Zeroth Moments Research Articles

Electro-optically modulated polarizing Fourier-transform spectrometer for plasma spectroscopy applications.

A new electro-optically modulated optical solid-state (MOSS) interferometer has been constructed for measurement of quantities related to the low-order spectral moments of line emission from optically thin radiant media such as plasmas. When Doppler broadening is dominant, the spectral moments give the Radon transform of corresponding moments of the velocity distribution function of the radiating species. The instrument, which is based on the principle of the Fourier-transform spectrometer, has high etendue and is rugged and compact. When electro-optical path-length modulation techniques are employed, the spectral information is encoded in the temporal frequency domain at harmonics of the modulation frequency and can be obtained by use of a single photodetector. Specifically, for a plasma in drifting local thermodynamic equilibrium the zeroth moment (brightness) is given by the average signal level, the first moment (shift) by the interferometric phase, and the second moment (linewidth) by the fringe visibility. To illustrate the MOSS performance, I present spectroscopic measurements of the time evolution of the plasma ion temperature and flow velocity for rf-heated discharges in the H-1 heliac, a toroidal plasma magnetic confinement at the Australian National University.

Solute transport in open channel flows and porous streambeds

Pore-water flow inside river sediment beds resulting from water flow in the overlying river channel can strongly affect the mass and momentum transfer processes across the channel-bed interface. An analytical model for transport in porous beds, with a two-layer, coupled transport across the interface is presented. Flow and transport in the porous bed and open channel are diffusion and advection driven. The variation in the velocity profile across the flow depth, in both open channel and porous bed, is consistent with experimental studies. A slip velocity at the interface links a parabolic velocity profile in the open channel with an exponentially decaying velocity profile in the porous bed. The contaminant exchange along the interface is accounted through the imposed boundary conditions of concentration and flux continuity. The initial conditions allow the presence of solute in either or both parts of the flow domain, in order to study the transfer of solute across the interface. The Aris' method of moments is used to derive analytical expressions for the zeroth and first moments of concentration and for the velocity and effective dispersion of the solute cloud. The role of dispersion in the bed relative to dispersion in the channel was found to greatly influence the transport characteristics in the river channel-bed interface. The depth of the porous bed relative to the depth of the open channel showed a strong effect on the mean velocity of the solutes particles and the effective dispersion of the solute cloud was found to be sensitive to the flow velocity profile in the open channel in a counter-intuitive way since the transport equations are expressed in terms of axes moving with the mean particle velocity. Finally, the model was validated against experimental data from A.H. Elliott and N.H. Brooks [Water Resour. Res. 33 (1997) 123, 137].

Split-Cell Exponential Characteristic Transport Method for Unstructured Tetrahedral Meshes

The nonlinear, exponential characteristic (EC) method is extended to unstructured meshes of tetrahedral cells in three-dimensional Cartesian coordinates. The split-cell approach developed for the linear characteristic (LC) method on such meshes is used. Exponential distributions of the source within a cell and of the inflow flux on upstream faces of the cell are assumed. The coefficients of these distributions are determined by nonlinear root solving so as to match the zeroth and first moments of the source or entering flux. Good conditioning is achieved by casting the formulas for the moments of the source, inflow flux, and solution flux as sums of positive functions and by using accurate and robust algorithms for evaluation of those functions. Various test problems are used to compare the performance of the EC and LC methods. The EC method is somewhat less accurate than the LC method in regions of net out leakage but is strictly positive and retains good accuracy with optically thick cells, as in shielding problems, unlike the LC method. The computational cost per cell is greater for the EC method, but the use of substantially coarser meshes can make the EC method less expensive in total cost. The EC method, unlike the LC method, may fail if negative cross sections or angular quadrature weights are used. It is concluded that the EC and LC methods should be practical, reliable, and complimentary schemes for these meshes.

