Article

Representation and Evaluation of Residence Time Distribution

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Abstract

A new method for the representation of residence time variability in continuous flow systems is presented. The method is based on the use of the intensity function. The advantage of this method is that it allows a physical insight into the mixing processes within the system and enhances the interpretability of experimental curves. The phenomenon of stagnancy is discussed and defined operationally. Theoretical and practical examples are used to illustrate the usefulness of the concepts introduced.

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... Thus, it is indicated that BF gas RTD study should be carried out under the specific operational conditions. Residence time intensity function which allows a physical insight of residence time was derived and proposed by Naor and Shinnar (1963). It is a direct and useful parameter to predict and evaluate the stagnancy in different vessels, even in some cases where their residence time distribution show little difference. ...
... Residence time distribution can comprehensively characterize molecules flow patterns in reactors, and thus is expected that the key information can be derived from the density distribution curve. The second central moment method is widely used to calculate the variance of residence time distribution as this indicator can represent the molecules dispersion from ideal flow patterns (Buffham and Mason, 1993;Naor and Shinnar, 1963). The following equation is used to derive the variance of molecules' residence time distribution. ...
... The physical meaning of residence time intensity function is the probability of escape for molecules which have stayed for a period of t (Naor and Shinnar, 1963). Generally, for piston-type flow, the residence time intensity is infinity at the ''pulsing" time, while for complete mixing in a single vessel, the residence time intensity is always unity due to the total mixing effect. ...
Article
Gas residence time distribution (RTD) is an effective and convenient indicator for evaluating the performance of complex multiphase chemical reactors including ironmaking blast furnaces (BFs). However, in the open literature, there lacks the systematic RTD research for BFs. In this study, an integrated mathematical model is developed for describing the gas RTD of a BF. The model combines a steady multi-fluid model for describing the in-furnace state of flow and thermo-chemical behavior of gas-solid-liquid phases and a transient model for describing the dynamic behavior of tracer materials. The results show that the gas flow field inside the BF is quite complex, resulting from many factors such as furnace geometry and coupled thermo-chemical behaviors of other phases. The tail of gas RTD curve resulted from the lagging phenomenon of tracer flow inside BFs, is captured. The gas RTD is discussed by using mean residence time, dispersion of molecules distribution, cumulative distribution function and residence time intensity function. Under the given BF conditions, mean residence time and space time of gas fluids are predicted as 13.5 s and 16.3 s, respectively. The existence of stagnant flow in the BF can be both derived and directly identified. Moreover, it is indicated the gas flow patterns in the BF are composed of piston-type flow, stagnant flow and limited mixing flow. This study provides a cost-effective tool for better understanding BF gas flow dynamics and optimizing BF operations.
... and F (t) may also be interpreted as probabilities [46]. F (t) is the probability of a single particle staying in the system for a time t or less and F (t) is the probability of the residence time exceeding t (notice the implicit assumption that both functions acquire values only in [0; 1]). ...
... Wang [53] mentions that mixing behaviour of a majority of the actual mixers deviates from the ideal mixer (exponential vessel). He argues that this deviation may be caused by non- Shinnar and Naor [46] have suggested the intensity function or escape probability density as a method of visualizing the features of RTD related to stagnancy. In the context of modelling based on Erlang and n-Hyperexponential distributions, stagnancy is generally associated with systems which total ‡ow may be decomposed into ‡ows connected in parallel and one of the components has a signi…cantly larger average residence time than the other. ...
... Thus any departure of (t) from constancy is an indication of ill-mixedness. Indeed, as Shinnar and Naor [46] indicate, a system with stagnancy is one in which the escape probability (or the intensity function) decreases in time over some interval. For example, imagine a system in which a considerable fraction of the particles moves in near plug ‡ow through it, whereas the remaining fraction is absorbed into a stagnant phase from which it is exuded later into the main stream. ...
