Dimensionless Design Equations and Operating Curves

The reaction-based design equations derived in the previous section are expressed in terms of extensive quantities such as reaction extents, reactor volume, molar flow rates, and the like. To describe flie generic behavior of chemical reactors, we would like to express the design equations in terms of intensive, dimensionless variables. This is done in two steps  

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, 

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Derived the dimensionless design equation for isothermal operation with single reactions and obtained the reaction operating curve. 

Solve the design equations for Z , s as functions of the dimensionless space time, T, and obtain the reaction operating curves. 

We solve the design equations simultaneously with the energy balance equation, subject to the initial condition that at t = 0, the extents of aU the independent reactions and the dimensionless temperature are specified. Note that we solve fiiese equations for a specified value of T t (or reactor volume). The reaction operating curves of plug-flow reactors with side injection are the final value of Z s and 9 for different values of Ttot-... 

Figure 7.16 plots Equation 7.64 for different values of B. When B = oo, Equation 7.64 reduces to Equation 7.25, and the curve for = 1.0 in Figure 7.4 is the same as the curv e for B = oo in Figure 7.16. With decreasing B, corresponding to slower external mass transfer, the concentration profile in the pellet becomes more uniform and the dimensionless surface concentration decreases. The lower concentration leads to lower reaction rates. Therefore one normally designs the reactor and chooses operating conditions, such as large gas velocities, to enhance external mass transfer and make the Biot number large.

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