A modified agglomeration kernel model used for particle agglomeration

Particle agglomeration in turbulent flows is an extremely complex process, and comprehensive study is beneficial to the development of turbulent agglomeration technology. The effects of vortex generator structure on particle agglomeration were investigated based on a combined computational fluid dynamics (CFD) and discrete element method (DEM) approach, including the geometric size, spanwise pitch, longitudinal pitch, row number, arrangement and form of the vortex generator. A dimensionless parameter K (the ratio of vorticity to strain rate, abbreviated as K) was defined to evaluate the agglomeration ability of the vortex generator, and the traditional agglomeration kernel function was modified by adding a correction coefficient α (agglomeration efficiency). By comparing the traditional agglomeration model and the modified agglomeration model with the experimental results, it showed that the modified agglomeration kernel model can simulate a result closer to the real agglomeration process.     

Volterra kernel identification of MIMO aeroelastic system through multiresolution and multiwavelets

In this paper a nonlinear multi-input multi-output (MIMO) aeroelastic systems’ Volterra kernel identification (VKI) is carried out by expansion of the Volterra kernels in terms of scale functions and multiwavelet functions employing multiresolution. The resulting system of discretized Volterra equations is solved through least square method employing singular value decomposition. A new algorithm for solution of the discretized Volterra equations is proposed which is beneficial for solution on computers with limited memory resources. A new input excitation signal is presented for simultaneous excitation of MIMO unsteady aerodynamic response. The identified MIMO Volterra kernels include first order and second order self-interaction and cross-interaction kernels of the system. The results from the VKI based reduced order model is compared with the coupled computational fluid dynamics and computational structural dynamics simulation of an aeroelastic wing.     

A multigrid accelerated hybrid unstructured mesh method for 3D compressible turbulent flow

 A cell vertex finite volume method for the solution of steady compressible turbulent flow problems on unstructured hybrid meshes of tetrahedra, prisms, pyramids and hexahedra is described. These hybrid meshes are constructed by firstly discretising the computational domain using tetrahedral elements and then by merging certain tetrahedra. A one equation turbulence model is employed and the solution of the steady flow equations is obtained by explicit relaxation. The solution process is accelerated by the addition of a multigrid method, in which the coarse meshes are generated by agglomeration, and by parallelisation. The approach is shown to be effective for the simulation of a number of 3D flows of current practical interest. Sponsored by The Research Council of Norway, project number 125676/410 Dedicated to the memory of Prof. Mike Crisfield, a respected colleague     

AN ADAPTIVE AGGLOMERATION METHOD FOR ADDITIVE CORRECTION MULTIGRID

In the computational simulation of fluid flow and scalar transport, multigrid iterative solution techniques often fail or stall when the discrete linearized equations have strongly anisotropic coefficients. In the present work, an adaptive agglomeration algorithm for forming coarse grids is presented that allows multigrid techniques to work efficiently for equation sets with anisotropic coefficients. The adaptive agglomeration is defined by two rules and several guidelines that follow from a physical interpretation of the performance of iterative solvers like Gauss–Seidel. The effectiveness of the adaptive agglomeration algorithm is demonstrated for a wide range of test cases. © 1997 by John Wiley & Sons, Ltd.     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.     

Multi-level residue harmonic balance solution for the nonlinear natural frequency of axially loaded beams with an internal hinge

In this article, an analytical approach, namely, multi-level residue harmonic balance is introduced and developed for the nonlinear free vibration analysis of axially loaded beams with an internal hinge. The main advantage of this method is that only one set of nonlinear algebraic equations is required to be solved for obtaining the zero level solution while the high accuracy of the higher level solutions can be obtained by solving a set of linear equations. The new approximate analytical solution method is developed for solving the governing differential equations. The accuracy and efficiency of the proposed method are verified by a numerical method. In the comparison, the results obtained from the proposed method well agree with those from other methods. The effects of vibration amplitude, axial force, and hinge location on the fundamental frequencies of various beam cases are investigated. The optimum and worst hinge locations are also studied.     

Turbulence-radiation interaction in confined combustion systems

A new approach for modeling turbulence-radiation interaction is proposed. The formulation is based upon equations for statistical moments. Additional to the balance equations for the velocity and mixture fraction, equations for the mean, variance, covariance of heat release rate, and mixture fraction is solved. The coupling with the chemistry model is formulated by means of a two dimensional pdf of mixture fraction and heat release rate. The proposed approach is open for improvement by more sophisticated submodels. A natural gas fired combustion chamber is designed and constructed, and the temperature field measured by CARS spectroscopy. The main features of the modified combustion system are discussed. The comparison of experimental temperatures with the numerical simulation of the combustion system shows the good quality of our approach. The modeling of the two dimensional pdf is found to be most suitable for the hot region near the burner, where most radiation effects take place.     

