Fiber mat deformation in liquid composite molding. II: Modeling

Mathematical models are proposed to simulate the flow induced fiber mat deformation during liquid composite molding. The fiber bed is treated as an elastic beam and the load acting on the bed causes its deformation. The lubrication approximation is used to simplify the resin flow equation in the fiber free region, while Darcy's law is used to calculate the pressure and velocity fields in the deformed fiber bed. The governing equations are solved using the control volume/finite element method. The numerical results show reasonable agreement with the experimental results from Part I.     

Numerical Simulation of the Lock-In Effect of a Fixed-Fixed Tube in Cross Flow

This paper deals with the development of a numerical calculation code that is able to simulate the three-dimensional flow through a heat exchanger tube bundle and therefore allows a coupled calculation of fluid-structure interaction between the flow and the tube bundle. The incompressible flow field is calculated by a Navier-Stokes solver using a first-order power law scheme, a SIMPLEC algorithm to calculate the pressure and velocity correction fields, and a line-by-line Gauss-Seidl tridiagonal algorithm to solve the linearised system of equations. The transient parts of the Navier-Stokes equations are discretised by a second-order forward finite differencing scheme. The turbulence is examined with the aid of a large-eddy turbulence model. The transient fluid forces acting on the tubes are calculated by integration of all local flow pressure values on the surfaces of the tubes. As an example a single fixed-fixed cylinder in a flow channel is considered using the structural calculation part already developed as well as the new flow field and flow forces subroutines. The time series of the tube's motion and the fluid forces acting on the tube are analysed by Fourier's transformation. The lock-in effect occurring when the vortex shedding frequency approaches the first natural frequency of the tube can be excellently demonstrated by varying the inflow velocity over a wide range of Reynolds numbers.     

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.     

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.     

Das Beschichten von Oberflächen

The coating of different substrates is an important part of many industrial manufacturing processes. The fluids used in these processes have very different rheological properties. They are coated to solid surfaces with high substrate velocities by dipping, spraying, casting or by the use of knives, blades and rollers. The field of flow between two rotating rollers is influenced by the free surface of the fluid and the contact point between the three phases, solid roller, coating fluid, and the surrounding gas. Viscous, inertial, gravitational and capillary forces control the velocity field between the rollers. Stable fluid coatings that satisfy the increasingly stringent demands on the quality of their surface structure have to be created within certain limits of the forces occurring. In order to predict these limits, the flow fields have to be completely calculated. Therefore all three conservation equations and an additional equation for the position of the free surfaces are solved with consideration of the constitutive equations and possible wall slip effects of the coating fluids.     

Hydromagnetic non‐Darcy flow,heat and mass transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

A numerical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics on steady two‐dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non‐Darcy porous medium in the presence of thermal radiation and viscous dissipation. The governing partial differential equations are convected into a system of nonlinear ordinary differential equations by similarity transformation and are solved numerically by using the Successive linearisation method, together with the Chebyshev pseudo‐spectral collocation method. The effects of various parameters on the velocity, temperature, and concentration fields as well as on the skin‐friction coefficient are presented graphically and in tabular forms.     

Fiber orientation in simple injection moldings. Part I: Theory and numerical methods

This paper sets out the theory and numerical methods used to simulate filling and fiber orientation is simple injection moldings (a film-gated strip and a center-gated disk). Our simulation applies to these simple geometry problems for the flow of a generalized Newtonian fluid where the velocities can be solved independently of fiber orientation. This simplification is valid when the orientation is so flat that the fibers do not contribute to the gapwise shear stresses. A finite difference solution calculates the temperature and velocity fields along the flow direction and through the thickness of the part, and fiber orientation is then integrated numerically along pathlines. Fiber orientation is three-dimensional, using a second-rank tensor representation of the orientation distribution function. The assumptions used to develop the simulation are not valid near the flow front, where the recirculating fountain flow complicates the problem. We present a numerrical scheme that includes the effect of the fountain flow on temperature and fiber orientation near the flow front. The simulation predicts that the orientation will vary through the thickness of the part, causing the molding to appear layered. The outer “skin” layer is predicted only if the effects of the fountain flow and heat transfer are included in the simulation.     

