets of closely packed particles. This leads to a surface renewal process whereby heat transfer occurs primarily by transient conduction between the heat transfer surface and the particle packets during their time of residence at the surface. Mickley and Fairbanks (1955) provided the first analysis of this renewal mechanism. Treating the particle packet as a pseudo-homogeneous medium with solid volume fraction, e, and thermal conductivity (kpa), they solved the transient conduction equation to obtain the following expression for the average heat transfer coefficient due to particle packets,... 

This packet renewal model has been widely accepted and in the years since 1955 many researchers have proposed various modifications in attempts to improve the Mickley-Fairbanks representation. Several of these modifications dealt with the details of the thermal transport process between the heat transfer surface and the particle packet. The original Mickley-Fairbanks model treated the packet as a pseudo-homogeneous medium with a constant effective thermal conductivity, suggesting that... 

An alternate to the concept of cluster renewal discussed above is the concept of two-phase convection. This second approach disregards the separate behavior of lean and dense phases, instead models the time average heat transfer process as if it were convective from a pseudo-homogeneous particle-gas medium. Thus h hcl, hh and hd are not... 

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