Analysis of wiped film reactors using the orthogonal collocation technique

Transport equation for ARB polymerization in wiped film reactors have been written. These have been reduced to the moment generation equations and using a suitable moment closure approximation, the zeroth and the second moments of the polymer have been numerically solved using the finite difference as well as the orthogonal collocation techniques. In the numerical solution by the finite difference technique, it is necessary to divide the dimensionless film thickness into at least 250 grid points to obtain stable results. The use of nine collocation points by the orthogonal collocation technique gives results close to those by the finite difference method and leads to considerable computational saving. The transport equations for the bulk and the film are found to involve four dimensionless parameters, and their effect upon the polymer formed at the end of the reactor has been studied.     

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.     

Emulsion polymerization of styrene: Simulation the effects of mixed ionic and non-ionic surfactant system in the presence of coagulation

A detailed model was developed for emulsion polymerization of styrene in batch reactor to predict the evolution of the product particle size distribution. The effect of binary surfactant systems (ionic/non-ionic surfactants) with different compositions was studied. The zero–one kinetics was employed for the nucleation rate, with the model comprising a set of rigorously developed population balance equations. The modeling incorporated particle formation by both nucleation and coagulation phenomena. The partial differential equations describing the particle population were discretized using finite volume elements. Binary surfactant systems, comprising sodium dodecyl sulfate (SDS) as anionic, and a commercial polyether polyol (Brij35^®) as non-ionic surfactants, were examined with different mass ratios. Increasing non-ionic surfactant mass fraction in binary surfactant system showed the decrease of particle number due to intensifying the coagulation between particles. Broader particle size distributions with greater average particle size were obtained with non-ionic surfactant comparing those obtained with anionic one.     

Response surface methodology approach to optimize coagulation-flocculation process using composite coagulants

Response surface method and experimental design were applied as alternatives to the conventional methods for optimization of the coagulation test. A central composite design was used to build models for predicting and optimizing the coagulation process. The model equations were derived using the least square method of the Minitab 16 software. In these equations, the removal efficiency of turbidity and COD were expressed as second-order functions of the coagulant dosage and coagulation pH. By applying RSM, the optimum condition using PFPD₁ was coagulant dosage of 384 mg/L and coagulation pH of 7.75. The optimum condition using PFPD₂ was coagulant dosage of 390 mg/L and coagulation pH of 7.48. Confirmation experiment demonstrated a good agreement between experimental values and model predicted. This demonstrates that RSM and CCD can be successfully applied for modeling and optimizing the coagulation process using PFPD₁ and PFPD₂.     

Coagulation of bipolarly charged ultrafine aerosol particles

Particle charging during coagulational growth is widely used in material synthesis processes as well as with industrial particle removal equipment. Coagulation behavior of charged particles is significantly different from that of neutral particles. To calculate the change in size/charge distribution of particles undergoing bipolar coagulation, a two-dimensional sectional model has been usually used. This method, however, needs considerable computation time although it gives very accurate prediction. In this study, the moment model, to solve the bipolar coagulation problem in the free-molecule regime, was developed to provide a time-efficient tool. Simultaneous particle charging by bipolar ions was also considered in this study. The developed model is based on the assumption that particles cannot have more than one unit charge and the particle size distribution remains log normal. The developed model was compared to the two-dimensional sectional model, with good agreement being shown. Some characteristics of bipolar coagulation were investigated using the developed model. The bipolar coagulation with simultaneous bipolar diffusion charging was shown to significantly increase the coagulation rate compared to the neutral Brownian coagulation. It was also shown from the simulation results that if one needs a higher coagulation rate in the initial stage, bipolar coagulation without ions is recommended, while bipolar coagulation with simultaneous charging by bipolar ions should be used if one wants a high coagulation rate for a long time.     

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.     

SIMULATION OF THE HEAT AND MOISTURE TRANSFER PROCESS DURING DRYING IN DEEP GRAIN BEDS

ABSTRACT

Grain drying is a simultaneous heat and moisture transfer problem. The modelling of such a problem is of significance in understanding and controlling the drying process. In the present study, a mathematical model for coupled heat and moisture transfer problem is presented. The model consists of four partial differential equations for mass balance, heat balance, heat transfer and drying rate. A simple finite difference method is used to solve the equations. The method shows good flexibility in choosing time and space steps which enable the simulation of long term grain drying/cooling processes. A deep barley bed is used as an example of grain beds in the current simulation. The results are verified against experimental data taken from literature. The analysis of the effects of operating conditions on the temperature and moisture content within the bed is also carried out     

SIMULATION OF THE HEAT AND MOISTURE TRANSFER PROCESS DURING DRYING IN DEEP GRAIN BEDS

Grain drying is a simultaneous heat and moisture transfer problem. The modelling of such a problem is of significance in understanding and controlling the drying process. In the present study, a mathematical model for coupled heat and moisture transfer problem is presented. The model consists of four partial differential equations for mass balance, heat balance, heat transfer and drying rate. A simple finite difference method is used to solve the equations. The method shows good flexibility in choosing time and space steps which enable the simulation of long term grain drying/cooling processes. A deep barley bed is used as an example of grain beds in the current simulation. The results are verified against experimental data taken from literature. The analysis of the effects of operating conditions on the temperature and moisture content within the bed is also carried out     

