Imperfect mixing

There are mixing-time sensitive reactions where fi/2 tmix will cause a difference in selectivity. A widely studied example is the Bourne reaction 

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Olah and Overchuk also attempted to discover evidence of slow mixing by carrying out reactions in high-speed flow systems. Evidence, including the isolation of dinitro compounds (> i %), was indeed found, but held to show that the effect of imperfect mixing was only minor. The reactions were, unfortunately, too fast to permit determinations of absolute rates (half-lives of about io s). 

This equation describes the change of population in a well-mixed system and is often used to model fully mixed crystallization and precipitation processes. If the system is imperfectly mixed, however, then the more complicated equation 2.88 can be used provided that the external flow field can be calculated e.g. by use of CFD (see later). 

A secondary particle formation process, which can increase crystal size dramatically, is crystal agglomeration. This process is particularly prevalent in systems exhibiting high levels of supersaturation, such as from precipitation reactions, and is considered along with its opposite viz. particle disruption in Chapter 6. Such high levels of supersaturation can markedly accentuate the effects of spatial variations due to imperfect mixing within a crystallizer. This aspect is considered further in Chapter 8. 

In the SFM the reactor is divided into three zones two feed zones fj and (2 and the bulk b (Figure 8.1). The feed zones exchange mass with each other and with the bulk as depicted with the flow rates mi 2, i,3 and 2,3 respectively, according to the time constants characteristic for micromixing and mesomix-ing. As imperfect mixing leads to gradients of the concentrations in the reactor, different supersaturation levels in different compartments govern the precipitation rates, especially the rapid nucleation process. 

There are many interesting reports in the literature where computer simulations have been used to examine not only idealized cases but have also been used in an attempt to explain segregation and viscosity effect in unperturbed polymerization reactors (6). Some experimental work has been reported (7, 8). It is obvious, however, that although there is some change in the MWD with conversion in the batch and tubular reactor cases and that broadening of the MWD occurs as a result of imperfect mixing, there is no effective means available for controlling the MWD of the polymer from unperturbed or steady-state reactors. 

Initial comparison of CFSTR runs with similar feed conditions indicates conditions for which the monomer conversion may be dependent on mixing speed. However, when the effects of experimental error in monomer conversion and differences in reaction temperature are considered, the monomer conversion is seen to be relatively independent of mixing speed for rpm equal to or greater than 500. Comparing Run 14 with Run 12 reveals a small decrease in monomer conversion in spite of a rise in reactor temperature of 2°C. This indicated the presence of a small amount of bypassing or dead volume at the lower mixing speed. This imperfect mixing pattern would also be present in Run 15. 

Finally, the reactor vessel has been assumed to be perfectly mixed. Imperfect mixing and a flow pattern created by different types of agitators, baffles, feed locations and other reactor vessel configurations will cause the performance to be below that indicated by perfect mixing. 

BYPRODUCT. This means that imperfect mixing favors the production of the BYPRODUCT. 

The imperfect mixing in a chemical reactor can be modeled by splitting the total volume into two perfectly mixed sections with circulation between them. Feed enters and leaves one section. The other section acts like a side-capacity element. 

A1, Fe, and Ca. Sulfur, sodium, and other minor species only accounted for a small fraction of these particles. Some carbon was also found in these large particles. This is not surprising imperfect mixing of fuel and air makes it unlikely that all of the char would be consumed at a fuel-air ratio so near stoichiometric. 

Unless carried out very carefully, data from flow reactors may be influenced by experimental uncertainties. Potential problems with the flow reactor technique include imperfect mixing of reactants, radial gradients of concentration and temperature, and catalytic effects on reactor walls. Uncertainties in induction times, introduced by finite rate mixing of reactants, presence of impurities, or catalytic effects, may require interpretation of the data in terms of concentration gradients, rather than just exhaust composition [442]. 

We first choose variables sufficient to describe the situation. This choice is tentative, for we may need to omit some or recruit others at a later stage (e.g., if V is constant, it can be dismissed as a variable). In general, variables fall into two groups independent (in our example, time) and dependent (volume and concentration) variables. The term lumped is applied to variables that are uniform throughout the system, as all are in our simple example because we have assumed perfect mixing. If we had wished to model imperfect mixing, we would have had either to introduce a number of different zones (each of which would then be described by lumped variables) or to introduce spatial coordinates, in which case the variables are said to be distributed.2 Lumped variables lead to ordinary equations distributed variables lead to partial differential equations. 

