Analytical Dispersion Model Research Articles

A simple and accurate model for attenuation and dispersion caused by squirt flow in isotropic porous rocks

Seismic waves propagating in fluid-saturated porous rocks exhibit attenuation and velocity dispersion in a broad range of frequencies. At sonic and ultrasonic frequencies, the attenuation is predominantly caused by fluid flow in cracks and grain contacts, so-called squirt flow. This physical mechanism for attenuation also may be relevant at seismic frequencies. We develop a simple and accurate analytical model for attenuation and dispersion caused by squirt flow in isotropic porous rocks. The input material properties for a specific rock model can be directly measured in a laboratory or calculated using analytical and numerical approaches. The results from our squirt flow model are compared with inherently accurate 3D numerical solutions for the same pore geometries. The analytical and numerical results are in good agreement. Furthermore, we observe that our analytical model is more accurate than the currently available analytical solution for squirt flow in isotropic porous rocks. MATLAB routines to reproduce the presented results are made available.

Lateral enhancement effect of reactive PELE: Two-step segmented simulation and analytical modeling

Comparing with traditional PELE, the reactive PELE shows a more significant lateral enhancement effect due to the deflagration reaction of the filling by impact. Here an analytical model for thin-walled jacket fragmentation and dispersion is presented based on the wave interaction and the reaction of reactive filling. Subsequently, a two-step segmented numerical simulation for the impact deflagration of reactive PELE is developed. Based on the numerical simulation, the wave interaction process and the internal stress distribution of the filling were given firstly, then the deflagration reaction of reactive filling, jacket fragmentation and dispersion were simulated. By the theory and simulation, the experiment results of reactive PELE impacting steel plate are analyzed. In the experiment, the thickness of the front steel target is 6mm-20 mm, and the multi-layer witness aluminum plate is set behind the steel target to obtain the lateral effect of the reactive PELE. The analysis results show that the released energy of the reactive filling increases the maximum radial velocity of the jacket fragments by more than 40 %, and the perforation dispersion on the witness plate significantly depends on the filling reaction level and the front steel plate thickness, the lateral enhancement mechanism of reactive PELE impacting plate is explained.

Effect of turbulence dispersion on bubble-particle collision efficiency

In this study, bubble-particle collision efficiency in a single-bubble system (bubble diameter ∼ 1 mm) for different turbulence intensity (Ti ∼ 4% − 20%) and various particle size (diameter ∼ 30 to 100 µm) were theoretically studied by injecting particles on a stationary bubble. A critical distance for particle injection upstream of the bubble was analytically determined ensuring no distortion of the flow streamlines at the particle release location. To quantify the turbulence collision efficiency, an analytical turbulence dispersion (TD) model was proposed. This model was also compared with a 3D computational fluid dynamics (CFD) model incorporating a random walk momentum exchange model. A statistical parameter, 90%-collision radius was introduced to represent effective particle releasing radius, which was shown to increase with the turbulence intensity and particle diameter. A threshold turbulence intensity of 7% was noted below which the analytical model agreed well with the CFD model, however, above this threshold, the analytical model was found to overestimate the collision efficiency and underestimate the 90%-collision radius. The CFD model demonstrated that with increasing turbulence intensity, energy dissipation rate in the system also increased and particles showed more stochastic pattern in their trajectories. A force analysis showed that larger diameter particles approaching the bubble experienced higher horizontal deflections in the higher turbulence intensity cases due to larger inertial forces acting on them, however such effect was not apparent farther downstream of the bubble.

