Reactor design equation Dimensionless forms

The viability of one particular use of a membrane reactor for partial oxidation reactions has been studied through mathematical modeling. The partial oxidation of methane has been used as a model selective oxidation reaction, where the intermediate product is much more reactive than the reactant. Kinetic data for V205/Si02 catalysts for methane partial oxidation are available in the literature and have been used in the modeling. Values have been selected for the other key parameters which appear in the dimensionless form of the reactor design equations based upon the physical properties of commercially available membrane materials. This parametric study has identified which parameters are most important, and what the values of these parameters must be to realize a performance enhancement over a plug-flow reactor. 

This chapter covers the second fundamental concept used in chemical reaction engineering—chemical kinetics. The kinetic relationships used in the analysis and design of chemical reactors are derived and discussed. In Section 3.1, we discuss the various definitions of the species formation rates. In Section 3.2, we define the rates of chemical reactions and discuss how they relate to the formation (or depletion) rates of individual species. In Section 3.3, we discuss the rate expression that provides the relationship between the reaction rate, the temperature, and species concentrations. Without going into the theory of chemical kinetics, we review the common forms of the rate expressions for homogeneous and heterogeneous reactions. In the last section, we introduce and define a measure of die reaction rate—the characteristic reaction time. In Chapter 4 we use the characteristic reaction time to reduce the reactor design equations to dimensionless forms. 

Before one can obtain a numerical value for t from the integral form of the plug-flow reactor design equation, given by (22-27), it is necessary to focus on the dimensionless kinetic rate law, which could be rather complex. 

Transformation of the independent variables to dimensionless form uses = r/R and jz = z/L. In most reactor design calculations, it is preferable to retain the dimensions on the dependent variable, temperature, to avoid confusion when calculating the Arrhenius temperature dependence and other temperature-dependent properties. The following set of marching-ahead equations are functionally equivalent to Equations (8.25)-(8.27) but are written in dimensionless form for a circular tube with temperature (still dimensioned) as the dependent variable. For the centerline. 

The tubular reactor, steady-state design equation is of interest here. The dimensional and dimensionless forms are compared for an nth-order reaction. 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

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. 

To reduce the design equation of an ideal batch reactor, Eq. 4.3.8, to dimensionless form, we first select a reference state of the reactor (usually, the initial state) and use the dimensionless extent, Z , of the mth-independent reaction, defined by Eq. 2.7.1 ... 

Equation 4.4.4 is die dimensionless, reaction-based design equation of an ideal batch reactor, written for die mth-independent reaction. The factor ( / Co) is a scaling factor that converts die design equation to dimensionless form. Its physical significance is discussed below (Eqs. 4.4.13-4.4.15). 

To reduce die design equations of flow reactors to dimensionless forms, we select a convenient reference stream as a basis for the calculation. In most cases, it is convenient to select die inlet stream into the reactor as the reference stream, but, in some cases, it is more convenient to select another stream, even an imaginary stream. There is no restriction on the selection of the reference stream, except that we should be able to relate the reactor composition to it in terms of the reaction extents. Once we select the reference stream, we use the dimensionless extent, Z, of the mth-independent reaction, defined by Eq. 2.7.2,... 

To reduce the reaction-based design equation of a plug-flow reactor to dimensionless form, we differentiate Eqs. 4.4.5 and 4.4.8,... 

In this chapter, the analysis of chemical reactors is expanded to additional reactor configurations that are commonly used to improve the yield and selectivity of the desirable products. In Section 9.1, we analyze semibatch reactors. Section 9.2 covers the operation of plug-flow reactors with continuous injection along their length. In Section 9.3, we examine the operation of one-stage distillation reactors, and Section 9.4 covers the operation of recycle reactors. In each section, we first derive the design equations, convert them to dimensionless forms, and then derive the auxiliary relations to express the species concentrations and the energy balance equation. 

To reduce the design equation to dimensionless form, we have to select a reference state and define dimensionless extents and dimensionless time. The reference state should apply to all operations, including those with an initially empty reactor, and should enable us to compare the operation of a semibatch reactor to that of a batch reactor. Therefore, we select the molar content of the reference state, (A tot)o. as the total moles of species added to the reactor. The dimensionless extent is defined by... 

The objective of this section is to begin with the generalized form of the dimensionless mass transfer eqnation, given by (22-1), and discuss the simplifying assumptions required to reduce this partial differential equation to an ordinary differential design eqnation for packed catalytic tubular reactors. It should be mentioned that the design equation for tubular reactors, which includes convection and chemical reaction, is typically developed from a mass balance over a differential control volume given by... 

Equations (3.103) and (3.104) are the design equations for this case. They can be put in dimensionless form as we did for the CSTR. On the same basis, the reader can develop these equation for the batch reactor case. 

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