Evaluation of the 1-point quadrature approximation in QMOM for combined aerosol growth laws

This work examines the applicability of various assumptions in implementation of the quadrature method of moments (QMOM) for solving problems in aerosol science involving simultaneous nucleation, surface growth and coagulation. The problem of aerosol growth and coagulation in a box and the problem of vapor condensation in a nozzle are reworked using quadrature method of moments. QMOM uses Gaussian quadrature to evaluate integrals appearing in the moment equations and therefore does not require any assumptions on the form of the size distribution function, the growth laws and coagulation kernels. Results are compared with calculations which assume a lognormal size distribution. The conditions for which one, two and higher quadrature points can be used in the quadrature formula and the issues regarding the accuracy are considered for combined aerosol nucleation, growth and coagulation processes. Results show that for these problems, the simplest 1-point quadrature scheme gives accuracy comparable with the lognormal calculations while using two and higher point quadrature gives highly accurate results. Some difficulties associated with the QMOM are discussed and some insights are provided.     

基于双流体模型的流化床模拟

从单相流体力学中描述气体流动的Navier-Stokes方程和单颗粒运动的Newton方程出发,使用比较严格的体积平均法推导出描述气固两相宏观流动的模型方程组并设法确定了模型参数,把该方程组加以简化,能得到Gidaspow、Blake和Davidson等人的模型方程,初步证明了模型的正确性,求解使用了经本文改进的基于控制容积有限差分的IPSA方法,编制了通用程序,模拟了二维射流流化床中浓密气固两相的流动,并计算出射流的穿透深度。     

A New Numerical Approach for a Detailed Multicomponent Gas Separation Membrane Model and AspenPlus Simulation

A new numerical solution approach for a widely accepted model developed earlier by Pan [1] for multicomponent gas separation by high‐flux asymmetric membranes is presented. The advantage of the new technique is that it can easily be incorporated into commercial process simulators such as AspenPlus^TM [2] as a user‐model for an overall membrane process study and for the design and simulation of hybrid processes (i.e., membrane plus chemical absorption or membrane plus physical absorption). The proposed technique does not require initial estimates of the pressure, flow and concentration profiles inside the fiber as does in Pan's original approach, thus allowing faster execution of the model equations. The numerical solution was formulated as an initial value problem (IVP). Either Adams‐Moulton's or Gear's backward differentiation formulas (BDF) method was used for solving the non‐linear differential equations, and a modified Powell hybrid algorithm with a finite‐difference approximation of the Jacobian was used to solve the non‐linear algebraic equations. The model predictions were validated with experimental data reported in the literature for different types of membrane gas separation systems with or without purge streams. The robustness of the new numerical technique was also tested by simulating the stiff type of problems such as air dehydration. This demonstrates the potential of the new solution technique to handle different membrane systems conveniently. As an illustration, a multi‐stage membrane plant with recycle and purge streams has been designed and simulated for CO₂ capture from a 500 MW power plant flue gas as a first step to build hybrid processes and also to make an economic comparison among different existing separation technologies available for CO₂ separation from flue gas.     

基于双流体模型的流化床模拟

从单相流体力学中描述气体流动的Navier-Stokes方程和单颗粒运动的Newton方程出发,使用比较严格的体积平均法推导出描述气固两相宏观流动的模型方程组并设法确定了模型参数,把该方程组加以简化,能得到Gidaspow、Blake和Davidson等人的模型方程,初步证明了模型的正确性,求解使用了经本文改进的基于控制容积有限差分的IPSA方法,编制了通用程序,模拟了二维射流流化床中浓密气固两相的流动,并计算出射流的穿透深度。     

Numerical solutions of population balance models in preferential crystallization

This article focuses on the implementation of numerical schemes to solve model equations describing preferential crystallization for enantiomers. Two types of numerical methods are proposed for this purpose. The first method uses high resolution finite volume schemes, while the second method is the so-called method of characteristics (MOC). On the one hand, the finite volume schemes which were derived for general system in divergence form are computationally efficient, give desired accuracy on coarse grids, and are robust. On the other hand, the MOC offers a technique which is in general a powerful tool for solving growth processes, has capability to overcome numerical diffusion and dispersion, gives highly resolved solutions, as well as being computationally efficient. Several numerical test examples for a preferential crystallization model with and without fines dissolution under isothermal and non-isothermal conditions are considered. The comparison of the numerical schemes demonstrates clear advantages of the finite volume schemes and the MOC for the current model.     

