The Macrofluid Model

The overall exit-age distribution for the system of two reactors is shown in the following figure. 

Part B What is the external-age distribution for the same CSTR followed by the same PFR  

The conceptual approach used in Part A will also be used here. 

When the CSTR is located ahead of the PFR, and a pulse of tracer is injected at r = 0, the tracer will emerge fixjm the CSTR with the exit-age distribution of an ideal CSTR. The tracer will enter the PFR as soon as it emerges from the CSTR. Each element of tracer will emerge from the PFR exactly rtime units after it enters. The effect of the PFR is to shift the exit-age distribution of the CSTR to later times by an amount x. The exit-age distribution will be exactly the same for the two configurations. 

The teachings of this simple example can be extended to any number of vessels in series. In general, E t)fora series of independent vessels will not depend on the order of the vessels. However, in Chapter 4, we learned that the final conversion from two different reactors in series may depend on the order of the two reactors. Although knowledge of the exit-age distribution is necessary to calculate the performance of a nonideal reactor, E(f) alone is not always sufficient for this purpose. 

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Estimating Reactor Performance ftom the Exit-Age Distribution— The Macrofluid Model 397... 

ESTIMATING REACTOR PERFORMANCE FROM THE EXIT-AGE DISTRIBUTION—THE MACROFLUID MODEL... 

This picture of fluid flow is referred to as the macwfluid" or segregatedflow model. It is an idealization, i.e., a hmiting case, when apphed to a gas or a low viscosity liquid, because it is difficult to imagine that there will be no exchange of mass between fluid elements. However, it can be a very reahstic model when apphed to some situations involving two-phase flow. For example, if the packets were solid particles, and the reaction took place only in the sohd phase, the macrofluid model should apply quite well. 

The macrofluid model is important because it permits reactor behavior to be estimated directly from the exit-age distribution, E(t), and the reaction kinetics. No other information is necessary. If all of the reactions taking place are first order, the macrofluid model provides an exact result. If the reactions are not first order, the macrofluid model provides a bound of reactor behavior. We shall deal with these issues a bit later, after learning how to use the macrofluid model. 

Predicting Reactor Behavior with the Macrofluid Model... 

APPROACH The conversion of Awill be predicted with the macrofluid model, EqrL (10-23). hr order to predict... 

B. If the reactor obeys the macrofluid model, what is the oudet conversion of A ... 

The macrofluid model (Eqn. (10-23)) will be solved for ca using E(t) for an ideal CSTR. 

Using the Macrofluid Model to Calculate Limits of Performance... 

For a first-order system, earliness or lateness of mixing does not affect reactor performance, for a given residence time distribution. Therefore, when the residence time distribution is known, the exact performance of a system of first-order reactions can be calculated from the macrofluid model. 

The macrofluid model represents the latest possible mixing/or a given residence time distribution. There is no mixing between fluid elements until the reaction is over, i.e., until the fluid has left the reactor. For a reaction with an effective order greater than 1, the macrofluid model represents the best possible situation. It provides an upper bound on conversion. If some mixing takes place before the fluid has left the reactor, the actual conversion will be less than predicted by the macrofluid model. 

However, the fluid leaving the first CSTR may not become mixed on a molecular level before it enters the second reactor. In the limit, the fluid elements in this stream may remain completely segregated between the two reactors. For this situation, the procedure described in the preceding paragraph is not appropriate because the stream entering the second CSTR is not uniform on a molecular level. Rather, the feed to the second CSTR consists of packets of fluid with different compositions. In this case, the macrofluid model must be used, with the measured RTD for the whole reactor, i.e., the reactor that was being modeled as a series of equal-volume CSTRs. In the case of a fluid that remains as a macrofluid, there is no need to fit the CIS model to the measured RTD. The macrofluid model must be applied directly. 

Since the effective reaction order for this example is greater than 1, we should have known a priori that the macrofluid model would predict a higher conversion than the microfluid model. Moreover, based on our discussion of CSTRs in series in Chapter 4, we should have known that N = 3 would give a higher conversion that N = 2. Therefore, we could have bracketed the above results with two calculations N = 2/microfluid (lowest conversion) and N = 3/macrofluid (highest conversion). 

If the Dispersion model cannot be used, or is not physically appropriate, the external-age distribution function E f) must be measured using tracer response techniques. Once E i) is available, the macrofluid model can be used to establish bounds on reactor performance. Moreover, the shape of the external-age distribution function can suggest various compartment models, including the CSTRs-in-series model, which can be used to explore reactor performance. 

It is useful to try to set some order in this host of models. For this purpose, the representation proposed by Spencer, Leshaw et al. (68) is helpful. It is essentially a two-environment model in which the assumption cited above Min. Mix. = macrofluid and Max. Mix. = microfluid is implicitly made. Along the axis of a small tube of the BPT model, the fluid gradually passes from a Min. Mix. state to a Max. Mix. state (Figure U). The residence time in this particular tube lies in the range tg, tg + dtg and the flow-rate is dQ = Q E(tg) dts. The flow transferred from the Entering Environment (E.E.) to the Leaving Environment (L.E.) in the interval... 

The simulations via the IEM model revealed an interesting property. A partially segregated fluid may be considered as a mixture of macrofluid (fraction 3) and microfluid (fraction 1-3). 

Figure 13. Microfluid/macrofluid volume ratio vs. reaction/diffusion time ratio. Key to curves 1 to U, simulation with the IEM model, tm = tD 1 to 3, second-order reaction k2CAo = 2(l), 5(2), 10(3) k9 second-order consecutive competing reactions = CRo, k.,/kp =...

As already mentioned above, the usual assumption is that the reactor comprises two Environments. The Entering (or Initial) Environment, which consist of a macrofluid, and the Leaving (or Final) Environment where the fluid behaves as a microfluid. The segregation function" presented above (Sec. 3-1) describes the transfer between these Environments. The first limiting case is that of Minimum Mixedness where the Entering Environment spreads out into the whole reactor. The tubes of the BPT-model remain insulated and (4-1) is still valid. The conversion for a single reaction is written... 

The competition between reaction and diffusion can be represented by the lEM-model. t is identified with a diffusion time t = yL /33 (see Sec. 3.2 above) where different diffusivities 3j and hence different micromixing times t- may be used for each species. This simple lumped parameter model gives results comparable to those of more sophisticated distributed models, at least for reaction systems which are not too "stiff". An interesting property is revealed by numerical simulations. The simplest way to represent partial segregation in a fluid is to consider that it consists of a mixture of macrofluid (fraction 3) and microfluid (fraction 1-3). It turns out that the ratio (1 - 3)/3 is always close to that of two characteristic times [28 QlSj. In the case of erosive mixing of two reactants in a CSTR (erosion controls mixing and the product of erosion is a microfluid), one finds... 

The number of CSTRs in series will be calculated fix>m the values of t and Act. The microfluid and macrofluid models will then be used to establish bounds of reactor behavior. If the number of CSTRs in series, as calculated fixjm t and Act, is not an integer, calculations will be performed for integral numbers of reactors that bracket the calculated value of N. 

Segregrated flow model The fluid in a flow reactor is assumed to behave as a macrofluid. Each clump functions as a miniature batch reactor. Mixing of molecules of different ages occurs as late as possible. 

Most of these models (except the last one) depend on a single parameter. As it is generally assumed that the E.E, consists of a macrofluid and the L.E. of a microfluid, the model also represents... 

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