Well-mixing assumption

Cross flow is a better reflection of practical configurations. The well-mixed assumption is only reasonable for low concentrations and low values of 9. As a comparison, seawater desalination involves a feed with a concentration of the order of 35,000 ppm. 

Differences were found in vertical spread as well. In the Lagrangian run (Fig. 20.5) the concentration fields in the lower free troposphere followed the surface-level concentrations in general. This reflected the simphfied vertical structure of this kernel, in particular, well-mixing assumption for the boundary layer and fixed diffusion term in the free troposphere. In the Eulerian run (Fig. 20.6), the higher-level modeled concentrations were patchy and less correlated with the nearsurface fields in comparison with the Lagrangian run. 

Note Gas just leaving system has the same thermodynamic properties as gas in the system by the well-mixed assumption. 

The generalization of these equations to multiple feed streams is simple, and is left to you.) In writing the energy balance equation, the kinetic and potential energy, shaft work, and P(dV/dt) terms have been neglected, because these terms are usually of little importance compared with heats of reaction and temperature change terms. Also, since the contents of the reactor are of uniform temperature and composition (by the well-mixed assumption), the species concentrations and temperature of the exit stream are the same as those of the reactor contents. 

Here (Ci)i and (//i)m are the concentration and partial molar enthalpy, respectjyely, of species i in the inlet stream, and C is the concentration of species i and H - its partial molar enthalpy in both the reactor and in the outlet stream (again, this is the well-mixed assumption). 

The continuous-stirred-tank reactor (CSTR) is shown in Figure 1.3. Reactants and products flow into and out of the reactor continuously, and the contents of the reactor are assumed to be well mixed. The well-mixed assumption can be realized more easily for liquids than gases, so CSTRs are often used for liquid-phase reactions. The fluid composition and temperature undergo a step change when passing from the feed stream into the interior of the reactor the composition and temperature of the effluent stream are identical to those of the reactor. 

Notice that in the total material balance the argument of the derivative involves this mixture density. This as we have seen is the sum of the two concentrations of the two components, not the density of the solid alone, nor that of the liquid alone. On the right-hand side of the same equation, we note that the two input terms do involve the densities of the solid and the liquid in their pure states. This is because they are being delivered to the system as pure "feeds." The outflow term, however, includes the mixture density, the same density that appears in the argument of the differential. This is critical to understand. It says that everywhere in the control volume the density is the same at any time and that the material exiting the control volume also has the same density as the material in the tank. This is the consequence of assuming the system is well-mixed. The same analysis can be made for the two component balances. They show the well-mixed assumption because they include the corresponding mixture concentrations in the differential and the out-flow term. 

Because of the well-mixed assumption, it is natural to think of the CSTR as a liquid phase reactor with a mixer as shown in Figure 1 ... 

Due to the assumption of well mixed air, detailed spacial zone air temperature distributions and ventilation efficiency studies cannot be performed. For this, CFD methods have to be applied (see Section 11.2). 

Rielly and Marquis (2001) present a review of crystallizer fluid mechanics and draw attention to the inconsistency between the dependence of crystallization kinetic rates on local mean and turbulent velocity fields and the averaging assumptions of conventional well-mixed crystallizer models. 

This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

In a formal sense, Equation (2.38) applies to all batch reactor problems. So does Equation (2.42) combined with Equation (2.40). These equations are perfectly general when the reactor volume is well mixed and the various components are quickly charged. They do not require the assumption of constant reactor volume. If the volume does vary, ancillary, algebraic equations are needed as discussed in Section 2.6.1. The usual case is a thermodynamically imposed volume change. Then, an equation of state is needed to calculate the density. 

Chapter 2 treated multiple and complex reactions in an ideal batch reactor. The reactor was ideal in the sense that mixing was assumed to be instantaneous and complete throughout the vessel. Real batch reactors will approximate ideal behavior when the characteristic time for mixing is short compared with the reaction half-life. Industrial batch reactors have inlet and outlet ports and an agitation system. The same hardware can be converted to continuous operation. To do this, just feed and discharge continuously. If the reactor is well mixed in the batch mode, it is likely to remain so in the continuous mode, as least for the same reaction. The assumption of instantaneous and perfect mixing remains a reasonable approximation, but the batch reactor has become a continuous-flow stirred tank. 

