Residence Time Distribution Function Research Articles

Residence time density (RTd) identification through deconvolution of finite time horizon input output data and its Application to a novel analytical dispersion process RTd model

This work presents first an approach to the identification of Residence Time density (RTd) and Residence Time Distribution (RTD) functions, through deconvolution of finite time horizon, input/output data generated by a zero-order hold input, using the novel concept of zero-order hold, average, impulse response coefficients. Subsequently, a novel RTd analytical expression is derived for an open-open axial dispersion process with a finite experimental section in a tube of infinite length, and an explanation is provided for this RTd model’s differences from the literature. This new RTd expression is then employed in two case studies to demonstrate the effectiveness of the proposed identification through a deconvolution conceptual framework in identifying RTd model parameters, and in assisting in the selection of RTd models which are compatible with experimental data.

Performance and hydrodynamic study of a new multi‐stage solar photoreactor with immobilized ZnO for synthetic wastewater treatment

AbstractThis study aims to design a new multi‐stage photoreactor with immobilized ZnO and assess its performance in reducing textile wastewater toxicity using solar energy. The electric conductivity measurement is retained as a detection method to obtain the residence time distribution (RTD) function. The RTD is then used to characterize the hydrodynamic behavior of the photoreactor and to evaluate its deviation from the distribution curves of ideal reactors. Solar experiments are conducted to demonstrate the performance of this multistage reactor towards the degradation of a textile azo dye namely Solophenyl Red 3BL (SR 3BL). The influence of the flow rate and the SR 3BL initial concentration (CO) are considered. The variation of the flow rate slightly influences the SR 3BL photodegradation efficiency and it is inversely proportional to its concentration. The study of the reuse of the immobilized catalyst shows that the degradation efficiency of 98% is reached even after multiple photocatalytic cycles. The figure of merit collector area per order was in the range of 35 to 110 . This result provides useful information for scaling up and estimating energy efficiency of the reactor.

Residence Time Distribution: Literature Survey, Functions, Mathematical Modeling, and Case Study—Diagnosis for a Photochemical Reactor

This paper aims to provide an overview of the fundamentals, development, and evolution of residence time distribution (RTD) methodology and its applications to the flow and mixing of fluids (and solid particles) modeling in different systems. A concise literature analysis is followed by a succinct presentation of RTD methodology’s experimental and theoretical foundations and RTD-based mathematical modeling, highlighting its importance. An experimental demonstration of RTD diagnostics on a photochemical reactor is performed to identify the most practical locations for the inlet/outlet pipes (axial or radial) and the photochemical reactor’s ideal working posture (horizontal, vertical, or inclined) and to understand the level of mixing and to determine the fluid flow defects. Using the relevant RTD functions and the corresponding central moments, it was possible to show that short circuits and dead zones occurred in each of the six considered reactor configurations. Following these investigations, design solutions were proposed to achieve a convenient exposure time, proper mixing, and uniform irradiation inside the reactor.

Subject-specific factors affecting particle residence time distribution of left atrial appendage in atrial fibrillation: A computational model-based study.

Atrial fibrillation (AF) is a prevalent arrhythmia, that causes thrombus formation, ordinarily in the left atrial appendage (LAA). The conventional metric of stroke risk stratification, CHA2DS2-VASc score, does not account for LAA morphology or hemodynamics. We showed in our previous study that residence time distribution (RTD) of blood-borne particles in the LAA and its associated calculated variables (i.e., mean residence time, tm , and asymptotic concentration, C ∞) have the potential to improve CHA2DS2-VASc score. The purpose of this research was to investigate the effects of the following potential confounding factors on LAA tm and C ∞: (1) pulmonary vein flow waveform pulsatility, (2) non-Newtonian blood rheology and hematocrit level, and (3) length of the simulation. Subject-Specific data including left atrial (LA) and LAA cardiac computed tomography, cardiac output (CO), heart rate, and hematocrit level were gathered from 25 AF subjects. We calculated LAA tm and C ∞ based on series of computational fluid dynamics (CFD) analyses. Both LAA tm and C ∞ are significantly affected by the CO, but not by temporal pattern of the inlet flow. Both LAA tm and C ∞ increase with increasing hematocrit level and both calculated indices are higher for non-Newtonian blood rheology for a given hematocrit level. Further, at least 20,000 s of CFD simulation is needed to calculate LAA tm and C ∞ values reliably. Subject-specific LA and LAA geometries, CO, and hematocrit level are essential to quantify the subject-specific proclivity of blood cell tarrying inside LAA in terms of the RTD function.

