Real reactors

How can we use age distribution information to predict the influence of such deviations on actual reactor performance  

As will be seen, a number of alternative approaches have been developed which provide a considerable degree of flexibility in the modeling of nonideal reactors. 

One of these approaches is to use the exit-age distribution directly. For ideal reactors this will allow determination of limits between which the actual conversion must lie. These two limits are, of course, those of complete segregation and maximum mixedness as described in Chapter 4. For nonideal reactors one follows the same procedure employing the experimentally determined exit-age distribution. 

A second approach is to use various combinations of the ideal reactor models in simulation of nonideal behavior. This may seem a bit contradictory at first, but hopefully our later discussion will be sufficient to illuminate the reasoning behind this method. A hint of this approach is given by the discussion in Chapter 4 on the comparisons between the conversions in PFR and CSTR sequences of the same total 

The concepts involved in the development and appHcation of these differing approaches are themselves quite different, and their successful application may well vary from case to case. To understand this better, let us first examine some of the kinds of deviation commonly encountered from ideal flow. 

---

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with re.sidence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

Real reactors may conform to some sort of ideal mixing patterns, or their performance may be simulated by combinations of ideal models. The commonest ideal models are the following ... 

In general, the optimum conditions cannot be precisely attained in real reactors. Therefore, the selection of the reactor type is made to approximate the optimum conditions as closely as possible. For this purpose, mathematical models of the process in several different types of reactors are derived. The optimum condition for selected parameters (e.g., temperature profile) is then compared with those obtained from the mathematical expressions for different reactors. Consequently, selection is based on the reactor type that most closely approaches the optimum. 

Since it is a conceptual study employing a theoretical reactor model, it is also important to appreciate the limits of this type of investigation. The advantage of the computer investigation over a pilot or production reactor investigation is the obvious cost and time saving over the real reactor experiment. 

The computer investigation can also yield a more definable relationship with fewer parameter excursions since the output will be free of scatter. In addition, excursions in reactor parameters can be taken which might be considered unsafe on or beyond the equipment limitations of an existing real reactor. 

The pitfalls of a computer model are obvious in that it is only a conceptual representation of the reactor and includes only as many aspects of the real reactor as present knowledge permits. In addition, even the most perfectly conceived description will still depend upon the accuracy of the physically measured constants used in the model for the quality of the process representation. The goal of this report is, however, only to show conceptual trends and the technological base is developed to the extent that the conceptual trends will be correct. In some respects the computer model is a better process development tool than the pilot plant used for the LDPE process since the pilot reactor does not yield directly scaleable information. The reader should take care to direct his attention to the trend information and conceptual differences developed in this work very little attention should be paid to the absolute values of the parameters given. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

The emphasis in this chapter is on the generalization of piston flow to situations other than constant velocity down the tube. Real reactors can closely approximate piston flow reactors, yet they show many complications compared with the constant-density and constant-cross-section case considered in Chapter 1. Gas-phase tubular reactors may have appreciable density differences between the inlet and outlet. The mass density and thus the velocity down the tube can vary at constant pressure if there is a change in the number of moles upon reaction, but the pressure drop due to skin friction usually causes a larger change in the density and velocity of the gas. Reactors are sometimes designed to have variable cross sections, and this too will change the density and velocity. Despite these complications, piston flow reactors remain closely akin to batch reactors. There is a one-to-one correspondence between time in a batch and position in a tube, but the relationship is no longer as simple as z = ut. 

The thoughtful reader may wonder about a real reactor with a high level of radial diffusion. Won t there necessarily be a high level of axial diffusion as well and won t the limit of oo really correspond to a CSTR rather than... 

The ideal flow reactors are the CSTR and the PFR. (This chapter later introduces a third kind of ideal reactor, the segregated CSTR, but it has the same distribution of residence times as the regular, perfectly mixed CSTR.) Real reactors sometimes resemble these ideal types or they can be assembled from combinations of the ideal types. 

Real reactors can have 0 < cr < 1, and a model that reflects this possibility consists of a stirred tank in series with a piston flow reactor as indicated in Figure 15.1(a). Other than the mean residence time itself, the model contains only one adjustable parameter. This parameter is called the fractional tubularity, Xp, and is the fraction of the system volume that is occupied by the piston flow element. Figure 15.1(b) shows the washout function for the fractional tubularity model. Its equation is... 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

Figure 8 shows the variation of the manipulated variables corresponding to the control problem of Figure 7. Except for some rather abrupt changes in I. and M. at the start of the operation, the functions are smooth and should not be difficult to produce in a real reactor system.

This work covers only open loop response. Use of this algorithm alone on a real reactor would presuppose that the model is very precise — that a desired MW and S can be obtained merely by... 

