A nonlinear mathematical model to study the interactions of hot gases with cloud droplets and raindrops

In this paper, a nonlinear mathematical model is proposed and analyzed to study the interactions of hot gases with cloud droplets as well as with raindrops and their removal by rain from the stable atmosphere. The atmosphere, during rain, is assumed to consist of five nonlinearly interacting phases i.e. the vapour phase, the phase of cloud droplets, the phase of raindrops, the phase of hot gaseous pollutants and the absorbed phase of hot gases in the raindrops (if it exists). It is further assumed that these phases undergo ecological type growth and nonlinear interactions. The proposed model is analyzed using stability theory of differential equations and by numerical simulation. It is shown that the cumulative concentration of gaseous pollutants decreases due to rain and its equilibrium level depends upon the density of cloud droplets, the rate of formation of raindrops, emission rate of pollutants, the rate of falling absorbed phase on the ground, etc. It is noted here that if gases are very hot, cloud droplets are not formed and rain may not take place. In such a case gaseous pollutants may not be removed from the atmosphere due to non-occurrence of rain.     

MODELING AND ANALYSIS OF THE ACID RAIN FORMATION DUE TO PRECIPITATION AND ITS EFFECT ON PLANT SPECIES

Abstract In this paper, a nonlinear mathematical model is proposed and analyzed to study the formation of acid rain in the atmosphere because of precipitation and its effect on plant species. It is considered that acid‐forming gases such as SO₂ , NO₂ emitted from various sources combine with water droplets (moisture) during precipitation and form acid rain affecting plant species. It is assumed that the biomass density of plant species follows a logistic model and its growth rate decreases with increase in the concentration of acid rain. The model is analyzed by using stability theory of differential equations and numerical simulation. The model analysis shows that as the concentration of acid rain increases because of increase in the cumulative emission rates of acid forming gases, the biomass density of plant species decreases. It is noted that if the amount of acid formed becomes very large, the plant species may become extinct.     

How artificial rain can be produced? A mathematical model

A non-linear mathematical model for rain making from water vapor in the atmosphere is proposed and analyzed. The model considers the process of artificial rain by introducing two kinds of aerosol particles conducive to nucleation of cloud droplets and formation of rain drops. The model analysis shows that, for uninterrupted rain, the water vapor in the atmosphere must be formed continuously with the required rate of rainfall. It is shown further that the intensity of rainfall increases as the concentrations of externally introduced aerosols, as well as the density of water vapor in the atmosphere, increase. Numerical simulation is also performed to see the effect of various parameters on the process of artificial rain making leading to rainfall.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

Eulerian Model of Diesel Fuel Sprays

The contribution reviews the research activities on modeling the fuel spray formation in combustion chamber of Diesel engines. A fully Eulerian code has been developed for computing the two-phase flows of drops dispersed in gaseous environment. Both the multi-component compressible gas and the drops are described using governing equations written in Eulerian coordinates. The basic laws of conservation are balanced on finite volumes with arbitrary movable boundaries. This facilitates the modeling of movable boundary problems (e.g. computations with moving engine piston). The model features full coupling between both phases in mass, momentum, and energy equations. The drop size distribution in sprays is taken into account using the multi-continua approach. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

UNSTEADY STATE DISPERSION OF AIR POLLUTANTS UNDER THE EFFECTS OF DELAYED AND NONDELAYED REMOVAL MECHANISMS

Abstract In this paper, we present a two‐dimensional time‐dependent mathematical model for studying the unsteady state dispersion of air pollutants emitted from an elevated line source in the atmosphere under the simultaneous effects of delayed (slow) and nondelayed (instantaneous) removal mechanisms. The wind speed and coefficient of diffusion are taken as functions of the vertical height above the ground. The deposition of pollutants on the absorptive ground and leakage into the atmosphere at the inversion layer are also included in the model by applying appropriate boundary conditions. The model is solved numerically by the fractional step method. The Lagrangian approach is used to solve the advection part, whereas the Eulerian finite difference scheme is applied to solve the part with the diffusion and removal processes. The solutions are analyzed to observe the effects of coexisting delayed and nondelayed removal mechanisms on overall dispersion. Comparison of delayed and nondelayed removal processes of equal capacity shows that the latter (nondelayed) process is more effective than the former (delayed removal) in the removal of pollutants from the atmosphere.     

Numerical simulation of the performance of a snow fence with airfoil snow plates by FVM

The aim of this work is to analyze the efficiency of a snow fence with airfoil snow plates to avoid the snowdrift formation, to improve visibility and to prevent blowing snow disasters on highways and railways. In order to attain this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (snow particles) and fluid phase (air). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, we have used a particle transport model termed as Lagrangian particle tracking model, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 15,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. Finally, the conclusions of this work are exposed.     

