Mathematical Model For Dispersion Research Articles

Concept for a System of Emergency Response During Dispersion of Radioactivity or Pollution in the Atmospheric Boundary Layer

The paper presents a self-consistent physical mathematical model for dispersion, transport, deposition and resuspension in the atmospheric boundary layer in which the turbulent diffusion coefficients are determined from the concentration distribution function. The available meteorological and local geographical information together with appropriate initial and boundary conditions are used to relate the field measurements to the concentration dispersion function and correspondingly, to the diffusion coefficients. This makes it possible to find parameters needed for parameterisation of turbulent diffusion coefficients from field experiments. A scheme is proposed for solving a self-consistent problem for various parameterisations of diffusion coefficients. The model can easily be used for a local monitoring network at any kind of facility or installation during emergencies because it uses real-time measurements of pollutant concentration near the surface and/or deposition as well as other diffusion characteristics. The model output permits prompt assessments of the emergency situation and provides aid in making decisions concerning mitigation of the consequences of the accident.

Mathematical model for dispersion with a moving front in groundwater : Strack, O D L Water Resour ResV28, N11, Nov 1992, P2973–2980

Mathematical model for dispersion with a moving front in groundwater : Strack, O D L Water Resour ResV28, N11, Nov 1992, P2973–2980

A mathematical model for dispersion with a moving front in groundwater

A new, non‐Fickian, constitutive equation is presented for the mass flux due to macroscopic dispersion in porous media. This constitutive equation contains the classical term of the diffusive type as well as a new term that may be viewed as an inertia term. Combination of the constitutive equation with the mass balance equation for a conservative contaminant yields a set of four first‐order partial differential equations in terms of the three components of the vector of dispersive mass flux and the concentration. For the case of longitudinal dispersion only, this system is reduced to a set of two first‐order partial differential equations in terms of the convective and dispersive mass fluxes as the two unknown functions. The sets of differential equations are hyperbolic and can be used to simulate and predict the movement of the front of a contaminant plume at finite velocity. An expression of the velocity of the front is presented in terms of the Darcian velocity and the ratio of the two dispersion coefficients that enter into the model. The effect of the non‐Fickian terms decreases for increasing times; the model reduces to the classical one as time increases. An exact solution is presented for the case of one‐dimensional flow and compared with the results of simulated column experiments.

CANCER NEAR THE THREE MILE ISLAND NUCLEAR PLANT: RADIATION EMISSIONS

As a public charge, cancers among the 159,684 residents living within a 10-mile (16-km) radius of the Three Mile Island nuclear plant were studied relative to releases of radiation during the March 28, 1979, accident as well as to routine plant emissions. The principal cancers considered were leukemia and childhood malignancies. Estimates of the emissions delivered to small geographic study tracts were derived from mathematical dispersion models which accounted for modifying factors such as wind and terrain; the model of accident emissions was validated by readings from off-site dosimeters. Incident cancers among area residents for the period 1975-1985 (n = 5,493) were identified by a review of the records at all local and regional hospitals; preaccident and postaccident trends in cancer rates were examined. For accident emissions, the authors failed to find definite effects of exposure on the cancer types and population subgroups thought to be most susceptible to radiation. No associations were seen for leukemia in adults or for childhood cancers as a group. For leukemia in children, the odds ratio was raised, but cases were few (n = 4), and the estimate was highly variable. Moreover, rates of childhood leukemia in the Three Mile Island area are low compared with national and regional rates. For exposure to routine emissions, the odds ratios were raised for childhood cancers as a whole and for childhood leukemia, but confidence intervals were wide and included 1.0. For leukemia in adults, there was a negative trend. Trends for two types of cancer ran counter to expectation. Non-Hodgkin's lymphoma showed raised risks relative to both accident and routine emissions; lung cancer (adjusted only indirectly for smoking) showed raised risks relative to accident emissions, routine emissions, and background gamma radiation. Overall, the pattern of results does not provide convincing evidence that radiation releases from the Three Mile Island nuclear facility influenced cancer risk during the limited period of follow-up.

