Residence Time Distribution Function Research Articles

The use of ozone as a gas tracer for kinetic modeling of aqueous environmental ozonation processes

Ozone has been used as a tracer for establishing the flow pattern of the gas phase in a bubble column. Negative step tracer experiments have been conducted to determine the residence time distribution function of the gas phase through a bubble column when flowing counter‐currently to a water stream. The results have been interpreted following the tank in series model and the dispersion model for non‐ideal flow. At the experimental conditions investigated the gas phase flow pattern can be characterized as the equivalent to two well mixed tanks in series. Values of the dispersion number were also calculated.

Flow patterns in twin-screw extruders

Flow visualisation studies in an intermeshing co-rotating twin-screw extruder operated under starve-fed conditions are presented. Different numbers of kneading discs and relative staggering angles were used in the definition of the mixing blocks. Conveying and left-handed elements were also utilised. The images recorded provided information on flow patterns, on the degree of filling of the screw channels and on the number of filled turns. Local flow times were estimated. Partial residence time distribution (RTD) functions were determined by inserting especially designed collecting devices along the barrel, and measuring the evolution with time of the concentration of a tracer.

One‐line measurement of the residence time distribution in screw extruders

AbstractA fluorescence monitoring device was constructed to measure in real time the residence time distribution (RTD) function in screw extruders. Its centerpiece was an optical fiber probe capable of transmitting optical excitation energy to the processed flow stream and detecting the subsequent fluorescence emission. The source of fluorescence emission was an anthracene‐bearing substance that was injected to the flow stream as a pulse (tracer) in very small amounts. This device was validated against an off‐line ultraviolet spectrophotometer and self‐checked as well. In addition to its great sensitivity to fluorescence emission and ease of implementation on an extrusion line, this device could be used to monitor in real time subtle variations of an extrusion operation. The influence of the nature of anthracene‐bearing compounds on the measured distribution and the effects of processing parameters on the RTD function were also investigated.

Residence time distribution of the liquid phase in a concentric-tube airlift reactor

The residence time distribution (RTD) analysis of liquid phase was performed in a concentric-tube airlift reactor of 0.07 m 3 nominal volume, regarded as a simple unit and discriminating its different sections (riser, downcomer, bottom zone and gas-liquid separator) using the tracer response technique. The reactor was operated in biphasic continuous flow of liquid and gaseous phase. The volumetric liquid inflow rate, Q 1, and gas superficial velocity in the riser, ν SGR, were chosen as independent variables. RTD functions in the reactor, namely: the RTD distribution functions at the reactor exit, inclusively in normalized form, E( θ), the distribution ages inside the reactor, I( θ), the intensity of distribution, λ( θ), respectively, as well as the mean liquid residence time, and the variance of distributions were essentially used for liquid flow diagnosis in the reactor in two operational modes: with and without recirculation. The experimental results revealed that the investigated airlift reactor had a more uniform flow than the tubular and bubble column reactors and the flow defects proved a significant attenuation. These conclusions were confirmed by the single parameter flow models: axial dispersion and tank-in-series models. This fact has a great practical importance, especially in the biotechnological applications of the airlift reactors, where the aerobic cultures can be affected greatly by the flow deficiencies.

Simulation of groundwater age distributions

The objective of our work is to examine how to simulate the age of groundwater in such a way that it can be compared to actual measurements. We start by showing that computation of kinematic age, the one obtained by tracking water along streamlines, is ill posed in heterogeneous aquifers. This, together with its inability to account for mixing processes, makes it inadequate for comparison with age measurements, which are the result of some averaging of the age distribution in the water sample (the type of averaging depends on the measurement procedure). Therefore we go on to write the equations for the cumulative distribution function of residence time under transient flow conditions. This allows us to derive transient equations for the mean age, as well as for the higher‐order moments of its distribution, which generalize previous results by others. These moments can be used for approximating age measurements, which need not be equal to the mean age of the water sample. Using both a synthetic and a real example, we show that mean age is an acceptable estimate of radiometric age measurements in many cases. Including a second‐order correction (variance of residence time distribution) always improves results. On the other hand, higher‐order approximations converge slowly for old (compared to half‐life) waters, to the point that third‐order approximations often worsen the results.

Distributive Mixing in an Axial Discharge Continous Mixer

Abstract The axial discharge continuous mixer-LCMAX 40, which combines the features of a continuous mixer and the co-rotating twin screw extruder, was analyzed in this work for its distributive mixing efficiency. We analyzed the distributive mixing performance for various rotor configurations. Four designs with either alternating or block arrangements of pushing and counter-pushing units were investigated. A fluid dynamics analysis package-FIDAP, based on the finite element method was used to model the flow behavior of a power law model fluid. Distributive mixing was quantified using particle distributions along the axial distance. Residence time distribution (RTD) functions were calculated and found to reveal interesting information related to design features. The influence of design on the evolution of the particle path length and shear strain distribution was also discussed. The counter-pushing units and their arrangements were found to play an important role in distributive mixing.

