Axial Dispersion Coefficient Research Articles

Industrially relevant Radioactive Particle Tracking study on the motion of adsorbent granules suspended in a pilot-scale water–air three-phase fluidized bed

The information obtained by one-dimensional Radioactive Particle Tracking is used to make an industrially relevant description of the macroscopic mixing of a three-phase bubble column. The objective is to characterize and compare the solid motion of granular activated carbon (dp = 1 mm) and calcium alginate beads (dp = 5 mm), widely used as adsorbents for pollutants and as support for catalysts, enzymes, and living organisms. The particles are suspended in water with upflowing air into a 0.1 m diameter and a 1 m height bubble column. The liquid–solid suspensions are in batch, and the air superficial velocity ranges within 0.01–0.12 m/s. The solid axial hold-up distribution of the carbon–water–air system is biased towards the bottom of the column, while the alginate–water–air system is more even. The axial dispersion coefficients obtained in both systems have a positive linear behavior within the operating range, which is uncannily marked in the carbon-water-air system, and it is an order of magnitude less than the alginate–water–air system. The contrasts of the two systems are mostly explained as a function of the solid densities. Axial mixing time distribution shows a sharp minimum at the second quartile of the column. Flow regime transition is assessed using information theory.

Liquid and solids phase backmixing in a bubble and slurry bubble column using a virtual tracer response methodology based on the trajectory data of the radioactive particle tracking (RPT) technique

AbstractVirtual tracer response methodology developed based on the trajectory data of the computer aided radioactive particle tracking (CARPT) technique was demonstrated. The demonstrated virtual tracer technique has advantages of non‐invasiveness, near perfect injection/sampling, and flexibility in choosing the sampling/injection boundaries in a specific spatial pattern. With the developed virtual tracer technique, liquid and solids backmixing were investigated at the conditions mimicking Fischer‐Tropsch synthesis. Experiments were conducted at different pressure, solids loading, and superficial gas velocity. The axial dispersion model (ADM) and recirculation and cross flow dispersion (RCFD) models were used to model the liquid mixing. Transient sedimentation dispersion model (SDM) was used to model the solids mixing. It was found that the measured axial dispersion coefficient (Dl) in the ADM model increases with increase in the pressure. The increase of dispersion coefficient was explained with the experimental values of mean axial diffusivity and mean recirculation velocity. Axial dispersion coefficients (Dz,uDz,d) in the RCFD model (compartment model) were apparently lower than the Dl, due to the decoupling of global recirculation from the dispersion coefficients in the RCFD. Further, it was found that the dispersion coefficients in the RCFD model follow the trend of the axial eddy diffusivity with change in the operating conditions revealing the dominance of the turbulence in the upflow and downflow compartments. From the solids backmixing study, axial dispersion coefficient (Ds) of solids was found to increase with increase in the solids loading, pressure, and superficial gas velocity at the studied conditions.

Experimental and modeling investigation of cobalt ion extraction in multistage extractor: Efficient evaluation of mass transfer coefficients using forward mixing approach

The current study develops comprehensive mass transfer models such as the plug flow model (PFM), backflow model (BFM), axial dispersion model (ADM), and forward mixing model (FMM). A new model of mass transfer was developed based on the size distribution of the dispersed phase droplets, and numerically solved using the technique of fitting concentration profiles. The studied process is a pilot-scale multistage extractor designed to provide proper contact between organic and aqueous phases for the recovery of Co(II) ions from dilute solutions. The great agreement of FMM was achieved with an average deviation (AD) of about 10.99%, and 10.50% for X and Y, respectively. The models were further used to evaluate process parameters and physical properties on the backflow, and axial dispersion coefficients. The analysis on the Ec of continuous phase indicates that the proportion of the axial dispersion coefficients transfer increased rapidly from 0.62 to 1.52 cm2/s as rotation speed increased from 160 to 220 rpm. The agitation rate has a positive influence on the mass transfer, while the effects of inlet flow rates are less than rotation speed. The FMM model has shown to be robust and reliable in the design of the multistage extractor.