Ab initio thermoelectric calculations of ring-shaped bands in two-dimensional Bi2Te3,Bi2Se3 , and Sb2Te3 : Comparison of scattering approximations

Materials with ring-shaped electronic bands are promising thermoelectric candidates, since their unusual dispersion shape is predicted to give large power factors. While previous calculations of these materials have relied on the assumption of a constant mean-free-path or relaxation time, recent first-principles modeling of electron-phonon scattering suggests that the scattering rates may be better approximated by the electron density-of-states (so-called DOS scattering model). In this work, we use density functional theory to investigate single and double quintuple-layer Bi$_2$Te$_3$, Bi$_2$Se$_3$ and Sb$_2$Te$_3$, with a focus on understanding how the three aforementioned scattering approximations impact thermoelectric performance -- emphasis is placed on the DOS scattering model. The single quintuple-layer materials possess two ring-shaped valence band maxima that provide an abrupt increase in conducting channels, which benefits the power factor. Additionally, below the band edge a ring-shaped minimum, located between the two maxima, is found to further enhance the thermoelectric performance but only with the DOS scattering model. This comes from a sharp drop in the DOS, and thus scattering, just below the ring-shaped minimum. An analytic octic dispersion model is introduced and shown to qualitatively capture the observed features. The double quintuple-layer materials display notably worse thermoelectric properties, since their dispersions are significantly modified compared to the single quintuple-layer case. The benefits of ring-shaped bands are sensitive to the alignment of the two ring maxima and to the degree of ring anisotropy. Overall, single quintuple-layer Bi$_2$Te$_3$ and Bi$_2$Se$_3$ are most promising, with the DOS scattering model giving the highest power factors.

Information-Driven Gas Source Localization Exploiting Gas and Wind Local Measurements for Autonomous Mobile Robots

Gas source localization (GSL) by an olfactory robot is a research field with a great potential for applications but also with numerous unsolved challenges, particularly when the search must take place in realistic, indoor environments that feature obstacles and turbulent airflows. In this work, we present a new probabilistic GSL method for a terrestrial mobile robot that revolves around the propagation of local estimations throughout the environment. By exploiting the geometry of the environment as the basis for this propagation, we avoid relying on analytical dispersion models, eliminating the need to assume controlled environmental conditions. Simulated and real experiments are presented in different indoor environments featuring multiple rooms and turbulent flows, demonstrating the suitability of our approach for locating the emitting gas source.

A novel analytical approach for advection diffusion equation for radionuclide release from an area source

Abstract The method of the Laplace transform has been used to obtain an analytical solution of the three-dimensional steady state advection diffusion equation for the airborne radionuclide release from any nuclear installation such as the power reactor in an area source. The present treatment takes into account the removal of the pollutants through the nuclear reaction. We assume that the pollutants are emitted as a constant rate from the area source. This physical consideration is achieved by assuming that the vertical eddy diffusivity coefficient should be a constant. The prevailing wind speed is a constant in x − direction and a linear function of the vertical height z. The present model calculations are compared with the other models and the available data of the atmospheric dispersion experiments that were carried out in the nuclear power plant of Angra dos Reis (Brazil). The results show that the present treatment performs well as the analytical dispersion model and there is a good agreement between the values computed by our model and the observed data.

Unified dispersion model for the electric susceptibilities of physically realizable materials

Based on the powerful method of network control theory, a complete, classically derived, broadband, analytic dispersion model is presented which describes electric susceptibility for a rigorously physically realizable, dielectric material system with a possible dc conductivity contributor. The analysis results in a susceptibility model constructed entirely from pole-residue groups. As a result, solitary Debye-type poles are not physically supported in the broadband analytic dispersion theory presented in this paper. While previous investigators have focused primarily on modeling susceptibility using only negative complex-conjugate poles, the type found naturally at optical frequencies and above, this analysis was conducted under no such restriction; hence, it is considered unified and should apply to all classical-type susceptibility behavior at all frequencies from dc to daylight, including some of the ultraviolet region where the material-continuum assumption begins to break down. Excluding conductivity due to the absence of any proof of magnetic monopole existence, the model should be equally applicable for modeling magnetic susceptibility dispersion. Based upon a rigorous emphasis on physical realizability, the complete, classical dispersion model is demonstrated by successfully modeling the complex permittivity of liquid water at 25 \ifmmode^\circ\else\textdegree\fi{}C using measurement data from 30 MHz to 1200 THz.