With a continuous flow method, flowing is the sole way the sample is mixed. Consequently, there may be imperfect mixing. Thus, the concentration of the adsorptive in the flow chamber may not equal the effluent concentration this is because transport and chemical kinetics phenomena are both occurring simultaneously. 

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the... 

The completely mixed model succeeds in representing part of the experimental data and predicts that at industrial conditions the reactor is open-loop unstable. Initiator productivity decreases are accounted quite accurately only by the second reactor model which details the mixing conditions at the initiator feed point. Independent estimates of the model parameters result in an excellent match with experimental data for several initiator types. Imperfect mixing is shown to have a tendency to stabilize the reactor. 

In a previous work (5), it was shown that the effect of mixing on reactor performances is very important. In particular, it was discussed that a) the mixing inside the reactor can be characterized by a recirculating flow rate caused by the impeller in the reaction zone, and that b) an imperfectly mixed reactor requires a higher initiator consumption per polymer produced than a perfectly mixed one operating at the same conditions. 

In the present communication it will be shown that most of these phenomena can be accounted for, quite accurately, by a simple "imperfectly mixed" reactor model. In order to stress the influence of mixing, two models have been developed. The first one simulates the vessel LDPE reactor as an ideal CSTR, while the second model accounts for mixing limitations. 

Figure 1. Schematic presentation of the two reactor models a, perfectly mixed model, and b, imperfectly mixed model.

The mathematical equations for this imperfectly mixed model consist of 12 differential equations similar to equations (l)-(4), four for each of the three CSTR s. At steady state, they reduce to 12 non-linear algebraic equations which are solved numerically in order to calculate the dependence of initiator consumption on polymerization temperature. An overall balance reveals that the monomer conversion and polymer production rate are still given by equations (5) and (8), while the initiator consumption is affected by the temperature and radicals distribution in the three CSTR s, so that equations (7) and (9) become much more complex. 

In Figure 5 the predictions of the second model are compared against the experimental data published in (2) and obtained in a small, well stirred, vessel reactor with 1 It total volume. The various initiators were tested under conditions representative of polymerization in commercial units, that is with 20 60 seconds residence time and an operating pressure between 1278 and 2352 atm. For the sake of convenience we will use here the same nomenclature and dimensions as in (2). The kinetic parameters used were those given in Table I. The relative size of the two small volumes and the recirculation rates were estimated once and for all cases from equations (13) and (15). The other parameter values, determined independently, were not changed in order to obtain a better fit with the data. As can be seen, the imperfectly mixed model is in excellent agreement with the experimental data, and accurately accounts for the effect of initiator type (Figure 5). 

Internal recycle reactors are designed so that the relative velocity between the catalyst and the fluid phase is increased without increasing the overall feed and outlet flow rates. This facilitates the interphase heat and mass transfer rates. A typical internal flow recycle stirred reactor design proposed by Berty (1974, 1979) is shown in Fig. 18. This type of reactor is ideally suited for laboratory kinetic studies. The reactor, however, works better at higher pressure than at lower pressure. The other types of internal recycle reactors that can be effectively used for gas-liquid-solid reactions are those with a fixed bed of catalyst in a basket placed at the wall or at the center. Brown (1969) showed that imperfect mixing and heat and mass transfer effects are absent above a stirrer speed of about 2,000 rpm. Some important features of internal recycle reactors are listed in Table XII. The information on gas-liquid and liquid-solid mass transfer coefficients in these reactors is rather limited, and more work in this area is necessary. 

Slow initiation alone leads to the maximal polydispersity of MJMn 1.35. This value decreases significantly when initiation is completed before all monomer is consumed. Slow and imperfect mixing can be another reason for higher polydispersities, as discussed by Szwarc [1],... 

Eurtaw, E.J., M.D. Pandian, D.R. Nelson and J.V. Behar (1996). Modeling indoor air concentrations near emission sources in imperfectly mixed rooms, 7. Air Waste Management Assoc., 46, 861-868. 

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