Residence-time distribution definition

RESIDENCE TIME DISTRIBUTIONS DEFINITIONS AND EXPERIMENTAL DETERMINATION FROM TRACER RESPONSES... 

Nonreacdive substances that can be used in small concentrations and that can easily be detected by analysis are the most useful tracers. When making a test, tracer is injected at the inlet of the vessel along with the normal charge of process or carrier fluid, according to some definite time sequence. The progress of both the inlet and outlet concentrations with time is noted. Those data are converted to a residence time distribution (RTD) that tells how much time each fracdion of the charge spends in the vessel. 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

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 design of the jet spouted bed requires the rigorous definition of the gas flow pattern in order for the residence time distribution to be considered. In previous papers, the regime of jet spouted bed and its hydrodynamics correlations have been defined [2-8]. The minimum jet spouting velocity is calculated by the following correlation [7]. 

At the instant r, the residence times of all the particles of the tracer A must be < f while the particles of B consist of two groups in the first group all the particles are fed at instant f0 = 0 or later and so also have residence times < t while the particles in the second group were fed into the device before tQ = 0 and so have residence times > t. If the particles in the first group are denoted by the superscript then, from the definition of the residence time distribution function, F, we have... 

According to the definition of the F-function, the residence time distribution functions of A and B for the case under consideration can be expressed by the corresponding ratios of the amounts of particles coming out from the device to those inputted in the time interval from 0 to t, i.e. 

A characteristic of micro channel reactors is their narrow residence-time distribution. This is important, for example, to obtain clean products. This property is not imaginable without the influence of dispersion. Just considering the laminar flow would deliver an extremely wide residence-time distribution. The near wall flow is close to stagnation because a fluid element at the wall of the channel is, by definition, fixed to the wall for an endlessly long time, in contrast to the fast core flow. The phenomenon that prevents such a behavior is the known dispersion effect and is demonstrated in Figure 3.88. 

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... 

It is possible to determine the cumulative residence time distribution function F(t) from either a tracer step-change or a tracer impulse response. From its definition, the properties of F(t) are ... 

The definition of symbols is in the Table of Nomenclature. Basically SD is a number proportional to the reactor length, made dimensionless by a proper combination of thermal and reaction kinetic paramters. t is proportional to the temperature rise, made dimensionless by a combination of inlet temperature and activation energy, y and a2 are the mean and variance, respectively, of the residence-time distribution in the reactor. 

The opposite of the large diameter pipeline with little axial or radial mixing is the perfect backmixed reactor with instantaneous mixing and uniformity. For polystyrene reactors with several hours of residence time, complete mixing in 1-2 min is usually adequate to satisfy a practical definition of perfectly mixed. The probability of exit of any fluid element from this type of reactor is independent of when it entered. The residence time distribution is exponential and the molecular weight distribution in the case of no termination is Mw/Mn = 2.0, which will spread out to 2.3 when chain transfer controls. If product requirements necessitate a narrower residence time distribution, one can utilize several of these reactors in series. This becomes necessary to control the grafting distribution in rubber modified polystyrene. 

The polymerization time in continuous processes depends on the time the reactants spend in the reactor. The contents of a batch reactor will all have the same residence time, since they are introduced and removed from the vessel at the same times. The continuous flow tubular reactor has the next narrowest residence time distribution, if flow in the reactor is truly plug-like (i.e., not laminar). These two reactors are best adapted for achieving high conversions, while a CSTR cannot provide high conversion, by definition of its operation. The residence time distribution of the CSTR contents is broader than those of the former types. A cascade of CSTR s will approach the behavior of a plug flow continuous reactor. 

The time it takes a molecule to pass through a reactor is called its residence time 6. Two properties of 6 are important the time elapsed since the molecule entered the reactor (its age) and the remaining time it will spend in the reactor (its residual lifetime). We are concerned mainly with the sum of these times, which is 6, but it is important to note that micromixing can occur only between molecules that have the same residual lifetime molecules cannot mix at some point in the reactor and then unmix at a later point in order to have different residual lifetimes. A convenient definition of residence-time distribution function is the fraction J ) of the effluent stream that has a residence time less than 0. None of the fluid can have passed through the reactor in zero time, so / = 0 at 0 = 0. Similarly, none of the fluid can remain in the reactor indefinitely, so that Japproaches 1 as 0 approaches infinity. A plot of J 6) vs 0 has the characteristics shown in Fig. 6-2a. 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

Convergence is obtained when the appropriate guess for d p./di at the reactor inlet predicts the correct Danckwerts condition in the exit stream, within acceptable tolerance. To determine the range of mass transfer Peclet numbers where residence-time distribution effects via interpellet axial dispersion are important, it is necessary to compare plug-flow tubular reactor simulations with and without axial dispersion. The solution to the non-ideal problem, described by equation (22-61) and the definition of Axial Grad, at the reactor outlet is I/a( = 1, RTD). The performance of the ideal plug-flow tubular reactor without interpellet axial dispersion is described by... 

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 primary advantage of such an approach is that it creates a flat flow profile, and thus allows definition of precise flow rates and narrow residence time distributions. Nevertheless, electroosmotic flow (EOF) pumps are severely limited in their widespread application to molecular synthesis due to a need for a conductive solvent and the fact that varying electrophoretic mobilities of reagents and products leads to time-dependent concentration gradients within the reactor that can degrade performance. 

A broad residence time distribution and the associated varying reaction times in the solution can also have negative effects on the product distribution. This is especially tme for fast multi-step reactions, which need a very definite reaction time for the first step. This second step can also be a precipitation or a quench (chemical or thermal). 

Monolith reactors are composed of a large number of parallel channels, all of which contain catalyst coated on their inner walls (Figure 1.9 [1]). Depending on the porosity of the monolith structure, active metals can be dispersed directly onto the inner channel walls, or the catalyst can be washcoated as a separate layer with a definite thickness. In this respect, monolith reactors can be classified among PER types. However, their characteristic properties are notably dhferent from those of the PBRs presented in Section 1.2.1. Monolith reactors offer structured, well-defined flow paths for the reactive flow, which occurs through random paths in PBRs. In other words, the residence time of the reactive flow is predictable, and the residence time distribution is narrow in monoliths, whereas in a PBR, different elements of the reactive mixture can pass through the bed at different rates, resulting in a wider distribution of residence times. This is a situation that is crucial for reactions where an intermediate species is the desired product and has to be removed from the reactor before it is converted into an undesired species. 

It should be noted that CARPT experiments in the gas—soHd riser produced for the first time the definitive soHds residence time distribution in the riser itself (Fig. 1.12). Precise monitoring of the time when the tracer particle enters the system across the inlet plane, and the time when it exits across either inlet or exit plane, provides its actual residence time in the riser. Ensemble averaging for several thousand particle visits yields the solids RTD. The first passage time distribution is also readily be obtained. This information cannot be obtained by measuring the response at the top of the riser to an impulse injection of tracer at the bottom. By using CARPT, true descriptions ofsoHds residence time distributions can be obtained in the riser (Bhusarapu et al., 2004, 2006). One task of CFD modelers is to develop codes that can predict the experimental observations of CARPT. Here, it... 

For an isothermal, homogeneous system of constant density with well-defined boundaries and a single inlet and outlet stream, a residence time distribution, F(t), is defined as the probability that a fluid element has a residence time less than t. At steady state F(t) is the fraction of the fluid in the outflow that has resided in the vessel less than time t. Here we assume that each fluid element has a definite point of entry into the vessel and that age (residence time) is acquired by the fluid element only while it resides within the system s boundaries. 

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