Finite element modeling of curing of epoxy matrix composites

A two‐dimensional finite element model is developed to simulate and analyze the mechanisms pertaining to resin flow, heat transfer, and consolidation of laminated composites during autoclave processing. The model, which incorporates some of the best features of models already in existence, is based on Darcy's law, the convection–diffusion heat equation, and appropriate constitutive relations. By using a weighted residual method, a two‐dimensional finite element formulation for the model is presented and a finite element code is developed. Numerical examples, including a comparison of the present numerical results with one‐dimensional and two‐dimensional analytical solutions, are given to indicate the accuracy the finite element formulation. Moreover, using the finite element code, the one‐dimensional cure process of a laminate made of 228 and 380 plies of AS4/3501‐6 unidirectional tape is simulated and numerical results are compared with available experimental results. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 2310–2319, 2007     

Modeling of structural reaction injection molding process. I. Mathematical model

A mathematical model of the infusion process in producing reinforced articles is proposed. The model is based on the analysis of flow of a Newtonian liquid inside a rectangular multilayer channel. According to the model, a liquid enters the central (feeding) layer, moves through this layer, and simultaneously impregnates peripheral layers. So, the flow is two‐dimensional. Flow inside the porous layers is treated in terms of the Darcy equation with different permeability coefficients in two directions. Principal solutions for the flow front development and pressure evolution were obtained and analyzed. Then the initial model, developed for a Newtonian liquid, is generalized for the so‐called “rheokinetic” liquid, which changes its rheological properties in time as a result of temperature variation and/or any possible chemical process, in particular, the reaction of curing of a binder. It was proven that in this case the solution is automodel. This means that the solutions obtained for a Newtonian liquid in the dimensionless form are valid for an arbitrary rheokinetic liquid.     

Non‐fickian three‐dimensional hindered moisture absorption in polymeric composites: Model development and validation

A new, three‐dimensional, anisotropic non‐Fickian diffusion model is developed to characterize moisture absorption in polymeric composites. The new hindered diffusion model extends the classical Fickian theory to include the effects of the interaction of diffusing molecules with the chemical and physical structure of polymeric composites. The numerical solution of the hindered diffusion model is obtained for a three‐dimensional, anisotropic domain by using a forward‐time, centered‐space finite difference technique. The moisture weight gain over time predicted by the model is shown to mimic a wide variety of anomalous absorption behavior, often exhibited by anisotropic composite laminates. The accuracy of the numerical solutions is verified by comparing the results to known analytical solutions of a one‐dimensional, “Langmuir‐type” diffusion model and for the limiting case of the three‐dimensional Fickian model. The utility of the proposed hindered diffusion model is demonstrated by accurately recovering the absorption behavior of three different material systems reported in literature. First, it is shown that the hindered diffusion model can accurately predict the moisture absorption data for unidirectional glass‐reinforced epoxy plates of varying dimensions exposed to a relative humidity of 80%. Second, the one‐dimensional version of the model is applied to experimental moisture absorption data for isotropic epoxy resin samples of different thicknesses. Anomalous effects due to sample thickness reported in the original article are accurately captured. Third, the proposed model is shown to be more accurate than a two‐stage diffusion model applied to moisture absorption data obtained from a woven 3‐ply carbon fiber reinforced bismaleimide composite. POLYM. COMPOS., 34:1144–1157, 2013. © 2013 Society of Plastics Engineers     

Modelling the Rotary Lime Kiln

A three‐dimensional steady‐state model to predict the flow and heat transfer in a rotary lime kiln is presented. All important phenomena are considered for the pre‐heat and calcination zones including turbulent gas flow, buoyancy, all modes of heat transfer, evolution and combustion of species and granular bed motion with calcination reaction. The model is based on a global solution of three sub‐models for the hot flow, the bed and the rotating wall/refractories. Information exchange between the models results in a fully coupled 3‐D solution of a rotating lime kiln. The overall model is validated using UBC's pilot kiln trials (5.5 m laboratory kiln). Results for this case are presented and potential implications are discussed.     

Mathematical Modeling of Coke Formation and Deposition due to Thermal Cracking of Petroleum Fluids

A two‐dimensional mathematical dynamics model is presented to predict coke formation due to thermal cracking inside the tubes of fired heaters on two types of petroleum fluid. The laminar and turbulent flows are analyzed for both petroleum fluids. The second‐order k‐? standard model is adopted to make this mathematical model more accurate than previous models of coke formation. The radial and axial variations for temperature, velocity, and concentration due to the high temperature gradients inside the tubes are considered in the model equations. The finite volume method is the numerical model used to discretize the conservation equations. The proposed model is suitable to predict coke formation inside heater tubes since it indicates operational conditions where coke formation is minimized.     