Molecular diffusion and pore-scale mechanical dispersion controls on time-variant travel time distribution in hillslope aquifers

The travel time distribution (TTD) is a lumped representation of the age composition of water leaving the system. Under nonstationary rainfall injection conditions, the TTD varies with transient groundwater flow, leading to a time-variant TTD. It reflects water and solute exports responding to rainfall. The time-variant TTD, time-variant residence time distribution (RTD) and storage selection function (SAS function) provide fundamental insights into transport and mixing processes in groundwater flow system. This study investigates the control of local-scale mixing (i.e., molecular diffusion and pore-scale mechanical dispersion) on time-variant TTD, RTD and SAS function. A solute transport model is constructed to simulate the time-variant TTD, RTD and SAS function under the influence of local-scale mixing; and a particle tracking model is applied to simulated time-variant TTD, RTD and SAS function without the influence of local-scale mixing. Generally, the TTD and cumulative TTD are found to be more sensitive to local-scale mixing than RTD and cumulative RTD due to the ‘volume effect’. The local-scale mixing smoothens the sharp fluctuations of backward TTD and SAS function. The differences of cumulative breakthrough curves and cumulative backward TTDs between two models deviate to negative value except the tails of the two functions, and this verifies the random velocity theory. The differences of cumulative RTDs between the two models indicate the hydrodynamic dispersion can result in more young water leaving the system and more old water remaining in the storage. Meanwhile, two rainfall time series, one with seasonality and one without seasonality, are applied in the two models separately; it is found that the hydrodynamic dispersion shapes the 2-dimensional backward TTD by increasing its continuity along the 45° direction. Finally, the spatial distribution of mean groundwater age is used to illustrate the local-scale mixing in the aquifer and further explain the controls of local-scale mixing on time-variant TTD, RTD and SAS function mechanistically.

Residence time distribution in RO channel

Residence time distribution (RTD) in a commercial Filmtec™ BW30-400 RO channel is determined using Computational Fluid Dynamics (CFD) and correlated with the Exponentially Modified Gaussian (EMG) model. The RTD, cumulative function of RTD, and flushing dynamics in RO channels are shown to be different from the analytical solutions derived for plane channels, due to the existence of spacer filaments in the former. The results for multiple channels connected in series are also acquired. The efficacy of purging after one displacement volume is estimated to be 90% for one RO element and 95% for seven RO elements connected in series.

Investigation of relationship between residence time distribution and size distribution of solid particles in tank leaching modelling

In the most important models of continuous tank leaching process, residence time distribution and particle size distribution are considered independently. While the studies by some researchers has shown that particle size distribution affect the residence time in the tank. However, no systematic study has been conducted on this issue yet. In this research, while studying the relationship between particle size and residence time, an empirical method for determining of residence time distribution based on inflow size composition to the agitated tank was presented. The relationship between average size of one size fraction of particles and its residence time was obtained by mean residence time and parameter (the slop of a fitted line to the data of a diagram that its horizontal axis is nondimensional time and its vertical axis is the natural logarithm of the ratio of solid concentration in time t to initial solid concentration). Results showed that with enhancing the particle sizes from -38 micron to the range of +125-150, mean residence time increases from 59.1 to 93 min and decreases from 0.95 to 0.3 .Also using calculated amount of for each particle fraction, residence time distribution function for inflow stream with different size distribution was determined. Furthermore, the effect of changing of parameter on the results of segregated flow model of leaching process was evaluated. Results showed that the decreasing in the from 1 to 0.7 leads to increasing the conversion from 0.71 to 0.78.