A time dependent mathematical model for dispersion of air pollutants from point sources

A comprehensive time‐dependent model for atmospheric pollution from a point source is presented in this paper. The effects of surface and inversion layers are taken into account. The pollutants emitted from an industrial stack are assumed to be situated in the surface layer. The atmospheric diffusion equation is solved analytically with appropriate boundary conditions in the atmospheric layer a ≥ z ≥ h, where h and a are the heights of the surface layer and the inversion layer respectively. The interaction of pollutants with the underlying surface has been included in this model through boundary conditions. It has been observed that the effect of leakage of pollutants from the top of the inversion layer is to decrease the pollutant concentration.

Ambient air concentrations of particulate matter from passenger cars

Using our measurement results on particulate emissions from passenger cars we have calculated ambient air concentrations for various US and European scenarios. This was carried out with the help of mathematical dispersion models for different traffic situations including street canyons and motorways. We have been very conservative in our choice of the scenarios, i.e. we have always used situations in which there are very high stress levels (e.g. constantly high traffic flow instead of average traffic flow). Finally, the thus determined air concentrations are compared with the corresponding air quality standard available from the literature.

Uncertainty in risk analysis: an alternative approach through pessimisation

Probabilistic risk analysis (PRA) was originally devised for situations where the spread of consequences is dominated by random variables, such as those describing the weather conditions when a release takes place, which belong to a well understood and measurable frequency distribution. Careful interpretation is required, however, in other applications, where the spread of the variables represents uncertainty due to lack of data and the probability distributions are to some extent arbitrarily assigned. Practical computational difficulties can also arise in situations where the risk appears to be concentrated into a narrowly defined region around a particular set of parameter values. An iterative approach to PRA is suggested, in which a readily available computer subroutine is first used to locate any such peaks and determine the sensitivity of the risk to each of the variables near to the peak. The results are then used as a guide for reappraisal of the input data and, in particular, to ensure that the final analysis correctly represents expert opinion in the most important areas that have been identified. The method is demonstrated with a simple mathematical model of dispersal from a nuclear waste repository.

Perilymph volume flow measured under physiological conditions

This study made use of an ionic tracer (trimethylphenylammonium, TMPA) to mark perilymph in one region of the cochlea. Perilymph flow was assessed by monitoring tracer movement to other regions with ion selective electrodes. Small, nontoxic, quantities of tracer (typically a 50‐nl bolus of 150 nM TMPACI) were required, since concentrations as low as 1 μM were readily detected. Stringent precautions were taken to seal the injection and recording electrodes into the perilymphatic scalae so that no artifactual flow was induced by perilymph leakage. Results were compared with a mathematical model of tracer dispersion by volume flow and passive diffusion. For scala tympani, TMPA dispersion corresponded to a flow rate of 1.6 nl/m in the apical direction. This rate was significant (p < 0.05), although the physiological effects of such a low flow rate are presumed to be negligible. Perforation of the cochlear apex resulted in flow rates of over 1 μ1/m, almost 1000 times the physiological rate. In scala vestibuli, no significant flow could be detected under normal conditions. These results demonstrate that in the sealed state, longitudinal volume flow of perilymph is extremely low. Artifactual volume flow is readily induced by procedures which involve perforation of the otic capsule. [This work supported by NIH.]

Atmospheric dispersion model of removal mechanism with integral term

A general mathematical model for atmospheric dispersion is suggest, in the form of an integro-partial differential equation, the integral term characterizing the removal mechanism which may arise due to absorption of air pollutant in fog or rain droplets present in the atmosphere. This model is applied to the study of the effects of instantaneous and delayed removal on dispersion of air pollutant from an elevated time-dependent point source. It has been shown that the concentration of the pollutant along the central line at a given time and point decreases due to the removal process and the concentration of pollutant with a delayed removal process may be greater than that for an instantaneous removal case.