A comparison of Newtonian and viscoelastic constitutive models for dry spinning of polymer fibers

A comparison is made of the predictions of one‐dimensional mathematical model simulations of dry spinning based on Newtonian and viscoelastic constitutive equations for the spin dope. The viscoelastic model is based upon a modified Giesekus constitutive equation with a temperature and composition‐dependent relaxation time. The simulation algorithm includes the effects of the glass transition on the expected solution viscosity and relaxation time behavior along the spinline. Predictions of axial velocity, tensile stress, and composition profiles for the two cases suggest the role of viscoelasticity in the locking‐in behavior associated with fiber solidification along the spinline. The effects of model parameters and processing conditions are also discussed. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 87: 2136–2145, 2003     

Modellierung der Einlaufstrecke und deren Auswirkungen auf die Verweilzeitverteilung in laminar durchströmten Rohren und Kanälen

A one‐range and a two‐range model for the laminar velocity distribution in the entrance region of tubes and ducts are presented. These allow the calculation of the residence time distribution under the impact of the flow development in the hydrodynamic entrance region. For the dispersion‐free case, an analytical solution is given. A cell model with place‐changing probability (ZEMP) is applied for the consideration of dispersion. This approach allows the fast quantification of the influence of different parameters on the residence time distribution for relatively short pipes and ducts. The numerical results are compared with earlier presented results of semi‐empirical models.     

Modeling of Multi‐Tubular Reactors for Fischer‐Tropsch Synthesis

The results of the simulation of multi‐tubular Fischer‐Tropsch reactors based on a two‐dimensional pseudo‐homogeneous model are presented. The model takes into account the intrinsic kinetics of two commercial iron and cobalt catalysts, intraparticle mass transfer limitations, and the radial heat transfer within the fixed bed and to the cooling medium (boiling water). The effective rate with Co is slightly higher than with Fe. Hence, a temperature level can be used for Co that is 20 °C lower compared to Fe. The conversion and product selectivies are then almost the same and the reactor can be operated safely without a temperature runaway. The results of the simulations are consistent with literature data and show that there is still room for improvement of fixed bed FT reactors, e.g., by an enhanced heat transfer.     

HEAT TRANSFER ENHANCEMENT WITH CHEMICAL REACTION-MODEL COMPARISONS

A one dimensional tubular reactor model which incorporates heat transfer enhancement calculations in the presence of homogenous chemical reactions is presented. The model compensates for the distortion in the actual radial temperature profile due to chemical reaction by calculating the temperature profile in the film next to the reactor wall which is then used to correct for the heat transfer coefficient based on the non-reactive case. The relative simplicity and rapidity of the model makes it a viable alternative to the two dimensional model     

A numerical model for flame spread along combustible flat solid with charring material with experimental validation of ceiling flame spread and upward flame spread

This paper gives a numerical model for flame spread along combustible flat solid with charring materials. The presented model consists of a one‐dimensional flame spread model coupled with a one‐dimensional pyrolysis model. The existing experimental data (the ceiling flame spread beneath medium density fibreboard) are used for comparison to validate the model. In addition, the model can also be used to predict upward flame spread; only some expressions are changed. A comparison with full‐scale experimental data on the upward flame spread over plywoods from the literature is performed. The results obtained from numerical simulations using the model are consistent with these two kinds of experimental tests. Thus, the presented model is appropriate for modelling not only the ceiling flame spread, but also the upward flame spread. Copyright © 2007 John Wiley & Sons, Ltd.     

Spectral Analysis of Uneven Data in a Bipolar Charging Agglomeration System

This paper presents particle dynamics analysis (PDA) results from three measurement sections of a bipolar charged agglomeration system. The periodogram model and the Burg algorithm for autoregressive spectral estimates were used for the spectral analysis of the unevenly spaced data. The one‐dimensional energy spectrum from the Burg algorithm is smoother than that from periodogram model. The spectrum of the main flow velocity with charging is lower than for the case without charging. The results show that the one‐dimensional kinetic energy distribution at all frequencies can be well represented by the one‐dimensional energy spectrum. The dissipation rate was then obtained from the energy spectrum. The results also show that there is a lower dissipation rate in measurement section 1 with charging than without charging.     

