Dimensionless Dispersion Coefficient Research Articles

Gas Dispersion Coefficient Test System and Dimensionless Inversion Method for Porous Media.

Due to the complex porous media structure of the longwall gob area, it has been difficult to determine the gas dispersion coefficient of oxygen when studying spontaneous coal combustion in the gob area. In this work, we first designed an experimental device for testing the gas diffusion coefficient of porous media. Then, the distribution law of gas concentration in porous media under different particle size conditions was obtained by experiments. Subsequently, we established a dimensionless mathematical model of gas dispersion in porous media and developed a corresponding numerical simulator based on the finite volume method (FVM). The influence of the dimensionless gas dispersion coefficient on the gas concentration distribution was analyzed, and then a dimensionless inversion method of the gas dispersion coefficient was summarized and put forward. Finally, we obtained the values of the gas dispersion coefficient in the experimental device under different particle size conditions by inversion and discussed its effect on the gas dispersion behavior in porous media. The results show that (1) the distribution of gas concentration obtained from the experimental test and numerical simulation is consistent, which verifies the reliability of our work; (2) the dimensionless gas concentration is the highest near the injection point and gradually decreases along the depth and both sides of the test container; (3) with the increase of the dimensionless gas dispersion coefficient, the distance required for uniform gas mixing in the test container is gradually shortened and the gas dispersion coverage is wider; and (4) the larger pore space facilitates the dispersion behavior of the gas, and the gas dispersion coefficient shows a parabolic trend with the increase of porous medium particle size.

Development of Simple Formula for Transverse Dispersion Coefficient in Meandering Rivers

This study aims to develop a straightforward and practical formula to estimate transverse dispersion coefficients in meandering natural rivers, a critical factor for predicting solute transport. We present a novel expression for the transverse dispersion coefficient based on dispersion and hydraulic data sets obtained from tracer experiments conducted in natural rivers. A distinctive feature of the formula is its reliance on one dimensionless hydraulic parameter, u¯u*hRc. To assess the effectiveness and accuracy of our proposed formula, we compare it with previously established equations commonly employed in the field. Furthermore, we apply the formula to natural river bends situated in the Nakdong River of Korea. This equation serves to estimate the initial value of the dispersion coefficient in two-dimensional solute transport modeling. As a result, the calibrated value of the dimensionless transverse dispersion coefficient is 0.97, which is only a 16% difference between the initial value of 0.81 as obtained from the formula. The formula presented in this study, simplifying and utilizing the dimensionless hydraulic parameter, offers a promising approach to estimating transverse dispersion in natural meandering rivers in cases where tracer and secondary flow data are unavailable. Additionally, the formula can be refined with more recent dispersion data, leading to a clearer, more straightforward, and validated formulation that captures the intricate interplay between topography and transverse dispersion.

Influences of Momentum Ratio on Transverse Dispersion for Intermediate-Field Mixing Downstream of Channel Confluence

This study aims to analyze the influences of momentum ratio (Mr) and confluence angle (α) on the transverse dispersion in an urban scale confluence channel from the numerical simulation results using the Environmental Fluid Dynamics Code model. By changing the momentum flux and confluence angle from the simulation results, the analysis focused on the relations between the vertical variations of transverse velocity and transverse dispersion. The high momentum tributary aligned the mixing interface toward the outer bank and created a strong helical motion, which transported the contaminated water along the channel bed and inflows into the recirculation zone. The high momentum ratio induced the large vertical shear in transverse velocity with a strong helical motion and increased the transverse dispersion. However, the helical motion persistence rapidly decreased as the flow reached downstream and led to a decrease in the transverse dispersion for the large confluence angle. Thus, the transverse dispersion coefficient increased with a high momentum ratio and low confluence angle, and the dimensionless transverse dispersion coefficient was in the range of 0.39-0.67, which is observed in meandering channels, for Mr > 1 and α = 45°.

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.