Experimental investigation and mass transfer modelling in rotating disc contactor with asymmetric configuration for zinc recovery

• Overall mass transfer coefficient was illustrated by PFM, BFM, ADM, and FMM models. • For the first time, FMM was used for the description of zinc extraction. • Maximum yield of zinc ions ~97.67% was obtained at 555 rpm of rotation speed. • The curve fitting of the concentration profile is incorporated in the modeling approach. • Agitation speed has a considerable impact on the magnitude of the mass transfer coefficients. • Results reveal that the lowest absolute relative errors belong to the FMM model. • Concept of the mass transfer rates for Zn(II) extraction applies to wastewater treatment. This work reports a detailed investigation on Zinc ion recovery in a rotating disc contactor with asymmetric configuration. The variables are the rotation speed and the liquid flow rates of organic and aqueous phases. The maximum yield of zinc ions ~97.67% was obtained at 555 rpm of rotation speed in the extraction stage. Most modeling investigations (plug flow, backflow, axial dispersion) on mass transfer coefficients assume uniform conditions for the dispersed phase. This assumption does not seem realistic, since the non-uniform condition and the limited contact area may result in an appreciable deviation from the ideal flow regime. In this study, the forward mixing model with drop size distribution is applied through experimental and modeling approaches. The curve fitting of the concentration profile is incorporated in the modeling approach to accurately extract zinc ions to estimate the mass transfer coefficient. According to the experimental data, the agitation speed has a considerable impact on the magnitude of the mass transfer coefficients, axial dispersion, and backflow coefficients. The experimental results also reveal that the lowest absolute relative error for X-values, and Y-values equal to 5.21, and 5.20, respectively belong to the forward mixing model. This paper provides case studies of the extraction of zinc ions that verify this concept of the mass transfer coefficients that applies to the refinery of wastewater treatment.

Study on Dispersed-Phase Axial Dispersion in an Agitated–Pulsed Solvent Extraction Column with a Step Tracer Injection Technique

In this work, the residence time distribution (RTD) of the dispersed phase (aqueous phase) and continuous phase (organic phase) in an agitated-pulsed extraction column (APC) was measured online with a step tracer injection technique and the axial dispersion was subsequently calculated via RTD analysis. It was found that both the agitation speed and the pulsation intensity had noticeable effects on the axial dispersion of the dispersed phase, which escalates with the increase in the agitation speed and this is especially authentic at high pulsation intensity. The dispersed-phase axial dispersion coefficient also becomes higher with the increase in the dispersed-phase velocity. Compared with the dispersed-phase axial dispersion in the APC, the continuous-phase axial dispersion is 2-3 times higher. Empirical equations have been proposed to correlate the dispersed- and continuous-phase axial dispersion coefficients for variation of both agitation and pulsation conditions in APC. This study could be valuable for assessing the dispersed-phase axial dispersion in liquid-liquid extraction columns.

Numerical approximation of nonlinear and non-equilibrium model of gradient elution chromatography

A nonlinear lumped kinetic model of liquid chromatography is formulated and solved numerically to theoretically investigate the effect of column overloading on gradient elution. Linear solvent strength (LSS) model is utilized for Henry’s constant, non linearity coefficient and axial dispersion coefficient. A semi-discrete high-resolution finite volume scheme is extended and applied to obtain the approximate solutions of the governing model equations. The effects of changing modulator concentration are examined on the single and two-component elution. The benefits of gradient elution over isocratic elution are thoroughly discussed. The influences of optimizing free-parameters available in gradient chromatography are analyzed on the efficiency of the column and on the production of targeted components. For instance, the results obtained are used to study the effects of gradient slope, modulator concentration, solvent strength, nonlinearity coefficient, mass transfer coefficient, and axial dispersion coefficient on the concentration profiles.

