Age Distribution of Fluid, the RTD

It is evident that elements of fluid taking different routes through the reactor may take different lengths of time to pass through the vessel. The distribution of these times for the stream of fluid leaving the vessel is called the exit age distribution E, or the residence time distribution RTD of fluid. E has the units of time  

We find it convenient to represent the RTD in such a way that the area under the curve is unity, or 

This procedure is called normalizing the distribution, and Fig. 11.6 shows this. 

We should note one restriction on the E curve—that the fluid only enters and only leaves the vessel one time. This means that there should be no flow or diffusion or upflow eddies at the entrance or at the vessel exit. We call this the closed vessel boundary condition. Where elements of fluid can cross the vessel boundary more than one time we call this the open vessel boundary condition. 

With this representation the fraction of exit stream of age between t and t + dt is 

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An alternative method to RTD theory for treating non-ideal reactors is the use of zone models. In this approach, the reactor volume is broken down into well mixed zones (see the example in Fig. 1.5). Unlike RTD theory, zone models employ an Eulerian framework that ignores the age distribution of fluid elements inside each zone. Thus, zone models ignore micromixing, but provide a model for macromixing or large-scale inhomogeneity inside the reactor. 

Each flow pattern of fluid through a vessel has associated with it a definite clearly defined residence time distribution (RTD), or exit age distribution function E. The converse is not true, however. Each RTD does not define a specific flow pattern hence, a number of flow patterns—some with earlier mixing, others with later mixing of fluids—may be able to give the same RTD. 

In constructing a flow model for a given reactor, we must know the flow pattern through the reactor. This can be conveniently achieved by determining the age distribution of the elements of the fluid in the exit stream or the RTD within the reactor. 

RTD methods are based on the concept of age distribution functions and make use of the experimentally measured or calculated residence time distribution of fluid elements in a reactor vessel (Figure 12.3-1, C and D). A Lagrangian perspective is taken and the age of a fluid element is defined as the time elapsed since it entered the reactor. In what follows, steady state operation of a vessel fed with a volumetric flow rate F is considered. A residence time distribution (RTD) experiment can be performed with inert tracers, such that at an instant of time all fluid elements entering a reactor or process vessel are marked. The injection of an impulse of tracer into the vessel at time zero can be mathematically represented by means of the Dirac delta function or perfect unit impulse function ... 

Next to the RTD, another age distribution of interest in some applications is the internal age distribution, I. I a) da is the fraction of the fluid inside the vessel with age in the range a, a + da), where age is the length of time that a fluid element has been in the vessel. I a) has properties similar to... 

The exit-age distribution function is a measure of the distribution of the ages of fluid elements leaving a vessel, and hence is an RTD function. As a function of time, f, it is defined as ... 

In the CRE literature, the residence time distribution (RTD) has been shown to be a powerful tool for handling isothermal first-order reactions in arbitrary reactor geometries. (See Nauman and Buffham (1983) for a detailed introduction to RTD theory.) The basic ideas behind RTD theory can be most easily understood in a Lagrangian framework. The residence time of a fluid element is defined to be its age a as it leaves the reactor. Thus, in a PFR, the RTD function E(a) has the simple form of a delta function ... 

When a steady stream of fluid flows through a vessel, different elements of the fluid spend different amounts of time within it. This distribution of residence times is denoted by a curve which represents, at any given time, the amount of fluid with ages between t and t- -dt flowing out in the exit stream. When normalized with respect to the total flow (i.e., expressed as fraction of the total flow), this distribution, known as the residence time distribution (RTD), satisfies the condition... 

The function E(t) is defined as function of RTD which represents the age distribution function of molecules in the fluid element derived from the cumulative distribution function. 

The function fit) is called the residence time distribution (RTD). It is denoted by a curve that represents, at any given time, the amount of fluid with ages between t and t + dt flowing out in the exit stream. The time... 

It is often convenient to use a dimensionless time 6 = 6 I x and a corresponding version of the RTD, E 6). The relationship between E 6) and E 6) follows from both distributions representing the same physical entity, i.e., the fraction of fluid leaving the vessel with age 6 ... 

The age a of a fluid (or solid) particle is defined as the time elapsed since it entered the reactor. Its life expectancy or residual lifetime X is the time it has still to spend before leaving the reactor. The residence time t = a + X is the age of the leaving particle in the outlet section. As the reactor contains particles with different histories, all these quantities are distributed. E(tg) is the RTD such that the fraction of leaving flow-rate Q containing particles with a residence time between tg and... 

Macromixing is the process whereby parts of a fluid having different histories come into contact and mix-up on a macroscopic scale. It is thus a consequence of the macroscopic hydrodynamic pattern. The development of powerful computer codes make it possible now to determine the average velocity pattern in any kind of equipment, at least in single phase newtonian fluids. This probably won t eliminate methods based on the characterization of complex flow by internal age distributions and residence time distributions (RTD), which can be determined with tracers. Such methods have proven their efficiency for building up flow models well adapted to scale up and to calculation of chemical conversion -... 

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