Flow of viscous fluid through a circular aperture

The creeping flow of a highly viscous incompressible fluid through a circular aperture located in an infinitely wide horizontal plate is analyzed by solving Navier-Stokes equations without inertia terms. Solutions for vertical and radial velocities as well as pressure have been obtained in terms of integral equations with an undetermined Kernal function. This function has been evaluated by assuming several different velocity distributions at the aperture, and the corresponding pressure drop for each case has been calculated. The results show that the pressure loss for a given flow rate goes through a minimum as the assumed velocity profile changes from flat to parabolic. Based on the minimum energy dissipation theorem of Helmholtz, the most appropriate velocity distribution is discussed. Experimental data obtained using sharp-edged orifices are compared with theoretical predictions.     

Optimization‐based design of polymer sheeting dies using generalized Newtonian fluid models

A polymer sheeting die design methodology is presented, which integrates finite element flow simulations, numerical optimization, and design sensitivity analyses to compute die cavity geometries capable of giving a near‐uniform exit velocity. This work extends earlier die design methods to include generalized Newtonian fluid (GNF) models that represent the shear‐thinning behavior of polymer melt. Melt flow computations and design sensitivity analyses are provided using the generalized Hele‐Shaw flow approximation with isothermal power‐law, Carreau‐Yasuda, Cross, Ellis, and Bingham fluid models. The nonlinear equations for die cavity pressure are solved using the Newton‐Raphson iteration method and design sensitivities are derived with the adjoint variable method. The die design method is applied to an industrial coat hanger die, in which a design parameterization is defined that allows for an arbitrary gap height distribution in the manifold of the die. In addition, die performance is assessed and compared for power‐law and Carreau‐Yasuda fluid flow over a range of die operating conditions. Pareto optimal die designs are also considered in this study. POLYM. ENG. SCI., 45:953–965, 2005. © 2005 Society of Plastics Engineers     

大型塑料件注塑模浇注系统设计

利用模流分析软件Moldflow Plastics Advisers(MPA)对空调机面板注塑模6种构型不同的浇注系统分别进行注射成型流动分析。通过模拟空调机面板在一定成型材料、成型工艺条件下注射成型流动时的温度场、压力场、速度场,获得熔体流动前沿温度、实际注射时间、实际注塑压力、气穴位置、熔接痕位置等参数;在此基础上比较流动结果,预测空调机面板成型质量,确定最优的注塑模浇注系统设计参数,成功地解决了大型塑料件成型时可能出现的充模不足、熔接痕明显等质量缺陷,为模具设计提供了科学依据。     

A finite element method for flow in compression molding of thin and thick parts

Based on Barone and Caulk's model and a generalized variational functional, a finite element simulation was developed for the compression molding of thin and thick parts. For solving the u-v-p type equations, an element-based penalty method and a mixed formulation were implemented. Numerical results show that the new model gives better accuracy in velocity and velocity gradient than the Hele-Shaw formulation for cases where both models are appropriate. Predictions of velocity and its gradient by the model are compared with other FEM results and BEM solutions. Using a fixed base mesh that covers the mold cavity, a new technique was developed for tracking the moving flow front. Temporary elements and nodes are generated for the filled part of elements intersected by the flow front. This method allows a smooth representation of the flow front and has exact boundary conditions on the flow front. The scheme is demonstrated for compression molding of an elliptical and an L-shaped charge.     

有限元方法求解化学气相淀积反应器模型(Ⅰ)——冷壁反应器模型及有限元解

建立了冷壁化学气相淀积反应器的数学模型,用Galerkin有限元方法对模型予以求解.计算中考虑了温度对物性参数的影响及自然对流因素,计算值和实验结果基本上一致.     

有限元方法求解化学气相淀积反应器模型(Ⅰ)——冷壁反应器模型及有限元解

建立了冷壁化学气相淀积反应器的数学模型,用Galerkin有限元方法对模型予以求解.计算中考虑了温度对物性参数的影响及自然对流因素,计算值和实验结果基本上一致.     

EFFECTS OF HALL CURRENT AND CHEMICAL REACTION ON OSCILLATORY MIXED CONVECTION-RADIATION OF A MICROPOLAR FLUID IN A ROTATING SYSTEM

This article is concerned with the analysis of the effects of thermal radiation on oscillatory mixed convection flow of a micropolar fluid in a rotating frame of reference in the presence of transverse magnetic field and Hall current. The influence of a first-order homogeneous chemical reaction and heat source effects is also analyzed. The governing partial differential equations with the appropriate boundary conditions are reduced to a set of ordinary differential equations using similarity transformations. The dimensionless governing equations for this investigation are solved analytically after using small perturbation approximation. The effects of various parameters on the velocity, temperature, and concentration fields as well as on skin-friction coefficient, Nusselt number, and Sherwood number with their amplitude and phase are discussed in detail. Numerical results are discussed with the help of graphs and tables. Present results are also compared with previously published work.     

