Home About us Contact
|
Home About us Contact | ||
|
|
||
Series Solution (series + solution)
Selected AbstractsA Simple Model of Soil-Gas Concentrations Sparged into an Unlined Unsaturated ZoneGROUND WATER MONITORING & REMEDIATION, Issue 2 2003David W. Ostendorf We derive an analytical model of soil-gas contamination sparged into an imlined unsaturated zone. A nonaqueous phase liquid (NAPL) source lies in the capillary fringe, with an exponential sparge constant within the radius of influence and a constant ambient evaporation rate beyond. Advection, diffusion, and dispersion govern the conservative soil-gas response, expressed as a quasi-steady series solution with radial Bessel and hyperbolic vertical dependence. Simulations suggest that sparged contamination initially spreads beyond the radius of influence down a negative gradient. This gradient eventually reverses, leading to a subsequent influx of ambient contamination. Soil-gas concentrations accordingly reflect slowly varying source conditions as well as slowly varying diffusive transport through the radius of influence. The two time scales are independent: One depends on NAPL, airflow, and capillary fringe characteristics, the other on soil moisture, gaseous diffusivity, and unsaturated zone thickness. The influx of ambient contamination generates an asymptotic soil-gas concentration much less than the initial source concentration. The simple model is applied to a pilot-scale sparging study at Plattsburgh Air Force Base in upstate New York, with physically plausible results. [source] Interpretation of the enhancement of field-scale effective matrix diffusion coefficient in a single fracture using a semi-analytical power series solutionHYDROLOGICAL PROCESSES, Issue 6 2009Tai-Sheng Liou Abstract A power series solution for convergent radial transport in a single fracture (PCRTSF) is developed. Transport processes considered in PCRTSF include advection and hydrodynamic dispersion in the fracture, molecular diffusion in the matrix, diffusive mass exchange across the fracture-matrix interface, and mixing effects in the injection and the extraction boreholes. An analytical solution in terms of a power series in Laplace domain is developed first, which is then numerically inverted by de-Hoog et al.'s algorithm. Four dimensionless parameters determine the behaviour of a breakthrough curve (BTC) calculated by PCRTSF, which are, in the order of decreasing sensitivity, the matrix diffusion factor, two mixing factors, and the Peclet number. The first parameter is lumped from matrix porosity, effective matrix diffusion coefficient, fracture aperture, and retardation factors. Its value increases as the matrix diffusion effect becomes significant. A non-zero matrix diffusion factor results in a , 3/2 slope of the tail of a log,log BTC, a common property for tracer diffusion into an infinite matrix. Both mixing factors have equal effects on BTC characteristics. However, the Peclet number has virtually no effect on BTC tail. PCRTSF is applied to re-analyse two published test results that were obtained from convergent radial tracer tests in a discrete, horizontal fracture in Silurian dolomite. PCRTSF is able to fit the field BTCs better than the original channel model does if a large matrix diffusion coefficient is used. Noticeably, the ratio of field-scale to lab-scale matrix diffusion coefficients can be as large as 378. This enhancement of the field-scale matrix diffusion coefficient may be ascribed to the presence of a degraded zone at the fracture-matrix interface because of karstic effects, or to flow channeling as a result of aperture heterogeneity. Copyright © 2009 John Wiley & Sons, Ltd. [source] Torsion of orthotropic bars with L -shaped or cruciform cross-sectionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001Y. Z. Chen Abstract For an orthotropic torsion bar with L -shaped or cruciform cross-section, the studied torsion problem can be reduced to a boundary value problem of elliptic partial differential equation. The studied region is separated into several rectangular sub-regions, and the series solution is suggested to solve the problem for the individual sub-region. By using the continuation condition for the functions on the neighbouring sub-regions, the investigated solution is obtainable. Finally, numerical results for the torsion rigidities of bars are given to demonstrate the influence of the degree of orthotropy. Copyright © 2001 John Wiley & Sons, Ltd. [source] Analysis of singular stress fields at multi-material corners under thermal loadingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006Chongmin SongArticle first published online: 7 SEP 200 Abstract The scaled boundary finite-element method is extended to the modelling of thermal stresses. The particular solution for the non-homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi-material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side-faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi-analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power-logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite-element method. Copyright © 2005 John Wiley & Sons, Ltd. [source] Synthesis of charged ultrafiltration poly(styrene- co -divinyl benzene) composite membraneJOURNAL OF APPLIED POLYMER SCIENCE, Issue 1 2008Sonny Sachdeva Abstract A ceramic supported crosslinked polystyrene composite membrane has been prepared from its monomers using a dual initiator system. The nonionic hydrophobic membrane so prepared has been chemically modified by a low temperature (50°C), single step reaction with chloroacetic