Continuously stirred tank reactor describing equations

When reactions 9.3.3 and 9.3.4 take place in a single continuous stirred tank reactor, the route to a quantitative relation describing the product distribution involves writing the design equations for species V and A. 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

If the points lie close to a straight line, this is taken as confirmation that a second-order equation satisfactorily describes the kinetics, and the value of the rate constant k2 is found by fitting the best straight line to the points by linear regression. Experiments using tubular and continuous stirred-tank reactors to determine kinetic constants are discussed in the sections describing these reactors (Sections 1.7.4 and 1.8.S). 

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

Many chemical and biological processes are multistage. Multistage processes include absorption towers, distillation columns, and batteries of continuous stirred tank reactors (CSTRs). These processes may be either cocurrent or countercurrent. The steady state of a multistage process is usually described by a set of linear equations that can be treated via matrices. On the other hand, the unsteady-state dynamic behavior of a multistage process is usually described by a set of ordinary differential equations that gives rise to a matrix differential equation. 

Example 4-8 An ideal continuous stirred-tank reactor is used for the homogeneous polymerization of monomer M. The volumetric flow rate is O, the volume of the reactor is V, and the density of the reaction solution is invariant with composition. The concentration of monomer in the feed is [M]o. The polymer product is produced by an initiation step and a consecutive series of propagation reactions. The reaction mechanism and rate equations may be described as follows, where is the activated monomer and P2, . . , P are polymer molecules containing n monomer units ... 

These are systems where the state variables describing the system are lumped in space (invariant in all space dimensions). The simplest chemical reaction engineering example is thp perfectly mixed continuous stirred tank reactor. These systems are described at steady state by algebraic equations while in the unsteady state they are described by initial value ordinary differential equations where time is the independent variable. 

Equations (4.8a), (4.9a), and (4.10b) in Example 4.10 describe the dynamic behavior of a continuous stirred tank reactor with a simple, exothermic and irreversible reaction, A - B. Develop a numerical procedure that solves these equations and can be implemented on a digital computer. Also, describe a numerical procedure for solving the algebraic steady-state equations of the reactor above. (Note For this problem you need to be familiar with numerical techniques for the solution of differential and algebraic equations on a computer.)... 

RTD experiments showed that the fixed-bed almost behaves like a plug-flow reactor and the infrared cell like a continuous stirred tank reactor. This fixed-bed is described by the tanks-in-series model, using 9 tanks for the catalyst compartment. The two kinetic models (Equations 1-6) are able to describe the stop-effect experiments at 180 and 200°C, and the considerations made in this work are valid for both temperatures. However, for the sake of clarity, only model discrimination at 180°C will be presented here. In the experimental conditions used here, both models can be simplified the first adsorption step is considered as irreversible, and instantaneous equilibrium is assumed for the second one. With these hypothesis the total number of kinetic parameters is reduced from five (ki, Li, k2, k.2 and ks) to three (ki, K2 and ks), and the models can be expressed as follows ... 

The performance equation for a continuous stirred tank reactor (CSTR) was developed in Chapter 4. We use the same equation now but with a complex rate equation replacing the simpler one of the earlier chapter. We describe the method for any complex reaction consisting of N components and M reactions. The following material balances can be written for the different constituents of the complex reaction at hand (see Figure 11.2) ... 

Consider the control of a jacketed, continuous, stirred-tank reactor (CSTR) in which the exothermic reaction A — B is carried out. This system can be described by 10 variables, as shown in Figure 20.7 h, T, Ca, Cai, T Fi, F Fc, T and T oy diree of which are considered to be externally defined C I and Tco- Its model involves four equations, assuming constant fluid density. 

Because component concentrations are uniform throughout a continuous stirred tank reactor, a mass balance can immediately be constructed over the entire reactor. At steady-state conditions, the following algebraic equation is obtained describing the mass balance equation for a component A over the entire continuous stirred tank reactor ... 

The stirred-tank reactor and the tubular reactor are two basic reactors used for continuous processes, so much of the experimental and theoretical studies pubhshed to date on continuous emulsion polymerization have been conducted using these reactors. The most important elements in the theory of continuous emulsion polymerization in a stirred-tank reactor or in stirred-tank reactor trains were presented by Gershberg and Longfleld [330]. They started with the S-E theory for particle formation (Case B), employing the same assumptions as stated in Sect. 3.3, and proposed the balance equation describing the steady-state number of polymer particles produced as ... 

We turn now to consider the principal types of reactors and derive a set of equations for each that will describe the transformation 5 of the state of the feed into the state of the product. The continuous flow stirred tank reactor is one of the simplest in basic design and is widely used in chemical industry. Basically it consists in a vessel of volume V furnished with one or more inlets, an outlet, a means of cooling and a stirrer which keeps its composition and temperature essentially uniform. We shall assume that there is complete mixing on the molecular scale. It would be possible to treat of other cases following the work of Danckwerts (1958) and Zweitering (1959), but the corresponding transformation is much less wieldy. If the reactants flow in and out at a constant rate q, the mean residence time T/g is known as the holding time of the reactor. 

A series of papers concerning the use of immobilized enzymes in industrial reactors has been published.The operational effectiveness factors of immobilized enzyme systems have been described.Analytical expressions have been developed that allow the generation of effectiveness graphs for immobilized whole-cell hollow-fibre reactors. A theoretical method of determining the kinetic constants of immobilized enzymes in continuous stirred tank and plug-flow reactors by transformation of rate-equation variables has been presented. 

These equations remain valid for bioreactors provided that one employs a suitable mathematical representation of the rate of disappearance of the substrate that is the limiting reagent. In Illustration 13.3 we employ an alternative form of the design equation to determine the holding time necessary to achieve a specified degree of conversion in a strictly batch bioreactor. This illustrative example also indicates how overall yield coefficients are employed as a vehicle for taking the stoichiometry of the reaction into account. Illustration 13.4 describes how one type of semibatch operation (the fed-batch mode) can be exploited to combine the potential advantages of batch and continuous flow operation of a stirred-tank reactor. 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

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