Parallel and high resolution numerical solution of concentration and temperature distributions in fluidized beds

In this article higher order in time numerical schemes with efficient time stepping for the solution of concentration and temperature distributions in fluidized beds using parallel computers are presented. The mathematical model equations consist of strongly coupled and semi linear convection-diffusion-reaction equations. Invariant regions for the model are derived to check the solution bounds. The numerical discretization for the space using the finite element method is presented and the numerical treatment is enhanced by using adaptive and higher order linearly implicit Runge–Kutta methods for the time discretization. For different time stepping methods and different spatial grid sizes numerical results are obtained and compared. The methods used show a clear improvement for the problem under consideration compared to previously presented results (Nagaiah, Warnecke, Heinrich, & Peglow, 2008). Additionally, the higher order time stepping methods yield a good parallel efficiency, paving the way for the efficient study of more complex phenomena.     

Research on a Numerical Schemes for Capturing Free Front during Injection Molding

In order to capture the free front in injection molding filling precisely, the numerical method is studied for the H-J (Hamilton Jacobi) equations. The central WENO (Weighted Essentially Non-Oscillatory) scheme for the level set equations, the third-order central WENO scheme in space respectively and the third-order TVD R-K (Total Variation Diminishing Runge–Kutta) scheme in time, are used for capturing the melt flow front in injection molding filling. The numerical results show the high-order WENO scheme can capture discontinuity precisely and the melt flow front can be captured precisely using this kind of numerical method during the injection molding filling.     

Studies in approximation methods—II: Initial value ordinary differential equations

A comparison of Runge—Kutta and orthogonal collocation methods is made for the solution of initial value ordinary differential equations. The direct connection between implicit Runge—Kutta and orthogonal collocation methods is shown for a certain class of initial value problems; this class being of importance to chemical engineering systems. A number of new semi-implicit Runge—Kutta methods which have an imbedded truncation error estimate feature and are A-stable are presented. Using some test systems these new algorithms are shown to be the most computationally efficient for initial value problems.     

A mathematical model of the calendered exiting thickness of micropolar sheet

A theoretical analysis based on the lubrication theory is presented to study the calendering mechanism. The material to be calendered is described by the constitutive relationship of a micropolar fluid. An exact solution and numerical solution of the problem is calculated. The roll‐separating force, power function and exiting sheet thickness are computed numerically using Runge‐Kutta method. The influence of the material parameters on the pressure distribution, pressure gradient and related quantities of engineering interest in calendering process is analyzed through graphs. POLYM. ENG. SCI., 58:327–334, 2018. © 2017 Society of Plastics Engineers     

Simple kinetic numerical model based on rheometer data for ethylene–propylene–diene monomer accelerated sulfur crosslinking

A simple numerical model for the interpretation of the reaction kinetics in ethylene–propylene–diene monomer (EPDM) vulcanized with accelerated sulfur is presented. The model is based on the assumption that during vulcanization, a number of partial reactions occurs, both in series and in parallel, which determine the formation of intermediate compounds, including activated and matured polymers. Once written a standard first‐order differential equation (DIFF‐EQ) for each partial reaction, an ordinary DIFF‐EQ system (ODEs), was obtained and solved through Runge–Kutta algorithms. Alternatively and more efficiently, a single second‐order nonhomogenous DIFF‐EQ with constant coefficients was deduced, for which a closed‐form solution was derived, provided that the nonhomogenous term was approximated with an exponential function. Kinetic constants were evaluated through experimental data fitting on standard rheometer tests. To assess model predictions, an experimental campaign at different temperatures on two EPDM compounds was performed. They exhibited moderate reversion at intermediate and high curing temperatures. A nonlinear least‐squares fitting was performed to evaluate unknown constants entering into the DIFF‐EQ model proposed. Scaled rheometer curves fit rather well, also in the presence of reversion. In addition, partial reaction kinetic constants were provided: this gave an interesting insight into the different reticulation processes occurring during vulcanization. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Successive linearisation analysis of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects

In this study, we investigate the application of the new successive linearisation method (SLM) to the problem of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects. The governing nonlinear momentum, energy and mass transfer equations are successfully solved numerically using the SLM approach coupled with the spectral collocation method for iteratively solving the governing linearised equations. Comparison of the SLM results for various flow parameters against numerical results and other published results, obtained using the homotopy analysis method (HAM) and Runge–Kutta methods, for related problems indicates that the SLM is a very powerful tool which is much more accurate and efficient than other methods. The SLM converges much faster than the traditional methods like the HAM and is very easy to implement. © 2011 Canadian Society for Chemical Engineering     

Stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition

The problem of a steady laminar two‐dimensional stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition is studied. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically using the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed. It is found that the skin friction coefficient and the heat transfer rate at the surface decrease with increasing values of the material parameter. Moreover, dual solutions are found to exist for the shrinking case, while for the stretching case, the solution is unique. © 2011 Canadian Society for Chemical Engineering     

Accurate solution for acceleration motion of a vertically falling spherical particle in incompressible newtonian media

In this paper, the acceleration motion of a vertically falling spherical particle in incompressible Newtonian media was investigated. The instantaneous velocity and acceleration were carried out using analytical solution technique i.e. the differential transformation method (DTM). Using the DTM, the non‐linear constrained governing equations were reduced to recurrence relations and related initial conditions were transformed into a set of algebraic equations. The solutions were subsequently solved by a process of inverse transformation. The results were also compared with those derived from the homotopy perturbation method (HPM), closed‐form solution, the established fourth order Runge–Kutta method and experimental data in order to verify the accuracy of the proposed method. It was found that this method can achieve more accurate results compared with the HPM. © 2012 Canadian Society for Chemical Engineering     

Novel solution for acceleration motion of a vertically falling non-spherical particle by VIM–Padé approximant

In this paper, the acceleration motion of a vertically falling non-spherical particle in incompressible Newtonian media was investigated. The velocity and acceleration were carried out using analytical solution techniques i.e., the variational iteration method (VIM) and a Padé approximant. The results were also compared with VIM and the established fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. It was shown that this method can lead into more accurate results compared with VIM.     

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.     

Diffusion coefficients of liquids in polymer membranes by a desorption method

A method to obtain the diffusion coefficients and free volume parameters from desorption data for organic liquids in polymer films is presented. The method consists of fitting a numerical solution of the diffusion equation to experimental desorption data. Some problems that arise in the solution of the diffusion equation are discussed and results for a benzene–polyethylene system are presented.     

On a numerical relaxation method for a chemical reaction-diffusion problem with an instantaneous and irreversible reaction

In this paper we deal with a numerical method for a free boundary value problem in 1D, describing a chemical diffusion problem accompanied by an irreversible and instantaneous reaction which gives rise to a moving internal boundary. Basically, the method consists of four steps: (1) a Landau transformation mapping each of the 2 time varying intervals on a fixed domain, which results in a strongly nonlinear boundary value problem; (2) a central difference method with respect to the space variable, that takes properly into account the various transition conditions; (3) the construction of an ODE containing a relaxation parameter ε to represent the movement of the internal boundary; (4) a time integration of the resulting stiff system of ODEs by suitable computer packages. The numerical method is evaluated by comparison with an analytical solution for a special but nontrivial case, and by a mass-balance argument. The presented method can be extended to the case of several irreversible and instantaneous reactions.     

Optimization and sensitivity analysis for multiresponse parameter estimation in systems of ordinary differential equations

Methodology for the simultaneous solution of ordinary differential equations (ODEs) and associated parametric sensitivity equations using the Decoupled Direct Method (DDM) is presented with respect to its applicability to multiresponse parameter estimation for systems described by nonlinear ordinary differential equations. The DDM is extended to provide second order sensitivity coefficients and incorporated in multiresponse parameter estimation algorithms utilizing a modified Newton scheme as well as a hybrid Newton/Gauss-Newton optimization algorithm. Significant improvements in performance are observed with use of both the second order sensitivities and hybrid optimization method. In this work, our extension of the DDM to evaluate second order sensitivities and development of new hybrid estimation techniques provide ways to minimize the well-known drawbacks normally associated with second-order optimization methods and expand the possibility of realizing their benefits, particularly for multiresponse parameter estimation in systems of ODEs.     

An ESDIRK method with sensitivity analysis capabilities

A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge–Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. A key feature is the reuse of information already computed for the state integration, hereby minimizing the extra effort required for sensitivity integration. Through case studies the new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present, which renders the new algorithm particularly suitable for dynamic optimization purposes.     