Thesis
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In this report the concepts of heterogeneity and homogeneity are studied in the context of mixture of particulate materials. It is shown that the function of a mixer, defined as a chaos enhancer device, is to reduce the axial distributional heterogeneity of a mixture. In order to characterize this property, several different modelling techniques are reviewed. Based on these techniques several steady state models are developed and discussed. A probabilistic approach to determination of residence time distribution (RTD) of a mixer is presented. Several possible RTD models and their physical significance are also explained. An outline of the most common experimental methods for determination of RTD is given. Moreover, it is shown that if the superposition condition is fulfilled, a continuous mixer can be modelled as a linear time invariant (LTI) system. In this context the RTD and the transfer function of the system should coincide. It is demonstrated that the escape probability function of a mixer is a valuable tool in visualizing some of the general characteristics of a mixer. The mean residence time and variance reduction ratio are other quantities which can be used to measure the ability of a mixer in reducing axial distributional heterogeneity of a mixture. It is also demonstrated how results from queuing theory can be use to determine some of the features of the mean residence time of a mixer. In case of steady state feeding, the ingoing axial heterogeneity can be modelled as a quasi-stationary stochastic process. The axial heterogeneity can be broken into two independent components, a deterministic and random component. An estimate of the variance of the random component can be found by studying small perturbations of the variogram functions of the input and output axial heterogeneity, around zero. This can also be used to show that the Danckwerts formula correctly describes the variance reduction ratio of a continuous mixer. Each particle path in a mixer can be modelled by a random walk. It is shown that the dispersion model relates the statistics of these paths to the performance of the mixer. This report is concluded by a sketch of the future plans for research on continuous mixing.
... As Shinnar and Naor (1963) have also noticed, the functions F (t) and F (t) may also be interpreted as probabilities. Therefore, F (t) can be considered as being the probability of a single particle staying in the system for a time t or less and F (t) the probability that the particle's residence time exceeds t. ...
... Little else exists in the literature on how to handle such complex models. Nonetheless, as Shinnar and Naor (1963) have pointed out, all actual distribution functions may be approximated by a theoretical model composed of a number of exponential and near plug ‡ow vessels connected in some network. This is of course mathematically equivalent to the statement that all well behaved functions can be approximated by some power series. ...
... These are evidently part of a reason for complicated modelling networks. Shinnar and Naor (1963) have suggested the intensity function or escape probability density as a method of visualizing the features of RTD related to stagnancy. In the context of modelling based on n-Erlang and n-Hyperexponential distributions, stagnancy is generally associated with systems in which total ‡ow may be decomposed into ‡ows connected in parallel where one of the components has a signi…cantly larger average residence time than the other. ...
Thesis
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A theoretical framework for sampling theory is developed. In this relation, concepts like mixture heterogeneity and representative samples are mathematically defined. Further, the relation between Gy's concepts of accuracy and reproducibility with mixture quality and the entropy of the sample distribution is established. Moreover, it is shown that within the developed framework, Lacey's conjecture is mathematically consistent. It is also shown that a consequence of the theory is the prediction of the number of key components of given size in random binary closed batch systems. It is also shown that this estimate is a function of microstructural properties of the mixture under study. Furthermore, this theory is used to develop a unifying approach to description of RTD of continuous systems. These results are further used to develop a model for RTD of a commercial twin screw extruder. A new theoretical approach to the dynamics of the mixing processes is developed. In this context, the concept of heterogeneity landscape is introduced. It is argued that the valleys in the heterogeneity landscape correspond to different equilibrium states of the mixture. Further, it is shown that the valleys in the heterogeneity landscape can mathematically be described by heterogeneity equation and this would allow for classification of all the valleys. The characteristic function of the general solution to the heterogeneity equation is also determined. Moreover, it is shown that based on the mathematical model for the valleys, one can deduce that in the case of insufficient information about the mixture structure, the normal distribution, up to the second order; is the best distribution in describing the mixture structure.
... Although it is difficult to develop an analytical correlation between the observable geometric quantities, such as the distribution of hollow fibers, and the extent of bypass flow in a dialyzer shell, the presence of the flow maldistribution can be experimentally characterized by a relatively simple technique: the tracer dilution analysis [30,31]. This technique gives a macroscopic description of the flow distribution in a dialyzer. ...
... Exit age distribution functions of two dialyzers obtained from eqn. (31) are plotted in Fig. 10. The earlier appearance of the tracer (channeling) and the longer tail of the curve (hole up) are found for the loosely packed (55% packing) dialyzer. ...
Article
Counter-current hollow fiber dialyzers were studied. Performance of loosely packed hollow fiber dialyzers deviated from the prediction based on ideal flow distribution while well packed dialyzers did not deviate. The flow maldistributions were characterized by tracer analysis. The behavior of dialyzers was simulated with a model based on bypass now.
