Mesh Length Research Articles

Assessing the Accuracy of Wall Concentration Estimation Based on Averaged Permeate Velocity in Spacer-Filled Reverse Osmosis (RO) Membrane Systems

The accuracy of estimations of wall concentration using the constant (averaged) permeate velocity method was assessed in both empty and spacer-filled reverse osmosis (RO) channels under various operation conditions, based on the simulation results of a two-dimensional (2-D) concentration polarization model. It was determined that, although it always results in substantially larger variations of wall concentrations, the constant permeate velocity method is able to mimic the general variation patterns of wall concentrations in both empty and spacer-filled channels. The constant permeate velocity method can reasonably estimate the averaged wall concentration. The error is usually <10%, under the simulation conditions. It was also determined that the constant permeate velocity method always overestimates the maximum wall concentration. For the membrane that is not in contact with transverse filaments, the error is usually within 20%, and increasing the mesh length or increasing the system value of Δp/Δπ0 woul...

Numerical study on permeate flux enhancement by spacers in a crossflow reverse osmosis channel

The impact of spacer configurations (i.e. cavity, zigzag and submerged) and mesh length on the alleviation of concentration polarization and the enhancement of permeate flux in the crossflow reverse osmosis membrane channels was investigated. In this study, the wall concentration and permeate flux were directly determined from the numerical solutions of the fully coupled governing equations of momentum and mass transfer in the feed channel. It was demonstrated that the average permeate flux could be significantly enhanced by the spacers, especially those with zigzag configuration. Simulations showed that the zigzag configuration was the most effective one to alleviate concentration polarization and to enhance permeate flux while the submerged configuration is the least. It was further found that an optimum mesh length (corresponding to the maximum permeate flux enhancement) existed for cavity and zigzag configurations and the optimum mesh length decreased with increasing salinity of the feed water. The results suggested that different mesh length should be used in membrane modules for feed waters of different salinities to obtain the maximum permeate flux enhancement.

Exponentially-fitted Numerov methods

The error in the estimate of the kth eigenvalue of a regular Sturm–Liouville problem obtained by Numerov's method with mesh length h is O ( k 6 h 4 ) . It is shown that the error can be reduced to O ( k 3 h 4 ) by using one of the three versions of the exponentially-fitted Numerov method. Numerical examples demonstrate the usefulness of this approach even for low values of k.

Frozen Rigging Model of the Energy Dominated Universe

Composite rigging systems, involving membranes that meet on strings that meet on monopoles, arise naturally by the Kibble mechanism as topological defects in field theories involving spontaneous symmetry breaking. Such systems will tend to freeze out into static lattice type configurations with energy contribution ultimately be provided by the membranes. It has been suggested by Bucher and Spergel that on scales large compared with the relevant (interstellar separation) distance characterizing the relevant mesh length, such a system may behave as a rigidity-stabilized solid, having an approximately isotropic stress energy tensor with negative pressure, as given by a polytropic index γ = w+1 = 1/3. It has recently been shown that such a system can be rigid enough to be stable if the number of membranes meeting at a junction is even (though not if it is odd). Using as examples an approximaely O(3) symmetric scalar field model that can provide an “8-color” (body-centered) cubic lattice, and an approximate U(1)× U(1) model offering a disordered “5-color” lattice, it is argued that such a mechanism can account naturally for the observed dark energy dominance of the universe, without ad hoc assumptions, other than that the relevant symmetry breaking phase transition should have occurred somewhere about the Kev energy range.

Novel spacer design improves observed flux

In spiral wound membrane (SWM) modules, spacers are essential for keeping the membrane leaves apart. Feed side spacers also enhance flux by promoting eddies, but the disadvantage of spacers is that they increase the pressure loss along the flow channel. The effect of the filament diameter, mesh length and arrangement of spacer filaments was investigated experimentally for ultrafiltration (UF) of dextran and whey solutions. For conventional 2-layer feed spacers, a reduction of the mesh length first increases flux and then flux decreases beyond some optimum mesh length while pressure loss increases continuously with a decrease of the mesh length. An advanced 3-layer spacer (A3LS) is developed with superior mass transfer characteristics and less fouling propensity when compared with conventional 2-layer spacers. The 3-layer spacer improves flux without covering additional membrane area by filaments adjacent to the membrane wall under both fouling and non-fouling conditions. However, the current A3LS design also increases the pressure loss. Under fouling conditions the A3LS has more than 20% higher flux when compared with the 2-layer spacer.

