Perfect mixing, reactor model

The name continuous flow-stirred tank reactor is nicely descriptive of a type of reactor that frequently for both production and fundamental kinetic studies. Unfortunately, this name, abbreviated as CSTR, misses the essence of the idealization completely. The ideality arises from the assumption in the analysis that the reactor is perfectly mixed, and that it is homogeneous. A better name for this model might be continuous perfectly mixed reactor (CPMR). 

The process is described by the mathematical model of a nonisothermal, unsteady state, continuous and perfectly mixed reactor. It is defined by the beloiv differential equations ... 

Several models have been suggested to simulate the behavior inside a reactor [53, 71, 72]. Accordingly, homogeneous flow models, which are the subject of this chapter, may be classified into (1) velocity profile model, for a reactor whose velocity profile is rather simple and describable by some mathematical expression, (2) dispersion model, which draws analogy between mixing and diffusion processes, and (3) compartmental model, which consists of a series of perfectly-mixed reactors, plug-flow reactors, dead water elements as well as recycle streams, by pass and cross flow etc., in order to describe a non-ideal flow reactor. 

The following configuration demonstrates a general cell model of a continuous flow system described in ref. [77]. There are two possibilities to arrive at state 4, i.e. directly and via the upper plug flow reactor. However, in order to materialize these possibilities it was necessary to add state 3-a perfectly mixed reactor 3. The residence time in this reactor is controlled by the quantity [13. 

The resulting model is shown on the RHS of Fig.4.5-1. Clearly each vessel represents a state in a Markov process vessel 4 is the collector of the particles from which only the carrying stream is leaving at flow rate 2Qi. Q12, Q21, Q23 and Q 32 are recycle streams. This is due to the penetration of particles from one stream into the other where the impingement zone is designated by 2 in Fig.4.5-1 and is simulated as a perfectly mixed reactor. The effect of penetration is emphasized by the fact that movement of particles to vessel 4 is only possible from vessels 1 and 3. Whenever a particle reaches vessel 4, it remains there, i.e. the vessel is a trapping state. 

In this configuration, a CSTER is imbedded between two perfectly-mixed reactors 1 and 3 in the forward loop. As in model A above, electrolyte recycling is represented by a perfectly-mixed reactor 4 in the feedback loop shown in Fig.5.2-3(6). Electrolysis takes place in reactor 2 and the collector is reactor 5. 

Analogous to the experimental approaches discussed in the previous section, mathematical models have been developed to describe mass transfer at all three levels—cellular, multi-cellular (spheroid), and tissue levels. For each level two approaches have been used—the lumped parameter and distributed parameter models. In the former approach, the region of interest is considered to be a perfectly mixed reactor or compartment. As a result, the concentration of each region has no spatial dependence. In the latter approach, a more detailed analysis of the mass transfer process leads to information on the spatial and/or temporal changes in concentrations. Models for single cells and spheroids were reviewed in Section III,A and are part of the tissue-level models (Jain, 1984) hence, we will focus here only on tissue-level models. 

Another way of assessing the effect of axial dispersion is to use the model of n perfectly mixed reactors in series. Since the theory for dispersion in packed beds predicts Pe = 2 if the gas is mixed between each layer of particles, n L/dp. For n > 10, the conversion is almost the same as for a plug-flow reactor, as was shown in Figure 3.9. 

A variety of reactor models have been proposed for the regenerator, including two-phase models, a perfectly mixed reactor, and two or three mixers in series [31]. These models have been combined with a... 

The simplest kinetic reactor model is the CSTR (continuous-stirred-tank reactor), in which the contents are assumed to be perfectly mixed. Thus, the composition and the temperature are assumed to be uniform throughout the reactor volume and equal to the composition and temperature of the reactor effluent However, the fluid elements do not all have the same residence time in the reactor. Rather, there is a residence-time distribution. It is not difficult to provide perfect mixing of the fluid contents of a vessel to approximate a CSTR model in a commercial reactor. A perfectly mixed reactor is used often for homogeneous liquid-phase reactions. The CSTR model is adequate for this case, provided that the reaction takes place under adiabatic or isothermal conditions. Although calculations only involve algebraic equations, they may be nonlinear. Accordingly, a possible complication that must be considered is the existence of multiple solutions, two or more of which may be stable, as shown in the next example. 

Where n is the single parameter of the cellular model, equal to the number of cells (devices) in a cascade of perfect mixing reactors. Plug flow mode is achieved at —> o<= [1]. It is assumed [124] that if the number of cells in a reactor n > 8, calculation methods for plug flow reactors can be applied to such a device with accuracy sufficient for industrial application. 

Nelson (24) studied theoretically the general case of countercurrent equilibrium stage separation with chemical reaction and applied his technique to describe distillation reactors. His model relied on the assumption of each stage being a perfectly mixed reactor and also an equilibrium stage. 

Briefly we treated the perfectly mixed reactor RTD in the mathematical analysis provided above. It is important to note from Example 3.1 that the RTD theory is not fully capable of explaining the behavior of the reactors, especially when the fluid elements are interacting. Thus, we give examples for a few other models here and refer the reader to an excellent text by Fox (2003) for a more in-depth analysis of these models in the turbulent flow regime. Four broad classes of micromixing models are sketched in Figure 110. 

In this chapter the most important operation modes of reactors are considered. Models are developed by combining simple reaction kinetics for single-phase reactions with mass balances for five ideal model reactors the ideal batch reactor the semi-batch reactor the plug flow reactor the perfectly mixed continuous reactor and the cascade of perfectly mixed reactors. For isothermal conditions, conversions can be calculated on the basis of chemical kinetics only. 

CSTRs (Figure 5.3) are usually used to handle liquid-phase reactions. The behavior of a CSTR is often modeled as an ideal perfectly mixed reactor. The CSTR model is often used to simplify engineering calculations. In practice, it can only be approached in industrial size reactors. 

In the second model (Fig. 2.16) the continuous well-stirred model, feed and product takeoff are continuous, and the reactor contents are assumed to he perfectly mixed. This leads to uniform composition and temperature throughout. Because of the perfect mixing, a fluid element can leave at the instant it enters the reactor or stay for an extended period. The residence time of individual fluid elements in the reactor varies. 

Mixing Models. The assumption of perfect or micro-mixing is frequently made for continuous stirred tank reactors and the ensuing reactor model used for design and optimization studies. For well-agitated reactors with moderate reaction rates and for reaction media which are not too viscous, this model is often justified. Micro-mixed reactors are characterized by uniform concentrations throughout the reactor and an exponential residence time distribution function. 

A real continuous-flow stirred tank will approximate a perfectly mixed CSTR provided that tmix h/i and tmix i. Mixing time correlations are developed using batch vessels, but they can be applied to flow vessels provided the ratio of throughput to circulatory flow is small. This idea is explored in Section 4.5.3 where a recycle loop reactor is used as a model of an internally agitated vessel. 

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