Residence time distributions in SX–EW equipment

Residence time distribution functions were determined and modelled for a pilot plant mixer–settler and an electrowinning cell. The mixer RTD is modelled by a small plug flow reactor, a small perfect mixer and a large size one. The settler RTD is modelled by Cholette-Cloutier units in parallel with a series of small perfect mixers, although frequently the settler is assumed to behave as a plug flow. Finally the electrowinning cell RTD is very close to being a perfect mixer.

Quantifying stochastic resonance in bistable systems: Response vs residence-time distribution functions

A wealth of research on stochastic resonance (SR) during the past decade has led to different ways of defining this phenomenon. Most prominent are measures based on the response function and on the residence-time distribution function. While the theory for the response functions is well developed, a first fully systematic theory for the residence-time distribution functions is developed in this paper. Subsequently we reconsider formerly introduced measures of SR based on the residence-time distribution and compare with those based on the response function.

Analysis on Fractal-Like Behaviour Expected for Migration of Radionuclides in Geologic Sorbing Media

In earlier work, we showed that within nonhomogeneous sorbing media the desorption process becomes fractal-like. In migration of radionuclides in geologic media, the adsorption is an essential factor retardating he migration. Moreover, geologic media is inherently nonhomogeneous. It is therefore probable that the migration is significantly influenced by the fractal-like feature. Based on this idea, we have analyzed migration behaviours by employing a new model and compared the results with those obtained using conventional models. The nuclides migrate in the media with the flow of ground water being continually trapped on adsorption sites and released (desorbed) to the flow. The concept of the overall residence-time distribution function for nuclides on the adsorption sites is introduced in the new model. This function obeys the power form, −t −1-α (α > 0) for sufficiently large t (t denotes time). The migration behaviours predicted by our theory are qualitatively different from those by conventional theories, and the details of the differences are greatly dependent on the exponent a. In particular, the migration behaviour in cases of 0<α<1 is characterized by far larger retardation effects.

Kinetic modelling of aqueous trichloroethylene direct ozonation in a continuous tank

The kinetics of continous direct ozonation of trichloroethylene, a water priority pollutant, is investigated. The influence of water flow type on the kinetics is studied by the use of the residence time distribution function. In accordance with this, the actual reactor used behaves as two perfectly mixed continous stirred reactors. A kinetic model accounting for rates of volatilization, direct ozonation and mass transfer together with non‐idealk water flow is presented. The results obtained indicate that water flow type should be considered to, predict more accurately removal rates of trichloroethylene during direct ozonation.

Inversion of the variance—Peclet relationship of the axial dispersion model

It is the purpose of this communication to present a simple formula for accurate inversion of the σ 2 (Pe) relationship, thus allowing the non-iterative calculation of the Peclet number for any value of the variance of the residence time distribution function.

Performance characteristics of a packed‐bed ozone contactor

Performance characteristics of a packed bed ozone contact was investigated. A yellow dye was employed as the tracer to test the dispersion characteristics of the ozone contactor in the single‐phase (liquid) and two‐phase (countercurrent liquid/air) flow systems. The one‐parameter (tanks‐in‐series) and two‐parameter (combined plug‐flow/tanks‐in‐series) models were employed to fit the observed residence time distribution (RTD) functions. It was found that for the one‐parameter model, at least twenty well‐mixed tanks are required to reasonably fit the observed RTD functions while for the two‐parameter model, only a fewer well‐mixed tanks plus a plug flow reactor are needed to yield the same fit. The dispersion characteristics of flow in the ozone contactor could also be represented by the dispersion model with appropriate dispersion numbers. The experimental residence time distribution functions were employed in conjunction with the reaction kinetics to determine the discolorization of dyestuff by ozonation in the packed bed reactor. The experimental observations appeared to reasonably confirm the theoretical predictions.

Continuous Flow Residence Time Distribution Function Characterization

Disinfection contactors are frequently characterized using tracer studies to determine macromixing patterns. For some applications, such as compliance with the surface water treatment rule, this information can be interpreted directly without much analysis. However, when tracer studies are performed to characterize residence time distributions for design purposes, this information is frequently analyzed using a method of moments approach. This paper shows that the method of moments approach, when compared to a nonlinear regression approach, produces a biased estimate of mean residence time and dimensionless variance, and one with a greater mean square error. The amount of bias in moments estimators is significant in a numerical sense, and leads to qualitatively poor estimation of the resulting residence time distribution. Examples are given of characterization of a pilot-scale chlorine contactor and a pilot-scale ozone contactor.

Performance Characteristics of A Packed-Bed Ozone Contactor

Abstract Performance characteristics of a packed bed ozone contact was investigated. A yellow dye was employed as the tracer to test the dispersion characteristics of the ozone contactor in the single‐phase (liquid) and two‐phase (countercurrent liquid/air) flow systems. The one‐parameter (tanks‐in‐series) and two‐parameter (combined plug‐flow/tanks‐in‐series) models were employed to fit the observed residence time distribution (RTD) functions. It was found that for the one‐parameter model, at least twenty well‐mixed tanks are required to reasonably fit the observed RTD functions while for the two‐parameter model, only a fewer well‐mixed tanks plus a plug flow reactor are needed to yield the same fit. The dispersion characteristics of flow in the ozone contactor could also be represented by the dispersion model with appropriate dispersion numbers. The experimental residence time distribution functions were employed in conjunction with the reaction kinetics to determine the discolorization of dyestuff by o...

