CSTR, definition

Aris (1969) pointed out that the mathematical definition of the CSTR stability problem and the catalyst particle problem cooled by the feed flow were essentially identical. 

Note that, because of the definition of /A, FAg is the original feed rate to the CSTR and not that to the ith stage. 

In segregated flow only the CSTR will be different since the PFR is segregated by definition. For the CSTR,... 

Consider again the CSTR where the consecutive reactions A B C take place. The required discrete nonlinear control law is found by obtaining a discrete version of the immersion developed in Example 3 and its associated exponential holder which has the form He, and in this particular example is a 1 X 5 dimension vector, where and H are presented in matrices (38). The exponential holder can be calculated by using the definition of an exponential matrix,... 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm =0,t < 0 Ct = A/At, 0 < t < At Cin = 0, and t > At. The outlet response is found from the dynamic model of a CSTR, Equation (14.2). The result is... 

The second issue for cooled tubular reactors is how to introduce the coolant. One option is to provide a large flowrate of nearly constant temperature, as in a recirculation loop for a jacketed CSTR. Another option is to use a moderate coolant flowrate in countercurrent operation as in a regular heat exchanger. A third choice is to introduce the coolant cocurrently with the reacting fluids (Borio et al., 1989). This option has some definite benefits for control as shown by Bucala et al. (1992). One of the reasons cocurrent flow is advantageous is that it does not introduce thermal feedback through the coolant. It is always good to avoid positive feedback since it creates nonmonotonic exit temperature responses and the possibility for open-loop unstable steady states. 

The polymerization time in continuous processes depends on the time the reactants spend in the reactor. The contents of a batch reactor will all have the same residence time, since they are introduced and removed from the vessel at the same times. The continuous flow tubular reactor has the next narrowest residence time distribution, if flow in the reactor is truly plug-like (i.e., not laminar). These two reactors are best adapted for achieving high conversions, while a CSTR cannot provide high conversion, by definition of its operation. The residence time distribution of the CSTR contents is broader than those of the former types. A cascade of CSTR s will approach the behavior of a plug flow continuous reactor. 

Next, the deficiency of the augmented network is calculated (with 0 a complex which, by definition, is compatible with A). If the deficiency is zero, the strong deficiency zero theorem (Feinberg, 1987) applies Provided the kinetics are of the mass action type, no matter what (positive) values the kinetic constants may have, the CSTR cannot exhibit multiple steady states, unstable steady states, or periodic orbits. The result is, in a sense, very strong because the governing differential... 

Earlier it was shown that generally the mean residence time in a reactor is equal to V/u, or t. This relationship can be shown in a simpler fashion for the CSTR. Applying the definition of a mean residence time to the RTD for a CSTR, we obtain... 

The RTD will be analyzed from a tracer pulse injected into the first reactor of three equally sized CSTRs in series. Using the definition of the RTD presented in Section 13.2. the fraction of material leaving the system of three reactors (i.e., leaving the third reactor) that has been in the system between time t and / + Ar is... 

While these optimization-based approaches have yielded very useful results for reactor networks, they have a number of limitations. First, proper problem definition for reactor networks is difficult, given the uncertainties in the process and the need to consider the interaction of other process subsystems. Second, all of the above-mentioned studies formulated nonconvex optimization problems for the optimal network structure and relied on local optimization tools to solve them. As a result, only locally optimal solutions could be guaranteed. Given the likelihood of extreme nonlinear behavior, such as bifurcations and multiple steady states, even locally optimal solutions can be quite poor. In addition, superstructure approaches are usually plagued by the question of completeness of the network, as well as the possibility that a better network may have been overlooked by a limited superstructure. This problem is exacerbated by reaction systems with many networks that have identical performance characteristics. (For instance, a single PFR can be approximated by a large train of CSTRs.) In most cases, the simpler network is clearly more desirable. 

The process view concept is illustrated by an example in Fig. 3.87. The top part of the figure shows a part of the Polyamide-6 process (used throughout the entire subsection) as it is seen from the perspective of the Chemical Company. In the middle part of Fig. 3.87, a process view definition named Re-actionSimulationTasks for this process is shown which contains two simulation tasks Simulate CSTR and Simulate PFR with their input and output parameters from the overall process, while the control flow between both tasks is not included in the process view. 

It is these same requirements that have to be met in all reactors designed for kinetic studies, with the added problem of finding an appropriate definition of space time so that the results obtained in flow reactors designed for kinetic studies can be compared to those from a BR. These requirements exclude most reactor configurations from use in kinetic studies and leave us with the fundamental trio the BR, the PFR and the CSTR. Other configurations can yield reliably reproducible data but fall short in one way or an other when used for kinetic studies. 

Equation (4-51) is the basic design equation for what is popularly called a continuously stirred tank reactor (CSTR). The derivation assumes equality of volumetric flow rate of feed and effluent as in the case of the PFR, the residence-time definition must be changed if this is not so. In most applications of the CSTR, however, reactions in the liquid phase are involved and volume changes with reaction are not important. 

An item of interest in the above is that PFR and CSTR selectivities are the same for the Type (II) reaction. This equality must pertain for the relative yields as well, as long as the comparison is made for the same conversion level, and follows from the definition... 

At the entrance to the CSTR where x = 0, equation (I) again specifies the concentrations of the various alkanolamines in the feed stream. Algebraic manipulation of equations (I) to (L) followed by introduction of the definition of the space time for the CSTR yields the following set of three algebraic equations in three unknowns ... 

In these definitions, Qr and Kr have been used as arbitrary normalizing parameters. They can be readily identified for a constant volume reactor Qr = Qo = Qf = Q< 3nd Vr = y = Vm where represents the maximum volume of the reactor (in other words, the volume at which the reactor would function as a constant volume CSTR). Clearly, in this case f = F/Q = Fq/Qo is the true residence time. However, for the variable volume reactor, Qr and Fr have to be selected carefully on a case by case basis, and the residence time r= Fr/Qr should be regarded as no more than a parameter whose dimension is time or simply as pseudoresidence time. Using these dimensionless parameters (with normalizing variables different from those for an MFR), the general material balance of Equation 10.22 can be expressed as... 

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