Multi-scale finite element modelling using bubble function method for a convection-diffusion problem

Due to the multi-scale nature of the convection-diffusion equation the standard Galerkin method fails to predict accurate and stable results. A multi-scale finite element method based on the bubble function approach has been developed and applied to this problem. Finite element formulations based on the discontinuous weighting functions including the streamline upwind Petrov-Galerkin (SUPG) method are also evaluated. The results show that the multi-scale method provides stable and accurate results and matches very well the analytical solution.     

Finite element analysis of thermoplastic melts flow through the metering and die regions of single screw extruders

This research work is devoted to the development of a mathematical model for the simulation of the flow of polymer melts through the metering and die regions of single screw extruders. The sets of the governing equations (flow and energy) are solved using the finite element method. The power‐law model is used to describe the non‐Newtonian rheological behavior of the fluid. The standard Galerkin technique is used in conjunction with the continuous penalty scheme to solve the flow equations. Due to the low thermal diffusivity of the polymer melts, a streamline upwinding Petrov–Galerkin method is used to obtain convergent and stable results for the energy equation. This method is based on the extension of a previously developed scheme. The overall solution strategy is based on the Picard iterative scheme. Simulation results are obtained for the flow of a polypropylene melt through the metering and die zones of a laboratory scale extruder. To validate the proposed model, the results of the computer simulations are compared with experimentally measured mass flow rate and pressure profile. These comparisons show that there is very good agreement between the model predictions and actual data. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 676–689, 1999     

Numerical simulation of moving free surface problems in polymer processing using volume‐of‐fluid method

We have developed a numerical algorithm based on 2D/3D finite element method for solving non‐Newtonian fluid flow with the moving free surface encountered in polymer processing. The power law model is considered as a rheological constitutive equation. The standard Galerkin finite element formulation/penalty formulation are applied to discrctize the governing equations, the volume‐of‐fluid (VOF) scheme is used to track the moving free surface, and the donor‐acceptor model introduced by Hirt and Nichols is modified and implemented on FEM. We applied the numerical scheme to simulate fountain flow and viscous buckling problems. For fountain flow, the numerical prediction of this study is in good agreement with the experimental results of other investigators. For viscous buckling, both 2D and 3D numerical simulations show that the shear thinning effect retards buckling. As this algorithm is very effective in treating moving free surface problems and requires less memory than previous algorithms, it may help solve challenging problems in polymer processing such as transient visroelastic flow simulations with moving free surfaces.     

Modeling of the impregnation process during resin transfer molding

A model has developed for simulating isothermal mold filling during resin transfer molding (RTM) of polymeric composites. The model takes into account the anisotropic nature of the fibrous reinforcement and change in viscosity of the polymer resin as a result of chemical reaction. The flow of impregnating resin through the fibrous network is described in terms of Darcy's law. The differential equations in the model are solved numerically using the finite element technique. The Galerkin finite element method is used for obtaining the pressure distribution. A characteristics based method is used to solve the non-linear hyperbolic mass balance equation. The finite element formulation facilitates computations involving the motion of the polymer resin front characterized by a free surface flow phenomenon.     

Simulation of the newtonian flow through an abruptly contracted tube using a mixed finite element method

The solutions of the Navier-Stokes equation for a Newtonian flow through a 4/1 contraction tube were obtained numerically using the Galerkin finite element method with the nine-node Lagrangian element which was believed to be one of the most accurate tools for mixed-type interpolating formulations. It was proved from this study that the vortex occurrence in the entrance corner region were confirmed but its size was gradually decreased with the increase of Reynolds numbers, and that the velocity profiles and pressure distributions along the applied mesh layers were in agreement with the experimental and the previously reported numerical results.     