Liquid mutual diffusivities of the H2O/D2O system

The liquid-phase mutual diffusivities of the water (H2O) and deuterium oxide (D2O) system at 298.2 K were measured using an instrument based on the Taylor dispersion technique. The instrument has been designed to match, as closely as possible, the mathematical model of ideal Taylor dispersion, minimizing all the departures from the ideal model. The diffusivities were measured over the entire concentration range and the results follow a linear dependence on molar fraction given by 109D12 =\(2.24 - 0.36x_{D_2 O} \), where D12 is in m2·s−1. Comparison with highly accurate data obtained by a Rayleigh interferometer seems to indicate that the accuracy of the present instrument is 1%. The hard-sphere model was applied to the estimation of the mutual diffusivities of this system and good agreement was found with experiment, deviations being ±3.5%.

A multidisciplinary approach to the assessment of ocean sewage sludge disposal

AbstractThis article presents a multidisciplinary approach to assess the hazards of ocean sewage sludge dumping. It combines physical and chemical sludge characteristics, estimated safe chronic levels and bioaccumulation potential with estimated environmental exposure (concentration and time) information. The hazard assessment was used to evaluate sewage sludges from 12 water pollution control plants operated by the New York City Department of Environmental Protection. Using a mathematical dispersion model, the worst‐case minimum dilution after 4 h was predicted to be 11,500:1. Of the three marine species tested, mysid shrimp were found to be the most sensitive. Using an application factor of 0.05, estimated chronic safe levels were estimated, the values do not exceed the limiting permissible concentrations, after allowing for initial dilution. Also, the concentrations of the chemicals in the liquid sewage sludge phase were below ambient water quality criteria. The results suggest that the ocean disposal of sewage sludge does not appear to present the environmental hazard originally perceived.

Dispersion Simulation in Two‐dimensional Tidal Flow

An accurate numerical method for the mathematical modeling of contaminant dispersion in two‐dimensional tidal currents is developed and applied. The method avoids the excessive numerical damping or oscillations associated with most finite difference and finite element schemes for advection by using a characteristics approach with high order Hermite bicubic interpolation. The split‐operator algorithm, incorporating a Crank‐Nicolson operator for diffusion, provides a relatively simple and economic method for accurate simulation of pollutant dispersion on a fixed, Eulerian mesh. A special procedure for Lagrangian calculation of the dispersion of concentration fields which are small compared to the mesh size simulates the early stages of growth of point source plumes. These various procedures are described in detail, and their performance is demonstrated by application to schematic test cases and to the Bay of Saint‐Brieuc, France.

Mathematical model of beam intensity dispersion at the detector plane in a molecular-beam apparatus

A mathematical model of intensity dispersion at the detector plane in a molecular-beam apparatus has been developed for the case of a ribbon-shaped beam effusing from a narrow rectangular slit. The model is developed in terms of position probability distribution functions and Maxwellian velocity distribution functions. The relation derived for the axial flux density is nv̄(Δx′)(Δx′)/4π2L2 in units of number of particles per unit area per unit time. A plot of flux density versus off-axis position agrees with the reported distributions obtained experimentally.

Erratum for “Mathematical Modeling of Dispersion in Mixed Estuaries”

Erratum for “Mathematical Modeling of Dispersion in Mixed Estuaries”

Mathematical modeling and computer simulation of radioactive and toxic chemical species dispersion in porous media in the vacinity of uranium recovery operations

The management of uranium recovery operations, such as mill production, tailings, heep leaching or solution mining (in-situ mining), has been one of the most challenging issues facing environmental engineers and scientists in the last decade. Almost any activity (mill production, evaporative and holding ponds, storage sites of chemicals, tailings ponds and transportation) results in radioactive and toxic chemicals contamination of surface and groundwaters. Current estimates based on observations of plant operations indicate that 70–80% of the water used in mill production is lost by seepage and 10–20% by evaporation. This large quantity of seepage raises serious questions as to the degree of migration of radioactive and potentially toxic chemical species from the tailings pond and their impact on the surrounding surface and subsurface environments. This paper discusses the major radioactive and toxic chemicals involved in the uranium recovery operations and presents computer simulation results of their hydrodynamic dispersion, migration and prediction of future material movements.

