ANSYS二次开发在RTM工艺注模过程数值模拟中的应用

本文利用ANSYS软件包求解压力场,以ANSYS软件环境下的参数设计语言编程,实现了有限单元/节点控制体积(FE/NCV)算法.编写了程序UCVFEM,UCVFEM可以处理矩形网格和三角形网格上的注模过程模拟问题.数值验证表明数值解和解析解有良好的一致性.数值实践表明本文的方法具有可充分利用ANSYS软件包的优点,缩短软件开发周期,便于软件维护等优点.     

A finite difference procedure for solving coupled,nonlinear elliptic partial differential equations

A finite difference procedure is presented for solving coupled sets of partial differential equations. For one dependent variable, the procedure consists of replacing the concept of a single unknown at multiple grid points with the concept of a line of node points with multiple unknowns at each node point. The procedure is illustrated first for a second order, linear elliptic partial differential equation and then for a coupled set of non-linear elliptic partial differential equations. The method is easier to use and requires less computer storage than a banded solver method such as IMSL's routine LEQT1B. The procedure could be extended to include three spatial coordinates and time.     

Accuracy in Numerical Solution of the Particle Concentration Field in Laminar Wall-Bounded Flows with Thermophoresis and Diffusion

The accuracy of numerical finite-difference solutions to the one-way-coupled Eulerian partial differential equations for particle concentration in the presence of thermophoresis and diffusion is explored at different Schmidt numbers in laminar boundary-layer flow of a hot gas over a cold wall. Crank-Nicolson and MacCormack space-marching solutions to the coupled partial differential equations are compared with essentially exact solutions to the self-similar ordinary differential equation problem to determine the requirements for achieving accuracy in numerical solutions. When the diffusion sublayer at the wall is to be resolved, in flows laden with nanometer particles, the cell “Peclet” number referenced to the thermophoretic velocity and grid spacing in the wall-normal direction, and particle diffusion coefficient, serves as a criterion for the accuracy of space-marching solutions and determines the required number of wall-normal grid points, which is proportional to the particle Schmidt number. This criterion should be a useful guide in computations of other wall-bounded flows with thermophoresis, for which no accuracy criterion exists. When the diffusion sublayer at the wall is too thin to be resolved, as in flows laden with micron-size or larger particles, outer solutions to the particle concentration equation with no Brownian particle diffusion give excellent predictions of both the particle concentration profile and the flux of particles to the wall.

Copyright 2014 American Association for Aerosol Research     

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.     

A collocation-finite difference procedure for an elliptic partial differential equation

A numerical technique to solve a second order, linear, elliptic partial differential equation (PDE) is presented. Collocation is used in one coordinate direction to reduce the PDE to a set of coupled ODE. A second order finite-difference procedure is used to solve this system of ODE. Collocation is useful in emphasizing selected areas of the field and in obtaining accurate integrations over the field. Extensions of the procedure to non-linear problems, coupled PDE, and solutions with analytically identifiable singularities are discussed.     

Epoxidation reaction of trimethylolpropane with epichlorohydrin: Kinetic study of chlorohydrin formation

The kinetics of the reaction between trimethylolpropane (TMP) and epichlorohydrin (ECH) were studied. A mechanism is proposed which involves the synthesis of chlorohydrins, a previous stage which then leads to epoxy resin formation. From this mechanism a set of three coupled non-linear differential equations, (each equation corresponds to each chlorohydrin) was derived and numerical solutions were obtained using a Monte-Carlo method. The concentrations of chlorohydrins determined by this method (41.7, 35.7, and 22.5%) compare well with the experimental GPC results (42.7, 34.7, and 22.6%). The corresponding rate constants for the formation of the chlorohydrins were obtained.     

A collocation solver for systems of boundary-value differential/algebraic equations

A spline collocation procedure has been implemented to solve systems of multi-point boundary-value problems for mixed order differential/algebraic equations (BVP-DAEs). This implementation is based on the modification of an existing code “COLSYS”, which solves boundary-value ordinary differential equations (BVP-ODEs) alone. With minor changes to the original COLSYS code, the resulting procedure offers robust solutions to systems of DAEs, in addition to ODEs. A numerical example with application in chemical engineering is shown to demonstrate the superior ability of this procedure over other software.     