Numerical study of BSA ultrafiltration in the limiting flux regime — Effect of variable physical properties

Ultrafiltration of BSA solutions in a parallel cell was studied by numerical methods taking into account concentration dependent properties. The Navier-Stokes equations, in the stream function-vorticity formulation, and the mass transport equation were solved simultaneously. The numerical code was validated by comparison with data from benchmark analytical solutions. The flow and concentration fields in the limiting flux regime were studied in detail and compared with the fields for constant properties. Numerical results were also compared with experimental data of the permeate velocity in the limiting flux regime published in the literature. This comparison shows that the concentration dependence of the viscosity, diffusivity and osmotic pressure are all necessary to explain experimental results published in the literature.     

Modeling heterogeneous downward dense gas‐particle flows

A novel approach is proposed to model heterogeneous downward dense gas‐particle flows. The homogeneous behavior of the flow is described by the mass and momentum transport equations of the gas and particulate phases solved using a mono‐dimension finite volume method on staggered grids. The heterogeneous features of the flow are predicted simultaneously using the bubble‐emulsion formalism. The gas compressibility is taken into consideration. The model is supplemented with a new correlation to account for the wall‐particle frictional effects. The predictions are compared with the vertical profiles of pressure and the amount of gas that flows up and down two standpipes and a cyclone dipleg of an industrial fluid catalytic cracking unit and of a cold small‐scale circulating fluidized bed. The trends are well predicted. The model gives further information and is thus an innovative starting point for downward dense gas‐particle flow hydrodynamics investigation. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

A model for the oxidation of carbon silicon carbide composite structures

A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of carbon silicon carbide (C/SiC) composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. The result of the theoretical formulation is a set of two coupled non-linear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The non-linear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The numerical method is demonstrated by utilizing the method to model the carbon oxidation and weight loss behavior of C/SiC specimens during thermogravimetric experiments. The numerical method is used to study the physics of carbon oxidation in carbon silicon carbide composites.     

下行床反应器内催化裂化过程的CFD模拟

耦合湍流气粒多相流模型和催化裂化集总动力学模型,建立了描述下行床内多相流动和催化裂化过程的反应器数学模型,并利用计算流体力学单元模拟软件CFX4.3对下行床内的催化裂化过程进行了数值模拟及分析.模型能预测出在工业应用中反应器内最受关注的诸多参数,如固含率、相间滑移速度、压降、气固相的加速区以及各组分浓度的分布情况.预测结果表明,气相反应的进行将导致反应器内的气粒流动行为发生较大变化,充分考虑反应与流动行为的耦合十分重要;而反应器床径的增大将导致转化率和各产物收率的下降.     

Coupled analysis of injection molding filling and fiber orientation,including in-plane velocity gradient effect

On injection molding of short fiber reinforced plastics, fiber orientation during mold filling is determined by the flow field and the interactions between the fibers. The flow field is, in turn, affected by the orientation of fibers. The Dinh and Armstrong rheological equation of state for semiconcentrated fiber suspensions was incorporated into the coupled analysis of mold filling flow and fiber orientation. The viscous shear stress and extra shear stress due to fibers dominate the momentum balance in the coupled Hele-Shaw flow approximation, but the extra in-plane stretching stress terms could be of the same order as those shear stress terms, for large in-plane stretching of suspensions of large particle number. Therefore, a new pressure equation, governing the mold filling process, was derived, including the stresses due to the in-plane velocity gradients. The mold filling simulation was then performed by solving the new pressure equation and the energy equation via a finite element/finite difference method, as well as evolution equations for the second-order orientation tensor via the fourth-order Runge-Kutta method. The effects of stresses due to the in-plane velocity gradient on pressure, velocity, and fiber orientation fields were investigated in the center-gated radial diverging flow in the cases of both an isothermal Newtonian fluid matrix and a nonisothermal polymeric matrix. In particular, the in-plane velocity gradient effect on the fiber orientation was found to be significant near the gate, and more notably for the case of a nonisothermal polymer matrix.