The model assumes a well-mixed gas phase composition in the recycle loop, a well justified assumption in view of the very high (10-200) recycle ratio values used in the present work. For the batch electrocatalytic version we also neglect volume changes and assume linear kinetics for steps 1,3 and 4 of the consecutive OCM network (1), i.e. ... 

The basic assumption of well-mixed fluid on the feed-side of the membrane does not reflect the flow patterns for the configurations used in practice. The assumption simplifies the calculations and allows the basic trends to be demonstrated. 

Partition coefficients can then be combined to describe the ecosystem, assuming all the compartments are well mixed such that equilibrium is achieved between them. This assumption is generally not true of an environmental system since transfer rates between compartments may be slower than transformation rates within compartments. Therefore, equilibrium is never truly approached, except for perhaps with very stable compounds. However, such simplifications can give an indication into which compartments a chemical will tend to migrate and can provide a mechanism for ranking and comparing chemicals. 

The next level of complexity is to maintain the assumptions of the fundamental model, that compartments are well mixed and rapidly equilibrated, and consider degradation rates within compartments. If this is done, the half-life of the chemical in the system can be estimated along with an estimated amount degraded in each compartment. 

As shown in Fig. 3, CHEMGL considers 10 major well-mixed compartments air boundary layer, free troposphere, stratosphere, surface water, surface soil, vadose soil, sediment, ground water zone, plant foliage and plant route. In each compartment, several phases are included, for example, air, water and solids (organic matter, mineral matter). A volume fraction is used to express the ratio of the phase volume to the bulk compartment volume. Furthermore, each compartment is assumed to be a completely mixed box, which means all environmental properties and the chemical concentrations are uniform in a compartment. In addition, the environmental properties are assumed to not change with time. Other assumptions made in the model include continuous emissions to the compartments, equilibrium between different phases within each compartment and first-order irreversible loss rate within each compartment [38]. 

The volumetric flow rate of the recycle stream is many many times those of the fresh feed and product streams, and the fresh feed and recycle streams are well mixed at the juncture point. If one uses a mole ratio of 3.4 hydrogen to 1 toluene in the fresh feed stream, what fraction of the toluene is converted to benzene under the previously specified conditions The average residence time of a fluid element is 30.1 sec. Explicitly state any assumptions that you make. In order to obtain a numerical answer, a trial and error solution will be necessary. 

According to this assumption the two streams are so well mixed that the compositions of each phase within the stage are uniform. Further, the mass transfer is so efficient that the compositions of the streams leaving the stage are in equilibrium. 

Mass and energy transport occur throughout all of the various sandwich layers. These processes, along with electrochemical kinetics, are key in describing how fuel cells function. In this section, thermal transport is not considered, and all of the models discussed are isothermal and at steady state. Some other assumptions include local equilibrium, well-mixed gas channels, and ideal-gas behavior. The section is outlined as follows. First, the general fundamental equations are presented. This is followed by an examination of the various models for the fuel-cell sandwich in terms of the layers shown in Figure 5. Finally, the interplay between the various layers and the results of sandwich models are discussed. 

A model based on the assumption that a metabolite is present within a single compartment with defined rate constants for absorption and elimination of the metabolite. The rate of appearance of a tracee and the infusion of tracer are assumed to take place in a single pool that is instantly well-mixed. Wolfe has described in detail how the constant tracer infusion method allows one to calculate half-life, pool size, turnover time, mean residence time, and clearance time. 

Important limitations of the PBPK approach are realized for class 3 and 4 compounds with significant active distribution/absorption processes, where biliary elimination is a major component of the elimination process or where the assumptions of flow-limited distribution and well mixed compartments are not valid and permeability-limited distribution is apparent. These drawbacks could be addressed by the addition of permeability barriers for some tissues and by the incorporation of a more complex liver model which addresses active uptake into the liver, active efflux into the bile, biliary elimination and enterohepatic recirculation. However, this improvement to current methodologies requires the availability of the appropriate input data for quantification of the various processes involved as well as validation of the corresponding in vitro to in vivo scaling approaches. 

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