Residence time distribution function defined

The cumulative residence-time distribution function F(t) is defined as the fraction of exit stream that is of age 0 to t (i.e., of age t) it is also the probability that a fluid element that entered at t = 0 has left at or by time t. Since it is defined as a fraction, it is dimensionless. Furthermore, since F(O) = 0, that is, no fluid (of age 0) leaves the vessel before time 0 and F( ) = 1, that is, all fluid leaving the vessel is of age 0 to or all fluid entering at time 0 has left by time then... 

The washout residence-time distribution function W(t) is defined as the fraction of the exit stream of age s t (and similarly for W(0)). It is also the probability that an element of fluid that entered a vessel at t = 0 has not left at time t. By comparison, F(t) (or F(6)) is the probability that a fluid element has left by time t (or 13) (Section 13.3.2.)... 

Residence time distribution functions were developed by Danckwerts [3] and are defined as external or internal RTD functions. The external RTD function /(f) is defined such that f(t)dt is the fraction of fluid exiting the system with a residence timebetween t and t + dt and the internal RTD function g(t) is defined such that g(t)dt is the fraction of the fluid in the system with a residence time between t and t + dt. 

If we accept the premise that the total strain is a key variable in the quality of laminar mixing, we are immediately faced with the problem that in most industrial mixers, and in processing equipment in general, different fluid particles experience different strains. This is true for both batch and continuous mixers. In the former, the different strain histories are due to the different paths the fluid particles follow in the mixer, whereas in a continuous mixer, superimposed on the different paths there is also a different residence time for every fluid particle in the mixer. To quantitatively describe the various strain histories, strain distribution functions (SDF) were defined (56), which are similar in concept to the residence time distribution functions discussed earlier. 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

The three ideal reactors form the building blocks for analysis of laboratory and commercial catalytic reactors. In practice, an actual flow reactor may be more complex than a CSTR or PFR. Such a reactor may be described by a residence time distribution function F(t) that gives the probability that a given fluid element has resided in the reactor for a time longer than t. The reactor is then defined further by specifying the origin of the observed residence time distribution function (e.g., axial dispersion in a tubular reactor or incomplete mixing in a tank reactor). 

At this point, the utility of this property with respect to (P2) deserves attention. A careful look at P2 reveals that the shaded region in the projected space (for example, the X -X space) is exactly the projection on the X -X space of the feasible region of P2. The concave PFR projection defines the concentrations in segregated flow, and the interior is a convex combination of all boundary points created by the residence time distribution function. This gives a new interpretation to the residence time distribution as a convex combiner. For any convex objective function to be maximized, the solution to the segregated flow model will always lead to a boundary point of the AR. 

The residence time distribution function E i) of the growing latex particles in a perfectly mixed stirred tank reactor is defined as follows ... 

One method of characterising the residence time distribution is by means of the E-curve or external-age distribution function. This defines the fraction of material in the reactor exit which has spent time between t and t -i- dt in the reactor. The response to a pulse input of tracer in the inlet flow to the reactor gives rise to an outlet response in the form of an E-curve. This is shown below in Fig. 3.20. 

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 ... 

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. 

Similarly, in continuous operations the residence times that exiting fluid elements experienced in the system are not necessarily uniform, but there is a distribution of residence times that we must take into account, and we must define residence time distribution (RTD) functions. 

Figure 8-38 shows the residence time distributions of some commercial and fixed bed reactors. These shapes can be compared with some statistical distributions, namely the Gamma (or Erlang) and the Gaussian distribution functions. However, these distributions are represented by limited parameters that define the asymmetry, the peak,... 

The last equations prove that the Markov chains [4.6] are able to predict the evolution of a system with only the data of the current state (without taking into account the system history). In this case, where the system presents perfect mixing cells, probabilities p and p j are described with the same equations as those applied to describe a unique perfectly stirred cell. Here, the exponential function of the residence time distribution (p in this case, see Section 3.3) defines the probability of exit from this cell. In addition, the computation of this probability is coupled with the knowledge of the flows conveyed between the cells. For the time interval At and for i= 1,2,3,. ..N and j = 1,2,3,..N - 1 we can write ... 

The residence time distribution (RTD), also referred to as the distribution of ages, is based on the assumption that each element traveling through the column takes a different route and will therefore have a different residence time. Different methods are developed to determine the RTD in a module or in a reactor [190]. The RTD of a chromatographic column is defined by a function E (Figure 3.20), such that E dt is the fraction of material in the exit stream with an age between t and f - - dt. The -curve lies between the extremes of plug flow and continuously stirred tank reactor. The surface below the curve between f = 0 and t = oo has to be equal to unity E t) dt = 1, because all elements that enter the module must also exit the module. 

The age of an atom or molecule in a reservoir is the time since it entered the reservoir. Age is defined for all molecules, whether they are leaving the reservoir or not. As with residence times, the probability density function of ages [ (r)] can have different shapes. In a steady-state reservoir, however, y>(r) is always a non-increasing function. The shapes of V(t) corresponding to the three residence time distributions discussed above are induded in Fig. 

Since the foundations of residence time theory are rigorously given elsewhere (5,6, 7), only those features which are essential to the present treatment will be given here. The residence time distribution (residence time frequency function exit age distribution), f(t), is defined such that f(t)dt is the fraction of fluid at any instant leaving the system, having spent time between t and t + dt within the system. The cumulative residence time distribution is... 

Thus, either pulse-response or step-function response experiments give sufficient information to permit evaluation of exit-age, internal-age, and residence-time distributions. The average age or mean residence time, which we have defined intuitively in equation (4-13), can be more precisely stated in terms of the time average of the exit-age distribution. 

This type of curve, then, has an ordinate that gives the fraction of fluid that has a certain residence time, which is plotted on the abscissa. In more formal terms, the curve defines the residence time distribution or exit age distribution. The exact definition uses the common symbol (0) for the exit age-distribution frequency function as defined by Danckwerts [6] (see Himmelblau and BischofT [4] for more details) ... 

The residence time distribution has been found (Stonestreet and van der Veeken, 1999) to be a function of the net flow Reynolds number, Re , and the oscillatory Reynolds number, Re< where they are defined as ... 

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 ... 

Extensions of residence time distributions to systems with multiple inlets and outlets have been described (27-29). If the system contains M inlets and N outlets one can define a conditional density function E. (t) as the normalized tracer impulse response in outlet j to input in inlet i as shown schematically in Figure 2. 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a dynamic model of the system. Table 1-1 lists the types of RTDs that can be measured using tracer experiments. The simplest case is a negative step change. Suppose that an inert tracer has been fed to the system for an extended period, giving Ci = Cout = Q for t < 0. At time t = 0, the tracer supply is suddenly stopped so that Cm = 0 for t > 0. Then the tracer concentration at the reactor outlet will decrease with time, eventually approaching zero as the tracer is washed out of the system. This response to a negative step change defines the washout function, W(t). The responses to other standard inputs are shown in Table 1-1. Relationships between the various functions are shown in Table 1-2. 

In studying the mixing characteristics of chemical reactors, a sharp pulse of a nonreacting tracer is injected into the reactor at time t = 0. The concentration of material in the effluent from the reactor is measured as a function of time c(f). The residence time distribution (RTD) function for the reactor is defined as... 

---