Self-trapped dynamics of a hollow Gaussian beam in metamaterials

We present a systematic investigation on the dynamics of a hollow Gaussian beam (HGB) in metamaterials. We predict self-trapped propagation of HGBs and evolution of the beam is highly influenced by dimensionless dispersion coefficient (κ), which determines the strength of dispersion over diffraction. The evolutions of HGBs such as disappearance of single ringed intensity pattern and appearance of patterns with a central bright spot are achievable with less propagation distance in metamaterials with higher values of κ. On the other hand, metamaterials with low values of κ can preserve single ring intensity distribution over a long propagation distance without focusing. When the strength of dispersion over diffraction increases, it significantly influences the evolution of the beam and may lead to the formation of tightly focussed beam with high peak intensity at the center. The phenomenon of tight focussing is found to have some applications in trapping of nanosized particles.

Longitudinal and transverse dispersion coefficients of 2D contaminant transport model for mixing analysis in open channels

In this study, longitudinal and transverse dispersion coefficients of the two-dimensional advection-dispersion model were calculated using both the velocity and concentration data obtained through tracer tests in large-scale meandering channels. The velocity-based dispersion coefficients calculated by applying the theoretical equations to the velocity data were compared with concentration-based dispersion coefficients that were calculated by applying the routing procedure to the measured concentration curves. The results showed that the cross-sectional averaged values of the dimensionless longitudinal dispersion coefficient, DL/HU∗, calculated from the velocity profile data ranged 4.1–6.5, which corresponded to the theoretical value, whereas DL/HU∗ by concentration data ranged 14.7–35.5, which was 4–6 times larger than the velocity-based coefficient. The cross-sectional maximum values of the dimensionless transverse dispersion coefficient, DT/HU∗, by the velocity-based method ranged 0.1–3.3, whereas DT/HU∗ by concentration-based method was 0.9–2.9. These results revealed that the velocity-based longitudinal dispersion coefficient is a fractional value of the concentration-based result, which contained mixing caused by channel irregularities throughout the reach, in addition to shear dispersion by vertical velocity variations, while transverse dispersion coefficients by both methods represented the same dispersion mechanism which was dominated by the secondary currents developed in highly sinuous channels.

Dispersion Process in Sewer Pipes with Sediments and Deposits

This paper describes the dispersion process in sewer pipes, which is from the hydraulic point of view a prismatic stream channel with relatively constant roughness of streambed. In such hydraulic conditions should the effect of “dead zones” not occur, but this effect was observed during the field experiments. The reason for this was the presence of bed sediments and deposits, which form together with other small obstacles irregularities in the sewer pipe such dead zones. Dead zones are areas with small flow velocities, which act as a zone with transient (temporary) storage, where the pollution is accumulated and released gradually later. This process modifies the dispersion process in sewer systems and causes irregularities in the transport process. Field experiments were performed in a straight sewer section and also in the part with directional changes of sewer line, both under dry weather flow conditions, i.e. with relatively low pipe filling, discharges and velocities. Sewer pipes had a low slope, so a lot of deposits and sediments were present. Paper presents the results of field experiments and analyse the impacts of sediments and deposits in sewer system on the transport and dispersion process, which is reflected in the value of dispersion coefficient. In the case of sewer pipelines, the most important is the longitudinal dispersion coefficient DL. Comparing the values of DL in the straight part and in the part with directional changes, DL values in the straight part were higher than from the section with directional changes. The maximal value of the dimensionless longitudinal dispersion coefficient p reached 25.2 in a straight section; in the part with the directional changes p it was up to 39.3. This result indicates that in the straight part of the sewer line, the longitudinal dispersion of the substance is dominant in the total dispersion process, whereas in the sewer part with directional changes, the transversal (lateral) dispersion contributes to the whole mixing process (dispersion process is also influenced by the velocity gradient in the transverse direction). Within the measured channel section, there were three directional changes - angles of 90°, 135° and 105°. The influence of the direction change angle to the longitudinal dispersion coefficient within the performed measurements has not been clearly determined yet.