Investigating particle-level dynamics to understand bulk behavior in a lab-scale Agitated Filter Dryer (AFD) using Discrete Element Method (DEM)

Agitated filter drying (AFD) is a complex physical-thermal separation process which involves isolating solutes from its mother liquor. In agro-chemical and pharmaceutical industry, filter-dryers are used for sequestering active ingredients (AIs) and key intermediates from the wet cake after the crystallization step. During the agitated drying phase, the mechanical agitation of the wet cake, implemented to enhance heat and mass transport, has been commonly observed to result in formation of undesired agglomerates that require further processing. Only relatively few experimental and computational studies of the effects of operating parameters and material properties on the drying and agglomeration growth kinetics have been described in the literature. In absence of robust predictive models, the go-to solution in order to avoid the agglomeration behavior of AIs has been to use minimal agitation which is not only suboptimal but also significantly increases the drying times.The simulation of drying and agglomeration behaviors in AFD is particularly challenging because the agitated drying processes are mechanistically governed by simultaneous heat, mass and momentum transfer equations. In addition, the behavior of agglomeration growth and drying pathway varies significantly with the physical properties of the residual solvents in the cake as well as the operating conditions of the agitated dryer. A comprehensive modeling approach to simulate both drying and agglomeration behavior in AFDs through implementation of mechanistic Discrete Element Modeling (DEM) simulations with coupled granular liquid bridge cohesion model, heat conduction model and evaporation kinetics is presented. Additionally, in-depth analysis of particle scale behavior which is responsible for drying and agglomerate growth kinetics are also studied with respect to different scaling criteria is also presented.     

A fast multipole method for Maxwell equations stable at all frequencies

The solution of Helmholtz and Maxwell equations by integral formulations (kernel in exp(i kr)/r) leads to large dense linear systems. Using direct solvers requires large computational costs in O(N(3)). Using iterative solvers, the computational cost is reduced to large matrix-vector products. The fast multipole method provides a fast numerical way to compute convolution integrals. Its application to Maxwell and Helmholtz equations was initiated by Rokhlin, based on a multipole expansion of the interaction kernel. A second version, proposed by Chew, is based on a plane-wave expansion of the kernel. We propose a third approach, the stable-plane-wave expansion, which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane-wave expansion. The computational complexity is Nlog N as with the other methods.     

The benefits of parallel multibody simulation

To exploit the benefits of parallel computer architectures for multibody system simulation, an interdisciplinary approach has been pursued, combining knowledge of the three disciplines of dynamics, numerical mathematics and computer science. An analysis of the options available for the formulation and numerical solution of the dynamical system equations yielded a surprising result. A method initially proposed to solve the inverse problem of dynamics is the best choice to generate the system equations required for solving the simulation problem, when relying on implicit integration routines. Such routines have the particular advantage of handling stiff systems, too. The new O(N)-residual formalism, generating the system equations in a form required for implicit numerical integration, has a high potential to benefit from parallel computer architectures. Two strategies of medium and coarse grain parallelization have been implemented on a Transputer network to obtain a package for parallel multibody simulation. An analysis of the performance of this package demonstrates for typical multibody simulation problems that the new code is five times faster than existing codes when implemented on a serial computer. An additional speed-up by the same order of magnitude is obtained when the code is implemented on a Transputer network.     