Predicting Concomitant Denaturation of Vitamin as Influenced by Combined Process Temperature and pH in Batch and Continuous Flow Sterilization of Liquids

Thermal sterilization is widely used in the food industries because of its reliability and, generally, economy. Process design is based on a knowledge of the inactivation rate of the contaminating micro-organism of interest. For many liquids, denaturation of vitamin will occur during the exposure time necessary for sterilization. A mathematical model is used to develop a practical procedure to predict the concomitant denaturation of vitamin during sterilization as influenced by combined temperature and liquid pH over a range of exposure times in batch thermal processing. The procedure is illustrated using published data for denaturation of thiamine (vitamin B 1 ), and ascorbic acid (vitamin C). The effect of processing is expressed as per cent denaturation. The model on which the procedure is developed explained more than 94% of the variance accounted for (% V ) in these independent data. The model has fitted all available data without exception. The continuous flow simulation consists of combined thermal-pH denaturation kinetics and a residence time distribution function of a non-Newtonian liquid in a tube. Simulation uses the spores of the pathogen Clostridium botulinum as the contaminant together with a sterilization process requirement (i.e. the reduction in the number of viable spores) of 10−12. The value of the flow behaviour index and shear regime simulate conditions used in commercial sterilization. The influence of pH on vitamin denaturation is shown to be highly variable, indicating no simple rules of thumb. The procedure could readily be applied to a range of vitamins in different liquids. It should provide a convenient tool for the assessment of vitamin denaturation during thermal processing and, the design and evaluation of sterilizer design and process operations.

Application of residence time distributions to stormwater treatment systems

Statistical measurements and qualitative assessments made by applying standard residence time distribution theory to non-steady-flow systems are of limited utility. A method is developed to enable non-steady-flow systems to be brought into the realm of steady-flow systems, so that residence time distribution (RTD) theory can be employed in a straight forward manner to characterize the non-ideal flows in these systems. Constructed wetland effluent concentrations resulting from impulse tracer inputs to the systems are considered. In addition to representing the concentrations with standard RTD functions, the concentrations are represented as functions of the cumulative discharge volume. This method provides the ability to easily analyze and compare data from steady and non-steady-flow systems, and thus to infer the degree of non-ideality of unsteady systems.

Characterization of mixing of suspension in a mechanically stirred precipitation system

In the case of precipitational crystallization, the particle size distribution of the resulting product is greatly influenced by the mixing rate of the system. We have worked out a method of characterizing the mixing of precipitated suspensions by applying a function of mean residence time and particle size distribution. For the experiments a precipitated suspension of β-lactam-type antibiotic has been used in a mechanically stirred tank.

The concept of conversion distribution in reactive processing

A new concept of conversion distribution applied to the continuous chemical modification and polymerization of polymers in a twin screw extruder was discussed. The kinetics and the residence time distribution function (RTD) are two undissociable parameters in reactive extrusion because their combination reflects the degree of homogeneity of the modified polymer at the die die exit of the extruder. A function G defining the distribution of the modification rate allowed us to evaluate the distribution of the extent of the conversion around the mean conversion. This study has underlined the predominant role of the kinetics on the structural homogeneity of the modified product. This new concept was developed, within a framework of reactive processing, from previous works on chemical modification of polymers and polycondensation in corotating twin screw extruder. © 1996 John Wiley & Sons, Inc.

The concept of conversion distribution in reactive processing

A new concept of conversion distribution applied to the continuous chemical modification and polymerization of polymers in a twin screw extruder was discussed. The kinetics and the residence time distribution function ( RTD ) are two undissociable parameters in reactive extrusion because their combination reflects the degree of homogeneity of the modified polymer at the die die exit of the extruder. A function G defining the distribution of the modification rate allowed us to evaluate the distribution of the extent of the conversion around the mean conversion. This study has underlined the predominant role of the kinetics on the structural homogeneity of the modified product. This new concept was developed, within a framework of reactive processing, from previous works on chemical modification of polymers and polycondensation in corotating twin screw extruder.

Dynamic modeling of a closed loop jet mill

Although jetmilling is a very energy consuming grinding process it is increasingly used in industry because very fine grinding product with a narrow size distribution is attained without contamination as the milling occurs by inter particle collisions. At Delft University a project has been started to achieve a considerable energy reduction of the milling process. Main feature of the improved grinding system will be an operation control based on in-line particle size measurement using the laser diffraction technique. The pilot plant is equipped with an external classifier for the removal and recirculation of larger particles leaving the jet mill. Phenomena occurring in the mill are not yet well modeled and thus no real optimization of the grinding process can be achieved. Operation conditions of jet mills are still determined by trial and error. This paper is to form the basis of a model which allows a dynamic modelling of the behavior of the jet mill/classifier system. The closed loop grinding system has been split in 3 subsystems (zones), two internal to the mill and one external through which the particles pass. Exchange of particles is not described by complicated residence time distribution functions but by “particle size” dependent probability functions.