Advances in numerical methods for the solution of population balance equations for disperse phase systems

Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system. A population balance equation (PBE), a non-linear hyperbolic equation of the number density function, is usually employed to describe the micro-behavior (aggregation, breakage, growth, etc.) of a disperse phase and its effect on particle size distribution. Numerical solution is the only choice in most cases. In this paper, three different numerical methods (direct discretization methods, Monte Carlo methods, and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation, computational load and numerical accuracy. Special attention is paid to the relatively new and superior moment methods including quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), modified quadrature method of moments (M-QMOM), adaptive direct quadrature method of moments (ADQMOM), fixed pivot quadrature method of moments (FPQMOM), moving particle ensemble method (MPEM) and local fixed pivot quadrature method of moments (LFPQMOM). The prospects of these methods are discussed in the final section, based on their individual merits and current state of development of the field. Supported by the National Basic Research Program of China (Grant No. 2004CB720208), the National Natural Science Foundation of China (Grant Nos. 40675011 & 10872159), and the Key Laboratory of Mechanics on Disaster and Environment in Western China     

凝并和成核机理下颗粒尺度分布的Monte Carlo求解

颗粒的凝并和成核现象影响其尺度分布,现有的MonteCarlo方法描述颗粒尺度分布的时间演变过程存在若干困难.提出了一种新的多重MonteCarlo(MMC)算法,基于时间驱动,利用加权的虚拟颗粒的思想,在模拟过程中保持虚拟颗粒总数不变和计算区域体积不变.利用该算法对“常凝并核,一阶成核”的情况下颗粒尺度分布的时间演变过程进行了数值求解,所得结果与数值解相符,表明MMC算法具有高且稳定的计算精度.另外,MMC算法由于跟踪比实际颗粒数目少得多的虚拟颗粒而具有较低的计算代价.     

Runge–Kutta–Nyström methods with equation dependent coefficients and reduced phase lag for oscillatory problems

A new family of one-parameter equation dependent Runge–Kutta–Nyström (EDRKN) methods for the numerical solution of second–order differential equations are investigated. The coefficients of new three-stage EDRKN methods are obtained by nullifying up to appropriate order of moments of operators related to the internal and external stages. A fifth-order EDRKN method that is dispersive of order six and dissipative of order five and a fourth-order EDRKN method that is dispersive of order four and zero-dissipative are derived. Phase analysis shows that there exist no explicit EDRKN methods that are P-stable. Numerical experiments are reported to show the high accuracy and efficiency of the new EDRKN methods.     

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrödinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrödinger equation and of coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known methods.     

Effect of brownian diffusion on chemical kinetics

A new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady-state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self-reaction of tert-butyl radicals in n-alkanes shows a good correspondence.     

Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation

In this paper we present a new method for the numerical solution of the time-independent Schrödinger equation for one spatial dimension and related problems. A technique, based on the phase-lag and its derivatives, is used, in order to calculate the parameters of the new Numerov-type algorithm. We study the relation of the local truncation error with the energy of the model of the radial Schrödinger equation and via this investigation we examine how accurate is the new method compared with other well known numerical methods in the literature. We present also the stability analysis of the new method and the relation of the interval of periodicity with the frequency of the test problem and the frequency of the new developed method. We illustrate the accuracy and computational efficiency of the new developed method via numerical examples.     

A new modified embedded 5(4) pair of explicit Runge–Kutta methods for the numerical solution of the Schrödinger equation

In this work a new modified embedded 5(4) pair of explicit Runge–Kutta methods is developed for the numerical solution of the Schrödinger equation. We investigate the error of the new pair, based on the error analysis we apply the higher order method to the resonance problem, also we apply the new embedded pair to elastic scattering phase-shift problem. The applications show the efficiency of our new developed embedded pair and the higher order method.     

Brownian Coagulation of Fractal Agglomerates: Analytical Solution Using the Log-Normal Size Distribution Assumption

An analytical solution to Brownian coagulation of fractal agglomerates in the continuum regime that provides time evolution of the particle size distribution is presented. The theoretical analysis is based on representation of the size distribution of coagulating agglomerates with a time-dependent log-normal size distribution function and employs the method of moments together with suitable simplifications. The results are found in the form that extends the spherical particle solution previously obtained by K. W. Lee (J. Colloid Interface Sci. 92, 315-325 (1983)). The results show that the mass fractal dimension has a significant effect on the size distribution evolution during coagulation. When the obtained solution was compared with numerical results, good agreement was found. The self-preserving size distribution of nonspherical agglomerates is discussed. Copyright 2000 Academic Press.     

Solution of the Schrödinger equation for two different molecular potentials by the Nikiforov-Uvarov method

In this study, the solution of the Schr?dinger equation by a method developed by Nikiforov and Uvarov which is not based on the manipulation of formal power series has been schematically presented. The method gives elegant, easy and exact solutions of the Schr?dinger equation. In order to demonstrate the applications of the method, solutions of the Schr?dinger equation for the well-known pseudo-harmonic oscillator and a new symmetrical potential proposed by the authors are given. The concrete energy spectra and corresponding wave functions are obtained. The superiority and the limitations of the method compared to other methods have also been emphasized. Received: 20 November 1996 / Accepted: 1 October 1997     

Incorporation of the electron energy-loss straggling into the Fermi–Eyges equation

A modified Fermi–Eyges equation has been derived from the linear Boltzmann equation by including a term for describing electron energy-loss straggling. The solution has been obtained by the use of a generalized Eyges' method, yielding the electron energy distribution expressed with moments method in addition to Eyges' original solution. The first- and second-order approximations of the spectrum give the well-known continuous-slowing-down approximation (CSDA) and Gaussian distribution, respectively. Inclusion of the third-order moment in the spectrum yields the Vavilov distribution approximated with the Airy function. The higher order approximations can be evaluated numerically.     