A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations

AbstractAn efficient numerical method is developed for the numerical solution of non‐linear wave equations typified by the regularized long wave equation (RLW) and its generalization (GRLW). The method developed uses a pseudo‐spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. =10pt An important advantage to be gained from the use of this method, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method is second order in time and all‐order in space. The method presented here is for the RLW equation and its generalized form, but it can be implemented to a broad class of non‐linear long wave equations (Equation (2)), with obvious changes in the various formulae. Test problems, including the simulation of a single soliton and interaction of solitary waves, are used to validate the method, which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Simulation of Unsteady Flow and Vortex Shedding for Narrow Spacer-Filled Channels

Insights into the effect of spacer filaments on the unsteady flow behavior in narrow spacer-filled channels were obtained using a computational fluid dynamics model. The flow was computed for different filament configurations for channel Reynolds numbers up to 2000. For the filament diameter, channel height, and spacer mesh length examined here, the transition to unsteady flow for channels obstructed with cylindrical spacer filaments occurs somewhere between Rech of 200 and 800, depending on the spacer configuration. When the flow is unsteady, vortex shedding occurs behind the spacer filaments and the shed vortices scour the channel walls as they move toward the next downstream filament, thereby increasing the wall shear stress. For each spacer configuration examined, a characteristic vortex formation pattern was found.

Flow phase diagrams for concentration-coupled shear banding

After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant systems, we present flow phase diagrams spanned by shear stress Sigma (or strain rate gamma) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS-phi) model. We also give results for the macroscopic flow curves Sigma(gamma,phi) for a range of (average) concentrations phi. For any concentration that is high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence "plateau" that slopes upwards, dSigma/dgamma>0. As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point [Sigmac,phic,gammac]. We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, Sigma(gamma,phi), measured for several different values of phi. This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS-phi model contains two different spatial gradient terms that describe the interface between the shear bands. The first is in the viscoelastic constitutive equation, with a characteristic (mesh) length l. The second is in the (generalised) Cahn-Hilliard equation, with the characteristic length xi for equilibrium concentration-fluctuations. We show that the phase diagrams (and so also the flow curves) depend on the ratio r congruent with l/xi, with loss of unique state selection at r=0. We also give results for the full shear-banded profiles, and study the divergence of the interfacial width (relative to l and xi) at the critical point.

Concentration polarization in spiral-wound nanofiltration membrane elements

In predicting the nanofiltration (NF) process, concentration polarization (CP) should be seriously considered. Since many empirical expressions for the mass transport in the CP layer were reported, it seems difficult to choose a proper one. This paper discusses the CP development along the bulk flow direction in crossflow NF using a theoretical model, and then compares the differences ofthe concerned empirical expressions for the mass transport across the CP layer. The differences that the expressions caused were also presented with the NF experiment of the dilute PEG600 solution. The empirical expressions and S-K model were coupled to fit the experiment result. In the regressions of S-K model parameters, the pressure loss along the bulk flow channel and the bulk concentration variation were both considered. It was found that the concentration polarization along the bulk flow direction increased greatly before the boundary layer was fully developed. However, the polarization degree within a mesh length of 0.003 m was not serious. On the other hand, the empirical correlations gave sharply different CP degrees. Schock's and Hickey's expressions gave a comparatively close mass transport coefficient only when the bulk flow rate are around 0.15 m·s −1 '. Eriksson's expression showed that the CP in the spiral-wound element was not serious only if the bulk flow rate was not much less than 0.1 m·s −1 '. The results suggested that Eriksson's expression was still the best choice for the polarization correction while the other candidates seemed to overestimate the polarization.

Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation. Part I: smoothness problems

The finite difference method (FDM) using the Shortley–Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When u∈C 3( S ̄ ) (i.e., u∈C 3,0( S ̄ ) ) and f∈C 2( S ̄ ) , the superconvergence rate O( h 2) of solution derivatives in discrete H 1 norms by the FDM is derived for rectangular difference grids, where h is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform. Comparisons are made on the analysis by the maximum principle and the FEM analysis, conversions between the FDM and the linear and bilinear FEMs are discussed, and numerical experiments are provided to support superconvergence analysis made.