Further Studies of the Gas Penetration Process in Gas‐Assisted Injection Molding

Abstract

Gas‐assisted injection molding is one of the innovation injection‐molding processes recently developed. The solution of gas penetration thickness interface is the key problem in the simulation of gas‐assisted injection molding, but also the puzzle. By applying the matching asymptotic expansion method, an analytical solution of the gas penetration interface is deduced. First, the governing equations and boundary conditions are transformed to be dimensionless. And then matching asymptotic expansion method is applied to solve the dimensionless equations, where capillary number Ca and Ca ^2/3 are used as perturbation parameters. Compared with experimental results, the presented mathematical model and solving method are proved to be correct.     

Mathematical modelling of an industrial furnace for high‐temperature heat treatment of wood

There are various high‐temperature treatment methods for wood. In the “Bois Perdure” process, the thermal treatment of wood is carried out in a furnace by contacting it with hot combustion gases over 200°C without the addition of any chemicals in order to improve its dimensional stability and durability. The treatment eliminates free and bound water in the wood and modifies its molecular structure. In this study, a mathematical model describing the industrial furnace has been developed. The overall model consists of a 3‐D unsteady‐state sub‐model which solves for the flow, heat, and mass transfer in the gas coupled with a 1‐D unsteady‐state sub‐model which calculates the heat and mass transfer in the wood. The 3‐D gas sub‐model was developed using the commercial CFD code CFX. The 1‐D wood sub‐model is based on the solution of simultaneous heat and mass transfer equations (Luikov equations) using the implicit finite difference formulation. The model predicts the temperature and moisture distributions in the wood as well as the flow, heat, and moisture profiles in the gas. The model results are compared with the data obtained from the industrial furnace, and a good agreement was found between them.     

Radiation and conduction heat transfer coupled with liquid water transfer,moisture sorption,and condensation in porous polymer materials

A new one‐dimensional mathematical model to simulate the complex coupled heat and moisture transfer in porous polymer materials is presented. The new model takes into account effects of multiple involved processes such as radiation and conduction heat transfer, liquid capillary action, moisture sorption, and condensation. The technique of volume of fraction (VOF) is used to model the dynamic distribution of moisture in different phases, that is, vapor and liquid. The finite volume method (FVM) is used to develop the numerical computational scheme to solve the model. The temperature change on the fabric surface derived from the computational results of the model is compared with experimental measurements with reasonable agreement between the two. Further numerical simulations were carried out to investigate the complex interactions and coupling effects among the various heat and moisture‐transfer processes involved. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 89: 2780–2790, 2003     

Consolidation simulation of composite laminates: Formulation and validation verification

Advanced fiber‐reinforced polymer composites have been increasingly used in various structural components. One of the important processes to fabricate high‐performance laminated composites is an autoclave‐assisted prepreg lay‐up. Since the quality of laminated composites is largely affected by the cure cycle, selection of the cure cycle for each application is important and must be optimized. Thus, some fundamental model of the consolidation and cure processes is necessary to properly select the suitable parameters for each application. This article is concerned with the “flow‐compaction” model during the autoclave processing of composite materials. By using a weighted residual method, a two‐dimensional finite element formulation for the consolidation process of thick thermosetting composites is presented and the corresponding finite element code is developed. Numerical examples, including comparison of the present numerical results with one‐dimensional and two‐dimensional analytical solutions, are given to indicate the accuracy and effectiveness of the finite element formulation. In addition, a consolidation simulation of AS4/3501‐6 graphite/epoxy laminate is performed and is compared with the experimental results available in the literature. POLYM. COMPOS., 26:813–822, 2005. © 2005 Society of Plastics Engineers     

A local power‐law approximation to a smooth viscosity curve with application to flow in conduits and coating dies

Coating dies distribute liquid into a uniform layer for coating onto a moving substrate. A die comprises one or two cavities spanning the coating width and adjoining narrow slots of much higher resistance to flow. In modeling coating dies, the flows in the slots and cavities are often approximated as one‐dimensional to achieve a fully geometrically parameterized model of low computational load suitable for optimizing design for multiple liquids and flow rates. The power‐law model is mathematically efficient for one‐dimensional flows of shear‐thinning liquids but does not include limiting viscosities at low and high shear rates that are frequently present. In previous work, the truncated power‐law model, which is terminated at the limiting Newtonian viscosities, was used to alleviate this shortcoming without sacrificing the mathematical advantages. In this work, the Carreau–Yasuda model replaces the truncated power‐law model as an improvement. For flows in slots and cavities, the Carreau–Yasuda model can be approximated accurately by a local power‐law model with little increase in computational load over the truncated power‐law model. In the transition regions of the Carreau–Yasuda model between Newtonian and power‐law behavior, the local power‐law model gives more accurate results than the truncated power‐law model. POLYM. ENG. SCI., 54:2301–2309, 2014. © 2013 Society of Plastics Engineers     