A numerical technique for the solution of integrodifferential equations arising from balances over populations of drops in turbulent flows

This paper describes a numerical technique designed to solve certain forms of partial differential equations. The method is applied to the partial integrodifferential population balance equations presented by Jairazbhoy [Jairazbhoy, V., Tavlarides, L. L., & Lewalle, J., (1995) A cascade model for neutrally buoyant two-phase homogeneous turbulence – part I. Model formulation. International Journal of Multiphase Flow, 21(3), 467] that describe the behavior of dense liquid dispersions of interacting drops in isotropic turbulence. In the successively contained semi-discretization scheme developed, the drop number density functions are discretized into non-uniform intervals corresponding to Gaussian quadrature points. The governing equations are assumed to hold identically at all the discretization points, generating a set of ordinary integrodifferential equations that are solved by an integrator package. The integrals in each function evaluation are calculated by Gaussian quadrature. The results show that, in some cases, as many as fifteen quadrature points are required to achieve grid independence. Each additional discretization point results in an additional ordinary integrodifferential equation. To achieve comparable accuracy with a uniform discretization scheme, many more discretization points would be required, resulting in an inordinately large number of ordinary integrodifferential equations. The computations also show that, in every run, there appears to be an optimum number of discretization intervals around which incremental increases in the resolution do not increase the CPU time or perceivable accuracy of the solution.     

Moving mesh generation for tracking a shock or steep moving front

For tracking a shock or steep moving front in the numerical solution of Partial Differential Algebraic Equations (PDAEs), an accurate spatial discretization method, Weighted Essentially Non-Oscillatory (WENO) scheme, is combined with moving grid techniques so that spacing of moving meshes is smoothed locally and globally. Several monitor functions, as metric criteria of node concentration, are examined. While the fixed grid method (uniform grid size) needs many mesh points to obtain enough solution accuracy, the moving grid method (non-uniform grid size) enhances accuracy even at small mesh numbers but it may be prohibitive owing to the addition of complex and non-linear mesh equations into physical PDAEs. The combination of the WENO scheme (based on an adaptive stencil idea) with the moving grid techniques improves stability and accuracy in the numerical solution over the commonly used moving grid method of central discretization. To locate adequate grid position in the moving mesh method, suitable monitor function according to problems must be selected.     

Solving population balance equations for two-component aggregation by a finite volume scheme

A conservative finite volume approach, originally proposed by Filbet and Laurençot [2004a. Numerical simulation of the Smoluchowski coagulation equation. SIAM Journal on Scientific Computing 25(6), 2004-2048] for the one-dimensional aggregation, is extended to simulate two-component aggregation. In order to apply the finite volume scheme, we reformulate the original integro-ordinary differential population balance equation for two-component aggregation problems into a partial differential equation of hyperbolic-type. Instead of using a fully discrete finite volume scheme and equidistant discretization of internal properties variables, we propose a semidiscrete upwind formulation and a geometric grid discretization of the internal variables. The resultant ordinary differential equations (ODEs) are then solved by using a standard adaptive ODEs-solver. Several numerical test cases for the one and two-components aggregation process are considered here. The numerical results are validated against available analytical solutions.     

A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion

A new numerical approach for solving coagulation equation, TEMOM model, is first presented. In this model, the closure of the moment equations is approached using the Taylor-series expansion technique. Through constructing a system of three first-order ordinary differential equations, the most important indexes for describing aerosol dynamics, including particle number density, particle mass and geometric standard deviation, are easily obtained. This approach has no prior requirement for particle size spectrum, and the limitation existing in the log-normal distribution theory automatically disappears. This new approach is tested by comparing it with known accurate solutions both in the free molecular and the continuum regime. The results show that this new approach can be used to solve the particle general dynamic equation undergoing Brownian coagulation with sufficient accuracy, while less computational cost is needed.     