A New Mathematical Model for the Prediction of Internal Recirculation in Impinging Streams Reactors

A mathematical model for the prediction of internal recirculation of complex impinging stream reactors has been presented. The model constitutes a repetition of a series of ideal plug flow reactors and CSTR reactors with recirculation. The simplicity of the repeating motif allows for the derivation of an algebraic relation of the whole system using the Laplace transform. An impinging streams reactor system with one axial and two tangential inlet fluid streams was constructed and considered as a case study. The model predicts satisfactorily the complex and flow rate dependent experimental residence time distribution functions obtained employing a pulse tracer method for different total flow rates of the incoming feed. The variation of the controlling parameters with changing the total inlet flow rate is discussed. The presented model can predict complex internal recirculation streams within the impinging streams reactor system.

Investigation of Mixing in Tanks of a Special Geometry

Mixing is one of the most crucial processes in the chemical industry. Homogeneity is a requirement for all feedstocks and industrial products. The degree of mixing depends on the hydrodynamic properties of the fluid in the units. The Residence Time Distribution (RTD) was investigated in a tank of a special geometry. Mixing was investigated using various geometries of the tank by applying the Heaviside function in step-response experiments. After obtaining experimental results, the RTD function was calculated. The flow structure in the tank was approximated by fitting black-box transfer function models onto the RTD function of the system. Two general model structures were defined and their fitness compared. By evaluating the fitted models, a relationship was established between the flow structure in the tank and its geometry.

Residence time distribution in a structured packing unit for monitoring aerosol emissions

Solvent absorption in a structured packed column has been demonstrated as a viable option for post-combustion carbon capture. Amine emissions in the form of aerosol particles pose a major challenge that severely impacts the separation efficiency of the packed column. The aerosol particles quickly achieve the motion of the gas flow. In this context, residence time distribution (RTD) of the gaseous phase will provide insights for managing aerosol particles, thereby controlling amine loss in the packed column. Computational fluid dynamics (CFD) modeling of such columns is a multiscale problem because of a wide range of disparity in length scales. Accordingly, three-dimensional CFD simulations in the representative elementary unit of a structured packing are conducted using OpenFOAM. The RTD is analyzed in terms of residence time distribution function, mean residence time and variance. Effects of the diffusivity and flow rates on RTD are extensively studied, and the dispersion coefficient is calculated using a one-dimensional axial dispersion model. RTDs computed in a smaller domain are extrapolated to a larger domain using the convolution integral. The CFD computed and corresponding CFD models derived from the convolution integral match well at different flow rates and domain sizes. A comparison of RTD among diff ;erent packing designs is presented at a fixed number of repeating units and different flow rates. The mean residence time is found to larger than the superficial time, which is a unique behavior of structure packing. For a fixed number of repeating units, a packed column having higher specific area showed broad dispersion and the axial Péclet number decreases.

Numerical analysis of mixing in block‐head mixing screws

The research reported here used 3D non‐Newtonian flow simulations to investigate the pumping and mixing capability of block‐head mixers. Block‐head mixers are distributive mixing screws that are widely used to homogenize the polymer melt and eliminate thermal gradients. The polymer‐processing industry employs a variety of block‐head mixers, with little consensus on design and distribution of screw flights and mixing blocks. This analysis addresses this issue based on a computational design study in which the influence of three geometrical parameters was examined: (1) the number of flights at a mixing block, (2) the number of blocks along the screw, and (3) the stagger angle between the blocks. To examine the flow behavior of the mixing screws, the pressure consumption and energy dissipation is evaluated. Distributive mixing is analyzed using residence time distribution functions, kinematic stretching parameters, and the scale of segregation. Dispersive mixing is assessed by means of the mixing index and the shear stress. The results of this design study increase the understanding of block‐head mixers and contribute to the design and optimization of such geometries. The findings can further be applied to mixing screws of similar geometry, including pin‐type and knob mixers. POLYM. ENG. SCI., 59:E88–E104, 2019. © 2018 The Authors. Polymer Engineering & Science published by Wiley Periodicals, Inc. on behalf of Society of Plastics Engineers.

