Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model(PBM). A typical population balance equation(PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods(QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function(NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments(QMOM), direct quadrature method of moments(DQMOM),extended quadrature method of moments(EQMOM), conditional quadrature method of moments(CQMOM),extended conditional quadrature method of moments(ECQMOM) and hyperbolic quadrature method of moments(Hy QMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics(CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

Measurement of remanent magnetic moment using a torsion pendulum with single frequency modulation method

In Tian Qin spaceborne gravitational-wave detectors, the stringent requirements on the magnetic cleanliness of the test masses demand the high resolution ground-based characterization measurement of their magnetic properties. Here we present a single frequency modulation method based on a torsion pendulum to measure the remanent magnetic moment mr of 1.1 kg dummy copper test mass, and the measurement result is(6.45 ± 0.04(stat) ± 0.07(syst)) × 10^(-8)A · m^(2). The measurement precision of the mr is about 0.9 n A · m^(2), well below the present measurement requirement of Tian Qin. The method is particularly useful for measuring extremely low magnetic properties of the materials for use in the construction of space-borne gravitational wave detection and other precision scientific apparatus.

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization(RBMDO)of engineering structures,the saddlepoint approximation(SA)method can be utilized to enhance the accuracy and efficiency of reliability evaluation.However,the random variables involved in SA should be easy to handle.Additionally,the corresponding saddlepoint equation should not be complicated.Both of them limit the application of SA for engineering problems.The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments.However,the traditional moment matching method is not very accurate generally.In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization,a fourth-moment saddlepoint approximation(FMSA)method is introduced into RBMDO.In FMSA,the approximate cumulative generating functions are constructed based on the first four moments of the limit state function.The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function.Furthermore,the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment,which is based on the performance measure approach strategy.Two engineering examples are introduced to verify the effectiveness of proposed method.

Two-Stream Approximation to the Radiative Transfer Equation:A New Improvement and Comparative Accuracy with Existing Methods

Mathematical modeling of the interaction between solar radiation and the Earth's atmosphere is formalized by the radiative transfer equation(RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γparameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate(Disort) method(δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation(F-TSA) developed in a previous work. Notably,New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0, but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method(R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.

Anti-plane deformations around arbitrary-shaped canyons on a wedge-shape half-space:moment method solutions

The wave propagation behavior in an elastic wedge-shaped medium with an arbitrary shaped cylindrical canyon at its vertex has been studied.Numerical computation of the wave displacement field is carried out on and near the canyon surfaces using weighted-residuals(moment method).The wave displacement fields are computed by the residual method for the cases of elliptic,circular,rounded-rectangular and flat-elliptic canyons,The analysis demonstrates that the resulting surface displacement depends,as in similar previous analyses,on several factors including,but not limited,to the angle of the wedge,the geometry of the vertex,the frequencies of the incident waves,the angles of incidence,and the material properties of the media.The analysis provides intriguing results that help to explain geophysical observations regarding the amplification of seismic energy as a function of site conditions.

Rolling Force and Rolling Moment in Spline Cold Rolling Using Slip-line Field Method

Rolling force and rolling moment are prime process parameter of external spline cold rolling. However, the precise theoretical formulae of rolling force and rolling moment are still very fewer, and the determination of them depends on experience. In the present study, the mathematical models of rolling force and rolling moment are established based on stress field theory of slip-line. And the isotropic hardening is used to improve the yield criterion. Based on MATLAB program language environment, calculation program is developed according to mathematical models established. The rolling force and rolling moment could be predicted quickly via the calculation program, and then the reliability of the models is validated by FEM. Within the range of module of spline m=0.5-1.5 mm, pressure angle of reference circle α=30.0°-45.0°, and number of spline teeth Z=19-54, the rolling force and rolling moment in rolling process （finishing rolling is excluded） are researched by means of virtualizing orthogonal experiment design. The results of the present study indicate that： the influences of module and number of spline teeth on the maximum rolling force and rolling moment in the process are remarkable; in the case of pressure angle of reference circle is little, module of spline is great, and number of spline teeth is little, the peak value of rolling force in rolling process may appear in the midst of the process; the peak value of rolling moment in rolling process appears in the midst of the process, and then oscillator weaken to a stable value. The results of the present study may provide guidelines for the determination of power of the motor and the design of hydraulic system of special machine, and provide basis for the farther researches on the precise forming process of external spline cold rolling.

