Size-dependent coalescence kernel in fertilizer granulation-A comparative study

Granulation is a key process in several industries like pharmaceutical, food, fertilizer, agrochemicals, etc. Population balance modeling has been used extensively for modeling agglomeration in many systems such as crystallization, aerosols, pelletisation, etc. The key parameter is the coalescence kernel, β（ij） which dictates the overall rate of coalescence as well as the effect of granule size on coalescence rate. Adetayo, Litster, Pratsinis, and Ennis （1995） studied fertilizer granulation with a broad size distribution and modeled it with a two-stage kernel. A constant kernel can be applied to those granules which coalesce successfully. The coalescence model gives conditions for two types of coalescence, Type I and II. A twostage kernel, which is necessary to model granule size distribution over a wide size distribution, is applied in the present fluidized bed spray granulation process. The first stage is size-independent and non-inertial regime, and is followed by a size-dependent stage in which collisions between particles are non-random, i.e. inertial regime. The present work is focused on the second stage kernel where the feed particles of volume i and j collide and form final granule ij instead of i ＋j （Adetayo et al., 1995） which gives a wider particle size distribution of granules than proposed earlier.     

SPH方法中常数一致性核函数的建立及公式化

施加边界条件的不足是SPH方法的一个棘手问题,因为在结构边界外没有颗粒的存在,使得在边界处核函数的单位特性不能得到满足。施加"伪"颗粒是目前通用的一种方法,但是对于不规则结构和复杂几何边界,确定这些"伪"颗粒非常困难。本文讨论通过使用满足常数一致性的核函数来改善边界的不足。文章首先通过三种方法推导了满足常数一致性条件的核函数及其函数梯度的表达式,发现了两个不同分母式的表达,分析了满足常数一致性的修正核函数的数学特性。开展了二维和三维的算例比较,结果发现使用修正的核函数对边界条件有明显改善,对计算精度和稳定性也有显著提高。     

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.     

基于拉氏变换的弹性动力学插值型重构核粒子法

插值型重构核粒子法的形函数具有离散点插值特性和不低于核函数的高阶光滑性，因而不仅可以直接施加本质边界条件，同时也保证了较高的计算精度．本文将弹性动力学方程作拉氏变换后，在变换域内用插值型重构核粒子法求解，最后再借助Durbin数值反演方法求得时间域的解．针对典型的弹性动力学问题，给出了插值型重构核粒子法的数值算例，并验证了本文方法的有效性．     

Self-similar behavior for multicomponent coagulation

Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by introduction of different generating functions. In these solutions, two size-scale variables are introduced to characterize the asymptotic distribution of total mass and individual masses. The result of product kernel （gelling kernel） is consistent with the Vigli-Ziff conjecture to some extent. Furthermore, the steady-state solution with injection for the constant kernel is obtained, which is again the product of a normal distribution and the scaling solution for the single variable coagulation.     

Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations

Volterra series is a powerful mathematical tool for nonlinear system analysis, which extends the convolution integral for linear system to nonlinear system. There is a wide range of nonlinear engineering systems and structures which can be modeled as Volterra series. One question involved in modeling a functional relationship between the input and output of a system using Volterra series is to identify the Volterra kernel functions. In this article, a wavelet balance method-based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different orders are first estimated with the wavelet balance method, and then the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval. The simulation studies verify the effectiveness of the proposed Volterra kernel identification method.     

Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary

In this paper, the problem of a crack embedded in a half-plane piezoelectric solid with traction-induction free boundary is analyzed. A system of singular integral equations is formulated for the materials with general anisotropic piezoelectric properties and for the crack with arbitrary orientation. The kernel functions developed are in complex form for general anisotropic piezoelectric materials and are then specialized to the case of transversely isotropic piezoelectric materials which are in real form. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors for transversely isotropic piezoelectric materials.     

广义概率密度演化方程的再生核质点加密算法

采用一般质点近似和再生核质点近似表示系统响应量,给出了动力系统响应量的一般表达式。在此基础上,发展了一类求解广义概率密度演化方程的再生核质点加密算法,给出了详细求解步骤。以单自由度系统为例,从响应概率密度的角度考察了再生核质点加密算法的精度。以多自由度框架结构为例,验证了再生核质点加密算法求取非线性随机动力系统响应概率密度的正确性。     

Multiwavelet Constructions and Volterra Kernel Identification

The Volterra series is commonly used for the modeling of nonlinear dynamical systems. In general, however, a large number of terms are needed to represent Volterra kernels, with the number of required terms increasing exponentially with the order of the kernel. Therefore, reduced-order kernel representations are needed in order to employ the Volterra series in engineering practice. This paper presents an approach whereby multiwavelets are used to obtain low-order estimates of first-, second-, and third-order Volterra kernels. A family of multiwavelets is constructed from the classical finite element basis functions using the technique of intertwining. The resulting multiwavelets are piecewise-polynomial, orthonormal, compactly-supported, and can be constructed with arbitrary approximation order. Furthermore, these multiwavelets are easily adapted to the domains of support of the Volterra kernels. In contrast, most wavelet families do not possess this characteristic. Higher-dimensional multiwavelets can easily be constructed by taking tensor products of the original one-dimensional functions. Therefore, it is straightforward to extend this approach to the representation of higher-order Volterra kernels. This kernel identification algorithm is demonstrated on a prototypical oscillator with a quadratic stiffness nonlinearity. For this system, it is shown that accurate kernel estimates can be obtained in terms of a relatively small number of wavelet coefficients. These results indicate the potential of the multiwavelet-based algorithm for obtaining reduced-order models for a large class of weakly nonlinear systems.     

