Single residence time distribution

The second approach to a solution is appropriate for other types of mixing. The mixing is characterized by a residence time distribution, E(t), which is the distribution of time that the material is in the mill. When linear breakage kinetics occur and all particle sizes are characterized by a single residence time distribution, a general relationship between input smd output of the mill can be established, using the following equation ... 

FIG. 7-3 Concentration profiles in fiatch and continuous flow a) fiatch time profile, (h) semifiatcli time profile, (c) five-stage distance profile, (d) tubular flow distance profile, (e) residence time distributions in single, five-stage, and PFR the shaded area represents the fraction of the feed that has a residence time between the indicated abscissas. 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

Two template examples based on a capillary geometry are the plug flow ideal reactor and the non-ideal Poiseuille flow reactor [3]. Because in the plug flow reactor there is a single velocity, v0, with a velocity probability distribution P(v) = v0 16 (v - Vo) the residence time distribution for capillary of length L is the normalized delta function RTD(t) = T 1S(t-1), where x = I/v0. The non-ideal reactor with the para-... 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

The variance approach may also be used to determine n. From Illustration 11.2 the variance of the response data based on dimensionless time is 30609/(374.4)2, or 0.218. From equation 11.1.76 it is evident that n is 4.59. Thus the results of the two approaches are consistent. However, a comparison of the F(t) curves for n = 4 and n = 5 with the experimental data indicates that these approaches do not provide very good representations of the data. For the reactor network in question it is difficult to model the residence time distribution function in terms of a single parameter. This is one of the potential difficulties inherent in using such simple models of reactor behavior. For more advanced methods of modeling residence time effects, consult the review article by Levenspiel and Bischoff (3) and textbooks written by these authors (2, 4). 

The dispersion and stirred tank models of reactor behavior are in essence single parameter models. The literature contains an abundance of more complex multiparameter models. For an introduction to such models, consult the review article by Levenspiel and Bischoff (3) and the texts by these individuals (2, 4). The texts also contain discussions of the means by which residence time distribution curves may be used to diagnose the presence of flow maldistribution and stagnant region effects in operating equipment. 

A residence-time distribution analysis of the hydrodynamics within the intestine in man during a regional single-pass perfusion with Loc-I-Gut in-vivo permeability estimation, J. Pharm. Pharmacol. 1997, 49, 682-686. 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

We focus attention in this chapter on simple, isothermal reacting systems, and on the four types BR, CSTR, PFR, and LFR for single-vessel comparisons, and on CSTR and PFR models for multiple-vessel configurations in flow systems. We use residence-time-distribution (RTD) analysis in some of the multiple-vessel situations, to illustrate some aspects of both performance and mixing. 

The TIS and DPF models, introduced in Chapter 19 to describe the residence time distribution (RTD) for nonideal flow, can be adapted as reactor models, once the single parameters of the models, N and Pe, (or DL), respectively, are known. As such, these are macromixing models and are unable to account for nonideal mixing behavior at the microscopic level. For example, the TIS model is based on the assumption that complete backmixing occurs within each tank. If this is not the case, as, perhaps, in a polymerization reaction that produces a viscous product, the model is incomplete. 

In the compound mill, the cylinder is divided into a number of compartments by vertical perforated plates. The material flows axially along the mill and can pass from one compartment to the next only when its size has been reduced to less than that of the perforations in the plate. Each compartment is supplied with balls of a different size. The large balls are at the entry end and thus operate on the feed material, whilst the small balls come into contact with the material immediately before it is discharged. This results in economical operation and the formation of a uniform product. It also gives an improved residence time distribution for the material, since a single stage ball mill approximates closely to a completely mixed system. 

Lennernas H, Lee I, Fagerholm U and Amidon GL (1997) A Residence-Time Distribution Analysis of the Hydrodynamics Within the Intestine in Man During a Regional Single-Pass Perfusion With Loc-I-Gut In-Vivo Permeability Estimation. J Pharm Pharmacol 49 pp 682-686. 

In a final RTD experiment, a sheet of dye was frozen as before and positioned in the feed channel perpendicular to the flight tip. The sheet positioned the dye evenly across the entire cross section. After the dye thawed, the extruder was operated at five rpm in extrusion mode. The experimental and numerical RTDs for this experiment are shown in Fig. 8.12, and they show the characteristic residence-time distribution for a single-screw extruder. The long peak indicates that most of the dye exits at one time. The shallow decay function indicates wall effects pulling the fluid back up the channel of the extruder, while the extended tail describes dye trapped in the Moffat eddies that greatly impede the down-channel movement of the dye at the flight corners. Moffat eddies will be discussed more next. Due to the physical limitations of the process, sampling was stopped before the tail had completely decreased to zero concentration. 

Since the dimensionless time for a first-order reaction is the product of the reaction time t and a first-order rate constant k, there is no reason why k(x)t should not be interpreted as k(x)t(x), that is, the reaction time may be distributed over the index space as well as the rate constant. Alternatively, with two indices k might be distributed over one and t over the other as k x)t(y). We can thus consider a continuum of reactions in a reactor with specified residence time distribution and this is entirely equivalent to the single reaction with the apparent kinetics of the continuum under the segregation hypothesis of residence time distribution theory, a topic that is in the elementary texts. Three indices would be required to distribute the reaction time with a doubly-distributed continuous mixture. 

The local aspects of liquid-liquid two-phase flow in RE has been the focus of CFD analysis by different research groups (123-126). In principle, all aspects concerning single-phase flow phenomena (residence time distribution, impeller discharge flow rate, etc.) can be tackled, even with complex geometries. However, the two-phase CFD is still a challenge, and the droplet interactions (breakup and coalescence) and mass transfer are not implemented in commercially available codes. Thus these issues constitute an open area for further research and development (127). 

The derivation of the residence time behavior of the single streamlines now allows the formulation of the cumulative residence-time distribution function F(t) according to the following formula ... 

A number of elegant studies over the past few years have also addressed the need to minimize particle size distributions through the use of segmented flow microfluidic systems. Such an approach removes the possibility of particle deposition on channels and eliminates the problems of residence time distributions that occur in single phase systems (where drag at the channel walls sets up a velocity distribution inside the channel). For example, Shestopalov et al. reported the two-step synthesis... 

Single-component reactions can occur throughout the bulk of the material. At the start of the reaction only one single component, monomer or prepolymer, is present or the components used are well miscible and premixed. For this group of reactions the temperature of the mixture plays an important role, as well as macro-mixing over the length of the extruder, which is related to the residence time distribution. Both parameters determine the progress of the reaction. 

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