The available models mostly refer to ideal reactors, STR, CSTR, continuous PFR. The extension of these models to real reactors should take into account the hydrodynamics of the vessel, expressed in terms of residence time distribution and mixing state. The deviation of the real behavior from the ideal reactors may strongly affect the performance of the process. Liquid bypass - which is likely to occur in fluidized beds or unevenly packed beds - and reactor dead zones - due to local clogging or non-uniform liquid distribution - may be responsible for the drastic reduction of the expected conversion. The reader may refer to chemical reactor engineering textbooks [51, 57] for additional details. 

Computer software allows the solution of more complex problems that require numerical, as opposed to analytical, techniques. Thus, a student can explore situations that more closely approximate real reactor designs and operating conditions. This includes studying the sensitivity of a calculated result to changing operating conditions. 

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

In addition, the PFR model assumes that mixing between fluid elements at the same axial location is infinitely fast. In CRE parlance, all fluid elements are said to be well micromixed. In a tubular reactor, this assumption implies that the inlet concentrations are uniform over the cross-section of the reactor. However, in real reactors, the inlet streams are often segregated (non-premixed) at the inlet, and a finite time is required as they move down the reactor before they become well micromixed. The PFR model can be easily... 

If ki is less temperature-dependent that 2, the optimum temperature profile is one that starts off at a high temperature to get the first reaction going but then drops to prevent the loss of too much B. Figure 3.10 sketches typical optimum temperature and concentration profiles. Also shown in Fig. 3.10 as the dashed line is an example of an actual temperature that could be achieved in a real reactor. The reaction mass must be heated up to 7. We will use the optimum temperature profile as the setpoint signal. 

Because the expert system was not connected to a real reactor, we built a small table-driven simulation to model the growth of cells in suspension. The graphical interface includes images representing the reactor itself, several feed bins and associated valves. Also shown in Figure 1 are several types of gauges, including a strip chart, monitors of various states and alarm conditions, temperature, and the on/off state of heaters and coolers. 

In any real reactor, the flow will not follow the plug-flow pattern precisely. Non-ideal flow in chemical reactors is the subject of Chap. 6 where the various models used to predict the performance of industrial reactors are discussed at some length. 

Chapter 1 reviews the concepts necessary for treating the problems associated with the design of industrial reactions. These include the essentials of kinetics, thermodynamics, and basic mass, heat and momentum transfer. Ideal reactor types are treated in Chapter 2 and the most important of these are the batch reactor, the tubular reactor and the continuous stirred tank. Reactor stability is considered. Chapter 3 describes the effect of complex homogeneous kinetics on reactor performance. The special case of gas—solid reactions is discussed in Chapter 4 and Chapter 5 deals with other heterogeneous systems namely those involving gas—liquid, liquid—solid and liquid—liquid interfaces. Finally, Chapter 6 considers how real reactors may differ from the ideal reactors considered in earlier chapters. 

Ideal reactors have three ideal flow or contacting patterns. We show these in Fig. 2.1, and we very often try to make real reactors approach these ideals as closely as possible. 

We particularly like these three flow or reacting patterns because they are easy to treat (it is simple to find their performance equations) and because one of them often is the best pattern possible (it will give the most of whatever it is we want). Later we will consider recycle reactors, staged reactors, and other flow pattern combinations, as well as deviations of real reactors from these ideals. 

These three ideals are relatively easy to treat. In addition, one or other usually represents the best way of contacting the reactants—no matter what the operation. For these reasons, we often try to design real reactors so that their flows approach these ideals, and much of the development in this book centers about them. 

Find the fraction of reactant unconverted in the real reactor and compare this with the fraction unconverted in a plug flow reactor of the same size. 

Equation 20 with Eq. 5.17 compares the performance of real reactors which are close to plug flow with plug flow reactors. Thus the size ratio needed for identical conversion is given by... 

Conversion in the real reactor is found from Fig. 13.19. Thus moving along the kr = (0.307)(15) = 4.6 line from C/Cq = 0.01 to DluL = 0.12, we find that the fraction of reactant unconverted is approximately... 

The dispersion model has the advantage in that all correlations for flow in real reactors invariably use that model. On the other hand the tanks-in-series model is simple, can be used with any kinetics, and it can be extended without too much difficulty to any arrangement of compartments, with or without recycle. 

There are many ways that two phases can be contacted, and for each the design equation will be unique. Design equations for these ideal flow patterns may be developed without too much difficulty. However, when real flow deviates considerably from these, we can do one of two things we may develop models to mirror actual flow closely, or we may calculate performance with ideal patterns which bracket actual flow. Fortunately, most real reactors for heterogeneous systems can be satisfactorily approximated by one of the five ideal flow patterns of Fig. 17.1. Notable exceptions are the reactions which take place in fluidized beds. There special models must be developed. 

---