Selection of kernel functions used in a new constitutive equation for viscoelastic fluids

A selection of kernel functions is given to be used in a new integral constitutive equation proposed by Piau whereby the deviatoric stress is calculated from the integral of the history of the past intrinsic rate of rotation and rate of deformation tensors through a representation theorem. Piau has demonstrated the objectivity of a frame moving with a given particle whose axis are directed along the eigenvectors of the rate of deformation tensor. The use of such a framework provides a new approach in the attempt to reduce the computational difficulties associated with conventional constitutive equations written in co-deformational or co-rotational reference frames.The shear and primary normal-stress material functions and the extensional (elongational) stress growth function are defined for the proposed integral constitutive equation. These material functions are used to calculate the kernel functions using steady state, stress relaxation and stress growth data of Attané in simple shear flow for monodisperse polystyrene solutions. The shear and extensional stress growth data of Meissner for a polyethylene melt are also used to show the flexibility of the rheological model.The material functions are first written in terms of five monotonically decreasing functions of the time lag between the past and the present time. Then kernel functions are chosen such that when substituted in the new integral constitutive equation they yield the functions used to describe the data. A further condition imposed on the normalized kernel functions is that they be decreasing functions of time lag.     

Calculation of expansion of a packed bed of a disperse material subjected to pulsed fluidization

Pulsed fluidization of a packed bed of a disperse material is studied experimentally and numerically. Mathematical modeling of the process is performed within the framework of the approaches of mechanics of heterogeneous media on the basis of an experimental study of the bed uplift height as a function of the pressure difference in the fluidized bed. A detailed dynamic flow pattern corresponding to one pulse of the experimental setup is obtained numerically, velocity and pressure distributions inside the bed are found, and the maximum height of the bed uplift is determined. The results obtained are demonstrated to be in reasonable agreement with experimental data on the uplift height as a function of the pressure difference.     

势问题的重构核粒子边界无单元法

将重构核粒子法和势问题的边界积分方程方法结合,提出了势问题的重构核粒子边界无单元法. 推导了势问题的重构核粒子边界无单元法的公式,研究其数值积分方案,建立了重构核粒子边界无单元法的离散化边界积分方程,并推导了重构核粒子边界无单元法的内点位势的积分公式. 重构核粒子法形成的形函数具有重构核函数的光滑性,且能再现多项式在插值点的精确值,所以该方法具有更高的精度. 最后给出了数值算例,验证了所提方法的有效性和正确性. }      

Theoretical analyses on bed topography responses in large depth-to-width ratio river bends with constant curvatures

Bed morphology is the result of a dynamic response to a complex meandering river system. It is an important factor for the further development of river. Based on the meandering river characterized by a large depth-to-width ratio, a theoretical model is established with the coupling of Navier-Stokes (N-S), sediment transport, and bed deformation equations. The flow characteristics and bed response of river are obtained with the perturbation method. The research results show that, under the effect of two-dimensional flow disturbance, the bars and pools present the regular response. For a given sinuousness, the amplitude of the bed response can be used as a criterion to judge the bedform stability. The effects of the Reynolds number, disturbance wavenumber, sinuousness, and bed morphology gradient on the bed response development are described.     

A moving particle semi‐implicit method for free surface flow: Improvement in inter‐particle force stabilization and consistency restoring

The moving particle semi‐implicit (MPS) method has been widely applied in free surface flows. However, the implementation of MPS remains limited because of compressive instability occurred when the particles are under compressive stress states. This study proposed an inter‐particle force stabilization and consistency restoring MPS (IFS‐CR‐MPS) method to overcome this numerical instability. For inter‐particle force stabilization, a hyperbolic‐shaped quintic kernel function is developed with a non‐negative and smooth second order derivative to satisfy the stability criterion under compressive stress state. Then, a contrastive study is conducted on the contradiction between the common understanding of the conventional MPS hyperbolic‐shaped kernel function and its performance. The result shows that the conventional MPS hyperbolic‐shaped kernel function can easily cause violent repulsive inter‐particle force and then lead to the compressive instability. Therefore, the first order derivative of the modified hyperbolic‐shaped quintic kernel function is recommended as the form of the contribution of the neighbor particles to achieve a more stable inter‐particle repulsive force. For consistency restoring, the Taylor series expansion and the hyperbolic‐shaped quintic kernel are combined to improve the accuracy of the viscosity and pressure calculation. The IFS‐CR‐MPS algorithm is subsequently verified by the inviscid hydrostatic pressure, jet impacting, and viscous droplet impacting problems. These results can be used for choosing kernel function and the contribution of neighbor particles in particle methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling fluid and heat transport in the reactive, porous bed of downdraft (biomass) gasifier

