Dimensionless Extents

The stoichiometric relations derived so far provide a glimpse at the key role the reaction extents play in the analysis of chemical reactors. Whenever the extents of the independent reactions are known, the reactor composition and all other stated variables (temperature, enthalpy, etc.) can be determined. Unfortunately, the extent has two deficiencies  

While the use of calculated quantities may seem, at first, cumbersome and even counterproductive, it actually simplifies the analysis of chemical reactors with multiple reactions. In fact, calculated quantities such as enthalpy and free energy are commonly used in thermodynamics resulting in simplified expressions. Here too, by using the extents of independent reactions, we formulate the design of chemical reactors by the smallest number of design equations. 

To characterize the generic behavior of chemical reactors, it is preferred to describe their operations in terms of intensive dimensionless quantities. To convert the reaction extents to intensive quantities, dimensionless extents are defined. For batch reactors, the dimensionless extent, Z , of the mth independent reaction is defined by 

Extent of the mth independent reaction Total number of moles of reference state (iVtot)o 

Total molar flow rate of reference stream 

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As before, we have dimensionless extent, concentration, and temperature... 

Here again, has been used for the dummy integration variable it is not the dimensionless extent. 

Deduce that can be written in terms of a dimensionless extent as... 

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 ... 

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,... 

We solve these equations for Z s and 0 as functions of t, subject to the initial condition that at T = 0, the dimensionless extents and the dimensionless temperature are specified. For isothermal operations, Eq. 4.4.17 reduces to design equations should be solved. 

We can construct the matrix of stoichiometric coefficients and reduce it to a diagonal form to determine the number of independent reactions. However, in this case, we have three reversible reactions, and, since each of the three forward reactions has a species that does not appear in the other two, we have three independent reactions and three dependent reactions. We select the three forward reactions as the set of independent reactions. Hence, the indices of the independent reactions are m = 1, 3, 5, and we describe the reactor operation in terms of their dimensionless extents, Zi, Z3, and Z5. The indices of the dependent reactions are = 2, 4, 6. Since this set of independent reactions consists of chemical reactions whose rate expressions are known, the heuristic rule on selecting independent reactions is satisfied. The stoichiometric coefficients of the selected three independent reactions are... 

To solve the design equations, we have to express the rates of the individual reactions in terms of the dimensionless extents of the independent reactions, Zi, Z2, and Zj. 

Note that, in this case, the design equations have seven terms (seven rate expressions), whereas in the formulation in part (a), design equations (d), (e), and (f), have only five terms. This illustrates that, by adopting the heuristic rule, we minimize the number of terms in the design equations. To solve the design equations, we have to express the rates of the individual reactions in terms of the dimensionless extents of the independent reactions, Z, Z5, and Ze. 

Equation 5.2.18 is the dimensionless, differential energy balance equation of ideal batch reactors, relating the reactor dimensionless temperature, 0(t), to the dimensionless extents of the independent reactions, Z (t), at dimensionless operating time T. Note that individual dZ /dfr s are expressed by the reaction-based design equations derived in Chapter 4. 

Recall fliat is the dimensionless extent of the mth-independent chemical reaction defined hy... 

To solve the design equation, we have to express the reaction rate r in terms of Z and, to do so we relate the species concentrations to the dimensionless extent From Eq. 6.1.11, for isothermal operations with single reactions, and when the reference state is the initial state ... 

Express the reaction rates in terms of the dimensionless extents of the independent reactions, Z s. 

To solve Eq. 7.2.2 or 7.2.4, we have to express the reaction rate r in terms of the dimensionless extent Z. To do so, we express the species concentrations in terms of Z. Selecting the inlet stream as the reference stream, Zi = 0, and for liquid-phase reactions, Eq. 7.1.11 reduces to... 

Equation 7.5.4 provides an approximate relation for the ehanges in pressure along a plug-flow reactor, expressed in terms of dimensionless extents and temperature. It is applicable when the velocity does not exceed 80-90% of the sound velocity. For these situations, we solve Eq. 7.5.4 simultaneously with the design 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... 

We use Eq. 9.3.14 to express the reactant concentrations in terms of the dimensionless extents ... 

To reduce the design equation to a dimensionless form, we select the feed stream to the system (stream 0) as the reference stream and define a dimensionless extent of the mth-independent reaction by... 

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