Modeling and analysis of the removal of an organic pollutant from a water body using fungi

In this paper, a non linear mathematical model for removing an organic pollutant such as a dye from a water body is proposed and analyzed. In the modeling process four variables are considered, namely, (i) the concentration of the dye, (ii) the density of fungus population, (iii) the concentration of a nutrient and (iv) the concentration of dissolved oxygen (DO). It is assumed that an organic pollutant is present in water with given concentration or discharged with a constant rate in water. It is assumed further that fungus population is kept alive and active due to supply of a nutrient. It is considered that nutrient and oxygen are supplied to the water body from outside with constant rates. The model is analyzed by using the stability theory of differential equations. The model analysis shows that organic pollutant can be removed from the water body by fungus population and the level of degradation depends upon the concentration of organic pollutant, the density of fungal population and the interaction processes involved.The simulation analysis of the proposed model confirms the analytical results. It is also found that these results are qualitatively in line with the experimental observations of one of the authors (Sanghi).     

Eulerian and Lagrangian predictions of particulate two-phase flows: a numerical study

To predict particulate two-phase flows, two approaches are possible. One treats the fluid phase as a continuum and the particulate second phase as single particles. This approach, which predicts the particle trajectories in the fluid phase as a result of forces acting on particles, is called the Lagrangian approach. Treating the solid as some kind of continuum, and solving the appropriate continuum equations for the fluid and particle phases, is referred to as the Eulerian approach.Both approaches are discussed and their basic equations for the particle and fluid phases as well as their numerical treatment are presented. Particular attention is given to the interactions between both phases and their mathematical formulations. The resulting computer codes are discussed.The following cases are presented in detail: vertical pipe flow with various particle concentrations; and sudden expansion in a vertical pipe flow. The results show good agreement between both types of approach.The Lagrangian approach has some advantages for predicting those particulate flows in which large particle accelerations occur. It can also handle particulate two-phase flows consisting of polydispersed particle size distributions. The Eulerian approach seems to have advantages in all flow cases where high particle concentrations occur and where the high void fraction of the flow becomes a dominating flow controlling parameter.     

Differential quadrature for multi-dimensional problems

The technique of differential quadrature for the solution of partial differential equations, introduced by Bellman et al., is extended and generalized to encompass partial differential equations involving multiple space variables. Approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented. Application of these formulae is demonstrated on the solution of the convection-diffusion equation for the two- and three-dimensional space dependent cases and for both the transient and steady-state dispersion of inert, neutrally buoyant pollutants from continuous sources into an unbounded atmosphere.     

Mathematical modeling and analysis of the depletion of dissolved oxygen in water bodies

In this paper, a nonlinear mathematical model is proposed to study the depletion of dissolved oxygen in a water body caused by industrial and household discharges of organic matters (pollutants). The problem is formulated as a food chain model by considering various interaction processes (biodegradation and biochemical) involving organic pollutants, bacteria, protozoa, an aquatic population and dissolved oxygen. Using stability theory, it is shown that as the rate of introduction of organic pollutants in a water body increases, the concentration of dissolved oxygen decreases due to various interaction processes. It is found that if the organic pollutants are continuously discharged into water body, the concentration of dissolved oxygen may become negligibly small, thus threatening the survival of aquatic populations. However, by using some effort to control the cumulative discharge of these pollutants into the water body, the concentration of dissolved oxygen can be maintained at a desired level.     

Ill-posedness in three-dimensional plastic flow

This paper examines partial differential equations for frictional materials flowing via plastic yield, including the equations given by the Critical State Theory of Soil Mechanics. In particular, the material density is considered as a dependent variable. In previous work we demonstrated that two-dimensional plastic flow may be ill posed due to an instability along two rays in Fourier transform space. In this paper, we show that in three dimensions the equations are linearly well posed provided all three strain rates are nonzero.     

MODELING AND ANALYSIS OF THE DEPLETION OF ORGANIC POLLUTANTS BY BACTERIA WITH EXPLICIT DEPENDENCE ON DISSOLVED OXYGEN

In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of dissolved oxygen caused by interactions of organic pollutants with bacteria in a water body, such as lake. The system is assumed to be governed by three dependent variables, namely, the cumulative concentration of organic pollutants, the density of bacteria and the concentration of dissolved oxygen. In the model, the coefficient of interaction of organic pollutants with bacteria depends upon the concentration of dissolved oxygen nonlinearly and explicitly, which is the main focus of this paper, has never been studied before. The stability theory of differential equations is used to analyze the model and to confirm the analytical results numerical simulation is performed. The model analysis shows that if the coefficient of interaction mentioned above depends upon dissolved oxygen explicitly, the decrease in its concentration is more than the case when the interaction does not depend on dissolved oxygen and consequently the depletion of organic pollutants is also more in such a case.     