Improving the accuracy of the moments method for solving the aerosol general dynamic equation

The General Dynamic Equation for aerosol evolution is converted into a set of ordinary differential equations for the moments M_m by multiplying by v^m and integrating over particle volume, v. Closure of these equations is achieved by assuming a functional form for the moments, instead of the usual assumption of a functional form for the size distribution itself. Specifically, it is assumed that In(M_m) can be expressed as a pth-order polynomial in m. The time-dependent coefficients in the polynomial are found by solving (p + 1) differential equations numerically. The case p = 2 corresponds to the assumption that the size distribution is always log-normal but comparison with accurate solutions shows that increasing p increases the accuracy of the method for all processes considered (removal, condensation and Brownian coagulation). Particle loss during evaporation and achievement of a self-preserving form for Brownian coagulation are also considered. Inversion of the moment expression to obtain the size distribution using the Mellin inversion formula is discussed.     

MICROGRAVITY NUCLEATION OF REFRACTORY MATERIALS: MODELING OF TRANSPORT PROCESSES IN THE NUCLEATION CHAMBER

A two-part mathematical model has been developed to describe the transport processes in a nucleation chamber designed for condensation of refractory vapors in a microgravity environment. The model solves the transient diffusion equations for temperature and concentration fields in cylindrical coordinates using finite differences and the alternating direction implicit method. Vapor supersaturation ratios are then computed from the evolving concentration profiles thus permitting one to estimate the conditions at the location in the chamber where nucleation is observed in experiments.     

Modelling of polymer fluid flow and residence time distribution in twin screw extruder using fictitious domain method

Abstract

The flow behaviour of a polymer melt in the conveying region of an intermeshing corotating twin screw extruder was studied using the combination of mixed finite element and fictitious domain method. The model was a combination of the governing equations of continuity and momentum with Carreau rheological model in a three-dimensional Cartesian coordinate system. The equations were solved by the use of a mixed Galerkin finite element technique. The Picard’s iterative procedure was used to handle the non-linear nature of the derived equations. The particle tracking technique was used to obtain residence time distribution and analyse distributive mixing in conveying region. The shear rate distribution was investigated as a criterion for dispersive mixing. The applicability of this model was verified by the comparison of experimentally measured pressure and simulation results for high density polyethylene melt. This comparison shows that there is a good adequacy between experimental data and model predictions.     

Modeling particle formation kinetics in mixed-phase clouds

A computational model for droplet and ice particle formation kinetics in mixed clouds is developed using the kinetic equations of condensation/coagulation formulated by Piskunov and Petrov (J. Aerosol Sci. 33 (2002) 647) for two-phase disperse systems. Alongside new kinetic equations, the model involves equations of mass and thermal balance, accounts for the difference in rates of vapor condensation above water and ice, uses actual condensation and coagulation growth rates, describes the phase transition dynamics.A method of x–T diagrams is suggested for visual treatment of the dynamics of condensation redistribution of material (distillation) from the water fraction to ice particles. The method allows easy graphic representation and study of the basic condensation process stages and prediction of the final state parameters. Criteria are found for the efficiency of methods for artificial action on a cloud system involving its seeding with ice-forming reagents, based on the effect of the redistibution of material.The developed numerical model is used for computation of the precipitation formation processes in the conditions of the Montana experiment of 07/19/1981 (J. Geophys. Res. 91 (01) (1986) 1231; J. Geophys. Res. 90 (D4) (1985) 60079).     

Modeling nonisothermal impregnation of fibrous media with reactive polymer resin

The manufacture of polymer composites through resin transfer molding (RTM) or structural reaction injection molding (SRIM) involves the impregnation of a fibrous reinforcement in a mold cavity with a reactive polymer resin. The design of RTM and SRIM operations requires an understanding of the various parameters, such as materials properties, mold geometry, and mold filling conditions, that affect the resin impregnation process. Modeling provides a potential tool for analyzing the relationships among the important parameters. The present work provides the physical model and finite element formulations for simulating the mold filling stage. Resin flow through the fibers is modeled using two-dimensional Darcian flow. Simultaneous resin reaction and heat transfer among resin, mold walls, and fibers are considered in the model. The proposed technique emphasizes the use of the least squares finite element method to solve the convection dominated mass and energy equations for the resin. Excellent numerical stability of the proposed technique provides a powerful numerical method for the modeling of polymer processing systems characterized by convection dominated transport equations. Results from example numerical studies for SRIM of polyurethane/glass fiber composites were presented to illustrate the application of the proposed model and numerical scheme.     

EXTERNAL FLUID HEATING OF A POROUS ED

This work deals with the thermal analysis of externally heated porous beds of finite length. A one dimensional model was developed that includes conduction and storage in both the fluid and bed, convective exchange between the fluid and bed, and the effect of adsorption/desorption in the bed. This model results in two coupled differential equations for the fluid and bed temperatures as functions of four independent dimensionless parameters. These equations were solved numerically using finite difference approximations. A truncation error analysis was carried out to maintain an accurate solution. The method of normalization is such that the results of this analysis are of use in bed design when the breakthrough characteristics in finite length beds are of interest. A method to measure bed thermal performance is defined and a means to optimize bed thermal performance is presented. An experiment was conducted to validate the numerically obtained results. A comparison of numerical to experimental results is presented     

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.