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Element of reactor volume

Reactants /

enthalpy enter

Element of reactor volume

Reactants /

enthalpy enter

FIGURE 1.3 Conservation over element of reactor volume.

> Reactants/ enthalpy leave

FIGURE 1.3 Conservation over element of reactor volume.

This general balance over an element of reactor volume can be represented mathematically as dP) + d(fiUtf) = d_ / + S

where p is the density of the fluid, <p is the concentration of any component, Ui is the local velocity in the xi direction, is the effective diffusivity of <p and S$ is a volumetric source term (rate of production of <p per unit volume) of <p. The terms appearing in Eq. (1.2) represent corresponding terms in Eq. (1.1). More complete and detailed mathematical formulation of general conservation-governing equations is discussed in Chapter 2 (and in references cited therein). It is important to remember that the source term, , will be equal to the rate based on intrinsic kinetics only if there is no physical resistance, i.e. there are no concentration or temperature gradients within the volume element under consideration. In non-isothermal operations, it is necessary to solve an energy balance equation along with the material balance equation. The form of the energy balance equation is similar to Eq. (1.2) with <p being an enthalpy content of the fluid. The material and energy balance equations are strongly coupled with each other since the source term of the energy balance equations depends on the rate of reaction. The material and energy balance equations are starting points for all reaction engineering analysis. Strictly speaking, it is necessary to know the velocity field at each point in the reactor in order to solve the material and energy balances discussed above. For any arbitrary reactor type, the velocity field can be predicted by solving corresponding momentum balance equations over the reactor. However, over the years, chemical reaction engineering analysis has made significant contributions by making judicious simplifications of these general equations to draw useful conclusions about the behavior of the reactor and to bring out the limiting behavior of reactors without solving the momentum balance equations.

The most fruitful and extensively used concept along these lines is the concept of an 'ideal' reactor. The simplest reactor, whose performance is governed by the so-called 'zero dimensional' equation is a 'completely mixed reactor'. The key assumption is that mixing in the reactor is complete, so that the properties of the

reaction mixture are uniform in all parts of the reactor vessel and are, therefore, the same as those in the 'exit' stream (Fig. 1.4a). It means that mixing is much faster than the reaction and residence time based on net flow. It also means that the differential volume element chosen for the microscopic balance can cover the entire reactor. This greatly simplifies the governing equations and allows a reaction engineer to evaluate different modes of reactor operation (batch, semi-batch and continuous) and to understand the key features of the chemical system under investigation. An analysis based on a completely mixed reactor concept provides one of the limiting solutions for the performance expected from a practical reactor.

The other ideal reactor concept, known as a 'plug flow reactor' is based on a 'one dimensional' approximation of the material and energy balance equations. In an ideal plug flow reactor, unidirectional flow through the reactor is assumed (like flow through a tube). The velocity is assumed to be uniform over all the planes normal to the flow direction. In addition, it is assumed that no mixing takes place in the direction of flow and there are no gradients in the planes normal to the flow direction. These assumptions allow maximum variation of concentrations from reactor inlet to reactor outlet in contrast to the mixed reactor concept (Fig. 1.4b). Therefore, reaction engineering analysis of a reactor using this approximation provides a second limit on the performance expected from a practical reactor.

It may not be an exaggeration to say that carrying out the analysis of an industrial reactor using these concepts of ideal reactors is one of the most important tasks of a reactor engineer. Such an analysis results in a crucial understanding and useful information about the sensitivity of reactor performance to underlying mixing. It also helps to identify the characteristics of desirable mixing within the reactor. Such information can be very useful in optimizing reactor performance while carrying out multiple reactions and catalytic or autocatalytic reactions. In addition, ideal reactor concepts can be extensively used to understand the interaction between chemical and thermal processes. The simplifications in the underlying flow and mixing allow reactor

Inlet -

Stagnant region

Short circuit /

Inlet

Mixing due to vortices & turbulence

Mixing due to non-uniform velocity profile

(a) Mixed Reactor

(b) Plug Flow Reactor

### FIGURE 1.5 Deviations from ideal reactors.

engineers to carry out sophisticated operability and stability analysis of reactors (see reviews by Razon and Schmitz, 1987; Morbidelli and Carra, 1987). The concepts of ideal reactors have also been extensively used to understand the behavior of a variety of multiphase reactors, which are discussed a little later in this section.

