Perfectly Mixed Reactor Systems

Perfectly mixed reactors are the key element for conducting chemical processes and in simulation of complex flow systems. Other synonyms are mixed reactor, back mix reactor, an ideal stirred reactor and the CFSTR (constant flow stirred tank reactor). As the name implies, it is a reactor in which the contents are well stirred and uniform throughout. Thus, the exit stream from this reactor has the same composition as the fluid within the reactor. 

In the following a variety of configurations will be treated, which find importance in practice and in simulation, as well as elaborate the application of the model in chapter 4.1. In some cases the derivations are also detailed. 

The flow configuration comprising of two perfectly-mixed reactors of volumes Vi and V2 is demonstrated in Fig.4.3-1. A tracer in a form of a pulse input is introduced into reactor 1 and is transferred by the flow Qi into reactor 2 where it is assumed to accumulate. Thus, this reactor is a dead or absorbing state for the tracer, i.e. C = 0 in Fig.4-1. 

In the numerical solution it was assumed that the reactors are of an identical volume, thus, jii = )J-2 = M- The transient response of Ci and C2 is depicted in Fig.4.3-la where the effect of J. = 1/tm is demonstrated. As seen, increasing p, (or decreasing the mean residence time in the reactor) brings the reactors faster to steady state. 

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Plug flow-perfectly mixed reactor systems (chapter 4.4). 

PLUG FLOW-PERFECTLY MIXED REACTOR SYSTEMS... 

The system is sketched in Fig. 3.1 and is a simple extension of the CSTR considered in Example 2.3. Product B is produced and reactant A is consumed in each of the three perfectly mixed reactors by a first-order reaction occurring in the liquid. For the moment let us assume that the temperatures and holdups (volumes) of the three tanks can be different, but both temperatures and the liquid volumes are assumed to be constant (isothermal and constant holdup). Density is assumed constant throughout the system, which is a binary mixture of A and B. 

Deflnitions. The basic elements of Markov chains associated with Eq.(2-24) are the system, the state space, the initial state vector and the one-step transition probability matrix. Considering refs.[26-30], each of the elements will be defined in the following with special emphasize to chemical reactions occurring in a batch perfectly-mixed reactor or in a single continuous plug-flow reactor. In the latter case, which may simulated by perfectly-mixed reactors in series, all species reside in the reactor the same time. 

Z+1 designates the number of states, i.e. Z perfectly mixed reactors in the flow system as well as the tracer collector designated by As shown later, the probabilities Si(0) may be replaced by the initial concentration of the fluid elements in each state, i.e. Cj(0) and S(0) will contain all initial concentrations of the fluid elements. The one-step transition probability matrix is given by Eqs.(2-16) and (2-20) whereas pjk represent the probability that a fluid element at Cj will change into Ck in one step, pjj represent the probability that a fluid element will remain unchanged in concentration within one step. 

In treating a certain configuration, the first step is to define the states that the system can occupy. By a state is meant, the concentration Ci in a perfectly mixed reactor i or at the inlet or the exit of a plug-flow reactor, that the system (fluid element) can occupy. The states will be designated by Ci, C2,... whereas the state space SS, will read ... 

The closed recirculation system shown below comprises of Z perfectly-mixed reactors of not the same volume. If a tracer is introduced in a form of a pulse input into the first reactor, the recorder will measure the tracer as it flows the first time, the second time, and so on. In fact, it measures a tracer which passed through Z reactors, 2Z reactors and so on, i.e. the superposition of all these signals. 

The following system comprises two plug flow reactors, perfectly mixed reactors and recycle streams Q45, Q53 and Q52. Reactors 2, 4 and 5 may be considered also as measurement points of the concentration, and in this case their i s are assigned a large value, say, 500. 