Mathematical modeling and numerical simulation of sulfamethoxazole adsorption onto sugarcane bagasse in a fixed-bed column

Having rigorous mathematical models is essential for the design and scaling of adsorption columns. In this study, the dynamic behavior of the sulfamethoxazole adsorption on sugarcane bagasse was studied and compared using analytical models and a theoretical mechanistic model. Initially, fixed-bed column tests were carried out at different flow rates and bed heights. Then, the experimental data were fitted with the most widely used analytical kinetic models, and their fit and fixed-bed parameters were compared with the mechanistic model. Of all analytical models analyzed, the Log-Gompertz model was the one that had the best agreed with experimental data. Although some analytical models fitted the experimental data accurately, their usefulness was questionable. Their parameters did not show a clear relationship with the change in operating conditions, and in certain cases had different behavior from that observed in experimentation. Conversely, the mechanistic model not only predicted the breakthrough curves with great accuracy in the initial and transition stage (R2 > 0.92; SSE < 0.06), but it also estimated relevant parameters. Additionally, the effects of the global mass transfer coefficient (Ki) and the axial dispersion coefficient (Dz) on breakthrough curves were studied using the mechanistic model. Increasing Ki increased the slope of the breakthrough curves with a faster adsorption rate. Similarly, high values of Dz produced lower adsorption capacities of the adsorbent; and it was established that the axial dispersion is relevant in SMX adsorption on SB. The theoretical model presented can be used for the design, scaling, and optimization of adsorption columns.

Particle-scale study of heat and mass transfer in a bubbling fluidised bed

In this study, the multiphase flow and thermochemical behaviours of char combustion in a bubbling fluidised bed (BFB) are simulated using CFD-DEM approach featuring particle size polydispersity and thermochemical sub-models. The model is first validated in terms of mixing index, particle temperature, and particle diameter. Then, it is applied to examine the contribution of each heat transfer mode and study particle-scale behaviours of char and sand comprehensively. The results show that the polydisperse drag model should be used to accurately reproduce bed hydrodynamics when simulating a BFB system with polydisperse particles. Under the simulation conditions, the particle-averaged heat fluxes to char particles through convection, conduction, radiation, and char reaction take 9.79%, 0.82%, 40.44%, and 48.95%, respectively; the particle-averaged heat fluxes to sand particles through convection, conduction, and radiation take 30.28%, 1.0%, and 68.72%, respectively. For reactive char particles, the radiation and heat of reaction are dominated, while for inert sand particles, the radiation and convection are dominated; and for both particle species, the conduction is negligible. Axial dispersion coefficient is one order of magnitude larger than the horizontal one, demonstrating the dominant role of the introduced gas flow in determining bed hydrodynamics.

Rotary drums with sectional internals: Experimental investigation on the influence of section number and section length

The impact of sectional internals, as often used in industrial rotary drums, on the axial solid transport in rotary drums is investigated experimentally under variation of the operation parameters solid feed rate, drum inclination and rotational speed. Different section numbers and internal-lengths have been analyzed, applying the axial dispersion model, in a fully optically accessible lab scale rotary drum apparatus.The mean residence times are increasing with the section numbers and their length up to 4 times compared to bare drum results. Capacity limits have been detected, resulting in higher back-spillage ratios dependent on the set of operational parameters. The impacts of parameter changes on the axial dispersion are decreasing with section number and length, resulting in a more constant axial dispersion coefficient of about 1 × 10−6 m2/s. A qualitative similarity of the axial solid bed profile proposed by the Saeman-model for bare drums has been found for sectional internals.

Mixing of nanoparticle agglomerates in fluidization using CFD-DEM at ABF and APF regimes

Mixing of nanoparticle agglomerates in a fluidized bed was investigated by CFD-DEM. A parallel program was used that simultaneously uses the computational resources of GPU and CPU on a single computer. The bed expansion obtained from simulation was compared with experimental and good agreement was observed. Bed expansion in agglomerate particulate fluidization (APF) was 2 to 5 times the initial bed height and final bed expansion in agglomerate bubbling fluidization (ABF) was 1 to 2 times the initial bed height by changing the gas velocity from 2 to 4 and 4 to 8cm/s for APF and ABF regimes respectively. Gas volume fraction distribution was investigated for both APF and ABF regimes, which showed a more uniform distribution in APF. Increasing the gas velocity led to a more homogenous in the APF bed but it did not change the homogeneity of the ABF bed. Besides, the axial and radial dispersion coefficients in ABF were greater than those in APF. Axial dispersion coefficient varied from 2×10−6 to 4×10−5m2/s. Lacey mixing index was investigated in both APF and ABF regimes and it showed that the quality of mixing in ABF was better than APF.