Sudden pollutant discharge in vegetated compound meandering rivers

In this study, the effects of different vegetation densities and different relative flow depths on the longitudinal dispersion coefficient are investigated in a compound meandering channel. Simulated vegetation with three different densities was placed over the floodplain and tracer was released in line simultaneously and equally in the main channel and flood plain. Digital image processing technique was used to measure the tracer concentration along the channel by analyzing a series of sequential images of the tracer cloud. Acoustic Doppler Velocimeter was used to measure 3-D velocity components. The results showed that the depth-averaged longitudinal velocity increases in the main channel due to the presence of vegetation but it declines in the floodplain. The maximum amount of the turbulent kinetic energy and the dimensionless longitudinal dispersion coefficient (K/U⁎H) were observed at the bend apex. Moreover, as the relative flow depth increases, K/U⁎H declines in the compound meandering channel for all the vegetated cases. Additionally, the longitudinal dispersion coefficient increases up to 59% in the main channel and decreases up to 42% in the floodplain by increasing the vegetation density in a specific relative flow depth.

Direct numerical simulation of hydrodynamic dispersion in open-cell solid foams

Fully resolved simulations of flow and mass transfer in a unit cell of structured open-cell foam catalysts are presented. Numerical studies are conducted on a uniform three-dimensional Cartesian grid where the fluid-solid interface coupling is enforced via a sharp interface Immersed Boundary technique. Several validation cases for the numerical method are presented followed by extensive calculations to quantify hydrodynamic dispersion in open-cell foams. In our study five different porosities of the idealized foam structure, represented by the spatially periodic Kelvin’s unit cell, were considered. Dimensionless dispersion coefficients were calculated for varying Péclet numbers and flow directions using volume-averaging theory. Our numerical studies indicate that Taylor dispersion is the dominant mechanism for structured porous media in the Darcy-Brinkman flow regime.

An experimental study on the performance of an electro-dialysis desalination using hollow cubic assembled porous spacers fabricated by a 3D printer

In order to improve the performance of a filter press type electro-dialysis (E.D.) system, 5 types of hollow cubic assembled porous spacer fabricated by a 3D printer were applied instead of the conventional mesh spacers. The effects of structure of porous spacers on the E.D. system performance were experimentally investigated in terms of limiting current density, stack voltage and pressure drop. It was found that the porous spacer with vertical staggered arrangement normal to membrane surface and with smaller interfacial surface area between spacer and membranes contributes a higher limiting current density, which is about 2.5–3.0 times higher than that of E.D. system without spacer. Moreover, this spacer filled in E.D. system results in a relative lower electrical resistance and a smaller pumping power compare to other spacers. Since the vertical staggered structure is able to mix the flow (i.e. mechanical dispersion), supplies ions in the flow towards membranes, and suppresses concentration polarization, so that increases the LCD at expense of slightly increase in electrical resistance. Moreover, dimensionless mechanical dispersion coefficient towards membranes was estimated by fitting the experimental data with the analytical solution proposed in previous paper. Finally, a suggestion for making a good porous spacer has been proposed.

A comparative study of longitudinal dispersion models in rigid vegetated compound meandering channels

Releasing and mixing of pollutants in rivers can cause serious threats for downstream users. Longitudinal dispersion is an important factor in describing pollutant transport process in rivers. The focus of this study is on the effect of floodplain rigid vegetation characteristics including its arrangement, density and relative depth (the ratio of the flow depth in the main channel to that over floodplain) on flow and longitudinal dispersion coefficient in a meandering channel with compound cross section. Digital image processing technique was used to measure the tracer concentration. Sequential images were continuously captured at seven sections downstream of release point. PVC cylinders were installed over the floodplain to simulate rigid vegetation in tandem and random arrangements. Acoustic Doppler Velocimeter (Micro-ADV) was used to measure 3-D velocity components. The results showed that roughening the flood plain with stems increases the longitudinal flow velocity and Reynolds shear stress in the main channel, while these parameters decrease in the flood plain compared to non-vegetated cases. As a result, in the presence of rigid vegetation, the travel time decreases in the main channel up to 20% compared to non-vegetated conditions. As the relative flow depth increases, the dimensionless longitudinal dispersion coefficient, K/U∗H, decreases up to 74%. Also, the dimensionless longitudinal dispersion coefficient increases up to 38% with stem density in a specific relative flow depth of Dr = 0.11. Finally, a comparison has been made between laboratory data and some well-known previously reported models. Later, the most accurate equations are presented for predicting the longitudinal dispersion coefficient in rigid vegetated compound meandering channels. The results of the present study reveal that rigid vegetation over flood plain can be used as an alternative method in river rehabilitation projects for environmental management purposes.