... In the first case (ideal mixer: IM), all material elements inside the mixer at a given time have the same chance of exit; in the second (plug flow: PF), material flows like plug from inlet to outlet. In dimensionless terms, the RTD functions corresponding to these two ideal models are (Danckwerts, 1953;Naor and Shinnar, 1963 where t* = t/t AVG is the dimensionless time and e* has been normalized so that the area under it (the cumulative RTD) goes to 1 as t* goes to ¥ ; δ(ξ) is Dirac's delta function (a spike at ξ = 0). ...
... The two ideal models can be combined to form "mixed models", widely used to study backmixing in complex flow systems. One of the simplest is composed of N ideal mixers (IM) units in series, each with the same effective volume φV F /N, so the mean residence time in any one of them is t AVG / N. The dimensionless RTD of this tanks-in-series model (N´IM) is given by (Naor and Shinnar, 1963 for the N´IM model. The coefficient of dispersion can be taken as a good measure of the degree of backmixing. ...
... The chemical reaction engineering literature has dealt extensively with analytical, statistical, and computational dynamics models that can simulate the complex flow processes common to many types of flow reactors, both single and multiphase [see for example Levenspiel (1999) and Fogler (2006)]. So the issue of modeling biomass particle RTDs in flash pyrolysis is, in fact, a subset of a much more general problem that has received considerable attention for over 50 years [e.g., see Van de Vusse (1962) and Naor and Shinnar (1963)]. Many of the general modeling approaches developed for chemical reactors have been adapted for modeling particle RTDs in bubbling and circulating beds. ...
... in situ optical probe, can be modeled in various ways. [27][28][29][30][31] Without loss of generality, we demonstrate the applicability of our approach by modeling the normalized RVDs using simple empirical equations based on transfer functions that represent a series of perfect batch mixers. 32,33 A secondorder transfer function that corresponds to the mixing occurring in the screw and the die zones of the extruder was used. ...
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Nanocomposites, with superior material properties, have promising potential applications in almost every field. The present work aims at developing a reproducible and continuous fabrication process to obtain the nanocomposites of aluminum nanoparticles that are uniformly dispersed in a polymer matrix and carbon nanotubes (CNTs) that are uni-directionally oriented in a polymer matrix. Due to its mixing potential, extrusion through a 28 mm Werner and Pfleiderer Twin Screw Extruder (TSE) was chosen to fabricate the nanocomposites. The aim of this research effort is to understand the nanoscale mixing characteristics of TSE using aluminum nanoparticles and CNTs, and to obtain the alignment of CNTs via extrusion through microchannels. Experimentally obtained residence time distributions of the mixing process were used to relate the mixing within the TSE to the microstructure of the extrudate. The microstructure of the nanocomposites has been characterized with scanning electron microscopy.
... Each particle that enters a continuous mixer spends a certain amount of time in the mixer. The duration of stay of each particle in the system, depending on the system parameters, follows a certain distribution that is known simply as the residence time distribution (RTD) of the mixer (Shinnar & Naor, 1963;Williams 145 & Rahman, 1971). The axial mixing capability of a mixer is often related to its RTD. ...
Article
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The concept of heterogeneity in relation to axial mixing in continuous mixers is investigated. The state of the mixture along the axial direction can be described by an axial heterogeneity function. It is shown that this function can be decomposed into two independent components. The first component describes the fluctuations caused by the feeding system. The second component is the fluctuations due to the particulate nature of the material. In addition, the second component, or the random component, can be modeled as a band-limited Gaussian white noise. Moreover, the variogram function is shown to be a useful tool in determining the variance of the random component. A linear time-invariant (LTI) model is proposed for continuous mixers. This model also implies that the Danckwerts-Weinekötter formula is applicable for the variance reduction ratio (VRR). However, it is shown that the Danckwerts formula for VRR is more appropriate for determination of mixer efficiency.