acid. The carboxylated membrane has acid functional groups on its surface making it negatively charged and highly hydrophilic in nature. The membranes (unmodified and carboxylated) have been used for the separation of hazardous chromium (VI) salt solution where observed and intrinsic rejection has been studied as a function of pressure and concentration of the feed solution. The intrinsic rejection has been determined by calculating the concentration at the membrane surface (Cm) using Speigler-Kedam model and osmotic pressure model. The observed rejection for the chemically modified membrane decreases with increasing pressure but the intrinsic rejection is found to be more than 80% for all concentrations in the range of study. The experimental results have been fitted using Space-Charge model to obtain the membrane wall potential and the membrane surface concentration which are difficult to measure directly. The transport through the membrane capillaries has been described by the two dimensional model using Nernst-Planck equation for ion transport, Navier-Stokes equation and Poisson-Boltzmann equation for the radial distribution of potential. We have then presented a semianalytical series solution to the highly nonlinear Poisson-Boltzmann equation to reduce the computational time required to solve the set of coupled differential equations. The effective wall potential of the carboxylated membrane was found to be ,28.07 mV. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008 [source] WATER DIFFUSION COEFFICIENT AND MODELING OF WATER UPTAKE IN PACKAGED YERBA MATEJOURNAL OF FOOD PROCESSING AND PRESERVATION, Issue 4 2007LAURA A. RAMALLO ABSTRACT Effective water diffusion coefficient (Deff) was determined from the kinetics of moisture gain in a yerba mate bed. A value of 1.5 × 10,9 ± 0.4 × 10,9 m2/s was obtained at 40C and 90% relative humidity, by fitting experimental data to the series solution of Fick's second law. A model was developed to predict moisture profile and water uptake in packaged yerba mate. In order to simulate moisture gain in the packaged food, the model considers that the global process of humidity gain is controlled by combined mechanisms of package permeability, product sorption balances and water diffusion within the food bed. The explicit finite difference method was used to numerically solve the resulting equations. The validity of the model was tested by comparing predicted and experimental moisture profiles for high (WVTR , 20 g/m2/day) and low (WVTR , 400 g/m2/day) barrier packages. The model was found to adequately predict the profile of moisture content. [source] Coaxially fed monopoles in shorted waveguidesMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 12 2007Mi Y. Park Abstract An efficient series solution for coaxially fed monopoles in a shorted parallel-plate waveguide is presented. The analysis is based on mode-matching and image method. A coaxially fed monopole in a shorted rectangular waveguide is considered as a special case. Computations are performed to verify the validity of solution. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 3145,3148, 2007; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.22959 [source] Stretching a plane surface in a viscoelastic fluid with prescribed skin frictionNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2009M. Sajid Abstract A model of forced convection flow due to stretching surface is derived to represent the physical system with prescribed skin friction. To achieve the similar solutions, the partial differential equations are reduced into ordinary differential equations. The analytic solutions of the resulting problems have been obtained by a homotopy analysis method. The convergence of the developed series solution is seen. Finally, the results of velocity, temperature, the stretching velocity, and Nusselt number are analyzed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] Analytical power series solutions to the two-dimensional advection,dispersion equation with distance-dependent dispersivitiesHYDROLOGICAL PROCESSES, Issue 24 2008Jui-Sheng Chen Abstract As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection,dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection,dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio eL exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large eL values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd. [source] Charged relativistic spheres with generalized potentialsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2009S. Thirukkanesh Abstract A new class of exact solutions of the Einstein,Maxwell system is found in closed form. This is achieved by choosing a generalized form for one of the gravitational potentials and a particular form for the electric field intensity. For specific values of the parameters it is possible to write the new series solutions in terms of elementary functions. We regain well-known physically reasonable models. A physical analysis indicates that the model may be used to describe a charged sphere. The influence of the electromagnetic field on the gravitational interaction is highlighted. Copyright © 2008 John Wiley & Sons, Ltd. [source] Spectral theory and iterative methods for the Maxwell system in nonsmooth domainsMATHEMATISCHE NACHRICHTEN, Issue 6 2010Irina Mitrea Abstract We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain , , ,3. By employing Rellich-type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L2(,,) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of ,. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | |||||||||||||