Numerical simulation of bubble growth in a limited amount of liquid

In this study, we examined the growth of a spherical bubble in a limited amount of liquid by using a finite‐element‐based numerical simulation method. The bubble growth was assumed to be controlled by both momentum and mass transfer. A truncated power‐law constitutive equation was used to describe the rheology of the melt. The gas inside the bubble followed the ideal gas law. The gas concentration at the bubble surface obeyed Henry's law. A computer code was programmed to solve the equations with the Galerkin method. A backward Euler scheme was used to discretize time. Grids were remeshed after each incremental time step to ensure the accuracy of the numerical results. The bubble growth process was simulated with the code. The numerical results, such as the instantaneous bubble size, gas pressure inside the bubble, and gas concentration profile in the liquid, were predicted. The influences of the liquid volume, initial gas pressure, temperature, and rheology of the melt on bubble growth were also studied. The results of the bubble growth simulation in this study were in satisfactory agreement with others' work. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

Part V: Dynamic evolution of the multivariate particle size distribution undergoing combined particle growth and aggregation

The present paper presents a study on the application of the Galerkin finite element method (FEM) for the solution of the dynamic multivariate population balance equation (PBE) in batch particulate systems undergoing aggregation as well as combined aggregation and growth. The performance of the Galerkin FEM in terms of accuracy and stability was assessed by a direct comparison of the calculated particle size distributions and/or their corresponding moments to available analytical solutions as well as by comparison to univariate numerical solutions. Numerical simulations were carried out for a variety of particle aggregation and growth mechanisms including constant, Brownian, and modified Brownian aggregation as well as constant and linear growth rate functions and for a wide range of values for the aggregation and growth rate coefficients. The simulation results revealed that the proposed Galerkin FEM produces very accurate numerical solutions but at significant computational cost.     

基于P1/PO四面体宏元的三维注塑充填速度场压力场有限元数值模拟

计算效率和解的稳定性是影响三维注塑充填有限元数值模拟的关键因素.针对黏性不可压缩聚合物熔体三维充填过程的速度场和压力场,以提高计算机求解速度为出发点,分析了采用P1/P0四面体单元(速度线性,压力常数)得到的有限元方程的解不收敛的原因,提出一种采用P1/P0四面体宏元离散空间域的求解方案,从而降低了求解的自由度数量,提...     

求解对流扩散问题的积分方程法

提出了一种求解对流扩散问题的积分方程法。在这一方法中,首先利用Laplace方程的级数形式的格林函数将对流扩散方程转化为积分方程,然后利用级数的正交性质,把积分方程进一步简化为代数方程组,求解该方程组即可得到对流扩散方程的级数形式的近似解。最后,分别利用Chebyshev多项式和Fourier级数求解了3个典型的一维和二维对流扩散问题。该方法和有限体积法、有限元法和迎风差分法相比,展现出非常高的精度并且避免了由解的不连续性造成的虚假振荡。     

Population balance modeling of polyurethane foam formation with pressure-dependent growth kernel

Polyurethane foams are widely used materials often chosen for their useful characteristics such as low thermal conductivity, ease of application, and high strength-to-weight ratios. Computational models are needed to predict the dynamics of the flow and expansion, and the resulting material properties, to improve manufacturing processes. In this paper, a model for PMDI, a water-blown polyurethane foam, is presented. By extending a kinetics-based approach by adding bubble-scale information via a population balance equation (PBE) using the quadrature method of moments, we can track bubble size distributions during foaming. We present results from a three-dimensional computational fluid dynamics model using arbitrary Lagrangian–Eulerian interface tracking implemented in finite element software. The model compares favorably with experimental data, including dynamics, bubble distributions measured by both camera and diffusion wave spectroscopy, and post-test bubble size from scanning electron microscopy and density measurements from x-ray computed tomography.     

Computer modelling and numerical analysis of hydrodynamics and heat transfer in non-porous catalytic reactor for the decomposition of ammonia

Chemical reactors exhibit very complex behaviours such as multiple steady states, oscillations, etc. resulting from complex linkage between the transport processes and the non-linear chemical reaction kinetics. Ammonia is a potential hydrogen source for a number of fuel cell applications for small scale power generation useful for portable equipments. In the present work, we analyse the fluid dynamics and heat transfer in catalytic microreactor systems for the decomposition of ammonia over a monolayer Ni non-porous catalyst. The overall model for this convective-diffusive-reactive system consists of a flow model, a mass transport model, an energy conservation model and a reaction kinetics model for ammonia decomposition. The flow model is described by the Stokes equation for a creeping flow regime. The mass transport and energy conservation models are based on convective-diffusion equations. The rate of ammonia decomposition can be measured as a function of the catalyst activity and ammonia concentration. A standard Galerkin finite element technique has been applied for the solution of the flow equations. A slightly perturbed form of the mass continuity equation is used to satisfy the Ladyzhenskaya-Babuška-Brezzi stability criterion. For the solution of convection-diffusion equations, a streamline inconsistent upwind finite element scheme has been chosen to avoid any spurious oscillations. C⁰-continuous 9-noded Lagrangian biquadratic isoparametric finite elements are used for the approximation of the field variables. A second-order Taylor-Galerkin time-stepping scheme has been chosen for the temporal discretisation of the flow equations whilst an implicit theta method has been used for convection-diffusion equations. The results are presented in the form of velocity vectors and concentration, temperature contours and are examined for stability, convergence and theoretical consistency.     