A Boltzmann-based mesoscopic model for contaminant transport in flow systems

The objective of this paper is to demonstrate the formulation of a numerical model for mass transport based on the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. To this end, the classical chemical transport equation is derived as the zeroth moment of the BGK Boltzmann differential equation. The relationship between the mass transport equation and the BGK Boltzmann equation allows an alternative approach to numerical modeling of mass transport, wherein mass fluxes are formulated indirectly from the zeroth moment of a difference model for the BGK Boltzmann equation rather than directly from the transport equation. In particular, a second-order numerical solution for the transport equation based on the discrete BGK Boltzmann equation is developed. The numerical discretization of the first-order BGK Boltzmann differential equation is straightforward and leads to diffusion effects being accounted for algebraically rather than through a second-order Fickian term. The resultant model satisfies the entropy condition, thus preventing the emergence of non-physically realizable solutions including oscillations in the vicinity of the front. Integration of the BGK Boltzmann difference equation into the particle velocity space provides the mass fluxes from the control volume and thus the difference equation for mass concentration. The difference model is a local approximation and thus may be easily included in a parallel model or in accounting for complex geometry. Numerical tests for a range of advection–diffusion transport problems, including one- and two-dimensional pure advection transport and advection–diffusion transport show the accuracy of the proposed model in comparison to analytical solutions and solutions obtained by other schemes.

Empirical Interatomic Potential from Depolarized Interaction – Induced Light Scattering Spectra for Mercury

Depolarized interaction-induced light scattering spectra of mercury between 793 K and 1073 K have been used for deriving the empirical multiparameter M3SV interatomic potential. Absolute zeroth and second moments have been measured and compared with theoretical calculations using various models for the interatomic potential. The results show that it is the most accurate potential yet reported for this system.

Effects of carboxylic additives on the polymerization characteristics of nylon-6 reactors with diffusional water removal

The effects of carboxylic acid on the polymerization characteristics of nylon-6 were investigated in reactor models that consist of a continuous-flow stirred tank reactor (CSTR) and a tubular reactor with a diffusional water-removal system, which are connected in series. Mathematical models for the CSTR and the tubular reactor were established and solved by numerical methods. In the CSTR, with an increase of the feed acetic acid content, the monomer conversion, and the molecular weights are increased. In the tubular reactor, the acid behaves like a catalyst and a modifier at the same time in the polymerization of nylon 6. The effects of the feed acetic acid content and the diffusional water removal on the zeroth, first, and second moments and the polydispersity index of the polymer were investigated. The polydispersity index is greatly affected by the feed content of carboxylic acid in the CSTR, but it finally approaches to values of ∼ 2 in the tubular reactor. The diffusional water removal is found to have little effect on the polydispersity index of the polymer. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 81: 1226–1237, 2001

On the late‐time behavior of tracer test breakthrough curves

We investigated the late‐time (asymptotic) behavior of tracer test breakthrough curves (BTCs) with rate‐limited mass transfer (e.g., in dual‐porosity or multiporosity systems) and found that the late‐time concentration c is given by the simple expression c = tad{c0g − [m0(∂g/∂t)]}, for t ≫ tad and tα ≫ tad, where tad is the advection time, c0 is the initial concentration in the medium, m0 is the zeroth moment of the injection pulse, and tα is the mean residence time in the immobile domain (i.e., the characteristic mass transfer time). The function g is proportional to the residence time distribution in the immobile domain; we tabulate g for many geometries, including several distributed (multirate) models of mass transfer. Using this expression, we examine the behavior of late‐time concentration for a number of mass transfer models. One key result is that if rate‐limited mass transfer causes the BTC to behave as a power law at late time (i.e., C ∼ t−k), then the underlying density function of rate coefficients must also be a power law with the form αk−3 as a α → 0. This is true for both density functions of first‐order and diffusion rate coefficients. BTCs with k < 3 persisting to the end of the experiment indicate a mean residence time longer than the experiment, and possibly an infinite residence time, and also suggest an effective rate coefficient that is either undefined or changes as a function of observation time. We apply our analysis to breakthrough curves from single‐well injection‐withdrawal tests at the Waste Isolation Pilot Plant, New Mexico.