Pyrite Oxidation Halts Migration of a Phosphorus Plume.

We have established a monitoring record of phosphate (PO4 3- ) migration in the Long Point, ON campground septic system plume that now spans 26 years. Previously, at year 16 (2006), a P plume 16 m in length was documented and provided a good fit with an analytical advection dispersion model when a P migration velocity of 0.8m/yr was used (retardation factor of 37) and when P behaved in an otherwise conservative manner (sorption only). However, between years 16 and 26 (2016), the P plume length expanded by only 2m (0.2m/yr) and increased in depth by only 0.5m. The zone of abrupt P depletion at depth occurs close to the zone where SO4 2- concentrations increase in response to NO3 - oxidation of pyrite. Scanning electron microscope images of sand grains from the nose of the P plume reveal abundant authigenic mineral coatings of considerable thickness (∼5 to 20 μm), with Fe as the dominant cation and containing 1 to 3wt % P. This evidence suggests that P is now being attenuated along a reaction front that coincides with the zone where pyrite oxidation is occurring. P migration may now be controlled by the rate of migration of the pyrite oxidation front and this is several times slower than the previously indicated rate in the shallower, sorption-controlled portion of the plume. Monitoring at Long Point has demonstrated the danger of embracing an overly simplistic conceptual model when attempting to predict wastewater P migration in groundwater and also highlights the unique insight provided by a long-term monitoring record.

Dispersion of traffic derived air pollutants into urban parks

Scarce land resources in dense cities means that small urban parks are important as a leisure and amenity resource for the urban population. However, streets with heavily traffic often surround these fragmented parks and increase the potential user exposure to air pollutants from vehicles. The dispersion profiles of PM2.5 and black carbon from roadside into urban parks at pedestrian level, in Hong Kong, were measured using mobile high time resolution instruments. In the downwind direction, pollutant concentrations decreased rapidly from roadside and by some tens of metres reached relatively constant values. An even sharper gradient is found in the upwind direction, with a rapid increase detected within 2m of the road edge. The distinct decay profiles were explained with an analytical dispersion model formulated based on the gradient transport theory using an Eulerian approach. The simulations using the dispersion model suggest 17m as a typical halving distance under normal urban conditions, which is introduced to simplify the description of dispersion profiles. Using Hong Kong as an example, ~90% of urban parks, to different extent, overlap with the 17m halving distance from roads, which means few urban parks in Hong Kong avoid the impact from nearby traffic emissions. Thus, from the perspective of human exposure to air pollutants in urban parks, this study provides observations of relevance for future park design in dense cities.

Development of an analytical Lagrangian model for passive scalar dispersion in low-wind speed meandering conditions

Describing the effects of wind meandering motions on the dispersion of scalars is a challenging task, since this type of flow represents a physical state characterized by multiple scales. In this study, a Lagrangian stochastic diffusion model is derived to describe scalar transport during the horizontal wind meandering phenomenon that occurs within a planetary boundary layer. The model is derived from the linearization of the Langevin equation, and it employs a heuristic functional form that represents the autocorrelation function of meandering motion. The new solutions, which describe the longitudinal and lateral wind components, were used to simulate tracer experiments that were performed in low-wind speed conditions. The results of the comparison indicate that the new model can effectively reproduce the observed concentrations of the contaminants, and therefore, it can satisfactorily describe enhanced dispersion effects due to the presence of meandering.

