BR and PFR

The performances of a BR and of a PFR may be compared in various ways, and are similar in many respects, as discussed in Section 15.2.2.1, since an element of fluid, of arbitrary size, acts as a closed system (i.e., a batch) in moving through a PFR. The residence time in a PFR, the same for all elements of fluid, corresponds to the reaction time in a BR, which is also the same for all elements of fluid. Depending on conditions, these quantities, and other performance characteristics, may be the same or different. 

For constant-density and isothermal operation, the performance characteristics are 

For an isothermal first-order reaction taking place in a constant-volume BR but at varying density in a PFR, it can be shown that the times are also equal this is not the case for other orders of reaction (see problem 17-8). 

---

The selectivity of R will be high if [A] and [5] are high, and [B] is low at high [/ ]. Since this pattern holds for BR and PFR, these are the preferred reactors for maximizing R. However, for an MRF, [B] within the reactor is uniformly high, being equal always to the exit concentration. Hence the first (desired) reaction is not favored, resulting in a lower selectivity for R than in a PFR or BR. 

We derived above the performance equations for the three ideal reactors batch, plug flow, and mixed flow. The BR and PFR are exactly comparable, with the reaaion time t in BR related to the residence time T at the corresponding axial position in PFR by... 

For BRs and PFRs, the momentaneous and integral yields naturally take different forms, since the reaction rates vary as a function of time or inside the reactor. By setting up the mass balances. 

The molar mass balances for a BR and PFR converge into similar mathematical expressions and can thus be treated simultaneously. The mass balances assume the following form (Equation 3.235) ... 

The expressions above, Equations 3.272 and 3.273, thus give the relationship between concentrations cr and ca, for BRs and PFRs. The reaction stoichiometry of the reactions implies that... 

FIGURE 3.20 A consecutive reaction A R — (left) and in BR and PFRs, respectively (right). 

Runaway criteria developed for plug-flow tubular reactors, which are mathematically isomorphic with batch reactors with a constant coolant temperature, are also included in the tables. They can be considered conservative criteria for batch reactors, which can be operated safer due to manipulation of the coolant temperature. Balakotaiah et al. (1995) showed that in practice safe and runaway regions overlap for the three types of reactors for homogeneous reactions (1) batch reactor (BR), and, equivalently, plug-flow reactor (PFR), (2) CSTR, and (3) continuously operated bubble column reactor (BCR). 

The calculation of time quantities half-life (r1/2) in a BR and a CSTR (constant density), problem 2-1 calculation of residence time t for variable density in a PFR (Example 2-3 and problem 2-5). 

The integrated form for constant density (Example 3-4), applicable to both a BR and a PFR, showing the exponential decay of cA with respect to t (equation 3.4-10), or, alternatively, the linearity of In cA with respect to t (equation 3.4-11). 

A second-order reaction may typically involve one reactant (A -> products, ( -rA) = kAc ) or two reactants ( pa A + vb B - products, ( rA) = kAcAcB). For one reactant, the integrated form for constant density, applicable to a BR or a PFR, is contained in equation 3.4-9, with n = 2. In contrast to a first-order reaction, the half-life of a reactant, f1/2 from equation 3.4-16, is proportional to cA (if there are two reactants, both ty2 and fractional conversion refer to the limiting reactant). For two reactants, the integrated form for constant density, applicable to a BR and a PFR, is given by equation 3.4-13 (see Example 3-5). In this case, the reaction stoichiometry must be taken into account in relating concentrations, or in switching rate or rate constant from one reactant to the other. 

We do this for isothermal constant-density conditions first in a BR or PFR, and then in a CSTR. The reaction conditions are normalized by means of a dimensionless reaction number MAn defined by... 

In Figure 4.4, similar to Figure 4.3, cA/cAo is plotted as a function of MAn. The behavior is similar in both figures, but the values of cA/cAo for a CSTR are higher than those for a BR or PFR (except for n = 0, where they are the same). This is an important characteristic in comparing these types of reactors (Chapter 17). Another difference is that cAlcAo approaches 0 asymptotically for all values of n > 0, and not just for n 1, as in Figure 4.3. 

In the examples in Sections 7.1 and 7.2.1, explicit analytical expressions for rate laws are obtained from proposed mechanisms (except branched-chain mechanisms), with the aid of the SSH applied to reactive intermediates. In a particular case, a rate law obtained in this way can be used, if the Arrhenius parameters are known, to simulate or model the reaction in a specified reactor context. For example, it can be used to determine the concentration-(residence) time profiles for the various species in a BR or PFR, and hence the product distribution. It may be necessary to use a computer-implemented numerical procedure for integration of the resulting differential equations. The software package E-Z Solve can be used for this purpose. 