Order-parameter model for unstable multilane traffic flow

We discuss a phenomenological approach to the description of unstable vehicle motion on multilane highways that explains in a simple way the observed sequence of the "free flow <--> synchronized mode <--> jam" phase transitions as well as the hysteresis in these transitions. We introduce a variable called an order parameter that accounts for possible correlations in the vehicle motion at different lanes. So, it is principally due to the "many-body" effects in the car interaction in contrast to such variables as the mean car density and velocity being actually the zeroth and first moments of the "one-particle" distribution function. Therefore, we regard the order parameter as an additional independent state variable of traffic flow. We assume that these correlations are due to a small group of "fast" drivers and by taking into account the general properties of the driver behavior we formulate a governing equation for the order parameter. In this context we analyze the instability of homogeneous traffic flow that manifested itself in the above-mentioned phase transitions and gave rise to the hysteresis in both of them. Besides, the jam is characterized by the vehicle flows at different lanes which are independent of one another. We specify a certain simplified model in order to study the general features of the car cluster self-formation under the "free flow <--> synchronized motion" phase transition. In particular, we show that the main local parameters of the developed cluster are determined by the state characteristics of vehicle motion only.

Colloid-facilitated tracer transport by steady random ground-water flow

We study the transport of reactive solute in a three-phase system (water–solid matrix-colloids) in natural porous media. Semianalytical (integral) solutions are derived for the first time, which can be used for computing expected concentration, mass flux, or discharge for the dissolved as well as for colloid-bounded tracer. The results are based on a few simplifying assumptions: advection-dominated transport, linear mass transfer reactions, and steady-state colloidal concentration. Derived semianalytical expressions capture the main features of colloid-facilitated transport (the reversible-equilibrium and irreversible-kinetic sorption of tracers on colloids), and are applicable for the general class of linear sorption processes on the porous matrix. Derived solutions account for spatial variability of flow and sorption parameters, which is relevant for field-scale applications. We apply the theoretical results to the transport of neptunium and plutonium, using flow and transport data from the alluvial aquifer near Yucca Mountain, Nevada. Based on the zeroth and first temporal moment, dimensionless indicators are proposed for assessing the potential impact of colloid-facilitated tracer transport in aquifers. Generic sensitivity curves show the importance of tracer-colloid kinetic rates. Even very low irreversible rates (which will generally be difficult to determine in the laboratory) may yield observable effects for sufficiently long transport times. The obtained results can be used for assessing the significance of colloid-facilitated tracer transport under field conditions, as well as for setting further constraints on relevant parameters which need to be estimated in the field.

Linearized segmentation method for probabilistic production cost simulation

The segmentation method is one of the basic algorithms for probabilistic production cost simulation. If the capacity block sizes are divisible by the segment size, it gives the exact expected energy and reliability values. In case a block size is not divisible by the segment size, the block size and the forced outage rate are suitably modified in the practice. However, this produces a loss in accuracy. In this paper, another approach for handling blocks with not divisible capacity sizes is presented. The basic idea is to divide each segment into two parts, temporarily, to perform the actual convolution or deconvolution computations. The zeroth and first moments of the parts are estimated based on a linear approximation. By our computational experience, at the expense of a slight increase of computational time, the accuracy can be significantly improved by the new method.

Vapor Nucleation and Droplet Growth: Cluster Distribution Kinetics for Open and Closed Systems

A theory based on cluster distribution kinetics for single-monomer addition and dissociation is presented as a framework for homogeneous and heterogeneous vapor nucleation and growth dynamics. For continuous cluster and monomer distributions in a well-mixed non-steady-state flow system, population (mass) balance equations yield moment equations for the cluster mass moments. Nuclei are either homogeneously generated or heterogeneously seeded, and subsequent cluster growth occurs by reversible condensation of vapor monomers. The zeroth moment is the number (or moles) of clusters, the first moment is cluster mass, and the second moment gives cluster-size variance. Solutions are proposed for steady-state flow (open) and non-steady-state batch (closed) systems. Experimental data are interpreted by recognizing that droplets typically observed in nucleation experiments have grown much larger than their nuclei. This allows resolution of the large temperature-dependent discrepancy between experiment and classical nucleation theory.

Macroscopic behavior and random‐walk particle tracking of kinetically sorbing solutes

Analytical expressions are derived for the zeroth, first, and second spatial moments of sorbing solutes that follow a linear reversible kinetic mass transfer model. We determine phase‐transition probabilities and closed‐form expressions for the spatial moments of a plume in both the sorbed and aqueous phases resulting from an arbitrary initial distribution of solute between the phases. This allows for the evaluation of the effective velocity and dispersion coefficient for a homogeneous domain without resorting to numerical modeling. The equations for the spatial moments and the phase‐transition probabilities are used for the development of a new random‐walk particle‐tracking method. The method is tested against three alternate formulations and is found to be computationally efficient without sacrificing accuracy. We apply the new random‐walk method to investigate the possibility of a double peak in the aqueous solute concentration resulting from kinetic sorption. The occurrence of a double peak is found to be dependent on the value of the Damköhler number, and the timing of its appearance is controlled by the mass transfer rate and the retardation factor. Two ranges of the Damköhler number leading to double peaking are identified. In the first range (DaI ≤ 1), double peaking occurs for all retardation factors, while in the second range (1 ≤ DaI ≤ 3), this behavior is most significant for R &gt; 12.