Hydraulic study of a non-steady horizontal sub-surface flow constructed wetland during start-up

This paper describes the hydraulic performance of a start-up, pilot-scale, horizontal sub-surface flow constructed wetland (CW), located outdoors at the Helmholtz UFZ, Leipzig. This paper aims to investigate the impact of the method of hydraulic calculation in a pilot-scale system. Impulse-response tracer tests were conducted at multiple depths and locations throughout the system and the uranine concentration was measured using a fluorometer. In addition, the volumetric flow rate was closely monitored and climatic data was gathered to support the hydraulic results. Werner and Kadlec's modified residence time distribution (RTD) theory (originally developed for systems with large flow rate and volume fluctuations) was applied and the results compared to those obtained using classic RTD theory.Progressive uranine dispersion, broadening of the RTD base, a change in peak shape and extended tailing were observed with increasing distance. All of these factors indicated deviation from plug flow and mixing effects with low-to-moderate dead volume. As this was a non-steady flow system, application of modified RTD theory ensured that the first moments of the normalized breakthrough curves and RTD functions were always unity. The Student's t-test (95% confidence) showed that the outlet RTDs calculated assuming steady-flow were significantly different, but those determined using the modified theory were closely comparable.In general, a decrease in flow rate from inlet to outlet was observed and fluctuations in the outflow were linked to climatic conditions. August was characterized by the highest temperatures, high global radiation and high rates of evapotranspiration. Low or no outflow was recorded in conjunction with high evapotranspiration. The lowest temperatures, low global radiation, low evapotranspiration and high humidity were recorded in October, as well as the second highest rainfall (82 mm) after June (115 mm). Surges in outflow were observed with rain events.

Spinning Disc Technology – Residence Time Distribution and Efficiency in Textile Wastewater Treatment Application

The spinning disc (SD) technology has received increased attention in the last years due to its enhanced fluid flow features resulting in improved property transfers. The actual study focuses on characterization of the flow within a spinning disc system based on experimental data used to establish the residence time distribution (RTD) and its dependence on the feeding liquid flowrate and the disc rotational speed. To obtain these data, an inert tracer (sodium chloride) was injected as a pulse input in the liquid stream entering the disc and the salt concentration of the liquid leaving the disc was continuously recorded. The obtained data indicate that an increase in the liquid flowrate from 10 L/h to 30 L/h determines a narrower RTD function. Also, at rotational speed of 200 rpm, the residence time distribution is broader than that for 500 rpm and 800 rpm. The RTD data suggest that depending on the needed flow characteristics, one can choose a certain flowrate and rotational speed domain for its application. Also, the SD technology was used to process textile wastewater treated with bentonite (as both coagulation and discoloration agent) in order to investigate whether the quality indicators such as the total suspended solid content, turbidity and discoloration, can be improved. The experimental results are promising since the discoloration and the removals of suspended solids attained values of over 40%, and respectively, 50 %, depending on the effluent flowrate (10 l/h and 30 L/h), and the disc rotational speed (200 rpm, 550 rpm and 850 rpm) without any other addition of chemicals, or initiation of other simultaneous treatment processes (e.g., advanced oxidative, or reductive, or biochemical processes). This recommends spinning disc technology as a suitable and promising tool to improve different wastewater characteristics.

Investigation of hydrodynamic non-idealities in a pilot-scale pond bioreactor

Pond-type reactors are widely used in biological wastewater treatment in view of their simplicity and low cost of construction. The knowledge of an approximate hydrodynamic behavior can contribute to the rational design of such operation. Thus, the aim of this work is to investigate, in a pilot scale pond bioreactor, the non-ideality through residence time distribution (RTD) technique and validation of a model for the homogeneous phase material balance. Experimental runs were carried out using the pulse injection technique of methylene blue tracer at constant water flow rate. The pilot-scale reactor constructed with a typical geometry design of those employed in animal waste treatment in southern Brazil. Residence time of 3, 6, and 9 days and a height/width ratio geometry parameter (H/D) of 0.25 and 0.18 were adopted. For each run normalized RTD functions and their moments (average residence time, variance and asymmetry) were evaluated. Variance analysis shows equivalent hydrodynamic behavior in the parameter range employed. To describe the non-idealities, we theorized several dimensionless RTD function models using a series of CSTRs (classified as conventional, bypass and recycle).The models were adjusted to the experimental data by non-linear regression and evaluated by the successive calculation method using the moments of RTD. The best model found is the representation of the pond reactor as two CSTRs in series with different volumes (first reactor with 91.5% of useful volume) and presence of dead zones (useful volume 84.9% of real volume).