Design formulae for reactive barrier membranes

Polymers films and membranes with immobile and irreversible reactive sites can provide significant barrier properties for packaging materials. The reactive term that consumes the mobile species in the governing transport equations for such materials is a function of both the mobile species and the immobilized reactive sites, leading to non-linear partial differential equations that typically have to be solved numerically. Here we present analytic design formulae to estimate the time varying flux, kill time and time lag through homogenous reactive membranes, obviating the need to numerically solve the model's non-linear equations.Modeling reveals three regimes for the time varying flux. For early times most reactive sites are still present, and an initial flux plateau is observed. For intermediate times, a moving reaction front is found to travel across the film coupled with an increasing flux of the mobile species. Finally, for long times when most reactive sites are consumed, the transient flux approaches its steady value. Analytic design equations characterizing the flux in these three regimes are developed based on perturbation theory and matched asymptotic expansions. They agree closely with the exact numerical solutions. Algebraic formulae for the kill time, time lag and speed of the reaction front also correspond to the full numerical solutions.     

A solution method for solving coupled nonlinear boundary-value problems

A Strum-Liouville integral transform technique is novelly applied to solve system of coupled nonlinear boundary-value problems approximately. The systems of differential equations consist of a linear differential operator and a nonlinear function of the dependent variables. To illustrate the potential of this technique we consider an example which comes from the modeling of diffusion and nonlinear chemical reaction systems in chemical engineering. The approximate solutions obtained by our technique agree surprising well with the numerically exact solutions obtained by the orthogonal collocation technique. To improve the approximation an iteration scheme in transform space is also defined.

Scope—Today, mathematical modeling of physical phenomena often produces (single or coupled) nonlinear differential equations. The true physical situation can, in many cases, be more closely described if the differential equations are allowed to be nonlinear. However, nonlinear differential equations are generally too difficult to be solved analytically apart from a few “tricks” or substitutions which apply only to a handful of equations [1]. An alternative approach is to look for a method which will reduce the problem, via analytical techniques, to a point where a “simple” computer program can solve the rest of the problem. The method introduced in this paper belongs to this class of solution techniques.

The method, which in this paper is applied to solving coupled nonlinear boundary-value problems, is a generalization of an idea in a paper by Do and Bailey [2] who apply it to a single nonlinear differential equation of boundary-value type. The equations, to which the technique is applied, arise from Fick's law diffusion into a porous solid and nonlinear reaction within the solid.

The solution method employs a Strum-Loiuville integral transform and to account for the nonlinear part an approximation is introduced. An iteration scheme is defined to improved the accuracy of the solution. The system of coupled nonlinear differential equations is reduced to a system of coupled nonlinear algebraic equations which is solved using a Newton-Raphson process. Finally, the solution is expressed as an infinite series, which is summed using a computer.

In response to papers by Do and Bailey [3] and Do and Weiland [4], Jerri [5] has tried to put this method on a more mathematical footing, and he shows that this method is a special case of a more general technique he has devised. Jerri uses the idea of Fourier transforms and convolution products to justify his method. The results for the example he considered are good, but he did not state how many iterations he required to obtain the solutions reported.

Conclusions and Significance—This paper has presented a very powerful method of solving boundary-value problems with linear operators and a nonlinear function of the dependent variable. The method works well for a single equation or coupled equations and can handle any kind of nonlinear function. We have shown through extensive numerical calculation the accuracy of this solution method, where the accuracy is measured in terms of a ratio of norms. In most cases an error of 4% can be achieved with just one iteration (Tables 2 and 3). Even though the present method has been applied to problems which have arisen from the modeling of chemical engineering problems, it would also be applicable to differential equations arising in other areas, provided they are of the same form.     