After establishing such understanding and analysis of the reaction system using ideal reactor concepts, the next most important question facing the reactor engineer is to evaluate the consequences of the assumptions involved in the concepts of ideal reactors to estimate the behavior of an actual reactor. The mixing in an actual reactor may deviate significantly from that assumed for ideal reactors. This deviation can be caused by channeling of fluid, by recycling of fluid or by the formation of stagnant regions within the reactor (Fig. 1.5). If pockets of stagnant fluid exist within the reactor, conversion will approach the upper limit in these regions, but this fluid does not leave the reactor. The fed reactants will flow through the remaining volume of the reactor and, therefore, will have less time to react. The result will be an average conversion lower than that for the ideal reactor. A similar result may occur if there is a short-circuit and fluid by-passes through the reactor without mixing. Deviations from ideal plug flow behavior in the form of some mixing in the direction of flow (instead of no mixing) and incomplete mixing in the plane normal to the flow direction may also occur. Bypassing and short-circuiting may also occur in a plug flow reactor. If one knows the complete history of all the fluid elements (velocity and mixing) flowing through the reactor, it is possible to solve the differential material and energy balances to quantitatively estimate the influence of such non-ideal behavior. In the absence of such knowledge, reaction engineers have devised ingenious tools to quantify the effects of non-ideal behavior.

The residence time distribution (RTD) and state of mixedness are the two most important concepts used for such analysis. RTD, as the name suggests, indicates the spread of residence time experienced by different fluid elements while flowing through the reactor. The response data or measurements of the variation of reactor outlet concentration for the known change of inlet concentration can be used to estimate the RTD of a given reactor. For reactions following other than first-order kinetics, knowledge of RTD will not be sufficient to estimate the reactor performance. It is necessary to know the state of mixing between fluid elements of different ages flowing through the reactor. Here again, it may be noted that completely segregated (assuming no mixing between fluid elements of different ages) and completely mixed fluid elements constitute the two limiting solutions. Obtaining the RTD of an actual reactor and applying these two limiting assumptions to obtain the bounds on performance of the reactor is a practical method for a useful reaction engineering analysis.

Several sophisticated techniques and data analysis methodologies have been developed to measure the RTD of industrial reactors (see, for example, Shinnar, 1987). Various different types of models have been developed to interpret RTD data and to use it further to predict the influence of non-ideal behavior on reactor performance (Wen and Fan, 1975). Most of these models use ideal reactors as the building blocks (except the axial dispersion model). Combinations of these ideal reactors with or without by-pass and recycle are used to simulate observed RTD data. To select an appropriate model for a reactor, the actual flow pattern and its dependence on reactor hardware and operating protocol must be known. In the absence of detailed quantitative models to predict the flow patterns, selection of a model is often carried out based on a qualitative understanding of flow patterns and an analysis of observed RTD data. It must be remembered that more than one model may fit the observed RTD data. A general philosophy is to select the simplest model which adequately represents the physical phenomena occurring in the actual reactor.

A flow model representing the actual flow patterns and mixing within the reactor is necessary for realistic description of reactor behavior. Such a flow model can even be just a qualitative understanding to guide model development or can be a model with varying degrees of sophistication. For example, Van de Vusse (1962) proposed a model for simulating a stirred tank reactor based on intuitive understanding of flow generated by an impeller. He visualized the flow pattern within the stirred reactor in the form of three loops (Fig. 1.6a) and constructed a mixing model based on this visualization (Fig. 1.6b). Such a model can predict the influence of flow patterns within the reactor on reactor performance. In order to relate reactor configuration (for example, degree of baffling) and operating conditions (for example, impeller rotational speed) with reactor performance, it is necessary to establish a relationship between these hardware/operating parameters and model parameters. This can be accomplished by fitting model parameters to simulate the observed RTD experiments conducted with different reactor configurations. Results by Takamatsu and Sawada (1968) relating type of impeller, degree of baffling and impeller Reynolds number (rotational speed) are shown in Fig. 1.6c. Although these experimental findings do not provide any generalized correlations for stirred reactors, they were useful to indicate the general trends and, probably, were among the first attempts to directly connect the reactor hardware and operating conditions to reactor performance via a reactor model.

Several such models with increasing complexity were developed. Mann and his coworkers developed models based on 200 to 400 completely mixed zones or cells with finite exchange between neighboring reactors connected in such a way as to represent the actual flow generated by an impeller (Mann and Mavros, 1982; Wang and Mann, 1992). In these models (shown schematically in Fig. 1.7), they used experimental measurements of velocity data for prescribing flow through different zones. They were reasonably successful in simulating reactor performance for fast, mixing controlled reactions. However, an approach which relies either on RTD data and qualitative understanding of the flow patterns or on experimental measurements of flow, to establish the relationship between reactor hardware and a reactor model or reactor performance has obvious limitations for general application. The extent of non-ideality, and therefore model parameters, will change with the reactor scale and operating conditions. These studies have increasingly pointed out the need for more rigorous flow modeling of chemical reactors, even for single-phase flows.

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