Theoretical analysis of acoustic and turbulent agglomeration of droplet aerosols

This study meticulously explores the agglomeration mechanisms in microscale droplet aerosols, specifically focusing on acoustic and turbulent agglomeration mechanisms. Our theoretical analysis reveals a significant impact of orthokinetic and hydrodynamic processes on acoustic agglomeration. The acoustic wake effect elucidates the swift replenishment of small particles subsequent to an orthokinetic phase. An optimal frequency, varying for different droplets, was identified in orthokinetic agglomeration within the 50–250 Hz range. Hydrodynamic agglomeration remained relatively stable at an acoustic frequency exceeding 1000 Hz. The aggregation kernel function, denoted as K_ij, exhibited a significant increase with increasing sound pressure levels, reaching up to 10⁻⁸ s⁻¹. Environmental temperature had a predominantly positive effect on orthokinetic and Brownian agglomeration, although it exhibited an inhibitory effect on hydrodynamic agglomeration. For raindrops, a correlation was identified between particle spacing and K_ij; a larger particle spacing corresponded to a smaller K_ij. Despite an increase in particle spacing to 50 times the particle diameter, the hydrodynamic effect persisted. The aggregation kernel function linked to Brownian thermal motion was found to be 3–4 orders of magnitude lower than that of orthokinetic and hydrodynamic interactions. Additionally, the turbulent agglomeration kernel function for fog, cloud, and rain droplets with corresponding parent nuclei of 100 μm was of the same order of magnitude as the acoustic agglomeration kernel function.     

Meshfree particle simulation of micro channel flows with surface tension

This paper presents a study of micro channel flows using a meshfree particle approach. The approach is based on smoothed particle hydrodynamics (SPH) and its variant, adaptive smoothed particle hydrodynamics (ASPH). The incompressible flow in the micro channels is modeled as an artificially compressible flow. The surface tension is incorporated into the equations of motion. The classic Poiseuille flow and a practical micro channel flow problem of flip-chip underfill encapsulation process are investigated. It is found that the adaptive kernel can well match the computational geometry with long channels and can greatly save computational time. The simulation results are in close agreement with the analytical solutions.     

A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built‐in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin‐based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher‐order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two‐dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

A stochastic simulation method for uncertainty quantification in the linearized inverse conductivity problem

This paper considers the inverse problem in electrical impedance tomography with non‐informative prior information on the required conductivity function. The problem is approached with a Newton‐type iterative algorithm where the solution of the linearized approximation is estimated using Bayesian inference. The novelty of this work focuses on maximum a posteriori estimation assuming a model that incorporates the linearization error as a random variable. From an analytical expression of this term, we employ Monte Carlo simulation in order to characterize its probability distribution function. This simulation entails sampling an improper prior distribution for which we propose a stable scheme on the basis of QR decomposition. The simulation statistics show that the error on the linearized model is not Gaussian, however, to maintain computational tractability, we derive the posterior probability density function of the solution by imposing a Gaussian kernel approximation to the error density. Numerical results obtained through this approach indicate the superiority of the new model and its respective maximum a posteriori estimator against the conventional one that neglects the impact of the linearization error. Copyright © 2011 John Wiley & Sons, Ltd.     

Torsional impact response of a penny-shaped crack lying on a bimaterial interface

The torsional impact response of a penny-shaped crack lying on a bimaterial interface is considered in this study. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution to the dual integral equations is expressed in terms of a Fredholm integral equation of the second kind with a finite integral kernel. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time and material constants is discussed.     

Large deformation analysis of rubber based on a reproducing kernel particle method

 A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.     

Solutions of population balance equations by double Legendre polynomial transformation

Abstract

First‐order partial differential equations of population balance are solved by employing the Legendre polynomials. The key of the method is that the dependent variable of the population density function is assumed to be expressed by a double series of Legendre polynomials with respect to time and space variables. The approach algorithm is that a series of ordinary differential equations are obtained by making the Legendre transformation with respect to the space coordinate. The series of time‐function ordinary differential equations are further transformed into algebraic equations of expansion coefficients with respect to time. The expansion coefficients of the Legendre polynomials are obtained by solving matrix equations which represent the series of algebraic equations. Illustrative examples are given, and the computational results are compared with those of other numerical values given in the literature. Satisfactory agreements are obtained.     

Crystal agglomeration of europium oxalate in reaction crystallization using double-jet semi-batch reactor

The particle agglomeration of europium oxalate was investigated in a double-jet semi-batch reactor over a wide range of operating variables, including the agitation speed, reactant feed rate, and reactant concentration. The size of the agglomerates was directly dictated by the particle collision and supersaturation promoting agglomeration and the fluid shear force inhibiting agglomeration. Thus, with a longer feeding time and higher feed concentration for the reaction crystallization, the mean particle size increased, while the corresponding total particle population decreased due to the enhanced chance of particle agglomeration, resulting from a longer residence time and higher supersaturation in the reactor. Agitation was found to exhibit a rather complicated influence on particle agglomeration. Although both particle collision and turbulent fluid shear were promoted by an increase in the mixing intensity, the crystal agglomeration of europium oxalate was maximized at around 500 rpm of agitation speed due to an optimized balance between particle aggregation and breakage.