Uses of Residence Time Distributions

Laminar Flow without Diffusion. Section 8.1.3 anticipated the use of residence time distributions to predict the yield of isothermal, homogeneous reactions, and... 

Shinnar, R. and D. Rumschitzki. The Use of Residence Time Distributions in Heterogeneous Reactor Modeling, Design and Scaleup. 76th AIChE Annual Meeting, San Francisco, November 1984. paper 139b. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Ross (R2) measured liquid-phase holdup and residence-time distribution by a tracer-pulse technique. Experiments were carried out for cocurrent flow in model columns of 2- and 4-in. diameter with air and water as fluid media, as well as in pilot-scale and industrial-scale reactors of 2-in. and 6.5-ft diameters used for the catalytic hydrogenation of petroleum fractions. The columns were packed with commercial cylindrical catalyst pellets of -in. diameter and length. The liquid holdup was from 40 to 50% of total bed volume for nominal liquid velocities from 8 to 200 ft/hr in the model reactors, from 26 to 32% of volume for nominal liquid velocities from 6 to 10.5 ft/hr in the pilot unit, and from 20 to 27 % for nominal liquid velocities from 27.9 to 68.6 ft/hr in the industrial unit. In that work, a few sets of results of residence-time distribution experiments are reported in graphical form, as tracer-response curves. 

Figure 3. Molecular weight distribution of polymer used in residence time distribution tests...

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

The characterization is performed by means of residence time distribution (RTD) investigation [23]. Typically, holdup is low, and therefore the mean residence time is expected to be relatively short Consequently, it is required to shorten the distance between the pulse injection and the reactor inlet. Besides, it is necessary to use specific experimental techniques with fast time response. Since it is rather difficult, in practice, to perfectly perform a Dirac pulse, a signal deconvolution between inlet and outlet signals is always required. 

In addition to the aforementioned slope and variance methods for estimating the dispersion parameter, it is possible to use transfer functions in the analysis of residence time distribution curves. This approach reduces the error in the variance approach that arises from the tails of the concentration versus time curves. These tails contribute significantly to the variance and can be responsible for significant errors in the determination of Q)L. 

The IEM model is a simple example of an age-based model. Other more complicated models that use the residence time distribution have also been developed by chemical-reaction engineers. For example, two models based on the mixing of fluid particles with different ages are shown in Fig. 5.15. Nevertheless, because it is impossible to map the age of a fluid particle onto a physical location in a general flow, age-based models cannot be used to predict the spatial distribution of the concentration fields inside a chemical reactor. Model validation is thus performed by comparing the predicted outlet concentrations with experimental data. 

Calculate the conversion of A B, r = kC in two CSTRs using the residence time distribution and compare the result with that obtained by integrating the CSTR mass balances. Repeat this problem for zeroth-order kinetics. 

We observe that it is the uniformity of the dense phase and the linearity on the bubble side that allows us the freedom to use any residence time distribution. The reaction term, which we made first order for simplicity, can be nonlinear. A discussion of various generalizations is given in [310]. 

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... 

Weiss, M., Generalizations in linear pharmacokinetics using properties of certain classes of residence time distributions. I. Log-convex drug disposition curves, Journal of Pharmacokinetics and Biopharmaceutics, Vol. 14, No. 6, 1986, pp. 635-657. 

The following treatment considers the effect of the residence time distribution on the size distribution of particles produced in a gas phase reactor. To do this we have to assume that the particles are produced by nucleation, either single point at the inlet of the reactor or multipoint through out the reactor, and particle growth is atom by atom with a growth rate G. Using the residence time distribution, the particle size distribution can be calculated for these two cases of nucleation [33]. 

Equations describing velocity profiles can be used, among other applications, to study the effect of different rheological models on the distribution of velocities and to understand the concept of residence-time distribution across the cross-section of a pipe or a channel. 

In the analysis of residence time distributions, it is conventional to normalize time using the mean residence time of a noneliminated bolus input. This makes the normalized mean residence time (f ) of the signal in the dispersion model fi=l. 

In calculations of this kind, two different equations of mixing according to Dankwerts131 are usually used one for micromixing and one for complete segregation of flows. The imperfect reactor operation can be taken into account by means of the curve of residence time distribution and various empirical combinations of elementary reactor volumes based on it. 

Using the residence-time distribution function for a laminar-flow reactor, compare the yield of B in n Type III reaction with that obtained in a PFR of the same average residence time. There is no B or C in the feed, and k = Ikj with kji = 1. 

Since the applications of chromatography were nurtured to a large extent by use in the field of separation processes, much early work was couched in terms arising from the concepts of equilibrium-stage separation units. Indeed, some of these ideas go back quite some time [A.J.P. Martin and R.L.M. Singe, Biochem. JL, 35, 1359 (1941)]. However, our approach throughout the text has been in terms of continuum balances (with the possible exception of CSTR sequences), and we will continue that approach. As in previous sections of this chapter, there is more than a superficial resemblance in the analysis to earlier considerations of residence-time distribution in fiow reactors. This presentation will largely follow the classical work of Lapidus and Amundson [L. Lapidus and N.R. Amundson, J. Phys. Chem., 56, 984 (1952)]. 

The concepts of residence time distribution and mean residence time can be employed in a wide range of disciplines other than chemical reactor design (e.g., in the analysis of equipment used in separation processes and sewage treatment plants). Another example is the analysis of the behavior of pharmaceuticals in humans and animals. Such information is important in determining the conditions necessary to maintain efficacy of these materials in vivo. The data from the pharmacokinetic study presented helow pertain to a drug that was being studied as a potential inhibitor of the HIV virus. 

One can hardly recognize the mechanism of processes occurring in apparatus using such approach because real speeds field in it is not known. That is in this case apparatus is considered as black box . At the same time such analysis method of flow structure in reaction zone is easy enough. Quantitative data processing is essentially simplified because they deflnite the function of only one variable - time. Furthermore, the data of residence times distribution guarantee reliable estimation of real flow structure in apparams, i.e. allow to look into black box [3,7,16]. 

Compared with Eq. (4.338), we conclude that the characteristic time corresponds to the residence time r for removal processes of the first-order rate. Using the residence time distribution approach, we define ... 

There is a long history of investigations of residence times in twin-screw extruders [124—136]. The first studies of residence time distribution in intermeshing corotating twin-screw extruders were reported in 1966 by Herrmann [124] of Werner Pfleiderer. Table 6.4 summarizes the many different techniques have been used to measure the residence time distribution in intermeshing co-rotating twin-screw extruders. 

In this analysis, the history of each particle can be monitored for such quantities as velocity and velocity gradients as it travels through the mixer. With this information, the distributive and dispersive mixing of the device can be evaluated qualitatively and quantitatively. For instance, the distributive mixing can be calculated using the residence time distribution by monitoring the time it takes for the parti-... 

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