Solute transport along a single fracture in a porous rock: a simple analytical solution and its extension for modeling velocity dispersion

A simple and robust solution is developed for the problem of solute transport along a single fracture in a porous rock. The solution is referred to as the solution to the single-flow-path model and takes the form of a convolution of two functions. The first function is the probability density function of residence-time distribution of a conservative solute in the fracture-only system as if the rock matrix is impermeable. The second function is the response of the fracture-matrix system to the input source when Fickian-type dispersion is completely neglected; thus, the effects of Fickian-type dispersion and matrix diffusion have been decoupled. It is also found that the solution can be understood in a way in line with the concept of velocity dispersion in fractured rocks. The solution is therefore extended into more general cases to also account for velocity variation between the channels. This leads to a development of the multi-channel model followed by detailed statistical descriptions of channel properties and sensitivity analysis of the model upon changes in the model key parameters. The simulation results obtained by the multi-channel model in this study fairly well agree with what is often observed in field experiments—i.e. the unchanged Peclet number with distance, which cannot be predicted by the classical advection-dispersion equation. In light of the findings from the aforementioned analysis, it is suggested that forced-gradient experiments can result in considerably different estimates of dispersivity compared to what can be found in natural-gradient systems for typical channel widths.

Study of fluid dynamic conditions in the selected static mixers part II‐determination of the residence time distribution

AbstractThe presented paper is part of the complete description of fluid‐dynamic conditions in the selected static mixers: Koflo and Kenics. The research was also performed for an empty pipe used as a reference state. The scope of work included the analysis of the residence time distribution in the mentioned devices for different Reynolds numbers for both laminar and turbulent flow regimes. The report contains a discussion about residence time distribution function E(t), cumulative distribution function F(t), as well as the parameters like the mean residence time tm and the variance σ2. The main aim of the presented work was to show the applicability of static mixers as chemical reactors and to present their operation characteristics to evaluate the derogations distinguishing them from well known ideal states (i.e. plug flow or ideal mixing). As a result of the accomplished study it was proved that Reynolds number increase results in narrower RTD in all of the tested devices, however the fluid motion is far from plug flow due to axial and radial mixing and the considered equipment should be treated as non‐ideal reactors requiring some additional models for the right description. What is more, in the case of laminar flow, the Kenics static mixer showed the narrowest spread and was considered as the best solution among the studied devices. In the turbulent flow, the difference between E(t) functions for all of the mentioned equipment was so small that the suggestion of small impact of insert type on the RTD was made.

Unified Analysis of Multi-Chamber Contact Tanks and Mixing Efficiency Evaluation Based on Vorticity Field. Part II: Transport Analysis

Mixing characteristics of multi-chambered contact tank are analyzed employing the validated three-dimensional numerical model developed in the companion paper. Based on the flow characterization, novel volumetric mixing efficiency definitions are proposed for the assessment of the hydrodynamic and chemical transport properties of the contact tank and its chambers. Residence time distribution functions are analyzed not only at the outlet of each chamber but also inside the chambers using the efficiency definitions for both Reynolds averaged Navier–Stokes (RANS) and large eddy simulation (LES) results. A novel tracer mixing index is defined to characterize short circuiting and mixing effects of the contact system. Comparisons of the results of these indexes for RANS and LES solutions indicate that mixing characteristics are stronger in LES due to the unsteady turbulent eddy mixing even though short circuiting effects are also more prominent in LES results. This result indicates that the mixing analysis based on the LES results simulates the mixing characteristics instantaneously, which is more realistic than that in RANS. Since LES analysis can capture turbulent eddy mixing better than RANS analysis, the interaction of recirculation and jet zones are captured more effectively in LES, which tends to predict higher turbulent mixing in the contact system. The analysis also shows that the mixing efficiency of each chamber of the contact tank is different, thus it is necessary to consider distinct chemical release and volumetric designs for each chamber in order to maximize the mixing efficiency of the overall process in a contact tank system.