3-D Non-isothermal die flow of power-law fluids with viscous heating

The nonisothermal flow of incompressible viscous non-Newtonian fiuids is analyzed numerically. A three-dimensional finite element code is developed for the flow simulation purpose. The energy equation is decoupled from the equations of motion and both the flow field and the thermal field are solved iteratively. Two dimesionless groups, the Peclet number and the Brinkman number, are introduced to represent the characteristics of the thermal field. Results are presented for velocity, pressure and temperature distributions in the entrance region, and comparisons made with various mesh layouts. The results provide new insight into the temperature regulation in the extrusion process.     

Computer simulation of membrane processes: ultrafiltration and dialysis units

An algorithm for computer simulation of membrane processes such as ultrafiltration and dialysis has been developed using a simplified finite volume approach. The technique used is slightly different from the standard finite difference, finite volume and finite element methods where all the parameters are considered at fixed nodal points. In the present approach the entire flow chamber is divided into a large number of volume elements and each element is considered to be an independent unit (similar to finite volume method). All mass flux and velocity components are calculated at the boundaries whereas concentration is considered at the center of the element. Thus, unlike FDM, FVM, and FEM, in the present approach nodal points for velocity and concentration are different. It has been observed that this method is more accurate and fast and requires less computational effort.     

HEAT TRANSFER TO SPHERES AT LOW TO INTERMEDIATE REYNOLDS NUMBERS

The Navier-Stokes equation and the energy equation are solved using the Galerkin finite element method for flow past a solid sphere at low to intermediate Reynolds numbers. The calculated results are compared with exact theories valid for small or large Peclet numbers. A correlation is provided to predict the numerical results for ranges of Prandtl number from 0.001 to 1000 and Reynolds numbers from 1 to 100. A new correlation is proposed that matches the theoretical results at low Peclet numbers, the numerical results at intermediate Peclet numbers, and the existing experimental data at intermediate to high Peclet numbers.     

气泡羽流气液两相流动特性的研究进展

气泡羽流是一种复杂的气液两相流,广泛应用于废水处理、石油加工、环保等工业领域。气泡羽流的流动特性对气液两相间质量、动量传递及工业应用至关重要。本工作总结了理论与实验研究等方面气泡羽流流动特性的研究进展。详细讨论了气泡羽流气液两相流体水力学特性、羽流运动行为的影响因素。根据气含率、气泡直径等水力学参数的预测模型和经验公式,归纳了不同液相物性和结构参数下羽流模型的适用范围,揭示了流动对传质的作用。总结了分层流体中气泡羽流流型变化规律、羽流去分层效果以及引起流型变化的影响因素。阐释了横向流动环境下羽流的偏移行为呈线性变化,该变化与横向流速及表观气速等因素有关。最后讨论了气泡羽流气液两相流动特性研究手段和理论方法的局限性,展望了气泡羽流运动规律多尺度研究的方向。     

Caractérisation viscoélastique du comportement d'une membrane thermoplastique et modélisation numérique de thermoformage

In this work, we are interested, on the one hand in the characterization of circular polymeric ABS membrane under biaxial deformation using the bubble inflation technique, on the other hand in modelling and numerical simulation of the thermoforming of ABS materials using the dynamic finite element method. The viscoelastic behaviour of the Lodge model is considered. First, the governing equations for the inflation of a flat circular membrane are solved using a variable‐step‐size‐finite difference method and a modified Levenberg‐Marquardt algorithm to minimize the difference between the calculated and measured inflation pressure. This will determine the material constants embedded within the model used. For dynamic finite elements method, we consider a nonlinear load in air flow which obeys the Redlich‐Kwong equation of state of the real gases. For numerical simulation, the lagrangian formulation together with the assumption of the membrane theory is used. Moreover, the influence of the viscoelastic model on the thickness and on the stress distribution in the thermoforming sheet are analysed for ABS material.     