... Such a generalization was introduced in chemical engineering long ago, because of its importance in designing flow chemical reactors and in evaluating their performance. The significance of the random nature of residence time in a continuous flow system was pointed out by Danckwerts as early as in 1953, and this yielded to the introduction of the concept of a residence time distribution (besides Dankwerts [1953], see e.g., Naor and Shinnar [1963], Wolf and Resnick [1963], and Nauman [1969]). This approach and the subsequent developments are concerned to various dilution schemes other than the ideal mixing and do not usually consider the variability of the inflow/outflow; in other words, in the chemical engineering literature the random nature of the response variable (the concentration at the outlet) is caused by a source within the system (e.g., mechanism of micromixing/macromixing, shape of the chemical reactor, flow pattern, and so on). ...
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Residence times of conservative pollutants in a lake or a reservoir are studied without the usual approximation of constant inflow/outflow. The effects of a random discharge flushing out a reservoir are investigated with the techniques of stochastic differential equations. Stochastic properties of the concentration in the reservoir are derived from stochastic properties of discharge (moments, autocorrelation function, and probability distribution function), and some approximations are analyzed. The major results are two simple and usable corrections for the residence time in two limiting cases: when autocorrelation time of discharge is much shorter than reservoir residence time and, at the opposite, when it is much longer.
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A systematic study of liquid phase axial dispersion was conducted in glass columns (inner diameters, 1 cm and 1.6 cm), packed randomly with granular sand, by varying the fluid flow rate, particle size and bed height. Pulse and step response techniques, with KCl as an inert tracer, were used. The resultant data, covering the Reynolds number range from 1 to 50, are presented as plots of the Peclet number based on particle diameter against Reynolds number.Inert tracer experiments were also carried out in a column (inner diameter, 1.6 cm) packed with activated carbon granules, using different particle sizes, fluid flow rates and bed heights, in order to estimate the effective intraparticle diffusivity. We show that flow maldistribution produces pulse response curves with sharp, narrow peaks which, when compared with theoretical curves, result in small intraparticle diffusivities.We illustrate how the outer-phase transfer function can be obtained from the overall transfer function of the activated carbon bed and we compare it with the transfer function obtained directly using impermeable particles similar to the activated carbon granules.
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Responses to a tracer dye have been measured in a laboratory scale stirred tank using fiber optics for water and aqueous polymer solutions. An empirica Short-circuiting becomes significant below a particular stirrer speed for a given system. The observable segregation decreases markedly as stirrer spee models are also evaluated and compared with those of the current study.RésuméA l'aide de la technique de l'optique des fibres et dans le cas de l'eau et de solutions aqueuses de polymères, on a étudié au laboratoire le de sortie lissées, et on présente des paramètres de mélangeage qui chiffrent l'importance de la ségregation et de la rétention du récipie mesure que la vitesse d'agitation croît, la diminution de la ségrégation est prononcée. Des courbes lissées et idéales de distribution de t certain nombre de modèles bien connus, et on les compare à ceux que l'on obtient dans le présent travail.ZusammenfassungAntwortfunktionen wurden durch farbigen Spurstoff in einem Laboratoriumsrührkessel durch die Anwendung von Glasfaseroptik und wässriger Polymer Entmischung und ‘Holdback’ angeben, wurden entwickelt. Der Kurzschluß wird unterhalb einer gegebenen Drehzahl des Rührers signifikant. Die beob denen Entmischung nachweisbar war. Halbempirische Parameter wurden aus einer Zahl gut bekannter Modelle ermittelt und mit denen der vorliegenden Arbeit
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The cycle time distribution (CTD) within closed, continuously circulating systems is defined and related to the residence time distributions of flow regions which make up such systems. Examples of the application of the CTD are noted and experimental methods for determining CTDs for various systems are summarized.
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Tracer experiments for studying transport processes in multiphase systems are discussed. Such experiments are useful in in vivo physiological studies of transport processes across membranes, in engineering studies of porous packed columns, and in chromatography. In all these processes, the main forward flow is in one continuous phase from which fluid diffuses into a stagnant outer phase (and back). A nondiffusible tracer is used to characterize the flow in the main phase (inside the capillaries of an organ or between the particles of a column), and diffusible tracers are used to study the transport through the interface (membrane or film) and in the outer phase (extravascular space or inside a porous particle). From the concentration history of the different tracers at the outlet, we can reconstruct sojourn time distributions in the different phases. The statistical properties and the relations between the distributions and their moments are discussed. Methods are given for estimating interfacial transport coefficients (or permeabilities) as well as the diffusion coefficients in the outer phase from the moments of the measured distributions. It is also shown that these relations simplify considerably the mathematical modelling of such systems.
