Batch reactor volume element

The batch reactor is assumed to be well stirred, so there are no concentration gradients anywhere in the reactor volume. In this case it is natural to consider the entire reactor contents to be the reactor volume element as in Figure 4.2, and V = Vr. Because the reactor is well stirred, the integrals in Equation 4.2 are simple to evaluate,... 

This equation can be used directly for any well-mixed, batch, semi-batch or continuous volume element. The term on the left-hand side represents the rate of energy accumulation. The first term on the right-hand side depicts the energy needed to raise the temperature of the incoming reactants, including inert material, to the reactor temperature. The second term describes the heat... 

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. 

Where the composition within the reactor is uniform (independent of position), the accounting may be made over the whole reactor. Where the composition is not uniform, it must be made over a differential element of volume and then integrated across the whole reactor for the appropriate flow and concentration conditions. For the various reactor types this equation simplifies one way or another, and the resultant expression when integrated gives the basic performance equation for that type of unit. Thus, in the batch reactor the first two terms are zero in the steady-state flow reactor the fourth term disappears for the semibatch reactor all four terms may have to be considered. 

Regarding reactor sizes, a comparison of Eqs. 5.4 and 5.19 for a given duty and for s = 0 shows that an element of fluid reacts for the same length of time in the batch and in the plug flow reactor. Thus, the same volume of these reactors is needed to do a given job. Of course, on a long-term production basis we must correct the size requirement estimate to account for the shutdown time between batches. Still, it is easy to relate the performance capabilities of the batch reactor with the plug flow reactor. 

In the ideal batch stirred-tank reactor (BSTR), the fluid concentration is uniform and there are no feed or exit streams. Thus, only the last two terms in the previous equation exist. For a volume element of fluid (VL), the mass balance for the limiting reactant becomes (Smith, 1981 Levenspiel, 1972)... 

If the compositions vary with position in the reactor, which is the case with a tubular reactor, a differential element of volume SV, must be used, and the equation integrated at a later stage. Otherwise, if the compositions are uniform, e.g. a well-mixed batch reactor or a continuous stirred-tank reactor, then the size of the volume element is immaterial it may conveniently be unit volume (1 m3) or it may be the whole reactor. Similarly, if the compositions are changing with time as in a batch reactor, the material balance must be made over a differential element of time. Otherwise for a tubular or a continuous stirred-tank reactor operating in a steady state, where compositions do not vary with time, the time interval used is immaterial and may conveniently be unit time (1 s). Bearing in mind these considerations the general material balance may be written ... 

There is, however, another way of looking at a tubular reactor in which plug flow occurs (Fig. 1.15). If we imagine that a small volume of reaction mixture is encapsulated by a membrane in which it is free to expand or contract at constant pressure, it will behave as a miniature batch reactor, spending a time r, said to be the residence time, in the reactor, and emerging with the conversion aA/. If there is no expansion or contraction of the element, i.e. the volumetric rate of flow is constant and equal to v throughout the reactor, the residence time or contact time... 

In order to And the overall relative yield batch reaction or at the outlet of a tubular reactor, consider an element of unit volume of the reaction mixture. If the concentration of A decreases by SCA either (a) with time in a batch reactor or (b) as the element progresses downstream in a tubular reactor, the amount of A transformed into P is - [Pg.59]

It can be readily discerned that the reactor equation for the batch reactor (5.12) and the plug-flow reactor (5.13) are identical. In the former, the concentration changes with time, in the latter, with location. In contrast to the situation in the other two ideal reactors, the residence time T in a CSTR is only an average, as every volume element has a different residence time throughout the reactor. 

Three forms of the reactor operator, R(Y), are shown in Figure 3. These are generally differential operators which operate on each monomer and polymer species to describe the effects of accumulation and the physical processes which move material in and out of the reactor or reactor element. The concentration of a specific species is given by the variable Y. In a simple batch reactor, the reactor operator, RB, is merely defined as the rate of accumulation of a certain species with time per unit volume of reactor—i.e., the rate of change of concentration of the species. 

Numerous reactions are performed by feeding the reactants continuously to cylindrical tubes, either empty or packed with catalyst, with a length which is 10 to 1000 times larger than the diameter. The mixture of unconverted reactants and reaction products is continuously withdrawn at the reactor exit. Hence, constant concentration profiles of reactants and products, as well as a temperature profile are established between the inlet and the outlet of the tubular reactor, see Fig. 7.1. This requires, in contrast to the batch reactor, the application of the law of conservation of mass over an infinitesimal volume element, dV, of the reactor. In contrast to a batch reactor the existence of a temperature profile does not allow us to consider the mass balances for the reacting components and the energy balance separately. Such a separation can only be performed for isothermal tubular reactors. 