CFD analysis and flow model reduction for surfactant production in helix reactor

Flow pattern analysis in a spiral Helix reactor is conducted, for the application in the commercial surfactant production. Step change response curves (SCR) were obtained from numerical tracer experiments by three-dimensional computational fluid dynamics (CFD) simulations. Non-reactive flow is simulated, though viscosity is treated as variable in the direction of flow, as it increases during the reaction. The design and operating parameters (reactor diameter, number of coils and inlet velocity) are varied in CFD simulations, in order to examine the effects on the flow pattern. Given that 3D simulations are not practical for fast computations needed for optimization, scale-up and control, CFD flow model is reduced to one-dimensional axial dispersion (AD) model with spatially variable dispersion coefficient. Dimensionless dispersion coefficient (Pe) is estimated under different conditions and results are analyzed. Finally, correlation which relates Pe number with Reynolds number and number of coils from the reactor entrance is proposed for the particular reactor application and conditions.

Pollutant Spreading in a Small Stream: A Case Study in Mala Nitra Canal in Slovakia

The Water Framework Directive requires as an obligatory goal to achieve and to keep “good water quality” status within the defined period (for Slovakia—up to the year 2015). For surface waters, the main criterion is the ecological and chemical status of the water. Mathematical and numerical modelling allows evaluating various situations of contaminants spreading in rivers (from everyday wastewater disposal through fatal accidents and discharges of the toxic substances) without immediate destructive impact to the environment. Determination of longitudinal and transverse dispersion coefficient values, as the main hydrodynamic characteristics of the dispersion, has the highest extent of uncertainty for hydrodynamic models simulating pollutant transport in streams. This paper deals with the determination of dispersion coefficients based on field tracer experiments performed in a small modified stream (basic hydrodynamic parameters during the experiments were: discharge Q = 0.138–0.553 m3.s−1, depth h = 0.29–0.48 m, width B = 5.2–5.9 m). During the experiments, various conditions and situations were taken into account, e.g., continuous and instantaneous pollution source, as well as various positions of pollution source along the river width, among others. Field measurements were evaluated using three different methods for dispersion coefficient determination: based on statistical evaluation, based on analytical solutions of advection–dispersion equation, and using numerical models. The dimensionless dispersion coefficients values were determined, which can be used for numerical simulation of pollutant transport in similar types of streams.

Modeling dimensionless longitudinal dispersion coefficient in natural streams using artificial intelligence methods

Learning the Longitudinal Dispersion (LD) mechanism in natural channel is vitally important to be able to control water pollution and to prevent different stratification in flow that crucial for water resources conservation for both human and aquatic life. Many related studies can be found in the existing literature. However, almost all studies aim to investigate the mechanism of the LD in natural channels or to model dimensional LD coefficient. The main goal of this work is to develop three models based on different artificial neural network techniques to predict dimensionless (not dimensional) LD coefficients in natural channels. Another goal of this study is to present a large and critical assessment on the existing studies made on both dimensional and dimensionless LD. The data sets obtained from the literature concerning more than 30 rivers at different times in the United States of America, include the depth, the width, and the mean cross-sectional velocity of the flow, shear velocity, and dimensionless longitudinal dispersion coefficient. The results have been compared with the data at hand, a fuzzy based model, and seven conventional equations proposed in literature by using statistical magnitudes, error modes, and contour map method. It is observed that the feed forward neural network yields the best reliable results.

Spatially Variable Dispersion Coefficients in Meandering Channels

To analyze the spatial variability of two-dimensional dispersion coefficients in meandering channels, an observed dispersion tensor was calculated by applying a theoretical equation to the measured velocity profiles of two laboratory channels. From the observed dispersion tensor, local longitudinal and transverse dispersion coefficients were estimated by the least squares method. In both channels, the spatial variation in the dispersion coefficients was found to be strongly dependent on the specific location within the channel, because alternating bends in channels induces the periodic generation of secondary currents that alter the magnitude of transverse dispersion. Dimensionless transverse dispersion coefficients of each measurement section vary from 0.33 to 1.14 for the M1 channel and from 0.63 to 3.40 in the M2 channel around the entrance and apex of the second bend. The maximum negative correlation occurred near the apex of the channels because of the separation of the point of maximum primary velocity and the center of the circulation cells by the complex flow structure in the meandering channel.