Ship emissions measurement in the Arctic by plume intercepts of the Canadian Coast Guard icebreaker <i>Amundsen</i> from the <i>Polar 6</i> aircraft platform

Abstract. Decreasing sea ice and increasing marine navigability in northern latitudes have changed Arctic ship traffic patterns in recent years and are predicted to increase annual ship traffic in the Arctic in the future. Development of effective regulations to manage environmental impacts of shipping requires an understanding of ship emissions and atmospheric processing in the Arctic environment. As part of the summer 2014 NETCARE (Network on Climate and Aerosols) campaign, the plume dispersion and gas and particle emission factors of effluents originating from the Canadian Coast Guard icebreaker Amundsen operating near Resolute Bay, NU, Canada, were investigated. The Amundsen burned distillate fuel with 1.5 wt % sulfur. Emissions were studied via plume intercepts using the Polar 6 aircraft measurements, an analytical plume dispersion model, and using the FLEXPART-WRF Lagrangian particle dispersion model. The first plume intercept by the research aircraft was carried out on 19 July 2014 during the operation of the Amundsen in the open water. The second and third plume intercepts were carried out on 20 and 21 July 2014 when the Amundsen had reached the ice edge and operated under ice-breaking conditions. Typical of Arctic marine navigation, the engine load was low compared to cruising conditions for all of the plume intercepts. The measured species included mixing ratios of CO2, NOx, CO, SO2, particle number concentration (CN), refractory black carbon (rBC), and cloud condensation nuclei (CCN). The results were compared to similar experimental studies in mid-latitudes. Plume expansion rates (γ) were calculated using the analytical model and found to be γ = 0.75 ± 0.81, 0.93 ± 0.37, and 1.19 ± 0.39 for plumes 1, 2, and 3, respectively. These rates were smaller than prior studies conducted at mid-latitudes, likely due to polar boundary layer dynamics, including reduced turbulent mixing compared to mid-latitudes. All emission factors were in agreement with prior observations at low engine loads in mid-latitudes. Ice-breaking increased the NOx emission factor from EFNOx = 43.1 ± 15.2 to 71.6 ± 9.68 and 71.4 ± 4.14 g kg-diesel−1 for plumes 1, 2, and 3, likely due to changes in combustion temperatures. The CO emission factor was EFCO = 137 ± 120, 12.5 ± 3.70 and 8.13 ± 1.34 g kg-diesel−1 for plumes 1, 2, and 3. The rBC emission factor was EFrBC = 0.202 ± 0.052 and 0.202 ± 0.125 g kg-diesel−1 for plumes 1 and 2. The CN emission factor was reduced while ice-breaking from EFCN = 2.41 ± 0.47 to 0.45 ± 0.082 and 0.507 ± 0.037 × 1016 kg-diesel−1 for plumes 1, 2, and 3. At 0.6 % supersaturation, the CCN emission factor was comparable to observations in mid-latitudes at low engine loads with EFCCN = 3.03 ± 0.933, 1.39 ± 0.319, and 0.650 ± 0.136 × 1014 kg-diesel−1 for plumes 1, 2, and 3.

The Mathematical Treatment for Air Pollutant Diffusion Using Laplace Adomain Decomposition Method with Ground Surface

The method of Laplace Adomain Decomposition has been used to obtain a semi-analytical solution of the threedimensional steady state advection diffusion equation for dispersion of air pollutant from a point source. The present treatment takes into account a realistic boundary condition which considers the ground surface as an absorberreflector surface for the pollutant, simultaneously. This physical consideration is achieved by assuming that the vertical eddy diffusivity coefficient should be non-zero at the ground surface for vertical diffusion to be possible. The wind prevailing speed is parameterized in terms of vertical height using the power law profile. An upper boundary condition assuming capping inversion is considered which means that pollutant is subjected to a boundary Condition of zero flux. The present model calculations are compared with the available data of the atmospheric dispersion experiments that were carried out in the Copenhagen area (Denmark) and the semi-empirical model for Gaussian plume model with the same input data. In both comparison tasks, the results are reasonably good which indicates that the present treatment performs well as a simple analytical dispersion model.

