Design Equations for Ideal Reactors

X is defined as space time, which is the mean time of residence of fluid in the reaction vessel. This is the quantum of time that is made available for the fluid to undergo reaction in the vessel. The larger the value of x, the larger the extent of conversion (X = 1 - (C /Qo)) achieved in the reactor. For a specified amount of fluid (flow rate q) processed in the reaction vessel, it is the volume V of the reactor that determines the space, time x(x = V/q) and the extent of conversion (X ) achieved. For any reaction, given the rate equation an ideal CSTR or an ideal PFR can be designed using Equation 3.1 or 3.2, respectively. According to the Arrhenius law, reaction rate constant A is a function of temperature and it increases with an increase in temperature  

For a reactor operating in an isothermal condition, the value of k is taken as a constant. Assumption of an isothermal condition is justified if the heat of reaction (AHjJ is neglected. The design of reactors operating in an isothermal condition is presented in the following sections. The design of non-isothermal reactors will be discussed later in Section 3.1.5. 

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

TABLE A.3a Design Equations for Ideal Reactors—Simplified Form"... 

TABLE A Jb Design Equations for Ideal Reactors—General Form ... 

Chapter 2 is an overview of rate equations. At this point in the text, the subject of reaction kinetics is approached primarily from an empirical standpoint, with emphasis on power-law rate equations, the Arrhenius relationship, and reversible reactions (thermodynamic consistency). However, there is some discussion of collision theory and transition-state theory, to put the empiricism into a more fundamental context. The intent of this chapter is to provide enough information about rate equations to allow the student to understand the derivations of the design equations for ideal reactors, and to solve some problems in reactor design and analysis. A more fundamental treatment of reaction kinetics is deferred until Chapter 5. The discussion of thermodynamic consistency... 

All the design equations for ideal catalytic or fluid-solid reactors can be obtained from their homogeneous reactor analogs merely by substituting the catalyst or solid weight, W, for the reactor volume, V. The reactor volume is merely the catalyst weight W divided by the bulk density of the catalyst pj,. In the catalytic or fluid-solid reactor design equations,, based on catalyst mass, must of course be used. 

The design equations for ideal tubular-flow reactors involve no new concepts but simply substitute a rate of reaction for a heat-transfer rate or mass-transfer-rate function. The increased complexity of reactor design in comparison with the design of equipment for the purely physical processes lies in the difficulty in evaluating the rate of reaction. This rate is dependent on more, and less clearly defined, variables than a heat- or mass-transfer coefficient. Accordingly, it has been more difficult to develop correlations of experimental rates, as well as theoretical means of predicting them. 

Comparison of Eqs. (4-2) and (4-5) shows that the form of the design equations for ideal batch and tubular-flow reactors are identical if the realtime variable in the batch reactor is considered as the residence time in the flow case. The important point is that the integral c/C/r is the same in both reactors. If this integral is evaluated for a given rate equation for an ideal batch reactor, the result is applicable for an ideal tubular-flow reactor this... 

Below, we reduce the design equations of the three ideal reactors to dimensionless forms. Dimensionless design equations for other reactor configurations are derived in Chapter 9. 

Formulate the dimensionless, reaction-based design equations for ideal batch reactor, plug-flow reactor, and CSTR using the heuristic rule. ... 

Solution This example illustrates how to apply the design equation for ideal batch reactors with reactions whose rate expressions are not power functions of the species concentrations. 

DIFFERENTIAL FORM OF THE DESIGN EQUATION FOR IDEAL PACKED CATALYTIC TUBULAR REACTORS WITHOUT INTERPELLET AXIAL DISPERSION... 

Most large reactors do not fit the foregoing criteria, but in many cases the deviations from ideal reactors are small, and the equations for ideal reactors can be used for approximate design calculations and sometimes for determining optimum reaction conditions. In this chapter, ideal stirred-tank reactors are considered first and then plug-flow reactors are discussed. The effects of heat transfer, mass transfer, and partial mixing in real reactors are treated in later chapters. 

Introduction to Reactor Design Fundamentals for Ideal Systems 267 Substituting Equation 5-10 into Equation 5-7 gives... 

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. 

At steady-state conditions, the mass balance design equations for the ideal tubular reactor apply. These equations may be expressed as... 

A simulation model needs to be developed for each reactor compartment within each time interval. An ideal-batch reactor has neither inflow nor outflow of reactants or products while the reaction is carried out. Assuming the reaction mixture is perfectly mixed within each reactor compartment, there is no variation in the rate of reaction throughout the reactor volume. The design equation for a batch reactor in differential form is from Chapter 5 ... 

There are many ways that two phases can be contacted, and for each the design equation will be unique. Design equations for these ideal flow patterns may be developed without too much difficulty. However, when real flow deviates considerably from these, we can do one of two things we may develop models to mirror actual flow closely, or we may calculate performance with ideal patterns which bracket actual flow. Fortunately, most real reactors for heterogeneous systems can be satisfactorily approximated by one of the five ideal flow patterns of Fig. 17.1. Notable exceptions are the reactions which take place in fluidized beds. There special models must be developed. 

The various types of reactors employed in the processing of fluids in the chemical process industries (CPI) were reviewed in Chapter 4. Design equations were also derived (Chapters 5 and 6) for ideal reactors, namely the continuous flow stirred tank reactor (CFSTR), batch, and plug flow under isothermal and non-isothermal conditions, which established equilibrium conversions for reversible reactions and optimum temperature progressions of industrial reactions. 

For an ideal batch reactor, the differential form of the design equation for... 

In the next sections, the design equations for the three ideal reactor t)q>es will be derived for isothermal conditions. In practice, the heat effects associated with chemical reactions usually result in non-isothermal conditions. The application of the law of conservation of energy leads to the so-called energy balance equation. This derivation is analogous to the derivation of the mass balance equations, and will not be treated here (see for instance, [4,5]). However, it should be noted that under non-isothermal conditions, the energy balance equation should always be solved simultaneously with the corresponding mass balance equation, since the reaction rate depends not only on composition but also on temperature. 

In Section 4.2, we derive the design equations for the first three ideal reactor models. Other reactor configurations are discussed in Chapter 9. 

Equation 4.2.2 is the integral form of the species-based design equation for an ideal batch reactor, written for species j. It provides a relation between the operating time, t, the amount of the species in the reactor, Nj(t) and Nj(0), the species formation rate, (rj), and the reactor volume, V. Note that when the reaetor volume does not change during the operation, Eq. 4.2.2 reduces to... 

To analyze and design chemical reactors more effectively and to obtain insight into the operation, we adopt reaction-based design formulation. In this section, we derive the reaction-based design equations for the three ideal reactor models. Reaction-based design equations of other reactor configurations are derived in Chapter 9. 

Adams et al. (/. Catalysis 3, 379, 1964) investigated these reactions and expressed the rate of each as second order (first order with respect to each reactant). Formulate the dimensionless, reaction-based design equations for an ideal batch reactor, plug-flow reactor, and a CSTR. 

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