Semianalytical solution for epoxy-amine curing in the presence of nonisothermal effects

The curing of epoxy-amine polymer has been modeled by the reaction of functional groups, and the mole balance of these is governed by a set of six nonlinear differential equations. In this work, we have first developed a complete analytical solution for isothermal curing. For nonisothermal curing, we have divided the domain of hydroxyl group concentration β into small increments Δβ and adopted our analytical results for this domain. In addition, we solved the energy balance equation analytically and obtained the temperature rise for Δβ. We have compared our results with those obtained from the Runge Kutta numerical solutions. We have shown that our semianalytical technique is about a thousand times more efficient and faster.     

瓶级PET连续式固相缩聚生产线的计算机模拟

利用可逆化学反应和小分子扩散共同控制瓶级PET固相缩聚过程的数学模型 ,采用了Runge Kutta法和有限差分方法进行求解 ,仿真研究结果表明模型与工业过程吻合 ,能准确对实际生产进行模拟 ,偏差小于 2 %     

Kinetic analysis of two DSC peaks in the curing of an unsaturated polyester resin catalyzed with methylethylketone peroxide and cobalt octoate

The curing of an unsaturated polyester resin catalyzed with methylethylketone peroxide and cobalt octoate as promoter is studied by DSC at different heating rates. A high promoter/peroxide ratio is used. The DSC curves show two exothermic peaks. It has been assumed that they represent two independent cure reactions. A set of kinetic parameters for each DSC peak has been obtained. They describe the overall cure process using an empirical kinetic model. Nth order and autocatalytic kinetic functions have been employed. A computer program was developed to calculate the degrees of conversion associated with each peak, and to evaluate the kinetic parameters. The calculation algorithm uses the Runge–Kutta numerical integration and the “downhill simplex method.” This methodology permits to find all kinetic parameters simultaneously. DSC experimental data are very well fitted. The simulated first peak occurs always before the second one, and the fraction of reaction heat associated with the first peak over the overall reaction heat decreases with heating rate. Moreover, activation energies are in good agreement with the tabulated values for typical free radical polymerization induced by thermal and redox decomposition of the peroxides. POLYM. ENG. SCI., 47:62–70, 2007. © 2006 Society of Plastics Engineers     

A method for efficiently solving the IAST equations with an application to adsorber dynamics

The Ideal Adsorbed Solution Theory (IAST) developed by Myers and Prausnitz and Radke and Prausnitz provides a powerful tool to calculate multicomponent adsorption equilibria based on single component adsorption isotherms. An important aspect of the application of IAST is that it requires the solution of an implicit algebraic system of equations. Analytical solutions can be derived only for few simple single component isotherm models. This work offers a new concept to solve the equations of the IAST for mixtures of N components characterized by nondecreasing single component adsorption isotherm behavior. The approach is based on transforming the algebraic system of IAST equations to a system of ODEs with one specified initial value. This work also provides analytical expressions for the partial derivatives of the predicted adsorption equilibria and increases the efficiency of numerical calculations for fixed‐bed adsorber dynamics. The strength of the solution method is illustrated in case studies. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1263–1277, 2013     

Semianalytical solution of isothermal nylon-6 polymerization in batch reactors

Nylon 6 polymerization in batch reactors has been assumed to consist of important reactions of ring opening, step growth and polyaddition steps and a set of ordinary differential equations (initial value problem) for the concentrations and moments of the various reacting species has been derived. We have proposed a series solution for these variables in terms of conversion of ε-caprolactam. A technique similar to the finite element method for boundary value problems is used to divide the conversion domain into subdomains. The size of these is determined by a convergence criterion and the results determined at the end of conversion domain through sequential computations. This technique of solution involves the evaluation of constant coefficients of the series only and gives comparable results with those of Runge Kutta or Gear's algorithm (which involves evaluation of functions) in fewer steps. The scheme can easily be implemented on a personal computer and is considerably faster and more efficient.     

Numerical Investigations of Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds in a Cylindrical Coordinates System

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through a random packing of non‐overlapping spheres at moderate Reynolds numbers, spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier‐Stokes equations in three‐dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study. This observation suggests that no transition to turbulence could occur in the range of the Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three‐dimensional bounded granular bed at moderate Peclet numbers. In addition to numerical calculations, to describe the concentration profile of solute, an approximate solution for the mass transport equation in a bounded granular bed in a cylindrical coordinates system is proposed. Lateral fluid dispersion coefficients are then calculated by fitting the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.