Mathematical Modeling of Dispersion in Mixed Estuaries

The use of the conventional dispersion coefficient equations has met with a limited success, and they do not seem to be appropriate in tidal estuaries. An original dispersion coefficient equation has been developed for mixed estuaries. The developed equation, as a function of real time flow depth and four-thirds power of real time tidal velocity, represents isotropic flow conditions. In mixed estuaries the concept of isotropic dispersion may be acceptable. The dispersion model established by this equation represents adequately the rates of dispersion in the Brisbane River estuary in Australia.

Statistical relationship between median visibility and conditions of worst-case impact on visibility

A study was conducted of the statistical relationships between median visibility and the levels of visibility associated with worst-case impacts. The data base for the study consisted of midday visibility recordings for the years 1974–1976 at 28 suburban/nonurban airports throughout the United States. The visibility recordings were converted to estimates of extinction coefficients with the use of the Koschmeider formula. The data were sorted according to meteorology in order to eliminate days that were obviously dominated by natural causes of poor visibility. Three approaches were used for relating worst-case (90th through 99th percentile) extinction to median extinction. The first approach was based upon frequency distribution functions. The second used observed ratios of upper percentile to median extinction. The third employed regression techniques. All of the relationships were formulated and evaluated with the 1974–1976 data on a national/annual basis as well as regional/quarterly basis. Performance tests were also run against 1954–1956 data at 11 of the 28 sites. Simple ratio relationships are recommended for use in translating median visibility impacts into worst-case impacts. The errors associated with these ratio models are approximately 30 %, which is less than the error typically associated with mathematical dispersion models.

Real-time forecast of air pollution episodes in the venetian region. Part 1: the advection-diffusion model

The paper describes a mathematical model of pollution dispersion in an airshed. The model is of the advection-diffusion type and is of rather general form, in spite of some simplifying assumptions concerning the meteorological inputs and the diffusion coefficients. The paper suggests a numerical solution algorithm of the advection-diffusion equation, which proves to be accurate, unconditionally stable and computationally efficient. In particular, such an algorithm (which is of the Carlson-Crank-Nicolson type) allows one to use a non-uniform grid. This characteristic leads to a relevant computational saving in the presence of non-uniformly distributed emission sources (pollution from an industrial area). The model and the solution scheme are applied to the Venetian lagoon sulphur dioxide pollution case. The model performance is good in normal pollution situations, but unsatisfactory in the presence of episodes (high concentrations) because of the input inaccuracies (mainly lack of information about actual emission schedulings). This is the reason why the model has been corrected through the ‘stochastic embedding’ described in Part 2.

Ocean-dumping research and monitoring: Strategies and tools

Ocean-dumping studies are designed to establish critical indexes of environmental quality to assess the impacts of ocean dumping. Strategies emphasized are: 1) source function characterization including determinations of toxicities of original wastes, 2) environmental characterization, 3) waste dispersion studies in the laboratory and at sea, and 4) waste biota interaction studies, both in the short term, within a life cycle of organisms affected, and in the long term, over many life cycles. Present tools include: 1) quantitative mapping of pollutant distribution by aerial remote sensing and moored and drifting arrays, 2) realtime tracking of waste dispersion in the ocean by 20- and 200-kHz acoustic records, 3) continuous multidepth water sampling to obtain sea truth, 4) in situ bioassay devices, and 5) mathematical dispersion models, some of which are similar to atmospheric chimney plume dispersion models.

A measure of class concentration in bibliometrics

AbstractAn index of concentration for rank‐frequency distributions is proposed which permits comparison of subject and journal concentration in various fields. A mathematical model of random dispersion (the Whit‐worth distribution) of articles is suggested. Applications of the measure to several different aspects of bibliometrics are suggested. The measure holds some promise of providing a common measure by which to compare the large number of specific usage and citation studies already completed, and providing a point of departure for new ones.