Numerical prediction of fiber orientation in injection molding

We develop a numerical method for calculating fiber orientation in the midsurface of a molded part of small thickness. Two-dimensional fiber orientation is predicted on the basis of either Jeffery's equation or a constitutive equation for the orientation tensor. The calculation is fully transient; it is performed on a time-dependent flow domain. The method is based on finite elements. Updated finite element meshes are generated at every instant of filling and allow one to perform an accurate calculation of the orientation even along the boundary of the flow domain. The method is applied to several examples in plane and three-dimensional geometries.     

Mathematical model to evaluate the ohmic resistance caused by the presence of a large number of bubbles in Hall-Héroult cells

Bubbles play an important role in the productivity of an electrolysis cell. They induce flow in the cell and increase the overvoltage, which is still two times greater than the thermodynamic voltage. Their contribution to the total electrical resistance of the electrolyte must be known for several reasons such as the energy efficiency and control. A computationally efficient mathematical model has been proposed that computes the total resistance of the electrolyte by using the concept of parallel-connected current tubes. The resistance of the individual current tubes has been determined earlier by the solution of the Laplace equation around the bubbles by the finite element method. Both electrical resistance models take into account the morphology (position, size and shape of each bubble) of the bubble layer. The current-tube model has been compared to the solutions obtained by a finite element method (FEM) for several real and hypothetical situations, using a large number of bubbles. The agreement between the results obtained by the proposed model and the FEM is very good. The difference between the two approaches is around 5% for a covering factor of 50%.     

MULTI-SCALE CHARACTERISTICS OF CHAOS BEHAVIOR IN GAS-LIQUID BUBBLE COLUMNS

The multi-value phenomenon of correlation dimension appearing in chaos analysis of time series of pressure fluctuation obtained from gas-liquid bubble columns was studied. Its relationship with multi-scale flow behavior and possible application in the identification of flow regime and regime transition in bubble columns were investigated. The results indicated that the multi-value phenomenon of correlation dimension results from the multi-scale behavior existing in the heterogeneous churn flow regime in bubble columns. When a bubble column is in the homogeneous flow regime, only one correlation dimension is found at a specified superficial gas velocity, indicating that single-scale behavior is dominant in the system. When a bubble column is in the heterogeneous churn flow regime, multi- (generally three) correlation dimensions can be obtained, showing the appearance of multi-scale behavior. Therefore, the formulation of an effective flow model depends on an appropriate multi-scale analysis for bubble columns. Flow regime and regime transition can be characterized by the structure and structure variation of the plot of the correlation integral versus radius of the hyper-sphere. On the basis of the above analysis, a complementary potential methodology called correlation integral analysis for the identification of flow regime and regime transition in gas-liquid bubble columns is recommended.     

MULTI-SCALE CHARACTERISTICS OF CHAOS BEHAVIOR IN GAS-LIQUID BUBBLE COLUMNS

The multi-value phenomenon of correlation dimension appearing in chaos analysis of time series of pressure fluctuation obtained from gas-liquid bubble columns was studied. Its relationship with multi-scale flow behavior and possible application in the identification of flow regime and regime transition in bubble columns were investigated. The results indicated that the multi-value phenomenon of correlation dimension results from the multi-scale behavior existing in the heterogeneous churn flow regime in bubble columns. When a bubble column is in the homogeneous flow regime, only one correlation dimension is found at a specified superficial gas velocity, indicating that single-scale behavior is dominant in the system. When a bubble column is in the heterogeneous churn flow regime, multi- (generally three) correlation dimensions can be obtained, showing the appearance of multi-scale behavior. Therefore, the formulation of an effective flow model depends on an appropriate multi-scale analysis for bubble columns. Flow regime and regime transition can be characterized by the structure and structure variation of the plot of the correlation integral versus radius of the hyper-sphere. On the basis of the above analysis, a complementary potential methodology called correlation integral analysis for the identification of flow regime and regime transition in gas-liquid bubble columns is recommended.