Design Equations for a PFR

Equation 4.2.10 is the species-based differential design equations for a PFR, written for species j. To obtain the reactor volume of a PFR, we integrate Eq. 4.2.10,... 

A differential balance is written for a vanishingly small control volume within which is approximately constant. Such a differential balance is needed to derive the design equation for a PFR. The differential volume element has volume AT, cross-sectional area Ac, and length Az. See Figure 1.4. The general component balance applied to this differential volume gives... 

A general procedure for solving the reactor design equations for a PFR using the marching-ahead technique (Euler s method) has seven steps. Note that this general procedure allows for variable temperature as well as variable pressure. The necessary ODE for temperature is given in Chapter 5. 

We will again start from the design equation for a PFR for A ... 

The long-time solution to these ODEs gives (a<,ai)i and bout), which are the inlet concentrations for the piston flow portion of the system. The design equations for the PFR are... 

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. 

Packed-Bed Reactor. The derivation of the differential and integral forms of the design equations for a packed-bed reactor are analogous to those for a PFR [cf. Equations (2-15) and (2-16)). That is, substituting for Fa Equation (1-11) gives... 

The PFR design equation for a recycle reactor is also developed in the CD-ROM. 

Figure 12. Graphical presentation of the PFR design equation for a single reaction. For gaseous reactions, assuming negligible pressure drop, Eq. 7.1.15 reduces to...

In ihe three idealized types of reactors just discussed (the perfectly mixed batch reactor, the plug-fiow tubular reactor (PFR). and the perfectly mixed con-tinuous-siirred tank reactor (CSTR), the design equations (i.e.. mole balances) were dei doped based on reactor volume. The deris ation of the design equation for a packed-bed catalytic reactor (PBR) will be carried out in a manner analogous to the development of the tubular design equation. To accomplish this derivation. we simply replace the volume coordinate in Equation (MO) with (he catalyst weight coordinate H (Figure - 4). 

Chapter 1 treated PFRs with constant axial velocity. The design equations for this type of PFR are directly analogous to the design equations for a constant-density batch reactor. What happens in time in the batch reactor happens in space in the PFR, and the transformation t = zfu converts one design equation to the other. For component A,... 

The design equation for a fixed-bed PFR is provided by equation 4.20, and if a differential reactor is assumed, then equation 4.21 is used ... 

The design equation for a heterogeneous catalytic reaction in an ideal PFR, in integrated form, is... 

Equation (4-41) is the design equation for a recycle reactor. This equation shows that the behavior of a recycle reactor can be varied continuously between an ideal PFR (R = 0) and an ideal CSTR (R— oo)by changing the recycle ratio. As the recycle ratio increases, the reactor behaves more like a CSTR,... 

These balances are identical to the PFR design equation for a constant-density, catalytic reaction (Eqn. (3-31)). The problem solution could have begun with these design equations, one for CO and one for H2. 

The units on A are mol/(m s). This is the convective flux. The student of mass transfer will recognize that a diffusion term like —3>Adaldz is usually included in the flux. This term is the diffusive flux and is zero for piston flow. The design equation for the variable-density, variable-cross-section PFR can be written as... 

Solution With Z>, = 0, a reaction wiU never start in a PFR, but a steady-state reaction is possible in a CSTR if the reactor is initially spiked with component B. An anal5dical solution can be found for this problem and is requested in Problem 4.12, but a numerical solution is easier. The design equations in a form suitable for the method of false transients are... 

The results of Example 5.2 apply to a reactor with a fixed reaction time, i or thatch- Equation (5.5) shows that the optimal temperature in a CSTR decreases as the mean residence time increases. This is also true for a PFR or a batch reactor. There is no interior optimum with respect to reaction time for a single, reversible reaction. When Ef < Ef, the best yield is obtained in a large reactor operating at low temperature. Obviously, the kinetic model ceases to apply when the reactants freeze. More realistically, capital and operating costs impose constraints on the design. 

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