GPROMS-driven modeling and simulation of fixed bed adsorption of heavy metals on a biosorbent: benchmarking and case study.

Adsorptive separation of heavy metals from wastewater is a viable approach to reuse it and avoid environmental pollution. The productive employment of adsorptive separation at a commercial scale, however, relies on the optimized conditions of an adsorber bed holding maximum and selective isolation of the heavy metals. The experimental route includes a significant trial and error approach, is time-consuming, involves operating cost, and remains economically unattractive. Contrarily, simulation of a mathematical model mimicking the adsorption system along with experimental validation can significantly minimize optimization efforts and suggests the best conditions of separation. In this work, a convective-dispersive model and adsorption model for fixed bed adsorption of copper (Cu), chromium (Cr), and cadmium (Cd) metals over wheat bran biosorbent are simulated using the gPROMS tool for benchmarking. The influence of feed flow rate, bed height, and metal concentration is studied, and breakthrough profiles of all heavy metals are predicted and matched with the literature. The error values (R2 and RMSE) and Chi-squared values determined from gPROMS simulations matched well with the previously available MATLAB-simulated data. After a successful benchmarking, we modeled pilot-scale adsorption of Cr on coconut coir (or Biosorbent) in a gPROMS simulation environment. A detailed method and algorithm of gPROMS simulation for Cr isolation is provided. The influence of feed flow rate, bed height, and initial metal concentration is studied on the breakthrough curves of the Cr. The optimum operating condition for the pilot-scale isolation of Cr from the water is suggested. The parameters, such as the axial dispersion coefficient and distribution coefficient, are determined.

Model representation of flow patterns in the displacement of non-Newtonian products from spiral-wound membranes

In the production of food concentrates with spiral-wound membranes, the product is usually displaced by water prior to chemical cleaning procedures or start-up/shut-down sequences. While it is known that the duration of the displacement depends on the flow rate, the properties of the involved fluids, and the geometry of the flow channels, the process itself, and in particular the flow patterns determining the residence time, are not well understood. Therefore, the aim of this study was to elucidate the flow distribution in the sections of a plant equipped with spiral-wound membranes and how displacement curves originate. To this end, cumulative residence time curves were first recorded from experiments with reconstituted milk protein concentrates and xanthan solutions as non-Newtonian model solutions. Then, two mechanistic models were developed and fitted to the experimental data. For the first model, it was assumed that the membrane module, its housing and the connecting pipes could be represented by a series connection of plug flow reactors with axial dispersion and stirred tank reactors. In a second, more detailed model, bypass and short-circuit flows were also considered. The results support the hypothesis that bypass and short-circuit flows exist in the plant. In addition, the models provide axial dispersion coefficients for the displacement of reconstituted milk protein concentrates and xanthan solutions from spiral-wound membranes by water.

Application of transfer function for quick estimation of gas flow parameters—A useful model‐based approach to enhancing measurements

AbstractAt the core of chemical engineering, process design is fundamental to reach a goal of productivity. Axial dispersion is one of the key phenomena that can affect productivity. Therefore, its characterization is necessary for adequate modeling and designing devices. Applications of the transfer function (TF) to the derivation of a high precision model of tracer flow and, next, determination of an axial dispersion coefficient in a commercial measurement system are presented. A TF concept makes easier customization of model ideas to different systems and consequently allows for obtaining a model that matches in the best way a physical system. The method takes advantage of the efficient numerical algorithm by den Iseger, that is; inverse Laplace transform to solve the model both at the stage of model development (boundary value problem) and model application (inverse boundary value problem). As a result, a very precise model of commercial measurement instrument was developed and, next, it was employed to the determination of axial dispersion coefficients for an empty tube and packed bed. The method presented is precise in a wide range of operating conditions. The paper shows that mathematical modeling can be exploited to enhance measurements for a commercial measurement instrument, that is, unlock the system's full potential with no equipment design changes. The method is also a good alternative to computational fluid dynamics for high precision calculations—it fits experimental results better than a CFD model.