At constant pressure and granted ideal plug flow, the behavior of a tubular reactor at steady state is mathematically analogous to that of a batch reactor A volume element of the reaction mixture has no means of knowing whether it is suspended tea bag-style in a batch reactor or rides elevator-style through a tubular reactor being exposed to the same conditions it behaves in the same way in both cases. As in a batch reactor, what is measured directly are concentrations—here in the effluent—and a finite-difference approximation is needed to obtain the rate from experiments with different reactor space times and otherwise identical conditions. For a reaction without fluid-density variation ... 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

The reactor models considering complete mixing may be subdivided into batch and continuous types. In the continuous stirred tank reactor (CSTR) models, an entering fluid is assumed to be instantaneously mixed with the existing contents of the reactor so that it loses its identity. This type of reactor operates at uniform concentration and temperature levels. For this reason the species mass balances and the temperature equation may be written for the entire reactor volume, not only over a differential volume element. Under steady-state conditions, the species mass and heat balances reduce to algebraic equations. 

The stirred-iank reactor may be operated as a steady-state flow type (Fig. 3-lu), a batch type (Fig. 3- b), or as a non-steady-state, or semibatch, reactor (Fig. 3-lc). The key feature of this reactor is that the mixing is complete, so that the properties of the reaction mixture are uniform in all parts of the vessel and are the same as those in the exit (or. product) stream. This means that the volume element chosen for the balances can be taken as the volume V of the entire reactor. Also, the composition and temperature at which reaction takes place are the same as the composition and temperature of any exit stream. 

For non-isothermal or non-linear chemical reactions, the RTD no longer suffices to predict the reactor outlet concentrations. From a Lagrangian perspective, local interactions between fluid elements become important, and thus fluid elements cannot be treated as individual batch reactors. However, an accurate description of fluid-element interactions is strongly dependent on the underlying fluid flow field. For certain types of reactors, one approach for overcoming the lack of a detailed model for the flow field is to input empirical flow correlations into so-called zone models. In these models, the reactor volume is decomposed into a finite collection of well mixed (i.e., CSTR) zones connected at their boundaries by molar fluxes.4 (An example of a zone model for a stirred-tank reactor is shown in Fig. 1.5.) Within each zone, all fluid elements are assumed to be identical (i.e., have the same species concentrations). Physically, this assumption corresponds to assuming that the chemical reactions are slower than the local micromixing time.5... 

In this section we must be careful to respect our prior concern about the definition of rate with regard to the volume of reaction mixture involved. Further, since we wish to concentrate attention on the kinetics, we shall study systems in which the conservation equation contains the reaction term alone, which is the batch reactor of equation (1-12). It is convenient to view this type of reactor in a more general sense as one in which all elements of the reaction mixture have been in the reactor for the same length of time. That is, all elements have the same age. Since the reactions we are considering here occur in a single phase, the relationships presented below pertain particularly to homogeneous batch reactions, and the systems are isothermal. 

Reactors with complete mixing may be subdivided into batch and continuous types. In a batch type reactor with complete mixing the composition is uniform throughout the reactor. Consequently, the continuity equation may be written for the entire contents, not only over a volume element The composition varies with time, however, so that a first-order ordinary differential equation is obtained, with time as variable. The form of this equation is analogous with that for the... 

In the analysis of batch reactors, the two flow terms in equation (8.0.1) are omitted. For continuous flow reactors operating at steady state, the accumulation term is omitted. However, for the analysis of continuous flow reactors under transient conditions and for semibatch reactors, it may be necessary to retain all four terms. For ideal well-stirred reactors, the composition and temperature are uniform throughout the reactor and all volume elements are identical. Hence, the material balance may be written over the entire reactor in the analysis of an individual stirred tank. For tubular flow reactors the composition is not independent of position and the balance must be written on a differential element of reactor volume and then integrated over the entire reactor using appropriate flow conditions and concentration and temperature profiles. When non-steady-state conditions are involved, it will be necessary to integrate over time as well as over volume to determine the performance characteristics of the reactor. 

For a batch reactor, the reaction time t is the natural performance measure. For flow reactors, the residence time r is used, which is defined as the ratio of the reactor volume to the volumetric flow rate at reaction conditions. In mixed flow reactors, r represents a mean value because the residence time of the fluid elements is distributed. Only for plug flow tubular reactors is the residence time the same for all fluid elements. For heterogeneously catalyzed or gas-solid reactions it is convenient to use a (mean) modified residence time related to the mass of catalyst or solid. 

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