Series expansion feasibility of singular integral in method of moments

When calculating electromagnetic scattering using method of moments （MoM）, integral of the singular term has a significant influence on the results. This paper transforms the singular surface integral to the contour integral. The integrand is expanded to Taylor series and the integral results in a closed form. The cut-off error is analyzed to show that the series converges fast and only about 2 terms can agree wel with the accurate result. The comparison of the perfect electric conductive （PEC） sphere＇s bi-static radar cross section （RCS） using MoM and the accurate method validates the feasibility in manipulating the singularity. The error due to the facet size and the cut-off terms of the series are analyzed in examples.

An analysis of the radar backscatter from oil-covered sea surfaces using moment method and Monte-Carlo simulation: preliminary results

An analysis of the radar backscattering from the ocean surface covered by oil spill is presented using a mi- crowave scattering model and Monte-Carlo simulation. In the analysis, a one-dimensional rough sea sur- face is numerically generated with an ocean waveheight spectrum for a given wind velocity. A two-layered medium is then generated by adding a thin oil layer on the simulated rough sea surface. The electric fields backscattered from the sea surface with two-layered medium are computed with the method of moments （MoM）, and the backscattering coefficients are statistically obtained with N independent samples for each oil-spilled surface using the Monte-Carlo technique for various conditions of surface roughness, oil-layer thickness, frequency, polarization and incidence angle. The numerical simulation results are compared with theoretical models for clean sea surfaces and SAR images of an off-spilled sea surface caused by the Hebei （Hebei province, China） Spirit oil tanker in 2007. Further, conditions for better oil spill extraction are sought by the numerical simulation on the effects of wind speed and oil-layer thickness at different inci- dence angles on the backscattering coefficients.

Quadrature Method of Moments for Nanoparticle Coagulation and Diffusion in the Planar Impinging Jet Flow

A computational model combining large .eddy simulation with quadrature moment method was em-ployed to study nanoparticle evolution in a confined impinging jet. The investigated particle size is limited in the transient regime, and the particle collision kernel was obtained by using the theory of flux matching. The simulation was validated by comparing it with the experimental results. The numerical results show coherent structure acts to dominate particle number intensity, size and polydispersity distributions, and it also induce particle-laden iet to be diluted by .the ambient.The evolution of particle dynarnics in.the impinging jet flow are strongly related to the Rey-nolds number and nozzle-to-plate distance, and their relationships were analyzed.

Nanoparticle coagulation in a planar jet via moment method

Large eddy simulations of nanoparticle coagulation in an incompressible planar jet were performed. The particle is described using a moment method to approximate the particle general dynarnics equations. The time-averaged results based on 3000 time steps for every case were obtained to explore the influence of the Schmidt number and the Damkohler number on the nanoparticle dynamics. The results show that the changes of Schmidt number have the influence on the number concentration of nanoparticles only when the particle diameter is less than 1 nm for the fixed gas parameters. The number concentration of particles for small particles decreases more rapidly along the flow direction, and the nanoparticles with larger Schmidt number have a narrower distribution along the transverse direction. The smaller nanoparticles Coagulate and disperse easily, grow rapidly hence show a stronger polydispersity. The smaller coagulation time scale can enhance the particle collision and coagulation. Frequented collision and coagulation bring a great increase in particle size. The larger the Damkohler number is, the higher the particle polydispersity is.

Nanoparticle Transport and Coagulation in Bends of Circular Cross Section via a New Moment Method

Transport of nanoparticles and coagulation is simulated with the combination of CFD in a circular bend. The Taylor-expansion moment method(TEMOM)is employed to study dynamics of nanoparticles with Brownian motion,based on the flow field from numerical simulation.A fully developed flow pattern in the present simulation is compared with previous numerical results for validating the model and computational code.It is found that for the simulated particulate flow system,the particle mass concentration,number concentration,particle polydispersity, mean particle diameter and geometric standard deviation over cross-section increase with time.The distribution of particle mass concentration at different time is independent of the initial particle size.More particles are concen-trated at outer edge of the bend.Coagulation plays more important role at initial stage than that in the subsequent period.The increase of Reynolds number and initial particle size leads to the increase of particle number concentration.The particle polydispersity,mean particle diameter and geometric standard deviation increase with decreasing Reynolds number and initial particle size.

A NEW MOMENT METHOD FOR THE FAST AND ACCURATE ANALYSIS OF NORMAL MODE HELICAL ANTENNAS

In this letter, a new moment method using helical segments is presented to model Normal Mode Helical Antenna (NMHA). Using this method, the NMHA can be modeled by a few segments. The current distributions and radiation patterns of some NMHAs are calculated.A comparison is made between results obtained using this helical segment algorithm and a linear segment algorithm, and the results of the two algorithms agree fairly well. When calculating the impedance matrix [Z], all the elements of the matrix can be obtained by only calculating a few elements with the application of the symmetric and periodic characteristics of the NMHA.Therefore, the CPU time and the memory storage are significantly reduced, with the accuracy and speed enhanced.