Numerical solution of solidification in a square prism using an algebraic grid generation technique

The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simultaneous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the computational grid structure is regenerated at every time step.

A CFD study of unsteady flow in narrow spacer-filled channels for spiral-wound membrane modules

In spiral-wound membrane modules, spacers are used to enhance wall shear stress and to promote eddy mixing, thereby reducing wall concentration and fouling. Insights into the effect of spacer filaments on flow patterns in narrow channels were obtained using a computational fluid dynamics (CFD) code. The flow patterns were visualized for different filament configurations incorporating variations in mesh length, filament diameter and for channel Reynolds numbers up to 1000. The simulated flow patterns revealed the dependence of the formation of recirculation regions on the filament configuration, mesh length, filament diameter and the Reynolds number. When the channel Reynolds number is increased above 300, the flow becomes super-critical showing time-dependent movements for a filament located in the center of a narrow channel; and when the channel Reynolds number is increased above 500, the flow becomes super-critical for a filament adjacent to the membrane wall. For multiple filament configurations, flow transition can occur at channel Reynolds numbers as low as 80 for the submerged spacer at a very small mesh length ( l m / h ch = 1) and at a slightly larger Reynolds number at a larger mesh length ( l m / h ch = 4). The transition occurs above Re ch of 300 for the cavity spacer ( l m / h ch = 4) and above Re ch of 400 for the zigzag spacer ( l m / h ch = 4).

Simulation of the Flow around Spacer Filaments between Channel Walls. 2. Mass-Transfer Enhancement

A computational fluid dynamics (CFD) code was used to study the effects of Reynolds number, mesh length, and filament diameter on mass-transfer enhancement for three spacer configurations, a cavity, a zigzag, and a submerged spacer. For the cavity and zigzag spacers, mass-transfer enhancement first increases with a decrease in the mesh length, reaches a maximum, and then decreases with a further decrease in the mesh length, while pressure loss showed a continuous increase with a decrease in the mesh length. The submerged spacer shows a continuous increase of the mass-transfer enhancement and pressure loss with a decrease in the mesh length. For all spacer types, mass transfer increases with the filament diameter. However, at a smaller filament diameter, the overall spacer performance increases, as indicated by a high mass-transfer enhancement to pressure loss ratio. Overall, the CFD simulations reveal that the zigzag spacer is the most efficient spacer type for a spiral-wound membrane module.

A microscopic dynamic Monte Carlo simulation for unidirectional fiber reinforced metal matrix composites

A dynamic Monte Carlo large-fine numerical microcomposite model, based on the modified shear-lag model with fine mesh, sufficient fibers and adequate length, was established to study the microscopic tensile failure process of unidirectional fiber reinforced metal matrix composites (composites wire) under different strain-rates. In this model, the strength of the fiber elements is randomly allocated by the Monte Carlo method, the elasto-plastic properties of the matrix elements, the friction of the interfaces and the residual stress of both the fiber and matrix elements are definitely allocated. Using this model, the deformation, damage and failure process of the composites is simulated on the microscopic level, the tensile stress–strain relationship is well predicted, the relationship of mechanical properties between the composites and the original fiber, in situ fiber and the interface is discussed. The analysis also shows that, compared with the experimental results, the simulated results using in situ parameters of fiber arrive at good correlation while those using original parameters have much difference (the predicted strength is obviously higher than the experimental strength). It also shows that the effect of interface strength on the strength of composites is limited, the composites strength is dominantly determined by the in situ fiber strength

Simulation of the Flow around Spacer Filaments between Narrow Channel Walls. 1. Hydrodynamics

Insights into the effect of spacer filaments in membrane systems on the flow pattern were obtained using a computational fluid dynamics code. The flow patterns were examined for a single filament adjacent to the wall and centered in the channel and for three different spacer configurations, the cavity, zigzag, and submerged spacers, with variations in both the mesh length and filament diameter for Reynolds numbers ranging from 90 to 768. Large recirculation regions were formed behind the filaments, and the flow around the filament increased the shear stress on the wall. For an identical Reynolds number and filament diameter, a single filament adjacent to a membrane wall produced a larger recirculation region than a single filament in the center of the channel. For the cavity and submerged spacers, above a critical Reynolds number or mesh length, the recirculation regions between sequential filaments influenced each other and merged to form one large recirculation region between sequential filaments. In co...