Solid Flow in Rotary Drums with Sectional Internals: An Experimental Investigation

AbstractThe influence of different internal geometries on the solid flow in rotary drums under variation of the operational parameters feed rate, slope, and rotational speed is investigated. Four different configurations of a fully optically accessible lab‐scale rotary drum are analyzed. The internals are built as sectional compartments of the cross section, referring to industrial applications. The axial dispersion model is used to calculate the mean residence times and axial dispersion coefficients. It can clearly be shown that the sectional internal geometries significantly increase mean residence times and holdups in dependence on the set of operational variables. The axial dispersion coefficients were in a comparable range with literature data.

Reversed-phase chromatographic separation and downstream precipitation of lupane- and oleanane-type triterpenoids: Experiments and modeling based on the method of moments

The reversed-phase chromatographic separation of two triterpenic acids (TTAs), betulinic and oleanolic acids, using a triacontyl (C30) stationary phase was addressed in this work. Methanol, water, acetonitrile, ethanol, isopropanol, ethyl acetate, acetone and mixtures thereof were tested, and the best mobile phase to conduct the separation was found to be methanol/acetonitrile 50/50 (%, v/v) at 23 °C, taking into account parameters like selectivity, resolution and TTAs solubility. The method of moments was used to determine the equilibrium constants of isotherms, the axial dispersion coefficients and the global linear driving force coefficients of pure betulinic and pure oleanolic acids. These parameters were then successfully validated by modeling unary and binary breakthrough curves. Simulated moving bed calculations showed that betulinic and oleanolic acids can be both obtained with purity of 99.2 % and productivity of 56.2 kg/(m3adsorbent day) using the packing material of an Acclaim C30 column with a 1-1-1-1 configuration with columns of 7.5 cm long. Finally, in order to recover the two TTAs from the SMB extract and raffinate streams, water was envisioned as a precipitation agent. Accordingly, the solubility of each TTA was measured in methanol/acetonitrile 70/30, 50/50, and 30/70 (%, v/v) modified with water. The obtained results showed that adding 65 % (%, v/v) of water it is possible to precipitate 98 % of the dissolved TTAs in all the tested methanol/acetonitrile mixtures.

Microcapillary film reactor outperforms single-bore mesocapillary reactors in continuous flow chemical reactions

Meso- and micro-flow reactors are routinely used in continuous flow chemistry, however the role of capillary diameter, D, on conversion and reaction rates is often overlooked during scale-up. Volume, pressured drop and diffusion distances/times must be delicately balanced to fully realize the hydrodynamic capabilities of continuous chemical flow reactors. We carried out a comprehensive Computational Fluid Dynamics analysis experimentally validated with detailed fluid tracing, residence time distributions and continuous chemical reactions (neutralization and 4th Bourne reaction) to fully elucidate the role of D and molecular diffusion in reagents dispersion and chemical conversion. To our understanding, we captured and reported both numerically and experimentally for the first time the transition from convective, segregated flow to plug flow and dispersed flow, which we propose is linked to a dimensionless ratio between the time scales of diffusion to convection, tdiff/tconv. We tested three tubular systems: a small-bore (i.d. ~1100 µm) and large-bore (i.d. ~2400 µm) capillary reactor and a novel multiplexed (10-bore) Microcapillary Film Reactor (MFR) with mean i.d. 363 ± 32.2 µm. In the MFR’s narrow microcapillaries we observed excellent radial diffusion linked to the small diffusion distance, with low dimensionless axial dispersion coefficient values (Dax/uL) ranging from 0.0015 ± 0.0005 to 0.0033 ± 0.0006 (for flow rates 0.5–5.0 mL/min), exhibiting all the desired features of a high-performance ‘plug’ flow system. Dax/uL remained mostly independent of the Reynolds number, whereas for the single, large bore capillary the Dax/uL values (0.032–0.057) increased linearly with the Reynolds numbers (19.4–48.5), shifting towards very dispersive flow. We propose splitting flow through multiple parallel microcapillaries as in the MFR is a superior strategy for scaling-up continuous flow reactions compared to increasing D, which neglects diffusive effects.