APPLICATION OF THE MOMENT METHODS TO ANALYSIS OF X-RAY DIFFRACTION LINE PROFILE FOR PA1010-BMI SYSTEM

Pure X-ray diffraction profiles have been analysed for polyamide 1010 and PA1O1O-BMI system by means of multipeak fitting resolution of X-ray diffraction. The methods of variance and fourth moment have been applied to determine the particle size and strain values for the paracrystalline materials. The results indicated that both variance and fourth moment of X-ray diffraction line profile yielded approximately the same values of the particle size and the strain. The particle sizes of (100) reflection have been found to decrease with increasing BMI content, whereas the strain values increased.

Simulation of the Brownian coagulation of nanoparticles with initial bimodal size distribution via moment method

The Brownian coagulation of nanoparticles with initial bimodal size distribution, i.e., mode i and j, is numerically studied using the moment method. Evolutions of particle number concentration, geometric average diameter and geometric standard deviation are given in the free molecular regime, the continuum regime, the free molecular regime and transition regime, the free molecular regime and continuum regime, respectively. The results show that, both in the free molecular regime and the continuum regime, the num- ber concentration of mode i and j decreases with increasing time. The evolutions of particle geometric average diameter with different initial size distribution are quite different. Both intra-modal and inter-modal coagulation finally make the polydispersed size distribution become monodispersed. As time goes by, the size distribution with initial bimodal turns to be unimoda/and shifts to a larger particle size range. In the free molecular regime and transition regime, the inter- modal coagulation becomes dominant when the number concentrations of mode i and j are of the same order. The effects of the number concentration of mode i and mode j on the evolution of geometric average diameter of mode j are negligible, while the effects of the number concentration of mode j on the evolution of geometric average diameter of mode j is distinct. In the free molecular regime and continuum regime, the higher the initial number concentration of mode j, the more obvious the variation of the number concentration of mode i.

THE CALCULATION METHOD OF FLYWHEEL MOMENT OFINERTIA OF CRANE MECHANISM

The origin and substance of the tlywheel moment calculation method of inertia mo ment of crane mechanism are set forth comprehensively and systematically in this paper, that is, the nucleus or the focus ofmoment of inertia calculation is the problem of calculating convertible tlywheelmoment. It is much better for the calculation of flywheel moment of loading move mass of lifting, traveling and rotating mechanism using the low of conservation of energy, the theorem of kinetic energyfor that of radius - changing mechanism, and the law of conservation of energy for that of all parts of gearing mechanism.

The b value spectrum and η value by the moment method

The earthquake size distribution is generally considered to obey the Gutenberg Richter (GR) law. We have introduced the concept of the b value spectrum based on the moment method to investigate the deviation of the actual magnitude distribution of earthquakes from this law. This enables us to describe characteristic features of the magnitude frequency distribution of earthquakes. We found also a simple relation between the η value and the b value spectrum. Analysis using this scheme showed that the actual size distributions of earthquakes have large variations from case to case and sometimes deviate considerably from the widely assumed the GR formula.

THE MOMENT METHOD ON RESONANT QUADRIFILAR HELICAL ANTENNAS

An efficient measure is taken to rearrange Nakano’s kernels of integral equationsfor an antenna system composed of arbitrarily bent wires.By means of the moment method,great efforts are made to analyze and compute the circularly polarized patterns,directivity,axialratio,front-to-back ratio and beam-shaping characteristics of resonant quadrifilar helical antennasas well as the feeding technique and the effect of an electrically large conducting body on theperformance of the antennas.

The moment-method form of Pocklington's integral equation above ground

Pocklington’s integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expressed as the elementary functions using the Fresnel plane wave reflection coefficients method; and the Pocklington’s integral equation will be rearranged into a linear equation with solution easily obtained by using the method of moments, when the sinusoidal sub domain expansion is chosen to express the current distributions.

A method of moments for calculating dynamic responses beyond linear response theory

A method of moments for calculating the dynamic response of periodically driven overdamped nonlinear stochastic systems in the general response sense is proposed, which is a modification of the method of moments confined within linear response theory. The calculating experience suggests that the proposed technique is simple and efficient in implementation, and the comparison with stochastic simulation shows that the first three orders of susceptibilities calculated by the proposed technique have high accuracy. The dependence of the spectral amplification parameters at the first three harmonics on the noise intensity is also investigated, and another observed phenomenon of stochastic resonance in the systems induced by the location of a single periodic orbit is disclosed and explained.