The CMC–MRB Global Environmental Multiscale (GEM) Model. Part III: Nonhydrostatic Formulation

Abstract An integrated forecasting and data assimilation system has been and is continuing to be developed by the Meteorological Research Branch (MRB) in partnership with the Canadian Meteorological Centre (CMC) of Environment Canada. Part III of this series of papers presents the nonhydrostatic formulation and some sample results. The nonhydrostatic formulation uses Laprise's hydrostatic pressure as the basis for its vertical coordinate. This allows the departure from the hydrostatic formulation to be incorporated in an efficient switch-controlled perturbative manner. The time discretization of the model dynamics is (almost) fully implicit semi-Lagrangian, where all terms including the nonlinear terms are (quasi-) centered in time. The spatial discretization for the adjustment step employs a staggered Arakawa C grid that is spatially offset by half a mesh length in the meridional direction with respect to that employed in previous model formulations. It is accurate to second order, whereas the interpolat...

Superconvergence of coupling techniques in combined methods for elliptic equations with singularities

Several coupling techniques, such as the nonconforming constraints, penalty, and hybrid integrals, of the Ritz-Galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities. Based on suitable norms involving discrete solutions at specific points, superconvergence rates on solution derivatives are exploited by using five combinations, e.g., the nonconforming combination, the penalty combination, Combinations I and II, and symmetric combination. For quasi-uniform rectangular grids, the superconvergence rates, O( h 2− δ ), of solution derivatives by all five combinations can be achieved, where h is the maximal mesh length of difference grids used in the finite difference method, and δ(> 0) is an arbitrarily small number. Superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods. Therefore, a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references. Moreover, the five combinations having O( h 2− δ ) of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces.

Mesh selectivity of a sweeping trammel net for Japanese whiting Sillago japonica

SUMMARY: To determine the mesh selectivity of a sweeping trammel net for Japanese whiting Sillago japonica, experimental fishing operations using three different inner net mesh lengths (27.5, 31.0 and 33.5 mm) were carried out in Tateyama Bay, Chiba Prefecture, Japan from August to November 1996 and from April to May 1997. The length mode of captured fish increased in proportion to the mesh length from 150–160 mm of the smallest mesh to 190–200 mm total length of the largest mesh. The master curve of selectivity in terms of length ratio to the mesh length (l/m) showed a wide selection range of 50% relative efficiency compared to the mesh selectivity curve of gill net. The curve had a peak at l/m of 6.3 with the selection range from 5.4 to 7.4. This suggested that the trammel net caught a large number of entangled or pocketed fish in comparison to the gilled fish due to the effect of inner net slackness. In terms of body girth ratio to mesh perimeter (G/P), the curve had a peak at G/P of 1.31, which reflected that relative efficiency of capture reached its maximum when the body girth of fish was 1.31 times that of the mesh perimeter.

Numerical Analysis of Stress Concentration for Brittle Matrix Composites

A FEM numerical model is constructed of SiCf/LASIII glass-ceramic uni-direction plate under tension loading. The smeared crack method is used to deal with the failure point of fiber and matrix elements. The solution of the model gives the distribution of meso-stress of SiCf/LASIII glass-ceramic uni-direction plate. The stress concentration factor K and the size of stress concentration effect zone are also analyzed. The average saturation cracking space is obtained by a numerical method. The mesh length of the elements is also studied.

Corrected finite difference eigenvalues of periodic Sturm–Liouville problems

Computation of eigenvalues of regular Sturm–Liouville problems with periodic boundary conditions is considered. We show that a proof similar to that given by Andrew (1989) can be used to prove that a correction technique applied to a finite difference scheme given by Vanden Berghe et al. (1995) reduces the error in the kth eigenvalue estimate from O( k 4 h 2) to O( kh 2), where h is the uniform mesh length. We also provide a significantly shorter proof of a slightly weaker result.