Fig.4.4-5a demonstrates the effect of p on the transient response of C2 and Cn, i.e. the first and the last perfectly mixed reactors in the upper plug flow reactor in Fig.4.4-5. The tracer was introduced into reactor 2. It is obsereved that by increasing p from 80 to 800, the number of oscillations for reaching the steady state concentration 1/22 in the system is increased. It should also be noted that for p = 800, the distance between two successive peaks corresponding to C2 and Cn is...

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. 

Vi/Qi in a perfectly mixed reactor. An additional key quantity is the holdup of the particles in the reactor, Vi (kg particles) instead of the volume of the reactor. Thus, in consistent units, Eqs.(4-1) to (4-11) hold. An important quantity used in this section for comparing various effects is the mean residence time tm of the particles in the system. 

The basic element in the flow system is the perfectly-mixed reactor. In the multi-reactor system heat and mass transfer operation (absorption, desorption, dissolution of solids, heat generation or absorption as well as heat interaction between the reactor and the surroundings etc.) as well as chemical reactions may occur simultaneously, or not. The processes are governed by Eqs.(5-8), (5-12), (5-16), (5-19), (5-23) and (5-25) in the following, on the basis of which transition probabilities are derived as well as the single step transition matrix. 

Definitions. Afj s fk designates a system with respect to its chemical formula f and location (reactor) k in the flow system. The system is a fluid element containing F chemical species for which f = 1, 2, 3,. F where each figure designates a certain species with its chemical formula. In addition, k = a, b,. Z, where each letter designates a reactor in the flow system composed of Z + 1 perfectly mixed reactors, including reactor... 

Consider a plug flow reactor, with mean residence time t, followed by a perfectly mixed reactor, with mean residence time, i2. The overall RTD for this system will merely be that for a perfectly mixed vessel, but with a time delay caused by the... 

Liquid residence-time distributions in mechanically stirred gas-liquid-solid operations have apparently not been studied as such. It seems a safe assumption that these systems under normal operating conditions may be considered as perfectly mixed vessels. Van de Vusse (V3) have discussed some aspects of liquid flow in stirred slurry reactors. 

The feed is charged all at once to a batch reactor, and the products are removed together, with the mass in the system being held constant during the reaction step. Such reactors usually operate at nearly constant volume. The reason for this is that most batch reactors are liquid-phase reactors, and liquid densities tend to be insensitive to composition. The ideal batch reactor considered so far is perfectly mixed, isothermal, and operates at constant density. We now relax the assumption of constant density but retain the other simplifying assumptions of perfect mixing and isothermal operation. 

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. 

Perfectly mixed stirred tank reactors have no spatial variations in composition or physical properties within the reactor or in the exit from it. Everything inside the system is uniform except at the very entrance. Molecules experience a step change in environment immediately upon entering. A perfectly mixed CSTR has only two environments one at the inlet and one inside the reactor and at the outlet. These environments are specifled by a set of compositions and operating conditions that have only two values either bi ,..., Ti or Uout, bout, , Pout, Tout- When the reactor is at a steady state, the inlet and outlet properties are related by algebraic equations. The piston flow reactors and real flow reactors show a more gradual change from inlet to outlet, and the inlet and outlet properties are related by differential equations. 

The molecules in the system are carried along by the balls and will also have an exponential distribution of residence time, but they are far from perfectly mixed. Molecules that entered together stay together, and the only time they mix with other molecules is at the reactor outlet. The composition within each ball evolves with time spent in the system as though the ball was a small batch reactor. The exit concentration within a ball is the same as that in a batch reactor after reaction time tf,. 

Part (c) in Example 15.15 illustrates an interesting point. It may not be possible to achieve maximum mixedness in a particular physical system. Two tanks in series—even though they are perfectly mixed individually—cannot achieve the maximum mixedness limit that is possible with the residence time distribution of two tanks in series. There exists a reactor (albeit semi-hypothetical) that has the same residence time distribution but that gives lower conversion for a second-order reaction than two perfectly mixed CSTRs in series. The next section describes such a reactor. When the physical configuration is known, as in part (c) above, it may provide a closer bound on conversion than provided by the maximum mixed reactor described in the next section. 

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