Hydraulic performance of a modified constructed wetland system through a CFD-based approach

Low-cost household technologies, as horizontal subsurface flow constructed wetlands, are important to address water and sanitation needs in the Asia-Pacific region in a more integrated and sustainable manner, and a better understanding of these technologies would benefit their engineering design. Computational Fluid Dynamics (CFD) simulations of a modified constructed wetland system (EvaTAC) were undertaken to determine empirical effects of geometric and flow parameters on the hydraulic performance and the effluent pollutant fraction. The CFD model was validated by comparing the computed residence time distribution (RTD) with experimental results. RTD functions were then used to quantify hydraulic indexes: short-circuiting, mixing, and moment. The EvaTAC is composed of an evapotranspiration and treatment chamber (CEvaT) and a horizontal subsurface flow constructed wetland (HSSF-CW). For the CEvaT, length and the interaction between length and flow rate were the most important factors for the hydraulic efficiency. For the effluent pollutant fraction, the most important factor was flow rate. For the HSSF-CW, the strongest influence on the hydraulic efficiency was the length. Baffles and the interaction between length and baffles also had significant statistical influence on the hydraulic efficiency. Furthermore, the results showed that flow rate, length, and the interaction between flow rate and length influenced the effluent pollutant fraction significantly. Finally, a poor correlation between hydraulic indexes and effluent pollutant fraction was obtained, indicating that the hydraulic indexes are not good predictors of the effluent pollutant fraction.

Transfer function approach for artificial tracer test interpretation in karstic systems

Summary A karstic formation consists in a three-dimensional hydrological system which involves horizontal and vertical, diphasic or saturated water transfers characterised by a large range of velocity. These subsurface flow processes correspond to various water pathways through fractured, fissured, and underground streams or conduits leading to a nonlinear global behaviour of the system. An efficient way of investigating of a karstic system behaviour consists in the injection of artificial tracer tests at loss points and in careful analysis of the recovery tracer fluxes at one or several outlets of the systems. These injections are also an efficient way of providing hypotheses on characteristic time of contaminant transfer in these type of aquifers. Here, we propose a Laplace-transform transfer function of the Residence Time Distribution function that allows to discriminate between a quick-flow advection-dominated component and a slow-flow advection–dispersion/dominated component in the artificial tracer transfer in the system. We apply this transfer function on five high resolution sampling rate artificial tracer tests operated on the Baget system in the Pyrenees (France) in order to illustrate the advantages and limitations of this approach. We provide then an interpretation of the relationship between tracer test recovery shape and karstic system organisation between inlet and outlet site.

Hydrodynamic analysis and simulation of a flow cell ammonia electrolyzer

The hydrodynamic analysis and simulation of a non-ideal single pass flow cell alkaline ammonia electrolyzer was performed after the scale-up of a well-characterized deposited polycrystalline Pt on Ni anode. The hydrodynamic analysis was performed using the residence time distribution (RTD) test. The results of the hydrodynamic investigation provide additional insights for the kinetic analysis of the ammonia electrooxidation reaction on polycrystalline Pt electrocatalysts -which are typically obtained under controlled flow regime, e.g., rotating disk electrode- by including the flow non-uniformity present in the electrolyzer. Based on the RTD function, the ammonia electrolyzer performance was simulated as a non-steady stirred tank reactor (CSTR) and the unknown kinetic parameters were obtained by fitting the simulation results with an experimental current profile, obtaining an adequate prediction of the ammonia conversion. This simplified approach for the simulation of the ammonia electrolyzer could be implemented in process simulation packages and could be used for the design and scale-up of the process for hydrogen production and wastewater remediation.