A fluid flow and heat transfer model has been developed for the reactive, porous bed of the biomass gasifier to simulate pressure drop, temperature profile in the bed and flow rates. The conservation equations, momentum equation and energy equation are used to describe fluid and heat transport in porous gasifier bed. The model accounted for drag at wall, and the effect of radial as well as axial variation in bed porosity to predict pressure drop in bed. Heat transfer has been modeled using effective thermal conductivity approach. Model predictions are validated against the experiments, while effective thermal conductivity values are tested qualitatively using models available in literature. Parametric analysis has been carried out to investigate the effect of various parameters on bed temperature profile and pressure drop through the gasifier. The temperature profile is found to be very sensitive to gas flow rate, and heat generation in oxidation zone, while high bed temperature, gas flow rate and the reduction in feedstock particle size are found to cause a marked increase in pressure drop through the gasifier. The temperatures of the down stream zones are more sensitive to any change in heat generation in the bed as compared to upstream zone. Author recommends that the size of preheating zone may be extended up to pyrolysis zone in order to enhance preheating of input air, while thermal insulation should not be less than 15 cm.     

Minimum fluidization velocities for supercritical water fluidized bed within the range of 633–693 K and 23–27 MPa

Supercritical water fluidized bed is a new reactor concept for biomass gasification. In this paper, an experimental study on the hydrodynamics of a supercritical water fluidized bed was conducted. The frictional pressure drops of a fixed bed and a fluidized bed were measured for a temperature ranging from 633 to 693 K and pressure ranging from 23 to 27 MPa. The results show that the Ergun formula for calculating the frictional pressure drop of a fixed bed can still be applied in supercritical water conditions. The average deviation between Ergun formula and experiment results is 13.3%. A predicting correlation for the minimum fluidization velocity in a supercritical water fluidized bed was obtained based on the experimental results of a fixed bed and the fluidized bed pressure drop. The average error between the correlation and experiment results was about 3.1%. The results in this paper are useful for the design of SCW fluidized bed.     

Influence of drag laws on pressure and bed material recirculation rate in a cold flow model of an 8 MW dual fluidized bed system by means of CPFD

A cold flow model of an 8 MW dual fluidized bed (DFB) system is simulated using the commercial computational particle fluid dynamics (CPFD) software package Barracuda. The DFB system comprises a bubbling bed connected to a fast fluidized bed with the bed material circulating between them. As the hydrodynamics in hot DFB plants are complex because of high temperatures and many chemical reaction processes, cold flow models are used. Performing numerical simulations of cold flows enables a focus on the hydrodynamics as the chemistry and heat and mass transfer processes can be put aside. The drag law has a major influence on the hydrodynamics, and therefore its influence on pressure, particle distribution, and bed material recirculation rate is calculated using Barracuda and its results are compared with experimental results. The drag laws used were energy-minimization multiscale (EMMS), Ganser, Turton–Levenspiel, and a combination of Wen–Yu/Ergun. Eleven operating points were chosen for that study and each was calculated with the aforementioned drag laws. The EMMS drag law best predicted the pressure and distribution of the bed material in the different parts of the DFB system. For predicting the bed material recirculation rate, the Ganser drag law showed the best results. However, the drag laws often were not able to predict the experimentally found trends of the bed material recirculation rate. Indeed, the drag law significantly influences the hydrodynamic outcomes in a DFB system and must be chosen carefully to obtain meaningful simulation results. More research may enable recommendations as to which drag law is useful in simulations of a DFB system with CPFD.     

SPH方法在模拟线弹性波传播中的运用

通过对固体中波动问题的模拟建立了一种光滑粒子法的新形式,一种运用SPH的核函数的类似有限体积法的计算方法。通过对统计体积的修正以及对边界粒子的核函数修正,较好地解决了SPH方法中长期以来制约其被广泛应用的主要问题之一边界条件的表述。在此基础上成功地在光滑粒子法中实现了透射边界条件的模拟。同时利用反卷积修正使得较大粒子间距下的计算结果的精度大大提高。这种方法不但保持了SPH的简单性,而且很容易实现应力边界条件。     

Some Specific Features of Integral Equations with the Cauchy Kernel on a Closed Contour in Hydrodynamic Problems

Singular integral equations of the first and second kind with the Cauchy kernel on a limiting narrow closed contour are theoretically considered. The initial equations are found to become different on the limiting contour. This singularity of integral equations with the Cauchy kernel does not allow boundary-value problems of the flow around thin airfoils to be solved correctly; therefore, a system consisting of integral equations of the first and second kind is proposed for solving such problems. The results of the present study are tested against an exact solution of the problem of the flow past a flat plate.