Effect of kinetic models on hot spot temperature prediction for phthalic anhydride production in a multitubular packed bed reactor

Although there are several kinetic models for the production of phthalic anhydride from the partial oxidation of o‐xylene, only few studies have compared the effect of the kinetic model on the prediction of the hot‐spot temperature. In this work, the predicted temperature profile for the partial oxidation of o‐xylene to phthalic anhydride in a multitubular packed bed reactor was obtained for different kinetic mechanisms using one‐dimensional pseudo‐homogeneous and heterogeneous models. The predicted temperature profile using the one‐dimensional heterogeneous model with the kinetic model of Calderbank et al. but with the adjusted kinetic and transport parameters proposed by Anastasov presented a good correlation with regard to experimental data. Nevertheless, in the hot‐spot zone deviations, up to 30 K were presented. In conclusion, the temperature performance in the production of phthalic anhydride is suitably predicted by the one‐dimensional heterogeneous model and the Calderbank et al.'s kinetic model. Though, prediction using bidimensional models should be done to establish the best correlation with experimental data.     

Application of support vector regression for developing soft sensors for nonlinear processes

The field of soft sensor development has gained significant importance in the recent past with the development of efficient and easily employable computational tools for this purpose. The basic idea is to convert the information contained in the input–output data collected from the process into a mathematical model. Such a mathematical model can be used as a cost efficient substitute for hardware sensors. The Support Vector Regression (SVR) tool is one such computational tool that has recently received much attention in the system identification literature, especially because of its successes in building nonlinear blackbox models. The main feature of the algorithm is the use of a nonlinear kernel transformation to map the input variables into a feature space so that their relationship with the output variable becomes linear in the transformed space. This method has excellent generalisation capabilities to high‐dimensional nonlinear problems due to the use of functions such as the radial basis functions which have good approximation capabilities as kernels. Another attractive feature of the method is its convex optimization formulation which eradicates the problem of local minima while identifying the nonlinear models. In this work, we demonstrate the application of SVR as an efficient and easy‐to‐use tool for developing soft sensors for nonlinear processes. In an industrial case study, we illustrate the development of a steady‐state Melt Index soft sensor for an industrial scale ethylene vinyl acetate (EVA) polymer extrusion process using SVR. The SVR‐based soft sensor, valid over a wide range of melt indices, outperformed the existing nonlinear least‐square‐based soft sensor in terms of lower prediction errors. In the remaining two other case studies, we demonstrate the application of SVR for developing soft sensors in the form of dynamic models for two nonlinear processes: a simulated pH neutralisation process and a laboratory scale twin screw polymer extrusion process. A heuristic procedure is proposed for developing a dynamic nonlinear‐ARX model‐based soft sensor using SVR, in which the optimal delay and orders are automatically arrived at using the input–output data.     

Using NSGA‐II and TOPSIS Methods for Interior Ballistic Optimization Based on One‐Dimensional Two‐Phase Flow Model

In order to meet the design demands of new gun systems or new types of projectiles, the interior ballistic charge design seems especially important. In this paper, a one‐dimensional two‐phase flow model is presented. The model describes the transient combustion of granular propellants in a gun, and pressure waves are considered as an objective. This study adopts a hybrid method to solve the problem. In the first stage, the non‐dominated sorting genetic algorithm (NSGA‐II) with “a “filter” is employed to approximate a set of Pareto‐optimal solutions. In the subsequent stage, a multi‐attribute decision‐making (MADM) approach is adopted to rank these solutions from the best to the worst. The ranking of Pareto‐optimal solutions is based on the technique ordered preference by similarity to ideal solution (TOPSIS) method. In TOPSIS method each objective needs a corresponding weight coefficient, and a practical problem is introduced. Both the entropy method and linear analysis method are adopted to get two sets of weights for the objectives, respectively. The two pairs of final, best compromise solutions are compared for satisfying the designer’s aim. For the analysis of the results, a two‐phase flow interior ballistic model for design optimization is established, and the hybrid approach could get a reasonable design scenario.