Weak solution of the merged mathematical equations of the polluted atmosphere

Considered as a geophysical fluid, the polluted atmosphere shares the shallow domain characteristics with other natural large-scale fluids such as seas and oceans. This means that its domain is excessively greater horizontally than in the vertical dimension, leading to the classic hydrostatic approximation of the Navier–Stokes equations. In the past there has been proved a convergence theorem for this model with respect to the ocean, without considering pollution effects. The novelty of this present work is to provide a generalization of their result translated to the atmosphere, extending the fluid velocity equations with an additional convection–diffusion equation representing pollutants in the atmosphere.     

An Eulerian–Lagrangian mixed discrete least squares meshfree method for incompressible multiphase flow problems

Mixed discrete least squares meshfree (MDLSM) method has been developed as a truly meshfree method and successfully used to solve single-phase flow problems. In the MDLSM, a residual functional is minimized in terms of the nodal unknown parameters leading to a set of positive-definite system of algebraic equations. The functional is defined using a least square summation of the residual of the governing partial differential equations and its boundary conditions at all nodal points discretizing the computational domain. Unlike the discrete least squares meshfree (DLSM) which uses an irreducible form of the governing equations, the MDLSM uses a mixed form of the original governing equations allowing for direct calculation of the gradients leading to more accurate computational results. In this study, an Eulerian–Lagrangian MDLSM method is proposed to solve incompressible multiphase flow problems. In the Eulerian step, the MDLSM method is used to solve the governing phase averaged Navier–Stokes equations discretized at fixed nodal points to get the velocity and pressure fields. A Lagrangian based approach is then used to track different flow phases indexed by a set of marker points. The velocities of marker points are calculated by interpolating the velocity of fixed nodal points using a kernel approximation, which are then used to move the marker points as Lagrangian particles to track phases. To avoid unphysical clustering and dispersing of the marker points, as a common drawback of Lagrangian point tracking methods, a new approach is proposed to smooth the distribution of marker points. The hybrid Eulerian and Lagrangian characteristics of the approach used here provides clear advantages for the proposed method. Since the nodal points are static on the Eulerian step, the time-consuming moving least squares (MLS) approximation is implemented only once making the proposed method more efficient than corresponding fully Lagrangian methods. Furthermore, phases can be simply tracked using the Lagrangian phase tracking procedure. Efficiency of the proposed MDLSM multiphase method is evaluated using several benchmark problems and the results are presented and discussed. The results verify the efficiency and accuracy of the proposed method for solving multiphase flow problems.     

A three-phase FE-model for the simulation of partially saturated soils

While for many numerical simulations in geotechnics use of a two-phase model is sufficient, separate consideration of all three phases is mandatory for numerical simulations of partially saturated soils subjected to compressed air. This is a common technique frequently applied for temporary ground support in tunnelling. For the numerical simulation of tunnelling using compressed air, a multiphase model for soft soils is developed, in which the individual constituents of the soil – the soil skeleton, the fluid and the gaseous phase – and their interactions are considered. The three phase model is formulated within the framework of the Theory of Porous Media (TPM), based upon balance equations and constitutive relations for the soil constituents and their mixture. Water is modelled as an incompressible and air as a compressible phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and simulation of phenol removal from wastewater using a membrane contactor as a bioreactor

Diverse methods have been introduced for phenol removal from wastewater. Among them, membrane bioreactors have attracted considerable attention in the last decade. In this study, modeling and simulation of a hollow fiber membrane contactor as a bioreactor was carried out. The effects of the various parameters such as flow rate, ratio of membrane porosity to tortuosity, membrane length, initial phenol concentration, number of fibers, and inner and outer radius of the membrane on phenol removal efficiency were studied. A proposed set of partial differential equations and related boundary conditions in the model were solved by the finite element method via simulation with computational fluid dynamics techniques. The simulation results were compared with existing empirical data from literature and acceptable agreement was observed. Based on the obtained results, increasing the initial concentration had a reverse impact on the phenol removal efficiency. However, increasing the cell phase flow rate slightly enhanced the removal efficiency. Moreover, extending the membrane length had a desirable effect. Also, augmentation of the number of fibers within the contactor initially increased and then decreased the efficiency.     

Shock-Induced Phase Transitions in Systems of Hard Spheres with Attractive Interactions

Shock-induced phase transitions are studied by adopting the recently-developed theoretical framework, which is applicable for shock waves in three phases (gas, liquid, and solid), based on the system of hard spheres with mutual attractive interactions. The Rankine-Hugoniot conditions derived from the system of Euler equations with caloric and thermal equations of state are studied, and the admissibility (stability) of a shock wave is analyzed. Two typical scenarios of the shock-induced phase transitions from gas phase to solid phase are found. A scenario of shock-induced phase transitions involving three phases simultaneously near the triple point is also found.