Mixing limits, ideal

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. 

In addition to the laboratory-scale reactors described here, there are numerous more specialized reactors in use. However, as mentioned previously, the performance of these reactors must lie somewhere between the mixing limits of the PFR and the CSTR. Additionally, when using small laboratory reactors, it is often difficult to maintain ideal mixing conditions, and the state of mixing should always be verified (see Chapter 8 for more details) prior to use. A common problem is that flow rates sufficiently large to achieve PFR behavior cannot be obtained in a small laboratory system, and the flow is laminar rather than turbulent (necessary for PFR behavior). If such is the case, the velocity profile across the reactor diameter is parabolic rather than constant. 

The time axis is the reduced time, which is the ratio of real time to the holding time tr = and the y-axis is the nondimensionalized dye concentration. If we were to plot the experimental change in nondimensionalized concentration versus reduced time, it should fall very near to this curve. The extent to which the real system deviates is a measure of the degree to which the system veers from the ideally "well-mixed" limit. 

Figures 4.3(b) and 4.4(b) gave some response curves intermediate between the ideal mixing limits, which obviously reflect the effects of some type of deviation from well-characterized mixing. Two questions arise ...

Multi-zone, Tanks-in-Series, and Axial dispersion models (Fig. 12.3-1 F) Other, less fundamental approaches accounting for mixing limitations in reactors are described in Section 12.7. They are based on simplified descriptions of the mixing pattern, e.g., a ID axial dispersion approach, or on the decomposition of the complex flow reactor into multiple interconnected regions or zones, each of these being described by a different idealized mixing pattern. Such semi-empirical models contain model parameters which have to be determined, experimentally or a posteriori from PDF, CFD, or RTD data. 

This equation immediately raises the closure problem uX> and R(X) cannot be expressed straightforwardly in terms of X, owing to the nonlinearities and the complex space-time dependence of n In the immediate proximity of the ideal mixing limit, however, the following closure is expected to provide a satisfactory zeroth order theory ... 

Thus (28) only amounts to a renormalisation of the heat transfer coefficient as a first order correction to the ideal mixing limit. In this latter limit (D-j — oo ) we would recover the classical Semenov equation. From Fig. 7 we see that the critical points for ignition and extinction ( and y C2 respectively) are both shifted to... 

Two consequences follow from these assumptions firstly, the convected (ensemble-) average field is close to homogeneity on a scale L 1q and secondly, the scalar fluctuations Xx X - X are much smaller then the mean value X. Moreover, the assumptions for u imply restriction to the domain close to the homogeneous limit (ideal mixing) ... 

The chief weakness of RTD analysis is that from the diagnostic perspective, an RTD study can identify whether the mixing is ideal or nonideal, bnt it is not able to uniquely determine the namre of the nonideality. Many different nonideal flow models can lead to exactly the same tracer response or RTD. The sequence in which a reacting fluid interacts with the nonideal zones in a reactor affects the conversion and yield for all reactions with other than first-order kinetics. This is one limitation of RTD analysis. Another limitation is that RTD analysis is based on the injection of a single tracer feed, whereas real reactors often employ the injection of multiple feed streams. In real reactors the mixing of separate feed streams can have a profound influence on the reaction. A third limitation is that RTD analysis is incapable of providing insight into the nature... 

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

Witlox, H. W. M., 1993, Thermodynamics Model for Mixing of Moist Air with Pollutant Consisting of HF, Ideal Gas, and Water, Shell Research Limited, Thornton Research Center, TNER.93.021,. 

Partial oxidations over complex mixed metal oxides are far from ideal for singlecrystal like studies of catalyst structure and reaction mechanisms, although several detailed (and by no means unreasonable) catalytic cycles have been postulated. Successful catalysts are believed to have surfaces that react selectively vith adsorbed organic reactants at positions where oxygen of only limited reactivity is present. This results in the desired partially oxidized products and a reduced catalytic site, exposing oxygen deficiencies. Such sites are reoxidized by oxygen from the bulk that is supplied by gas-phase O2 activated at remote sites. 

The implications for films cast from mixtures of enantiomers is that diagrams similar to those obtained for phase changes (i.e., melting point, etc.) versus composition for the bulk surfactant may be obtained if a film property is plotted as a function of composition. In the case of enantiomeric mixtures, these monolayer properties should be symmetric about the racemic mixture, and may help to determine whether the associations in the racemic film are homochiral, heterochiral, or ideal. Monolayers cast from non-enantiomeric chiral surfactant mixtures normally will not exhibit this feature. In addition, a systematic study of binary films cast from a mixture of chiral and achiral surfactants may help to determine the limits for chiral discrimination in monolayers doped with an achiral diluent. However, to our knowledge, there has never been any other systematic investigation of the thermodynamic, rheological and mixing properties of chiral monolayers than those reported below from this laboratory. 

The performance of adsorption processes results in general from the combined effects of thermodynamic and rate factors. It is convenient to consider first thermodynamic factors. These determine the process performance in a limit where the system behaves ideally i.e. without mass transfer and kinetic limitations and with the fluid phase in perfect piston flow. Rate factors determine the efficiency of the real process in relation to the ideal process performance. Rate factors include heat-and mass-transfer limitations, reaction kinetic limitations, and hydro-dynamic dispersion resulting from the velocity distribution across the bed and from mixing and diffusion in the interparticle void space. 

Finally, to conclude our discussion on coupling with chemistry, we should note that in principle fairly complex reaction schemes can be used to define the reaction source terms. However, as in single-phase flows, adding many fast chemical reactions can lead to slow convergence in CFD simulations, and the user is advised to attempt to eliminate instantaneous reaction steps whenever possible. The question of determining the rate constants (and their dependence on temperature) is also an important consideration. Ideally, this should be done under laboratory conditions for which the mass/heat-transfer rates are all faster than those likely to occur in the production-scale reactor. Note that it is not necessary to completely eliminate mass/heat-transfer limitations to determine usable rate parameters. Indeed, as long as the rate parameters found in the lab are reliable under well-mixed (vs. perfect-mixed) conditions, the actual mass/ heat-transfer rates in the reactor will be lower, leading to accurate predictions of chemical species under mass/heat-transfer-limited conditions. 

---