An Analytical Dispersion Model for Sources in the Atmospheric Surface Layer with Dry Deposition to the Ground Surface

ABSTRACTThis study presents a dispersion model based on an analytical solution of two-dimensional advection-diffusion equation to simulate the crosswind integrated concentration in the bounded atmospheric boundary layer. The dispersion model considers the power law profiles of wind speed and vertical eddy diffusion coefficient and deposition velocity at the ground surface. A closed form analytical solution of the advection-diffusion equation for these parameterizations of wind and eddy diffusivity was derived by considering the deposition through lower boundary condition at the ground surface. A sensitivity analysis of the dispersion model for ground level concentrations with varying deposition velocities at the ground surface was performed. Hanford diffusion experiment for a depositing tracer was used to evaluate the dispersion model in stable condition. Model was evaluated by accounting both deposition and non-deposition at the ground surface. The simulated concentrations from the proposed dispersion model with dry deposition are in good agreement with those observed and the model predicts ~72% cases within a factor of two. It was shown that the consideration of no-deposition condition in the model for a depositing tracer gives rise to a larger over-prediction to the observations in comparison to with deposition.

Analytical dispersion model for the chain of primary and secondary air pollutants released from point source

Abstract An analytical model for dispersion of air pollutants released from a point source forming a secondary pollutant (e.g. chemical transformation or parent-daughter radionuclide chain) is formulated considering the constant wind speed and eddy diffusivities as an explicit function of downwind distance from the source in Cauchy (reflection-deposition type) boundary conditions. The dispersion of pollutants has been investigated by using the Gaussian plume dispersion parameters σ y and σ z instead of the diffusivity parameters K y and K z . For primary pollutant it was proposed to use the derived dry deposition factor instead of the source depletion alternative. An analytical solution for steady-state two-dimensional pollutant transport in the atmosphere is presented. Derived formulas include dependency from effective release height, gravitational and dry deposition velocities of primary and secondary pollutants, advection, surface roughness length and empirical dispersion parameters σ y and σ z . Demonstration of analytical solution application is provided by calculation of 135 Xe and 135 C air activity concentrations and the applicability of the model for the solution of atmospheric pollution transport problems.

Assimilation of concentration measurements for retrieving multiple point releases in atmosphere: A least-squares approach to inverse modelling

The study addresses the identification of multiple point sources, emitting the same tracer, from their limited set of merged concentration measurements. The identification, here, refers to the estimation of locations and strengths of a known number of simultaneous point releases. The source–receptor relationship is described in the framework of adjoint modelling by using an analytical Gaussian dispersion model. A least-squares minimization framework, free from an initialization of the release parameters (locations and strengths), is presented to estimate the release parameters. This utilizes the distributed source information observable from the given monitoring design and number of measurements. The technique leads to an exact retrieval of the true release parameters when measurements are noise free and exactly described by the dispersion model. The inversion algorithm is evaluated using the real data from multiple (two, three and four) releases conducted during Fusion Field Trials in September 2007 at Dugway Proving Ground, Utah. The release locations are retrieved, on average, within 25–45 m of the true sources with the distance from retrieved to true source ranging from 0 to 130 m. The release strengths are also estimated within a factor of three to the true release rates. The average deviations in retrieval of source locations are observed relatively large in two release trials in comparison to three and four release trials.

Spectral dispersion modeling of virtually imaged phased array by using angular spectrum of plane waves.

We present an analytical treatment for the relatively new spectral disperser termed virtually imaged phased array (VIPA). Angular spectrum representation of the input Gaussian beam helps us obtain an exact analytic dispersion model and a dispersion law for a general VIPA by using the principle of multiple-beam interference. The consideration of the optical aberrations caused by refractions makes our model more accurate and practical than previous models. The validity of the proposed dispersion law has been validated theoretically by comparing with previous results. Some considerations of using a VIPA are also provided.

