Batch reactors describing equations

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

Kinetic Model Discrimination. To discriminate between the kinetic models, semibatch reactors were set up for the measurement of reaction rates. The semi-batch terminology is used because hydrogen is fed to a batch reactor to maintain a constant hydrogen pressme. This kind of semi-batch reactor can be treated as a bateh reactor with a constant hydrogen pressme. The governing equations for a bateh reactor, using the product formation rate for three possible scenarios, were derived, as described in reference (12) with the following results ... 

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. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

One of the last two equations gives the relationship between Cv and CA at any time in a plug flbw or batch reactor. The stoichiometry of the system provides the additional information necessary to describe completely the system composition. 

A set of first order differential equations descriptive of the molar concentrations for polymeric species is given. Species, less than size n, are saturated and, therefore, accmulate only within the batch reactor and do not participate in branch/cross-llnk reactions. Molecules greater in size than n are unsaturated and will experience the reaction described by branch formation. 

We need reaction-rate expressions to insert into species mass-balance equations for a particular reactor. These are the equations from which we can obtain compositions and other quantities that we need to describe a chemical process. In introductory chemistry courses students are introduced to first-order irreversible reactions in the batch reactor, and the impression is sometimes left that this is the only mass balance that is important in chemical reactions. In practical situations the mass balance becomes more comphcated. 

Figure 2-9 Sketch showing correspondence between time in a batch reactor and position a in a plug-flow tubular reactor. The mass-balance equations describe both reactors for the constant-density situations.

We can therefore replace dt by dz/u in all of the preceding differential equations for the mass balance in the batch reactor and use these equations to describe reactions during flow through a pipe. This reactor is called the plug-flow tubular reactor, which is the most important continuous reactor encountered in the chemical industry. 

We have gone about as far as is useful in finding closed-form analytical solutions to mass-balance equations in batch or continuous reactors, described by the set of reactions... 

However, the solution for the CSTR obtained by the RTD equation is correct only for first-order kinetics. For other rate expressions the conversion predicted by the RTD is incorrect for a mixed reactor because molecules do not simply react for time t, after which they leave the reactor. Rather, the fluid is continuously mixed so that the history of the fluid is not describable in these terms. This expression for conversion in the CSTR is applicable for segregated flow, in which drops of fluid enter the reactor, swirl in the reactor, and exit after time t because then each drop behaves as a batch reactor with the RTD describing the probability distribution of the drops in the CSTR. 

On the basis of a mass balance the following general equations describing the basic reactor concepts have been derived in the last section. For the batch reactor the time tj, (s) needed for a conversion of a component i from C at t=0 to C is given by ... 

Also in case of diffusion-limited reactions where the overall effectiveness factor is used to describe the effect of diffusion on the rate of biocatalysis, the mathematics are the same as in the case of the batch reactor. Substitution of Equation (11.56) in Equation (11.23) thus yields ... 

We have seen that the basic P model has the form of a first-order partial differential Eq. (22) describing each narrow slice as a little batch reactor being transported through the reactor at constant speed. This equation was so elementary that it could be solved at sight in Eq. (30). When we added a longitudinal dispersion term governed by Fick s law and took the steady state, Eq. (40), we had a second-order o.d.e. with controversial boundary conditions. This is the model with ( ) = c(z)lcm and Pe = vLID, Da = kL/v,... 

A limitation of the methods described so far is that they have assumed a constant overall yield coefficient and do not allow the endogenous respiration coefficient kd (or alternatively the maintenance coefficient, m) to be evaluated. Equation 5.54 shows that the overall yield, as measured when monitoring a batch reactor, is affected by the growth rate and has the greatest impact when the growth rate is low. Consequently, it is desirable to be able to estimate the values of kd or m, so that the yield coefficient reflects the true growth yield. An equivalent method would be one where the specific rates of formation of biomass and consumption of substrate were determined independently, again without the assumption of a constant overall yield-coefficient. 

Equations 7.4 and 7.5 form a system of differential equations for which no analytical solution is known. Thus, the description of the behavior of the semi-batch reactor with time requires the use of numerical methods for the integration of the differential equations. Usually, it is convenient to use parameters which are more process-related to describe the material balance. One is the stoichiometric ratio between the two reactants A and B ... 