A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems

The presentation, development and analysis of a new two-stages tenth algebraic order symmetric six-step method is introduced, for the first time in the literature, in this paper. More specifically, we present the development of the new method (requesting the highest algebraic order and the elimination of the phase-lag and its first and second derivatives), the analysis (error analysis and stability and interval of periodicity analysis) and the evaluation of the new developed method comparing its efficiency with the efficiency of well known methods and very recently produced methods in the literature on the approximate solution of the resonance problem of the one dimensional (or radial) Schrödinger equation. From the developments achieved and the results presented, we prove that the new obtained method is most more effective than other well known or recently developed methods of the literature.     

Exponentially - Fitted Multiderivative Methods for the Numerical Solution of the Schrödinger Equation

In this paper exponentially fitted multiderivative methods are developed for the numerical solution of the one-dimensional Schrödinger equation. The methods are called multiderivative since uses derivatives of order two and four. An application to the the resonance problem of the radial Schrödinger equation indicates that the new method is more efficient than other similar well known methods of the literature.     

New two stages high order symmetric six-step method with vanished phase–lag and its first,second and third derivatives for the numerical solution of the Schrödinger equation

In the present paper, we obtain and analyze, for the first time in the literature, a new two-stages high order symmetric six-step method. The specific characteristics of the new proposed method are the highest possible algebraic order, the elimination of the phase–lag and its first, second and third derivatives. Additionally, for the new method we give the analysis of the method (both error and stability and interval of periodicity analysis) and the comparison of the effectiveness of the new developed method with the effectiveness of well known methods and very recently produced methods in the literature. The comparison is based on the numerical solution of the Schrödinger equation. The theoretical achievements and the numerical results show the effectiveness of the new developed method in comparison with other well known or recently developed numerical methods.     

Multi-Monte Carlo method for coagulation and condensation/evaporation in dispersed systems

A new multi-Monte Carlo (MMC) method is promoted to consider general dynamic equation (GDE) for particle coagulation and condensation/evaporation. MMC method introduces the concept of a "weighted fictitious particle" and is based on time-driven Monte Carlo technique, constant number of fictitious particles technique, and constant volume technique. MMC method for independent coagulation, for independent condensation/evaporation, and for simultaneous coagulation and condensation/evaporation are validated by some special cases in which analytical solutions exist, in which numerical results agree with corresponding analytical solutions well. Furthermore, the computation cost of MMC method is low enough to be applied in engineering computation and general scientific quantitative analysis.     

A new explicit hybrid four-step method with vanished phase-lag and its derivatives

A hybrid explicit sixth algebraic order four-step method with phase-lag and its first, second and third derivatives vanished is obtained in this paper. We present the development of the new method, its comparative error analysis and its stability analysis. The resonance problem of the Schrödinger equation, is used in order to study the efficiency of the new developed method. After the presentation of the theoretical and the computational results it is easy to see that the new constructed method is more efficient than other well known methods for the approximate solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

New P-Stable Eighth Algebraic Order Exponentially-Fitted Methods for the Numerical Integration of the Schrödinger Equation

A family of P-stable high algebraic order exponentially-fitted methods for the numerical solution of the Schrödinger equation is developed in this paper. Numerical illustration to the resonance problem of the radial Schrödinger equation indicates that the new proposed methods are generally more efficient than the previously developed exponentially-fitted methods of the same kind.     

Integral equation for calculation of distribution function of activation energy of shear viscosity

A new technique of calculation of a distribution function of activation energy (f(E)) of shear viscosity based on a regularization procedure applied to the Fredholm integral equation of the first kind has been developed using the Baxter-Drayton and Brady model for concentrated and flocculated suspensions. This technique has been applied to the rheological data obtained at different shear rates for aqueous suspensions with fumed silica A-300 and low-molecular (3,4,5-trihydroxybenzoic acid and 1,5-dioxynaphthalene) or high-molecular (poly(vinyl pyrrolidone) of 12.7 kDa and ossein of 20-29 kDa) compounds over a wide concentration range (up to 25 wt% of both components) and at different temperatures. Monomodal f(E) distributions are observed for the suspensions with individual A-300 or A-300 with a low amount of adsorbed organics. In the case of larger amounts of nanosilica and organics the f(E) distributions are multimodal because of stronger structurization and coagulation of the systems that require a high energy to break the coagulation structures resisting to the shear flow.     

Excited state polarizabilities from solvatochromic shifts

A new method is proposed to estimate the polarizability (α_e) of a molecule in an excited state using solvatochromic shift measurements and McRae's equation. In the earlier methods the contribution due to polarizability was not considered. In view of this, the proposed method is also expected to give a better estimation of excited state electric dipole moment (μ_e) and the (θ) angle between excited and ground state electric dipole moments, μ_e and μ_g apart from giving values of polarizability of the molecules in the excited state. This method has been applied in the case of the La band of p-nitro aniline and the results for all the parameters are found to be satisfactory and of right order in comparison with that reported in literature.     

Kinetic study of slow coagulation

Summary Two equations for the slow rate of coagulation of colloid have been verified experimentally. The kinetic equation ofSmoluchowski has been modified byGhosh and a spectrophotometric technique developed byMukherjee has been adopted for the verification of this modified equation with ThO₂ sol system. The second equation relating the time of coagulation with concentration of electrolyte originally proposed byBhattacharya and derived byGhosh fromReerink's equation has been verified with AlAsO₄ sol following the method ofBhattacharya. The autocatalytic nature (S-shape) has not been found; phenomenon of Entlastungseffekt has been observed with ThO₂ sol.