Approximate shock layer solutions for the separation of multicomponent mixtures with different axial dispersion or mass transfer coefficients

In this paper new approximate shock layer solutions for the separation of mixtures with different axial dispersion or mass transfer coefficients are presented and analyzed. These solutions are useful for understanding the dynamics of this kind of separation processes including fixed bed and countercurrent adsorption as well as distillation processes. Further, they can be used for model reduction using so called wave models.

Computational Fluid Dynamics Modelling to Predict Axial Dispersion in Pulsatile Liquid-liquid Two-phase Flow in Pulsed Sieve Plate Columns

ABSTRACT Computational Fluid Dynamics (CFD) is used to predict continuous phase axial dispersion coefficient for liquid-liquid two-phase flow in a pulsed-sieve plate column (PSPC). The CFD model is based on Euler-Euler model in which the drop diameter used in the interphase momentum exchange term is obtained by coupled solution of the population balance equations. Species transport is additionally solved for the continuous phase to obtain its residence time distribution (RTD) curve which is further processed to estimate axial dispersion coefficient of the continuous phase. Experimental studies are also conducted in a 3-inch diameter PSPC to obtain axial dispersion coefficient in the continuous phase for different operating conditions. The continuous phase is water. 30% (v/v) tributyl phosphate mixed in dodecane is the dispersed phase. Validation of the CFD model is done with the experimental data of axial dispersion coefficient of the continuous phase. The average absolute relative error between estimated and predicted axial dispersion coefficient of continuous phase is less than 3%. The validated model is used to understand two-phase hydrodynamics in the column.

Continuous adsorptive removal of glimepiride using multi-walled carbon nanotubes in fixed-bed column.

Water pollution by emerging pollutants such as pharmaceutical and personal care products is one of today's biggest challenges. The presence of these emerging contaminants in water has raised increasing concern due to their frequent appearance and persistence in the aquatic ecosystem and threat to health and safety. The antidiabetic drug glimepiride, GPD, is among these compounds, and it possesses adverse effects on human health if not carefully administered. Several conventional processes were proposed for the elimination of these persistent contaminants, and adsorption is among them. Therefore, in this study, the adsorptive removal of GPD from water using multi-walled carbon nanotubes (MWCNT) supported on silica was explored on a fixed-bed column. The effects of bed-height, solution pH, and flow rate on the adsorptive removal of GPD were investigated. The obtained adsorption parameters using Sips, Langmuir, and Freundlich models were used to investigate the continuous adsorption. The results showed that the drug removal is improved with the increasing bed height; however, it decreased with the flow rate. The effect of pH indicated that the adsorption is significantly affected and increased in acidic medium. The convection-dispersion model coupled with Freundlich isotherm was developed and used to describe the adsorption breakthrough curves. The maximum adsorption capacity (qm) was 275.3 mg/g, and the axial dispersion coefficients were ranged between 3.5 and 9.0 × 105 m2/s. The spent adsorbent was successfully regenerated at high pH by flushing with NaOH.

Determination of Bodenstein number and axial dispersion of a triangular external loop airlift reactor

In this research, the residence time distribution (RTD) approach was used for the downcomer of the special-shaped triangular external loop airlift reactor (EL-ALR) by applying a modified downcomer to describe the flow pattern and dispersion behavior. Moreover, the effects of the three parameters of the superficial gas velocities of the downcomer and riser (Vgs1 and Vgs2, respectively), and the angle of the hypotenuse (α) were studied. Response surface methodology (RSM) was applied to analyze the results statistically and also to optimize the mixing conditions. In addition, the Bodenstein number (Bo), liquid velocity (UL), and axial dispersion coefficient (Dax) were determined. A mathematical relationship with a high accuracy was developed for the prediction of Bo number in the reactor downcomer. Also, it was shown that the reactor configuration was the most significant factor among all the studied parameters that affected the reactor performance. As a result of the RTD analysis, the flow pattern had been tended to be perfectly mixed. In this regard, the obtained values of Bo number were within the range 7.18–11.21 that confirmed this finding. The predicted optimum conditions (α = 60°, Vgs1 = 0.005 m/s, and Vgs2 = 0.019 m/s) were validated using the experimental control test, indicating a small deviation of ±0.39.