Analysis of Surfactants Adsorbed onto the Surface of Latex Particles by Small-Angle X-ray Scattering

A comprehensive study of the process of adsorption of a nonionic surfactant C18E112 onto poly(styrene) (PS) latex particles by small-angle X-ray scattering (SAXS) is presented. The PS latexes (particle radii 35 and 71 nm) employed in this investigation bear no chemically bound surface charges. The analysis of the process of adsorption by SAXS demonstrates that the point of saturation of the surface may be determined directly from the scattering curves. Free micelles are formed beyond saturation, and no second layer of the surfactant is built up at higher concentrations of the surfactant. Any association of the micelles with the covered latex particle can be ruled out as well. Moreover, an analysis of the radial structure of the surface layer in terms the zeroth and the second moment of the electron density of the layers along the radial direction is given. Both moments can directly be obtained from the SAXS data. The zeroth moment of the excess electron density corroborates the finding that the surfactant is firmly adsorbed onto the surface of the particles. The average extension of the adsorbed layer (2−4 nm) as express through the second moment increases with the amount of adsorbed surfactant. This points to a stretching of the chains due to their mutual interaction when the point of saturation of the surfaced is approached.

A Comparison of Angular Difference Schemes for One-Dimensional Spherical Geometry SN Equations

To investigate errors caused by angular differencing in approximating the streaming terms of the transport equation, five different approximations are evaluated for three test problems in one-dimensional spherical geometry. The following schemes are compared: diamond, special truncation error minimizing weighted diamond, linear continuous (the original SN scheme), linear discontinuous, and new quadratic continuous. To isolate errors caused by angular differencing, the approximations are derived from the transport equation without spatial differencing, and the resulting coupled ordinary differential equations (ODEs) are solved with an ODE solver. Results from the approximations are compared with analytic solutions derived for two-region purely absorbing spheres. Most of the approximations are derived by taking moments of the conservation form of the transport equation. The quadratic continuous approximation is derived taking the zeroth moment of both the transport equation and the first angular derivative of the transport equation. The advantages of this approach are described. In all of the approximations, the desirability is shown of using an initializing computation of the μ = -1 angular flux to correctly compute the central flux and of having a difference approximation that ensures this central flux is the same for all directions. The behavior of the standard discrete ordinates equations in the diffusion limit is reviewed, and the linear and quadratic continuous approximations are shown to have the correct diffusion limit if an equal interval discrete quadrature is used.In all three test problems, the weighted diamond difference approximation has smaller maximum and average relative flux errors than the diamond or the linear continuous difference approximations. The quadratic continuous approximation and the linear discontinuous approximation are both more accurate than the other approximations, and the quadratic continuous approximation has a decided edge over the linear discontinuous approximation in relative flux errors. The diamond, weighted diamond, and linear continuous approximations show quadratic system absorption and system leakage error reduction behavior with increasing N. The linear discontinuous and quadratic continuous approximations show fourth-order error reduction in these quantities. In one of the two-region test problems, the slope of the exact angular flux changes from nearly vertical to nearly horizontal at those points in the exterior region at which the interior region source just becomes visible. At these spatial points, errors in the continuous approximations propagate to each successive outgoing direction, leading to an oscillatory spatial error. The discontinuous approximation does not propagate these errors, although errors near the point of rapid slope change are larger than in the quadratic continuous approximation.