Solving Initial Value Problem by Matching Asymptotic Expansions

An asymptotic expansion in powers of small dimensionless dispersion coefficient is sought to solve an initial value problem posed for an advection diffusion equation modeling orientation of pulp fibers in a steady fully turbulent flow. The regular expansion is shown to be nonuniform in a small neighborhood of $\phi=0$. Although the highest order derivative with respect to the orientation angle $\phi$ is multiplied by the small parameter, application of matched asymptotic expansions to obtain the inner solution in a small neighborhood of $\phi=0$ matchable with the regular expansion turned out to be unsuccessful. The multiple scales do not lead to the solution either. The problem is solved by matching two asymptotic expansions, one solving the initial value problem in a small neighborhood of the initial point, while another one solves the equation at large distances from the initial point. Thus, this is an example of using the method of matched asymptotic expansions to satisfy the given initial condition by the long-distance approximation of the solution to a nonsingular partial differential equation.

Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids

A theory is developed to describe a structural instability that has been observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.

Pollutant dispersion in water currents under wind action

A closed expression for the dimensionless dispersion coefficient due to differential advection in the presence of wind is obtained, through integration of the associated velocity and concentration vertical profiles. Tsuruya's model is used for the velocity distribution, while the concentration profile is obtained from a simplified pollutant transport equation explicitly deduced. The expression reduces to Elder's value (≈5.9) for the particular case of no wind. Higher values arise for winds acting in the same direction as the flow; and vice versa, at least for a practical range of wind velocities. For the case in which the wind acts along a direction non-coplanar with the mean flow direction, it is proposed to apply the analysis independently to the coplanar direction and its normal, where the mean current is, then, null. For this normal direction, the dimensionless dispersion coefficient value is around 24.9. In this way, the theory can be applied to 2D-horizontal pollutant transport modeling.

On Mixing and Segregation in Binary Solid−Liquid Fluidized Beds

A conceptual effective voidage model of a binary solid−liquid fluidized bed proposed by Bhattacharyya and Dutta (Ind. Eng. Chem. Res. 2002, 41, 5098) and successfully used to interpret the phase inversion behavior has been applied to the mixing and dispersion phenomena. A computational procedure for the determination of the axial concentration profiles of the two types of particles varying in size, density, or both has been developed. The values of the axial dispersion coefficients of the particles are evaluated simultaneously, satisfying the boundary conditions for the fluidized bed. Computed axial concentration profile and dispersion coefficients compared pretty well with available experimental data. Also, the experimental observations on the characteristics of liquid−solid fluidized beds as reported by different workers could be explained using the present model. A correlation for the dimensionless axial dispersion coefficients of the particle species was also developed. The most important feature of the model is that it relies upon the basic properties of the fluid and the solid particles and fitting of experimental axial concentration data are not required.

An Impedance Probe for the Measurements of Flow Characteristics and Mixing Properties in Stirred Slurry Reactors

Anew probe based on the measurement of electrical impedance has been developed to measure solid/liquid hold-up in stirred tank reactors working with high solid concentration. The main advantage of the probe is that it is fairly non-intrusive and can be used for both laboratory-scale and industrial stirred tank reactors. The relationship between impedance and liquid hold-up closely follows the predictions estimated with the Bruggeman’ s model. The solid concentration was measured in a 0.5-m diameter vessel mechanically stirred with a Lightnin A310 impeller, set at an off-bottom clearance C = 0.25 T. Three sizes of Lightnin A310 impeller were used corresponding to impeller to tank diameter ratios of D/T = 0.346, 0.386 and 0.5. The liquid phase was tap water, whereas two different sets of spherical glass beads (of density ps = 2500 kg/m3) with narrow size distributions and mean diameters dp = 90 and 600 μm, respectively, were used as solid phase. Solid loads up to 40% by weight were investigated. In addition to solid concentration, power consumption was measured by means of strain gauge technique, and interface height was determined. Preliminary results show that dimensionless axial dispersion coefficient, Des/ND2, decreases with increasing solid load.