Scale-Dependent Dispersion within the Stratified Interior on the Shelf of Northern Monterey Bay

Abstract Autonomous underwater vehicle measurements are used to quantify lateral dispersion of a continuously released Rhodamine WT dye plume within the stratified interior of shelf waters in northern Monterey Bay, California. The along-shelf evolution of the plume’s cross-shelf (lateral) width provides evidence for scale-dependent dispersion following the 4/3 law, as previously observed in both surface and bottom layers. The lateral dispersion coefficient is observed to grow to 0.5 m2 s−1 at a distance of 700 m downstream of the dye source. The role of shear and associated intermittent turbulent mixing within the stratified interior is investigated as a driving mechanism for lateral dispersion. Using measurements of time-varying temperature and horizontal velocities, both an analytical shear-flow dispersion model and a particle-tracking model generate estimates of the lateral dispersion that agree with the field-measured 4/3 law of dispersion, without explicit appeal to any assumed turbulence structure.

Air Quality Prediction of PM10 through an Analytical Dispersion Model for Delhi

An analytical dispersion model for point, line and area sources is formulated in this work. The analytical solution of an advection-diffusion equation with the Neumann (total reflection) boundary conditions for a bounded domain is obtained for point sources using the separation of variable and wind speed as a power law profile of vertical height above the ground. The downwind and vertical eddy diffusivities are considered as an explicit function of downwind distance and vertical height. The formulations for line and area sources are obtained by integrating the point source formulation in crosswind, crosswind and downwind directions, respectively. A gridded emissions inventory of Delhi City for the year 2008–09 has been developed to estimate the strength of emissions from various sources, namely vehicles, industries, power plants and domestic sources, and this has been made using primary and secondary data of PM10 (particulate matter with aerodynamic diameter ≤ 10 µm). Dispersion models generally require steady and horizontally homogeneous hourly surface and upper air meteorological observations. However, hourly meteorological observations are not easily available for most of the locations in India. To overcome this limitation, meteorological variables are computed using the Weather Research and Forecast (WRF) model (version 3.1.1) developed by the National Center for Atmospheric Research (NCAR). The performance of this analytical model is evaluated through concentrations of PM10 monitored at different locations in Delhi. The observed data for the period December 2008 has been obtained from the Central Pollution Control Board (CPCB). The model’s predicted values are found to be in good in agreement with the observed values. However, the model results depend on the accuracy of source strength data, i.e., the estimated values of emission rates for the various air pollutants.

Development and evaluation of a ground-level area source analytical dispersion model to predict particulate matter concentration for different particle sizes

The dispersion of particulate matter downwind of a biosolids applied site was modeled incorporating dry deposition. For this purpose, a ground-level area source analytical dispersion model known as particulate matter deposition – PMD was developed. The PMD model includes dry deposition which involves particulate deposition velocities in predicting concentrations. The performance of this model was statistically evaluated after applying it to particulate matter emitting from an agricultural field applied with biosolids. The results of the statistical evaluation are presented for different particle sizes. The statistical evaluation results showed that the PMD model satisfied the criteria for most of the statistical performance measures considered. This study concludes that the PMD model can be used effectively in predicting particulate matter concentrations with dry deposition in unstable conditions (B and C) at different wind speed categories for near-field downwind distances of an area source.

An analytical model for solute transport through a GCL-based two-layered liner considering biodegradation

An analytical model for solute advection and dispersion in a two-layered liner consisting of a geosynthetic clay liner (GCL) and a soil liner (SL) considering the effect of biodegradation was proposed. The analytical solution was derived by Laplace transformation and was validated over a range of parameters using the finite-layer method based software Pollute v7.0. Results show that if the half-life of the solute in GCL is larger than 1year, the degradation in GCL can be neglected for solute transport in GCL/SL. When the half-life of GCL is less than 1year, neglecting the effect of degradation in GCL on solute migration will result in a large difference of relative base concentration of GCL/SL (e.g., 32% for the case with half-life of 0.01year). The 100-year solute base concentration can be reduced by a factor of 2.2 when the hydraulic conductivity of the SL was reduced by an order of magnitude. The 100-year base concentration was reduced by a factor of 155 when the half life of the contaminant in the SL was reduced by an order of magnitude. The effect of degradation is more important in approving the groundwater protection level than the hydraulic conductivity. The analytical solution can be used for experimental data fitting, verification of complicated numerical models and preliminary design of landfill liner systems.