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A mathematical model for gas-fluidized beds is proposed that allows for a randomly fluctuating flow pattern. It is shown how mean first-order conversion is related to contact time distribution for arbitrary models of this type. A simplified version of the model is then studied, and it is found that the effect of fluctuating flow is similar to that of stagnancy in steady systems. This effect is inconsistent with the usual steady-state models, but it is shown that some published data on conversion in fluidized beds [6] exhibit this effect.RésuméOn propose un modèle mathématique pour des couches de gaz fluidisées permettant un courant présentant des fluctuations au hasard. On démontre la relation d'une conversion moyenne de premier ordre à la distribution des temps de contact, pour des modèles arbitraires de ce type. Une version simplifiée du modèle est ensuite étudiée et on trouve que l'effet du courant de fluctuation est similaire à celui de la stagnation dans les systèmes stables. Cet effet est contradictoire aux modèles habituels à l'état stable, mais l'on montre que certaines informations publiées sur la conversion des couches fluidisées [6] présentent cet effet.ZusammenfassungEs wird ein mathematisches Modell für durch Gas betätigte Wirbelschichten vorgeschlagen, das ein zufallsmässigen Schwankungen unterworfenes Strömungsbild in Betracht zieht. Die Beziehung der durchschnittlichen Umsetzung einer Reaktion erster Ordnung zur Verteilung der Berührungszeit für willkürlich gewählte Modelle dieser Art wird dann eine vereinfachte Version des Modells untersucht, und es wird festgestellt, dass der Effekt einer schwankenden Strömung ähnlich dem einer Stagnation in stationären Systemen ist. Dieser Effekt ist mit den üblichen Modellen des stationären Zustandes unvereinbar, doch wird gezeigt, dass verschiedene über die Umsetzung in Wirbelschichten gemachte Angaben [6] diesen Effekt aufzeigen.
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A general method for calculating residence time distributions for systems with internal reflux is described. The method allows the derivation of the Laplace transform of any system composed of mixed vessels with both forward and backward flow between them. In particular, the properties of a linear cascade of mixed vessels with forward and backward flow between the vessels is discussed.RésuméUne méthode générale du calcul des répartitions du temps de résidence pour des systèmes ayant un reflux interne, est décrite. La méthode permet la dérivation de la transformation de Laplace de tout système composé de récipients mixtes ayant entre eux un courant dans les deux sens. En particulier, les propriétés d'une cascade linéaire de récipients mixtes avec courant dans les deux sens entre eux, est discuté.ZusammenfassungEine allgemeine Methode zur Berechnung der Verweilzeitverteilungen für Systeme mit innerem Rückfluss wird beschrieben. Die Methode gestattet die Ableitung der Laplace-Transformierten für jedes System, das sich aus Gefässen mit Vorwärts- und Rückwärtsfluss zwischen ihnen zusammensetzt. Im besonderen werden die Eigenschaften und das Verhalten einer Linearkaskade gemischter Gefässe mit Vorwärts- und Rückwärtsfluss zwischen den Gefässen besprochen.
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The mixing plays a fundamental role in domains as fluid dynamics, chemical engineering, environmental studies and pharmacology. Discrete and continuous time models, based on model categorification method have been developed. The residence time distributions, RTD, for multi-scale imperfect mixing are expansions in terms of Meixner and Laguerre polynomials. The resulting RTD are compared to different models of imperfect mixing. Local anesthetic effects on membranes are presented in the general PSM framework. The SDG solution for imperfect mixing is exposed.
Chapter
IntroductionCharacteristic PropertiesGlobal Measuring TechniquesLocal Measuring TechniquesConclusions and Future TrendsReferences
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Turbulent chemical reactors are modeled by networks of stirred tanks, with the stochastic nature of the mixing introduced by taking the interstage flows to be stationary Markov processes. Some general features of tracer experiments in these quasi-steady flows are discussed, together with their relation to residence time distributions. The statistics of tracer experiments are analyzed, and related on the one hand to the esti-mation of mixing parameters, and on the other hand to the forecast of average yield from the reactor system under first-order kinetics. The variability of the reactor performance and the general story of more complicated kinetic mechanisms are deferred for a later report.
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