Other Residence Time Distribution Functions

Sometimes it is not convenient, or even possible, to inject a sharp pulse of trac right at the inlet to a vessel. An alternative approach is to use a step input of tracer. For example, consider a vessel with a constant-density fluid flowing through it at steady state. There is no traco- at all in the fluid entering the vessel. Then, at some time designated r = 0, the concentration of tracer in the feed is abmptly changed to a value of Co and is maintained at this concentration. 

The concentration of tracer in the effluent stream is measured continuously. If we wait long enough, the effluent tracer concentration will be Co- However, a good deal of information can be obtained from the measured tracer concentration during the period between r = 0 and the time required for the effluent tracer concentration to approach Cq. This type of step input experiment is illustrated in the following figure. 

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The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

In a laminar flow reactor (LFR), we assume that one-dimensional laminar flow (LF) prevails there is no mixing in the (axial) direction of flow (a characteristic of tubular flow) and also no mixing in the radial direction in a cylindrical vessel. We assume LF exists between the inlet and outlet of such a vessel, which is otherwise a closed vessel (Section 13.2.4). These and other features of LF are described in Section 2.5, and illustrated in Figure 2.5. The residence-time distribution functions E(B) and F(B) for LF are derived in Section 13.4.3, and the results are summarized in Table 13.2. 

The degree of conversion inside this volume is constant, but the MWD function qw(n, r), where n is the degree of polymerization, depends on r. This is a reflection of different reaction time in the various layers of the polymer. The residence time distribution function f(r) for the reactive mass in a reactor is determined from rheokinetic considerations, while the MWD for each microvolume qw(n,t) is found for various times t from purely kinetic arguments. The values t and r in the expressions for qw are related to each other via the radial distribution of axial velocity. 

On the other hand, one of the essential conditions for correct measurement of RTD is that the tracer particles have the properties, including RTD characteristics, very close to those of the process particles. This implies that, in the time domain of t > 0, the residence time distribution functions of particles A and B in the same device should be approximately equal to each other, i.e. 

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. 

The movement of the particles in this stage is very complex and extremely random, so that to determine accurately the residence time distribution and the mean residence time is difficult, whether by theoretical analysis or experimental measurement. On the other hand, the residence time distribution in this stage is unimportant because this subspace is essentially inert for heat and mass transfer. Considering the presence of significant back-mixing, the flow of the particles in this stage is assumed also to be in perfect mixing, as a first-order approximation, and thus the residence time distribution probability density function is of the form below ... 

It is well known that the measurement of residence time distribution usually employs the dynamic method [54], the so-called input-response technique. However, for measuring RTD of solid particles the input signal is a difficult and troublesome problem. The author of the present book employs an arbitrary known function as the input signal so that this problem is solved. This procedure is also applicable, in principle, to the measurements of RTD of solid materials in other devices. 

In a conventional design, each unit operation is modeled using terms and values which do not depend on the location inside the device. On the one hand, convection and diffusion in a micro structured device strongly influence the functioning of the device, and on the other hand the convection and diffusion conditions are affected by the shape of the device. To obtain an optimized fluidic micro device, some constraints on the shape of the device are necessary. These are constraints, e.g., on the average residence time, the residence time distribution and the temperature distribution [13]. 

It is interesting to note that the residence time distribution may also be used as a generating function for the tracer exchange curve as introduced in Sect. 3.2. The relative amount of tracer exchange at a certain time t is simply the sum over all molecules which have entered and remained in the system between time zero and the present instant, or, in other words, the residence... 

The residence time distribution (RTD) is a probability distribution function used to characterize the time of contact and contacting pattern (such as for plug-flow or complete backmixing) within the reactors. Excessive retention of some elements and shortdrcuiting of others due to backmixing and other dispersive phenomena lead to a broad distribution in the residence times of individual molecules in the reactor. This tends to decrease conversion and exerts a negative influence on product selectivity/yield. The RTD depends on the flow regime and is characterized by Reynolds (Re) and Schmidt (Sc) numbers. 

There is another practical method for estimating conversions in reactors with residence time distribution, for perfect micro-mixing, that is also applicable to other reaction orders. To this end the reactor is simulated by a model that consists of a cascade on N perfectly mixed equal reactors (section 3.3.3). The RTD-function of the cascade with total residence time x can be calculated ... 

In continuous operation mode, both feed and effluent streams flow continuously. The main characteristic of a continuous stirred tank reactor (CSTR) is the broad residence time distribution (RTD), which is characterized by a decreasing exponential function. The same behavior describes the age of the particles in the reactor and hence the particle size distribution (PSD) at the exit. Therefore, it is not possible to obtain narrow monodisperse latexes using a single CSTR. In addition, CSTRs are hable to suffer intermittent nucleations [89, 90) that lead to multimodal PSDs. This may be alleviated by using a tubular reactor before the CSTR, in which polymer particles are formed in a smooth way [91]. On the other hand, the copolymer composition is quite constant, even though it is different from that of the feed. 

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. 

As an idealization of the classified-fines removal operation, assume that two streams are withdrawn from the crystallizer, one corresponding to the product stream and the other a fines removal stream. Such an arrangement is shown schematically in Figure 14. The flow rate of the clear solution in the product stream is designated and the flow rate of the clear solution in the fines removal stream is set as (R — 1) - Furthermore, assume that the device used to separate fines from larger crystals functions so that only crystals below an arbitrary size are in the fines removal stream and that all crystals below size have an equal probabiHty of being removed in the fines removal stream. Under these conditions, the crystal size distribution is characterized by two mean residence times, one for the fines and the other for crystals larger than These quantities are related by the equations... 

As with HjS, the distribution of sulfur among the other FCC products depends on several factors, which include feed, catalyst type, conversion, and operating conditions. Feed type and residence time are the most significant variables. Sulfur distribution in FCC products of several feedstocks is shown in Table 2-4. Figure 2-9 illustrates the sulfur distribution as a function of the unit conversion. 

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