STOCHASTIC MODELING OF NON-LINEAR SIEVING KINETICS

The feed to a sieve is classified as oversize particles, undersize particles, and near mesh size particles. The sizes of the near mesh size particles vary around that of the sieve opening; the passage of these particles through the sieve effects the separation. A stochastic approach is employed for analyzing and modeling the sieving kinetics of the near mesh size particles, usually constituting the bulk of the material required to be separated. The master equation of the process is formulated based on probabilistic considerations, describing the passage of the particles in the presence of sieve blinding. However, an analytical solution cannot be obtained directly and hence a rational approximation technique, the system size expansion, is utilized to solve the master equation. This results in a system of non-linear, coupled, ordinary differential equations for the various statistical quantities characterizing the number of particles retained on the sieve and the number of blinded apertures. These equations are then solved numerically. From these general equations, specific cases as the first order kinetics law, applicable to the terminal stages of sieving, are easily obtained. In the absence of oversize particles, the process is described by a master equation which is solved directly, yielding an explicit analytical solution. The numerical solutions agree at least qualitatively with the available experimental observations     

WENO scheme with static grid adaptation for tracking steep moving fronts

The numerical solution of a hyperbolic or a convection-dominated parabolic partial differential equation is challenging due to the large local gradients that are present in the solution. A possible method to track the sharp fronts that are associated with large gradients is to adapt the grid and this can be done dynamically or statically, i.e. at discrete points of time during the simulation. In this paper, a novel approach that is based on combining the high-order WENO scheme with a static moving grid method is presented. The proposed algorithm is tested on the viscid Burgers’ equation, the linear advection equation and the population balance equation that describes particle growth in emulsion polymerization. Enhancements in the performance are observed in all case studies when compared with the conventional WENO scheme on a uniform grid making it a promising alternative when dealing with similar problems.     

CONJUGATE ANALYSIS OF NATURAL CONVECTIVE DRYING OF BIOLOGICAL MATERIALS

ABSTRACT

This paper presents a numerical analysis of heat and mass transport during natural convective drying of an extruded com meal plate. The conjugate problem of drying and natural convection boundary layer Is modeled. The finite volume technique was used to discretize and solve the highly nonlinear system of coupled differential equations governing the transport inside the plate. The boundary layer solution was obtained by means of a finite difference software package that utilizes Runge-Kutta's 5th order method to solve the inherent transport equations. A methodology for evaluating the heat and mass transfer coefficients during the numerical simulation was developed and successfully implemented. The results showed that there is no analogy between heat and mass transfer coefficients for this type of problem.     

微分进化算法应用于换热网络全局最优化

前言换热网络是过程系统中实现能量回收和高效利用的重要环节,其性能直接关系到整个系统的能量利用水平。目前,换热网络优化设计的方法主要分为三类,分别是夹点法[1-2]、数学规划法[3-4]与启发式方法[5-7]。     

Numerical Analysis of the Coupling Between the Flow Kinematics and the Fiber Orientation in Eulerian Simulations of Dilute Short Fiber Suspensions Flows

This paper focuses on an accurate evaluation of short fibers suspensions models coupling the flow kinematics with the fiber orientation evolution. In coupled models the flow kinematics is usually solved using the finite element method, where the fiber orientation is introduced in the constitutive equation through its value in some points (nodes or integration points). In this paper we will compare in a simple steady shear flow, the exact solutions of the extra‐stresses associated with the fibers' presence with the numerical simulations obtained using both the method of characteristics and the discontinuous Galerkin's method to solve the equation governing the generalized gradient evolution, in order to avoid the introduction of any closure relation. The error introduced if a quadratic closure relation is considered in the constitutive equation will be also quantified.     

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] to solve n-dimensional population balance equations (PBEs) with preservation of (n+1) instead of 2ⁿ properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.     

Three-dimensional numerical study of heat and mass transfer in fluidized beds with spray nozzle

The numerical computations of temperature and concentration distributions inside a fluidized bed with spray injection in three-dimensions are presented. A continuum model, based on rigorous mass and energy balance equations developed from Nagaiah et al., is used for the three-dimensional simulations. The three-dimensional model equation for nozzle spray is reformulated in comparison to Heinrich. For solving the non-linear partial differential equations with boundary conditions a finite element method is used for space discretization and an implicit Euler method is used for time discretization.The time-dependent behavior of the air humidity, air temperature, degree of wetting, liquid film temperature and particle temperature is presented using two different sets of experimental data. The presented numerical results are validated with the experimental results. Finally, the parallel numerical results are presented using the domain decomposition methods.