On the magnification relations in quadruple lenses: a moment approach

We present a new method of studying quadruple lenses in elliptical power-law potentials parameterized by (x,y) ∝ (x 2 + y 2 /q 2 ) �/2 /� (0 ≤ � < 2). For this potential, the moments of the four image positions weighted by signed magnifications (magnification times parity) have very simple properties. In particular, we find that the zeroth moment – the sum of four signed magnifications satisfies ≃ 2/(2 −�); the relation is exact for � = 0 (point-lens) and � = 1 (isothermal potential), independent of the axial ratio. Similar relations can be derived when a shear is present along the major or minor axes. These relations, however, do not hold well for the closely-related elliptical density distributions. For a singular isothermal elliptical density distribution without shear, the sum of signed magnifications for quadruple lenses is ≈ 2.8, again nearly independent of the ellipticity. For the same distribution with shear, the total signed magnification is around 2-3 for most cases, but can be significantly different for some combinations of the axial ratio and shear where more than four images can appear.

Continuous-Distribution Kinetic Analysis for Asphaltene Hydrocracking

Kinetic analysis of macromolecular conversion has many difficulties such as definition of rate of conversion and molecular-weight determination. Continuous-distribution kinetic analysis has been proposed for kinetic evaluation of asphaltene hydrocracking. Asphaltene has the molecular-weight distribution (MWD) spanning continuously in the wide range. A gel permeation chromatography (GPC) is a practical and convenient method to observe MWDs of reaction mixtures during the conversion. A mathematical model was presented using simple schemes of chemical reactions representing overall reactions to interpret experimental results. The weights and MWDs of asphaltene components were converted into the corresponding zeroth moment (molar concentration) and first moment (mass concentration), and the time-dependent changes of these moments gave the apparent kinetic parameters of the overall reactions.

Similarity of percolation thresholds on the hcp and fcc lattices

Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (p hcp c, S =0.199 255 5±0.000 001 0) and bond (p hcp c, B =0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of p c are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.

Analysis of Mean Disintegration Time and Mean Dissolution Time by Moment Analysis Using Microcalorimetric Curves

The mean disintegration time (MDGT; mean time required for disintegration of tablets) and mean dissolution time (MDST; mean time required for drug dissolution) of water-soluble drugsfrom solid dosage forms were determined by moment analysis using microcalorimetric curves. Microcalorimetric curves for heat of dilution and for heat of dissolution of the drug were prepared, and the zeroth and first moments of the calorimetric curves were then calculated. The difference between the first moments of the curves for powder dissolution and tablet dissolution was taken to be the MDGT. The difference between the first moment of the curve for heat of dilutionand that of the curve for heat of dissolution was taken to be the MDST. Nicotinic acid and D-mannitol were used as model drugs. The dissolution rate was determined by the conventional beaker method and also by the deconvolution method. The dissolution process could betraced well by moment analysis, as well as by the other methods employed. Moment analysis has some advantages: (a) both the MDGT and the MDST can be determined simultaneously; (b) it is applicable to many drugs that are soluble with heat evolution without the need for quantitative analysis of the drug.

Hyperbranched Polymers and Aggregates: Distribution Kinetics of Dendrimer Growth

We present an analytic solution for growth of branched aggregates or polymers with distributed cluster (dendrimer) size. Monomer addition to each branch follows first-order polymerization kinetics leading to a distribution of branch lengths. The rate constant for monomer addition is considered diffusion-dependent. Deterministic branching occurs so that at prescribed times tj (j ≥ 1), p branches emanate from the tip of a branch that began to grow at tj−1. When the ratio of average branch lengths is constant, Lj/Lj−1 = a, the fractal dimension of branches is ln(p)/‖ln(a)‖. Closed expressions for cluster mass moments show unbounded growth with time unless ap < 1. The number of clusters (zeroth moment) is constant during growth and equal to the number of initiating buds. Expressions for the number of branches and clusters, average cluster mass, volume, density, and viscosity in solution are functions of p and a. Density and molecular weight show features similar to observed behavior of dendrimers and hyperbranched polymers.

Sum rules and beyond

Abstract Sum rules relate the integrated signals of the spin-orbit split core levels to ground state properties, such as the spin-orbit coupling, magnetic moments and charge distribution. These rules give the zeroth moments of the spectral distribution. It is shown how this can be generalized to higher-moment statistical analysis. Within the independent-particle model an expression is given which relates the spectral shape of the core-level spectrum to the expectation values of ground-state operators. Another way to detect the core hole multipole moment is to probe the core hole polarization by spin polarization and magnetic dichroism in resonant photoemission.