Thus, the equations describing the thermal stability of batch reactors are written, and the relevant dimensionless groups are singled out. These equations have been used in different forms to discuss different stability criteria proposed in the literature for adiabatic and isoperibolic reactors. The Semenov criterion is valid for zero-order kinetics, i.e., under the simplifying assumption that the explosion occurs with a negligible consumption of reactants. Other classical approaches remove this simplifying assumption and are based on some geometric features of the temperature-time or temperature-concentration curves, such as the existence of points of inflection and/or of maximum, or on the parametric sensitivity of these curves. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

Batch Reactor Equations The dynamic equations describing the batch reactor are similar to those of a CSTR except that the feed and product streams are missing ... 

The CSTR operator, Rc, has an identical term to describe accumulation under transient operation. The algebraic sum of the two other terms indicates the difference of in-flow and out-flow of that species. This operator also describes semibatch or semicontinuous operation in cases where the volume can be assumed to be essentially constant. In the more general case of variable volume, V must be included within the differential accumulation term. At steady state, it is a difference equation of the same form as the differential equation for a batch reactor. 

When such a stirring is absolutely absent in a continuous flow system, as it takes place in the piston reactor (PR), regularities of the batch processes with the same residence time 0 are realized. This implies that in order to describe copolymerization in continuous PR one can apply all theoretical equations known for a common batch process having replaced the current time t for 0. As for the equations presented in Sect. 5.1, which do not involve t al all, they remain unchanged, and one can employ them directly to calculate statistical characteristics of the products of continuous copolymerization in PR. It is worth mentioning that instead of the initial monomer feed composition x° for the batch reactor one should now use the vector of monomer feed composition xin at the input of PR. In those cases where copolymer is being synthesized in CSTR a number of specific peculiarities inherent to the theoretical description of copolymerization processes arises. 

Various levels of models can be used to describe the behavior of pilot-scale jacketed batch reactors. For online reaction calorimetry and for rapid scale-up, a simple model characterizing the heat transfer from the reactor to the jacket can be used. Another level of modeling detail includes both the jacket and reactor dynamics. Finally, the complete set of equations simultaneously describing the integrated reactor/jacket and recirculating system dynamics can be used for feedback control system design and simulation. The complete model can more accurately assess the operability and safety of the pilot-scale system and can be used for more accurate process scale-up. 

Pseudo- and overall reaction orders. Kinetic textbooks describe other, more complicated methods applicable to other forms of proposed rate equations, mostly for evaluation of results from batch reactors. However, if the development chemist or engineer can commission experiments—as opposed to having to evaluate existing data—he can often save himself much effort by determination of pseudo- and overall reaction orders. For example, for a reaction A + B — product(s) and power-law rate equation —rk = kmCfCB, three series of experiments suggest themselves ... 

To describe this process, material and energy balances are required. Recall that the mass balance on a batch reactor can be written as [refer to Equation (3.2.1)] ... 

Consider an isothermal batch reactor, in which the concentrations Ui t) are described by the following system of ordinary differential equations ... 

The describing equation for chemical reaction mass transfer is obtained by applying the conservation law for either mass or moles on a time rate basis to the contents of a batch reactor. It is best to work with moles rather than mass since the rate of reaction is most conveniently described in terms of molar concentrations. The describing equation for species A in a batch reactor takes the form... 

Tubular flow reactors are usually operated under steady conditions so that, at any point, physical and chemical properties do not vary with time. Unlike the batch and tank flow reactors, there is no mechanical mixing. Thus, the state of the reacting fluid will vary from point to point in the system, and this variation may be in both the radial and axial direction. The describing equations are then differential, with position as the independent variable. 

The equation describing temperature variations in batch reactors is obtained by applying the conservation law for energy on a time-rate basis to the reactor contents. Since batch reactors are stationary (fixed in space), kinetic and potential effects can be neglected. The equation describing the temperature variation in reactors due to energy transfer, subject to the assumptions in its development, is... 

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