From the discussion in Section 15.2.2.1 comparing the performance of a BR and a PFR for a constant-density system, it follows that... 

We focus attention in this chapter on simple, isothermal reacting systems, and on the four types BR, CSTR, PFR, and LFR for single-vessel comparisons, and on CSTR and PFR models for multiple-vessel configurations in flow systems. We use residence-time-distribution (RTD) analysis in some of the multiple-vessel situations, to illustrate some aspects of both performance and mixing. 

Equation 17.1-16 can be obtained from equation 16.2-13, together with the result for cA/cAo f°r a BR (or PFR) obtained by integration of, for example, equation 15.2-16. Note that the latter places restrictions on the values for MA1I2, as noted in equation 17.1-13, and this has an implication for the allowable upper limit in equation 16.2-13. 

For a gas-phase reaction represented by A - B + C carried out (separately) isotheimally in a constant-volume BR, and isothermally and isobarically in a PFR, show that t r and tpF, for a feed of pure A,... 

Some aspects of reactor behavior are developed in Chapter 5, particularly concentration-time profiles in a BR in connection with the determination of values of and k2 from experimental data. It is shown (see Figure 5.4) that the concentration of the intermediate, cB, goes through a maximum, whereas cA and cc continuously decrease and increase, respectively. We extend the treatment here to other considerations and other types of ideal reactors. For simplicity, we assume constant density and isothermal operation. The former means that the results for a BR and a PFR are equivalent. For flow reactors, we further assume steady-state operation. 

As discussed in Sect. 2.1, physical and mathematical models of ideal chemical reactors are based on two very simplified fluid dynamic assumptions, namely perfect mixing (BR and CSTR) and perfect immiscibility (PFR). On the contrary, in real tank reactors the stirring system produces a complex motion field made out of vortices of different dimensions interacting with the reactor walls and the internal baffles, as schematically shown in Fig. 7.2(a). As a consequence, a complex field of composition and temperature is established inside the reactor. 

Table 7-11 summarizes laboratory reactor types that approach the three ideal concepts BR, CSTR and PFR, classified according to reaction types. 

In this section we focus on the three main types of ideal reactors BR, CSTR, and PFR. Laboratory data are usually in the form of concentrations or partial pressures versus batch time (batch reactors), concentrations or partial pressures versus distance from reactor inlet or residence time (PFR), or rates versus residence time (CSTR). Rates can also be calculated from batch and PFR data by differentiating the concentration versus time or distance data, usually by numerical curve fitting first. It follows that a general classification of experimental methods is based on whether the data measure rates directly (differential or direct method) or indirectly (integral of indirect method). Table 7-13 shows the pros and cons of these methods. 

Integral Data Analysis Integral data such as from batch and PFR relate concentration to time or distance. Integration of the BR equation for an nth-order homogeneous constant-volume reaction yields... 

It is these same requirements that have to be met in all reactors designed for kinetic studies, with the added problem of finding an appropriate definition of space time so that the results obtained in flow reactors designed for kinetic studies can be compared to those from a BR. These requirements exclude most reactor configurations from use in kinetic studies and leave us with the fundamental trio the BR, the PFR and the CSTR. Other configurations can yield reliably reproducible data but fall short in one way or an other when used for kinetic studies. 

As previously noted, the raw data collected in a kinetics experiment consists of the time, the temperature and the composition of the output, usually in mol fractions. The mol fractions at the outlet do not correspond to fractional yields or conversions. They must be converted to fractional conversions or to concentrations before the data is used for fitting in rate expressions. By plotting the composition at the output in terms of mol fraction converted or fractional yield of products (or the corresponding concentrations) against time, we obtain a figure that, in the case of the BR and the PFR, will let us calculate rates of reaction. To make the data in this plot compatible for purposes of conventional data analysis we keep the third variable, temperature, constant. 

As it is with all TS techniques, rate data acquisition using a TS-PFR can be very fast and a vast amount of data can be obtained from a single experiment. Unfortunately, in the TS-PFR rates cannot be calculated in real time, as was the case with the TS-BR and will be the case with the TS-CSTR. Instead, rate data is obtained after all the readings from the several runs of a TS-PFR experiment are available. This post-experiment processing of the raw data will, however, produce the same X-r-T triplets as we discussed above. The triplets will be available over the whole range of X-T conditions covered by the experiment, just as they were for the TS-BR. 

---