1
Nonideal Flow

1.1 Introduction

Basic chemical reactors (plug flow reactor or PFR, andcontinuously stirred tank reactor or CSTR) are studied considering their behavior is that of an ideal reactor. Unfortunately, in practice, we often find behaviors that are far from that considered ideal. Consequently, working with them, the chemical engineer must be able to handle and diagnose the behavior of these reactors. At the time of describing the nonideal behavior of a reactor, three concepts are introduced: the residence time distributions (RTDs), the quality of the mixture (not discussed in this book), and the models that can be used to describe the reactor. These three concepts are used to describe the deviations of the mixing assumed in the ideal models and are considered as attributes of the mixture in nonideal reactors.

One way of approaching the study of nonideal reactors is to consider them, in a first approximation, as if the flow model were the one corresponding to a CSTR or a PFR. However, in real reactors, the nonideal flow model implies a minor conversion, so a method that allows for this conversion loss to be considered must be available. Therefore, a higher level of approximation implies the use of information about the RTD.

1.2 Residence Time Distribution (RTD) Function

The idea of introducing the RTD in the analysis of the behavior of reactors occurred thanks to MacMullin and Weber (in 1935), although it was Danckwerts (later, in 1953) who structured this analysis and defined most of the distributions of interest.

In an ideal PFR, all the particles (or units) of material that leave the reactor have remained in it the same time. Analogously, in an ideal (well-mixed) batch reactor, all particles are in the reactor the same period of time. The time that these units have remained in the reactor is what we call the residence time of those particles in that reactor.

The ideal reactor's PFR and batch are the only ones in which all the portions of reactants present in the reactor have the same residence time. In all other reactors, the particles entering the reactor vessel remain inside the reactor for different periods of time; that is, there exists a RTD inside the reactor.

For example, consider an ideal CSTR; the input flow that is introduced to the reactor at a given moment mixes instantaneously and completely with the rest of the material that already exists inside the reactor. In this way, some of the particles that enter the reactor abandon this one almost immediately with the exit current, whereas other atoms remain of almost indefinite form, since all the material is never dragged. Of course, many of the particles leave the reactor after a period close to the average residence time.

The RTD of a reactor is a feature of the mixture that is taking place inside the reactor. Thus, in an ideal PFR there is no axial mixing, and this absence is reflected in the RTD that this type of reactors exhibit. In contrast, in an ideal CSTR there is a great degree of mixing, so the RTD that these reactors exhibit is very different from that of the plug flow. However, not all RTDs are unique to one type of reactor; reactors with marked differences can give identical RTDs. Despite this, the RTD of a certain reactor presents distinctive keys with respect to the type of mixture that is taking place inside it and is one of the ways to characterize the reactor that provides more information.

1.2.1 Measurement of the RTD

The experimental measurement of the RTD is done by injecting a tracer into the reactor at a definite time (t = 0). This tracer is a chemical, molecule, or an inert atom. The tracer concentration is then measured at the outlet at different times. The tracer must be inert, easily detectable, with physical properties similar to those of the substances present in the reaction mixture and easily soluble in it. In addition, it should not be adsorbed on walls or other surfaces on the reactor. The objective is to reflect, as best as possible, the behavior of the substances that are flowing through the reactor. The most commonly used tracers are dyes and radioactive material, while the most commonly used injection methods are pulse input ("Dirac delta function") and step input (suddenly increase of the tracer concentration).

1.2.1.1 Pulse Input

This kind of experiment consists of introducing into the current entering the reactor, quickly and at once, an amount N0 of tracer. The output concentration is subsequently measured as a function of time. Concentration-time curve characteristics for the input and exit of an arbitrary reactor can be observed in Figure 1.1. The concentration-time curve corresponding to the effluent is called curve C in RTD analysis.

Figure 1.1 Measurement of the RTD.

Let us consider a system with a single input and a single output, in which a tracer is injected in pulse and in which it is transported, exclusively because of the flow (not because of dispersion), through the system. If a time increment ?t small enough is chosen so that the concentration of tracer, C(t), which leaves the system between the instants "t" and "t?+??t" is constant, we can express the amount of tracer (in moles or grams), ?M, that leaves the reactor between "t" and "t?+??t" as

That is, ?M is the amount of tracer (in moles or grams, for example) that has remained in the reactor for a time interval comprising between "t" and "t?+??t." Dividing by the total amount of tracer injected into the reactor (M0), we obtain the tracer fraction whose residence time in the reactor is between "t" and "t?+??t":

For a pulse injection, we define the RTD function, E(t), as

This expression describes in a quantitative way how long the different elements of fluid have passed inside the reactor. In consequence:

If? M0 is not directly known, it may be obtained from the output concentrations, adding the different amounts of the tracer that have exited the reactor between 0 and infinity. Expressing in differential form:

and integrating, we obtain

Since the volumetric flow (Q) is generally constant, we will define E(t) as

In that expression, the integral in the denominator is the area under the curve, C(t). In this way, from the tracer concentration, C(t), it is possible to find the curve E(t), as long as that curve is obtained from a perfect pulse of the input tracer.

Another way to interpret the function of residence times is in its integral form:

Now, the fraction of tracer that passed inside the reactor a time t, between 0 and infinity is equal to 1; therefore:

The main drawback of the use of the pulse technique lies in the difficulty in achieving a tracer input to the reactor that is reasonably pulsed (as we explain in Chapter 2, deconvolution of curves). The injection should take place in a very short period compared to the residence times and the tracer dispersion between the injection point and the reactor inlet should be negligible. If these conditions are satisfied, the technique is a simple and direct way to obtain the RTD.

1.2.1.2 Step Input

During a step input experiment, a tracer is added at a steady rate to the reactor feed to give a steady input concentration of C0 (see Figure 1.1). The concentration of the tracer in the effluent is then monitored from the time of adding the tracer until it reaches a concentration approximating that of C0.

As stated before, the RTD curve can be easily obtained by injecting a tracer in a pulse input, but now we will formulate a more general relationship between a tracer injection (not necessarily a pulse input) and the corresponding tracer concentration in the effluent (current leaving the system).

Chapter 2 presents a more general equation establishing the relation of the concentration of a tracer leaving a reactor (Cout) and the input concentration (Cin). This is presented as the convolution integral:

In this equation, E(t) is the corresponding RTD for the reactor. The concentration at the input is evaluated at different times and the resulting signal Cout is a convolution of Cin and the RTD in the vessel.

In this way, in the following chapters we are able to calculate the E(t) curve given any function of concentration input Cin and measuring Cout. For now, we continue with more simple techniques, considering that the inlet concentration is introduced in the form of a step input (Figure 1.1).

Let us consider a system of constant volumetric flow (Q) in which a tracer input is introduced in the form of a step, with a rate of addition of tracer to the constant input flow and which begins to occur at time...