Ideal reactors equations

How a differential equation is formulated for some lands of ideal reactors is described briefly in Sec. 7 of this Handbook and at greater length with many examples in Walas Modeling with Differential Equations in Chemical Engineering, Butterworth-Heineman, 1991). 

For the ideal reactors considered, the design equations are based on the mass conservation equations. With this in mind, a suitable component is chosen (i.e., reactant or product). Consider an element of volume, 6V, and the changes occurring between time t and t + 6t (Figure 5-2) ... 

The rate constants are given by Equation (6.3), and both reactions are endothermic as per Equation (6.4). The flow diagram is identical to that in Figure 6.1, and all cost factors are the same as for the consecutive reaction examples. Table 6.1 also applies, and there is an interior optimum for any of the ideal reactor t5qjes. 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

Table 8.1 summarizes the fundamental design relationships for the various types of ideal reactors in terms of equations for reactor space times and mean residence times. The equations are given in terms of both the general rate expression and nth-order kinetics. 

This expression may be combined with equations 10.1.4 and 10.1.8 in order to analyze the different situations that may arise in operating the various types of ideal reactors. These analyses are the subject of Sections 10.2 to 10.4. 

Equation 13.5-2 is the segregated-flow model (SFM) with a continuous RTD, E(t). To what extent does it give valid results for the performance of a reactor To answer this question, we apply it first to ideal-reactor models (Chapters 14 to 16), for which we have derived the exact form of E(t), and for which exact performance results can be compared with those obtained independently by material balances. The utility of the SFM lies eventually in its potential use in situations involving nonideal flow, wheic results cannot be predicted a priori, in conjunction with an experimentally measured RTD (Chapters 19 and 20) in this case, confirmation must be done by comparison with experimental results. 

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

Summary of design equations for ideal reactors (continued)... 

Ridelhoover and Seagrave [57] studied the behaviour of these same reactions in a semi-batch reactor. Here, feed is pumped into the reactor while chemical reaction is occurring. After the reactor is filled, the reaction mixture is assumed to remain at constant volume for a period of time the reactor is then emptied to a specified level and the cycle of operation is repeated. In some respects, this can be regarded as providing mixing effects similcir to those obtained with a recycle reactor. Circumstances could be chosen so that the operational procedure could be characterised by two independent parameters the rate coefficients were specified separately. It was found that, with certain combinations of operational variables, it was possible to obtain yields of B higher than those expected from the ideal reactor types. It was necessary to use numerical procedures to solve the equations derived from material balances. 

In this chapter we develop the performance equations for a single fluid reacting in the three ideal reactors shown in Fig. 5.1. We call these homogeneous reactions. Applications and extensions of these equations to various isothermal and noniso-thermal operations are considered in the following four chapters. 

Tables 5.1 and 5.2 present the integrated performance equations for single ideal reactors.

In this chapter we deal with single reactions. These are reactions whose progress can be described and followed adequately by using one and only one rate expression coupled with the necessary stoichiometric and equilibrium expressions. For such reactions product distribution is fixed hence, the important factor in comparing designs is the reactor size. We consider in turn the size comparison of various single and multiple ideal reactor systems. Then we introduce the recycle reactor and develop its performance equations. Finally, we treat a rather unique type of reaction, the autocatalytic reaction, and show how to apply our findings to it. 

Table 16.2 Conversion Equations for Macrofluids and Microfluids with 8 = 0 in Ideal Reactors...

An ideal reactor model is composed of complete mix reactors with leaky dead zones that are also complete mixed. Assuming a pulse input to reactor 1, derive the concentration versus time equation for the third reactor-dead zone combination. Derive the concentration versus time equation for the nth reactor-dead zone combination. 

The distinguishing feature of the fed-batch reactor is that there is only an ingoing flow and no outgoing flow (Figure 11.13). Assuming ideal mixing equation (11.5) thus... 

The equations controlling the operation of ideal reactors are the energy balance and a mass balance for each reactant and product. 

Idealized Reactor with Instantaneous Mixing. For a reactor of the second type, a homogeneous system, where the composition, temperature, and pressure is everywhere the same, Equation 11 is immediately intcgrable to... 

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. 

Integrated Michaelis-Menten Equation in Ideal Reactors... 

The semibatch reactor is a cross between an ordinary batch reactor and a continuous-stirred tank reactor. The reactor has continuous input of reactant through the course of the batch run with no output stream. Another possibility for semibatch operation is continuous withdrawal of product with no addition of reactant. Due to the crossover between the other ideal reactor types, the semibatch uses all of the terms in the general energy and material balances. This results in more complex mathematical expressions. Since the single continuous stream may be either an input or an output, the form of the equations depends upon the particular mode of operation. 

Using this rate expression and the constant density ideal batch reactor equation gives... 

A batch reactor has no inlet or outlet flows, so n, o = = 0. Perfect mixing is assumed for this ideal reactor, and the rate r is independent of position. This changes our generation term in the general reactor design equation to... 

Then, by the general design equation, our ideal batch reactor equation becomes... 

Our ideal batch reactor equation, written in terms of any reactant A, can be changed to reflect a change in volume. For example,... 

This changes the ideal batch reactor equation to... 

Upon simplification the resulting ideal plug flow reactor equation is... 

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