Mixing - Isothermal Flow Problems

The basic concepts of mixing have already been discussed in Chapter 3 of this book. This section presents various examples and models that show how a mixing process can be analyzed and assessed. 

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Mixed convection flows are important as they are found in many practical situations- in nature and man-made devices. For problems of the continuum they are as relevant for atmospheric d3mamics at planetary scale to that for electronic devices at micro-scale. Mixed convection differs from isothermal flow due to the induced buoyancy effects via heat transfer. It... 

The same reactions considered in Problem 2.13 are now carried out in a single, perfectly mixed, isothermal continuous reactor. Flow rates, volume, and densities are constant. 

At some point in most processes, a detailed model of performance is needed to evaluate the effects of changing feedstocks, added capacity needs, changing costs of materials and operations, etc. For this, we need to solve the complete equations with detailed chemistry and reactor flow patterns. This is a problem of solving the R simultaneous equations for S chemical species, as we have discussed. However, the real process is seldom isothermal, and the flow pattern involves partial mixing. Therefore, in formulating a complete simulation, we need to add many additional complexities to the ideas developed thus far. We will consider each of these complexities in successive chapters temperature variations in Chapters 5 and 6, catalytic processes in Chapter 7, and nonideal flow patterns in Chapter 8. In Chapter 8 we will return to the issue of detailed modeling of chemical reactors, which include all these effects. 

Sorption of Cu(tfac)2 on a column depends on the amount of the compound injected, the content of the liquid phase in the bed, the nature of the support and temperature. Substantial sorption of Cu(tfac)2 by glass tubing and glass-wool plugs was observed. It was also shown that sorption of the copper chelate by the bed is partialy reversible . The retention data for Cr(dik)3, Co(dik)3 and Al(dik)3 complexes were measured at various temperatures and various flow rates. The results enable one to select conditions for the GC separation of Cr, Al and Co S-diketonates. Retention of tfac and hfac of various metals on various supports were also studied and were widely used for the determination of the metals. Both adsorption and partition coefficients were found to be functions of the average thickness of the film of the stationary phase . Specific retention volumes, adsorption isotherms, molar heats and entropy of solution were determined from the GC data . The retention of metal chelates on various stationary phases is mainly due to adsorption at the gas-liquid interface. However, the classical equation which describes the retention when mixed mechanisms occur is inappropriate to represent the behavior of such systems. This failure occurs because both adsorption and partition coefficients are functions of the average thickness of the film of the stationary phase. It was pointed out that the main problem is lack of stability under GC conditions. Dissociation of the chelates results in a smaller peak and a build-up of reactive metal ions. An improvement of the method could be achieved by addition of tfaH to the carrier gas of the GC equipped with aTCD" orFID" . ... 

Given the reaction stoichiometry and rate laws for an isothermal system, a simple representation for targeting of reactor networks is the segregated-flow model (see, e.g., Zwietering, 1959). A schematic of this model is shown in Fig. 2. Here, we assume that only molecules of the same age, t, are perfectly mixed and that molecules of different ages mix only at the reactor exit. The performance of such a model is completely determined by the residence time distribution function,/(f). By finding the optimal/(f) for a specified reactor network objective, one can solve the synthesis problem in the absence of mixing. 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

Simple equations can be derived to estimate the mendirane area for a given gas separation separation problem. Here it is assumed that the penneability coefficients remain constant and the separation occurs under isothermal condition. The calculations are dependent on the flow pattern in the module. The most simple equations are obtained by assuming complete mixing both in feed and penneate. This concept may be found in systems which operate at low recovery. Most gas separation systems resembles cross-flow conditions, i.e. plug flow at the feed side and complete mixing at the permeate side. These two concepts will be discussed here. In case of counter-current and co-current flow conditions the equations are somewhat different and the (terivations applicable for these systems can be found in literature. For vapour penneation the same approach can be used, however the... 

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